aainstitutetext: Department of Physics, Arizona State University, Tempe, AZ
85281, USAbbinstitutetext: Quantum Information Center, Department of Computer
Science, The University of Texas at Austin, 2317 Speedway, Austin, TX 78712,
USA

# An Entropic Lens on Stabilizer States

Cynthia Keeler a William Munizzi b and Jason Pollack<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The $n$-qubit stabilizer states are those left invariant by a $2^{n}$-element
subset of the Pauli group. The Clifford group is the group of unitaries which
take stabilizer states to stabilizer states; a physically–motivated generating
set, the Hadamard, phase, and CNOT gates which comprise the Clifford gates,
imposes a graph structure on the set of stabilizers. We explicitly construct
these structures, the “reachability graphs,” at $n\leq 5$. When we consider
only a subset of the Clifford gates, the reachability graphs separate into
multiple, often complicated, connected components. Seeking an understanding of
the entropic structure of the stabilizer states, which is ultimately built up
by CNOT gate applications on two qubits, we are motivated to consider the
restricted subgraphs built from the Hadamard and CNOT gates acting on only two
of the $n$ qubits. We show how the two subgraphs already present at two qubits
are embedded into more complicated subgraphs at three and four qubits. We
argue that no additional types of subgraph appear beyond four qubits, but that
the entropic structures within the subgraphs can grow progressively more
complicated as the qubit number increases. Starting at four qubits, some of
the stabilizer states have entropy vectors which are not allowed by
holographic entropy inequalities. We comment on the nature of the transition
between holographic and non-holographic states within the stabilizer
reachability graphs.

##  1 Introduction

Are all quantum states in a Hilbert space created equal? Viewed at the most
abstract level, every pure quantum state can be rotated to any other by a
unitary change of basis. But when the Hilbert space is endowed with some
structure, different states play different roles with respect to that
structure. One natural structure is given by specifying a _factorization_ of
the Hilbert space: an isomorphism between the abstract $\mathcal{H}$ and the
tensor product Hilbert space $\bigotimes_{i=1}^{N}\mathcal{H}_{i}.$ For finite
Hilbert spaces of composite dimension, the factorization associates to each
pure state ${|{\Psi}\rangle}\in\mathcal{H}$ an _entropy vector_ , the
collection of von Neumann entropies of the $2^{N}-1$ reduced density matrices
formed by tracing out each possible tensor product formed from factors
$\mathcal{H}_{i}$.

The entropy vectors provide a classification of the states in a tensor product
Hilbert space, but because every state has an associated entropy vector they
do not, by themselves, pick out any states as special. One way to accomplish
this is by fixing one or more preferred operators acting on the Hilbert space.
As a familiar example, choosing a particular Hermitian operator acting on a
Hilbert space to be the Hamiltonian, the generator of time translations,
specifies a basis of energy eigenstates, and every state in the Hilbert space
can then be expanded in the energy basis. Most states are superpositions of
more than one energy eigenstate, but not all: fixing a Hamiltonian picks out
the basis of eigenstates of the Hamiltonian, the energy eigenstates
themselves, which are the states where the Hamiltonian acts trivially, as
scalar multiplication. The particular scalar is just given by the eigenvalue
of the energy eigenstate, and if we just want to find the set of energy
eigenstates this is unimportant: we could instead say that the energy
eigenstates are the states where the projector onto the energy eigenspaces
acts as the identity.

A similar procedure applies when instead of specifying a single Hermitian
operator we pick a (multiplicative) group of Hermitian operators. Given the
group $G\in L(\mathcal{H})$, we can classify every state
${|{\Psi}\rangle}\in\mathcal{H}$ by the number of group elements that act
trivially on this state: the dimension of the stabilizer subgroup
$G_{|{\Psi}\rangle}$. Almost every state will have
$\mathrm{dim}\>G_{|{\Psi}\rangle}=1$: the only group element that acts
trivially is the identity operator. But some will have more, and the states
that have the largest stabilizer subgroups relative to $G$ are called the
_stabilizer states_ (which we will define more precisely below). For example,
if the Hilbert space is that of a qubit, the stabilizer states of the Pauli
group generated by $\langle X,Y,Z\rangle$ are the six states stabilized (up to
sign) by the identity and one additional Pauli operator. Stabilizer states,
especially with respect to the Pauli group on $n$ qubits, play an important
role in the theory of quantum error correction, and in the fundamentals of
quantum computing Gottesman:1997zz ; Gottesman:1998hu ; aaronson2004improved ;
knill2004fault ; bravyi2005universal . But here they emerged directly as
“generalized eigenstates,” the states which play nicely with a specified group
of operators.

This paper initiates a research program aimed at combining these two
classifications of states in Hilbert space. We seek an understanding of the
stabilizer states, picked out by their interaction with a specified group of
operators, in terms of their entropic structure, given by the underlying
factorization of the Hilbert space. In particular, we will focus on the
stabilizer states with respect to the Pauli group acting on $n$ qubits. In
this setting, any stabilizer state can be reached by starting with any other
stabilizer state and applying a unitary quantum circuit comprised of the
Clifford gates: two one-qubit gates, the Hadamard and phase gates, and a
single two-qubit gate, the controlled NOT gate. Since unitary operations on a
single tensor factor do not change the entropy vector, it is already clear
that moving between entropy vectors can only be accomplished via a CNOT gate.
But not all such gate applications alter the entropic structure, and the
picture we will find will be far richer than simply counting CNOT gates.

Besides understanding the general entropic structure of stabilizer states, we
are additionally motivated by the connections between stabilizer states and
holography. Stabilizer error-correcting codes satisfy a complementary recovery
property which implies, and is equivalent to, an operator-algebraic version of
the Ryu-Takayanagi formula Ryu:2006bv  relating entropies of states in the
boundary/physical Hilbert space to the expectation value of an “area” operator
acting on the bulk/logical Hilbert space Harlow:2016vwg ; Pollack:2021yij . A
holographic bulk geometry can be discretized into a graph Bao:2015bfa ; in the
limit of large bond dimension a tensor network built from random stabilizer
tensors saturates the RT formula with probability one Hayden:2016cfa ;
Nezami:2016zni .

Hence all holographic entropy vectors can be represented by stabilizer states,
but the converse is not true: the holographic _entropy cone_ consisting of
space of all allowed holographic entropy vectors is contained within the
stabilizer entropy cone, and this containment is strict starting at three
regions. Because the holographic entropy cone is well–characterized—explicitly
at up to five regions HernandezCuenca:2019wgh , and implicitly at arbitrary
finite region number by the methods pioneered in Bao:2015bfa —but the
stabilizer entropy cone, and even the larger quantum entropy cone of all
quantum states, are poorly understood, it is of great interest to understand
in more detail how the holographic states are embedded into the larger space
of stabilizer states. The stabilizer graph constructions presented in this
paper allow this question to be attacked.

###  1.1 Summary of Results

(a)

(b)

(c)

(d)

Figure 1: The collection of subgraph structures up through five qubits. (a)
1-Qubit reachability diagram displaying all single-qubit operations. (b)
2-Qubit restricted graph consisting of subgraph structures $g_{24}$ and
$g_{36}$. (c) Two new subgraph structures occur at three qubits, $g_{144}$ and
$g_{288}$. (d) At four qubits we witness the arrival of a new subgraph
$g_{1152}$, as well as an increase of entropy vectors on $g_{144}$. No new
subgraph structures emerge beyond $g_{1152}$, but the number of different
entropy vectors present in each subgraphs continues to increase for higher
qubit number. At four qubits we witness the first instance of non-holographic
states, appearing on subgraph $g_{144}$.

Figure 1 presents a preview of our results. All of the objects in the figure
are “reachability graphs,” which display the structure of a (subset of) the
stabilizer states. Each vertex is a particular stabilizer state, colored by
its entropy vector. The edges connecting them correspond to the action of a
particular Clifford gate. The first panel presents the full reachability graph
at one qubit, showing the action of the two one-qubit Clifford gates, Hadamard
and phase, on the six one-qubit stabilizer states. The remaining panels show
reachability graphs at higher qubits, constructed from a subset of the
Clifford gates consisting of the Hadamard and CNOT gates applied to two
particular qubits; we will argue below that this restricted set of gates is
sufficient to capture the changes in entropic structure as we move between
different stabilizer states. The second panel presents the two subgraphs found
in the restricted two-qubit reachability graph: one with 24 vertices and one
with 36.

At three qubits, the restricted graph has 16 connected components, made up of
copies of the 24- and 36-vertex subgraphs, as well as two additional
structures, with 144 and 288 vertices, respectively, which are presented in
the third panel. We will show how these more complicated structures, as well
as an 1152-vertex subgraph that appears at four qubits, can be constructed in
a simple way by certain “lifts” of states in the structures found at two
qubits to stabilizer states in a larger Hilbert space. By understanding the
lifts, we will also understand how to map entropy vectors at lower qubit
numbers to entropy vectors at higher qubit numbers: for example, although only
three distinct entropy vectors appear in the 144-vertex subgraphs at three
qubits, at four qubits there are versions of this subgraph with four distinct
entropy vectors, as seen in the final panel of the Figure.

At four qubits, the stabilizer states can have any of eighteen different
entropy vectors (see Table 5), and one of these entropy vectors, shown in blue
in the fourth panel of Figure 1, is not holographic. The reachability graphs
allow us to see how, at four and five qubits, applications of Clifford gates
can move states out of, and back into, the holographic entropy cone. We see,
for example, that moving from the inner octagonal structure to the outer ring
cannot be accomplished without passing through non-holographic states.

###  1.2 Structure of the Paper

The remainder of this paper is organized as follows. Section 2 reviews the
notions of stabilizer states and the entropy cone, which are of fundamental
importance for the rest of the paper. Section 3 initiates the study of the
intersection of these two concepts by presenting the two-qubit stabilizer
graph colored by entropy vector. We illustrate that additional insight can be
gained into the graph structure by considering the _restricted graphs_
generated by only a subset of the Clifford gates; in particular, we highlight
the special role of the restricted graph generated by only Hadamard and CNOT
gates.

Section 4 extends the discussion to the three-qubit stabilizer graph. We show
that much of the three-qubit graph can be understood as the natural extension
of the two-qubit graph, but that nontrivial new structures appear in addition.
We argue that because the Clifford gates act on at most two qubits it is
natural to consider the restricted graphs generated by gates that act on any
two of the three qubits. In Section 5, we pass to a discussion of the
situation for higher qubit numbers, illuminated by some direct results at four
and five qubits. We show that no fundamentally new objects appear as we go to
higher qubit number, but the existing objects develop progressively more
complicated entropic structures. At higher qubit numbers, entropy vectors
appear which cannot be represented by holographic states, and we comment on
how they fit into the stabilizer graph. Finally, in Section 6 we discuss and
conclude. Additional results and graphs are presented in appendices.

##  2 Reminder: Stabilizer States and the Entropy Cone

###  2.1 Review of Stabilizer States

The Pauli matrices

$I=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\,\,\sigma_{X}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\,\,\sigma_{Y}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\,\,\sigma_{Z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$ (1)

are a set of four Hermitian and unitary matrices with eigenvalues $\pm 1$,
which, given a fixed basis {${|{0}\rangle},{|{1}\rangle}\\}$, can be
interpreted as operators acting on the Hilbert space $\mathbb{C}^{2}$ of a
single qubit. The (nontrivial) Pauli operators
$\\{\sigma_{X},\sigma_{Y},\sigma_{Z}\\}$ generate the full algebra of linear
operators $L(\mathbb{C}^{2})$. More importantly for our purposes, they
generate a 16-element multiplicative matrix group, the Pauli group on one
qubit

$\Pi_{1}\equiv\langle\sigma_{X},\sigma_{Y},\sigma_{Z}\rangle=c\\{I,\sigma_{X},\sigma_{Y},\sigma_{Z}\\},c\in\\{\pm
1,\pm i\\}.$ (2)

Recall that for a pure state ${|{\Psi}\rangle}\in\mathcal{H}$ and a group of
operators $G\subset L(\mathcal{H})$, the stabilizer group of
${|{\Psi}\rangle}$ is defined by

$G_{|{\Psi}\rangle}\equiv\\{g\in G\>|\>g{|{\Psi}\rangle}={|{\Psi}\rangle}\\}.$
(3)

That is, the stabilizer group of ${|{\Psi}\rangle}$ consists of the operators
in $G$ for which ${|{\Psi}\rangle}$ is an eigenvector with eigenvalue one. By
inspection, the Pauli group on one qubit $\Pi_{1}$ has one element, $I$, with
two unit eigenvalues, which therefore stabilizes all states in
$\mathbb{C}^{2}$; one element with two negative eigenvalues, $-I$, which
stabilizes no states; eight elements with purely imaginary eigenvalues, which
also stabilize no states; and six elements with one unit eigenvalue,
$\\{\pm\sigma_{X},\pm\sigma_{Y},\pm\sigma_{Z}\\}$, which therefore each
stabilize one state in $\mathbb{C}^{2}$. Hence there are six states in the
Hilbert space, the _one-qubit stabilizer states_ , which are stabilized by a
two-element subgroup:

$S_{1}\equiv\left\\{{|{0}\rangle},{|{1}\rangle},{|{\pm}\rangle}\equiv\frac{1}{\sqrt{2}}({|{0}\rangle}\pm{|{1}\rangle}),{|{\pm
i}\rangle}\equiv\frac{1}{\sqrt{2}}({|{0}\rangle}\pm i{|{1}\rangle})\right\\},$
(4)

and all other states in the Hilbert space are stabilized by only the identity.
For example, ${|{+}\rangle}$ is stabilized by $I$ and $\sigma_{X}$, while
${|{1}\rangle}$ is stabilized by $I$ and $-\sigma_{Z}$.

If we fix a state ${|{\Psi}\rangle}\in S_{1}$, then there are a limited number
of operations we can do that will map ${|{\Psi}\rangle}$ to some other
${|{\Psi^{\prime}}\rangle}\in S_{1}$. In particular, we can ask what unitary
operators $U\in L(\mathbb{C}^{2})$ are guaranteed to map _any_ state in
$S_{1}$ back into $S_{1}$. Since we defined the stabilizer states by their
property of having the largest stabilizer group with respect to $\Pi_{1}$, we
can equivalently ask which unitaries $U$ _normalize_ the Pauli group, i.e.
take group elements to group elements under conjugation by U. Clearly every
element of the Pauli group itself has this property, but in general there is a
larger group of unitaries which do as well. The group of unitaries which
normalize the Pauli group is called the Clifford group,

$C_{1}=\left\\{U\in L(\mathbb{C}^{2})\>|\>UgU^{\dagger}\>\forall
g\in\Pi_{1}\right\\}.$ (5)

It suffices to check that a unitary takes $\sigma_{X}$ and $\sigma_{Z}$ to
elements of $C_{1}$. For example, the Hadamard sylvester1867lx ;
hadamard1893resolution  and phase gates,

$H\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix},\qquad
P\equiv\begin{pmatrix}1&0\\\
0&e^{\frac{i\pi}{2}}\end{pmatrix}=\begin{pmatrix}1&0\\\ 0&i\end{pmatrix},$ (6)

are both elements of the Clifford group, since
$H\sigma_{X}H^{\dagger}=\sigma_{Z}$, $H\sigma_{Z}H^{\dagger}=\sigma_{X}$, and
$P\sigma_{X}P^{\dagger}=\sigma_{Y}$, $P\sigma_{Z}P^{\dagger}=\sigma_{Z}$. In
fact, these two gates suffice to generate the Clifford group, $C_{1}=\langle
H,P\rangle$. We see, in particular, that because $PP=\sigma_{Z}$, we can
easily construct the Paulis themselves out of $H$ and $P$.

\begin{overpic}[width=390.25534pt]{OneQubitReachabilityDiagram.pdf}
\put(5.0,2.5){${|{0}\rangle}$} \put(66.0,2.5){${|{1}\rangle}$}
\put(20.5,3.5){${|{+}\rangle}$} \put(50.5,9.5){${|{-}\rangle}$}
\put(38.8,0.0){${|{i}\rangle}$} \put(32.8,14.2){${|{-i}\rangle}$}
\end{overpic} Figure 2: Complete one-qubit reachability diagram. Because the
Hadamard gate is its own inverse we have depicted edges corresponding to
Hadamard gate applications as undirected, but gates corresponding to phase
gates as directed.

Given the set of stabilizer states $S_{1}$ and a set of generators of the
Clifford group $C_{1}$, which we call _Clifford gates_ Gottesman:1997zz , we
can arrange the states into a _reachability graph_ , with each vertex labeled
by a stabilizer state and each edge between vertices labeled by the Clifford
gate which maps one vertex to the other. (The Hadamard gate is its own
inverse, so it can be represented by an undirected edge, but the phase gate is
not, so it is represented by a directed edge.) The reachability graph for
$S_{1}$ is shown in Figure 2. Note that the reachability graph depends on a
particular choice of generators of $C_{1}$, e.g. the Clifford gates. The
significance of choosing $H$ and $P$, in particular, as our generators is that
both operators have physical significance and can be implemented
experimentally (relatively) easily.

Note that the reachability graph consists of a single component: as implied by
the definition of the Clifford group, we can use the Clifford gates to get (in
some number of steps) from any initial stabilizer state, e.g. ${|{0}\rangle}$,
to any other stabilizer state. Furthermore, the graph contains many cycles:
trivial ones, where a Clifford gate acts as the identity on a particular
stabilizer state, but also longer ones. For example, because $H^{2}=P^{4}=I$,
every edge corresponding to a Hadamard application is part of a cycle of
length two, and every edge corresponding to a nontrivial phase application is
part of a cycle of length four, such as the diamond representing a nontrivial
cycle consisting of four phase gates we see at the center of Figure 2. More
interestingly, we have cycles consisting of more than one type of gate, such
as the triangles with two phase gates and one Hadamard.

To better understand these cycles, we can apply two basic group-theoretic
results. First, Lagrange’s theorem says that the order of any subgroup $H$ of
a finite group $G$ gives an integer partition of that group Alperin1995 :
explicitly,

$|G|=[G:H]\cdot|H|,\hskip 14.22636pt\forall H\leq G,$ (7)

with $[G:H]$ the index of $H$ in $G$. Second, the orbit-stabilizer theorem
says that, when $H$ is a stabilizer subgroup of $G$ with respect to some state
$x\in X$, i.e. $H=G_{x}$, there exists a bipartition between the orbit of x in
G, $|G\cdot x|$, and the set of cosets of the stabilizer subgroup in $G$,
$G/H$. Hence these two objects have the same dimension:

$|G\cdot x|=\frac{|G|}{|H|}=[G:H],$ (8)

where in the last equality we have used (7).

The Clifford group $C_{1}$ is a group of operators acting on the one-qubit
Hilbert space $\mathbb{C}^{2}$. As we have discussed, for each
${|{\Psi}\rangle}\in S_{n}\subset\mathbb{C}^{2}$, there exists a subgroup
$G_{|{\Psi}\rangle}$. Hence we can apply this group-theoretic machinery to our
case of interest. Substituting ${|{\Psi}\rangle}\in S_{1}$ for $x\in X$,
$C_{1}$ for $G$, and $G_{{|{\Psi}\rangle}}$ for $H$ in (8) gives

$|C_{1}\cdot{|{\Psi}\rangle}|=\frac{|C_{1}|}{|G_{{|{\Psi}\rangle}}|},$ (9)

where $|C_{1}\cdot{|{\Psi}\rangle}|$ denotes the length of the orbit of
${|{\Psi}\rangle}$. When we represent the action a of group element by a
graph, the orbit length is the largest number of vertices in any connected
component of the graph.

Explicitly, consider the set of one-qubit stabilizer states $S_{1}$ defined in
4. The one-qubit Clifford group $C_{1}$ is constructed from the generating set
$\langle H_{1},P_{1}\rangle$. The orbit of each ${|{\Psi}\rangle}\in S_{1}$
can be computed directly using Equation (9), with results shown in Table 1.
One can easily verify these results by comparing with the reachability diagram
in Figure 2.

${|{\Psi}\rangle}$ | $|C_{1}|$ | $|C_{1_{{|{\Psi}\rangle}}}|$ | $|C_{1}\cdot{|{\Psi}\rangle}|$
---|---|---|---
${|{0}\rangle}$ | 192 | 32 | 6
${|{1}\rangle}$ | 192 | 32 | 6
${|{+}\rangle}$ | 192 | 32 | 6
${|{-}\rangle}$ | 192 | 32 | 6
${|{i}\rangle}$ | 192 | 32 | 6
${|{-i}\rangle}$ | 192 | 32 | 6
Table 1: Orbit lengths for each single-qubit stabilizer state under the one-
qubit Clifford group $C_{1}$.

We can also use this machinery to consider the orbits of states
${|{\Psi}\rangle}\in S_{1}$ under subgroups of $C_{1}$, see Table 2. Let
$G_{P}<C_{1}$ denote the subgroup generated by only the phase gate. This group
contains the $4$ unique elements $P_{1},P_{1}^{2},P_{1}^{3},$ and $P_{1}^{4}$
(recall $P_{1}^{4}=I$). Each element acts trivially on ${|{0}\rangle}$ and
${|{1}\rangle}$, and thus these two states are stabilized by all elements of
$G_{P}$. The remaining states
$({|{+}\rangle},{|{-}\rangle},{|{i}\rangle},{|{-i}\rangle})$ form a cycle of
length $4$ under operation of $P_{1}$, each stabilized only by the identity
$P_{1}^{4}$. These orbits manifest as subgraphs of the reachability graph as
seen in Figure 3.

${|{\Psi}\rangle}$ | $|G_{P}|$ | $|G_{P_{{|{\Psi}\rangle}}}|$ | $|G_{P}\cdot{|{\Psi}\rangle}|$
---|---|---|---
${|{0}\rangle}$ | 4 | 4 | 1
${|{1}\rangle}$ | 4 | 4 | 1
${|{+}\rangle}$ | 4 | 1 | 4
${|{-}\rangle}$ | 4 | 1 | 4
${|{i}\rangle}$ | 4 | 1 | 4
${|{-i}\rangle}$ | 4 | 1 | 4

${|{\Psi}\rangle}$ | $|G_{H}|$ | $|G_{H_{{|{\Psi}\rangle}}}|$ | $|G_{H}\cdot{|{\Psi}\rangle}|$
---|---|---|---
${|{0}\rangle}$ | 2 | 1 | 2
${|{1}\rangle}$ | 2 | 1 | 2
${|{+}\rangle}$ | 2 | 1 | 2
${|{-}\rangle}$ | 2 | 1 | 2
${|{i}\rangle}$ | 2 | 1 | 2
${|{-i}\rangle}$ | 2 | 1 | 2

Table 2: Orbit lengths for each single-qubit stabilizer state under subgroups
$G_{P}<C_{1}$ and $G_{H}<C_{1}$, generated by only the phase gate and Hadamard
gate respectively.
\begin{overpic}[width=390.25534pt]{OneQubitH1OnlyP1Only.pdf}
\put(8.0,26.5){${|{0}\rangle}$} \put(63.5,26.5){${|{1}\rangle}$}
\put(32.5,36.0){${|{i}\rangle}$} \put(1.7,6.0){${|{0}\rangle}$}
\put(62.8,6.2){${|{1}\rangle}$} \put(29.0,15.0){${|{i}\rangle}$} \end{overpic}
Figure 3: The one-qubit reachability diagram restricted to only phase (Top)
and only Hadamard (Bottom) operations reveals disconnected orbits of varying
length for states ${|{\Psi}\rangle}\in S_{1}$.

Similarly, consider the subgroup $G_{H}<C_{1}$, generated by only the Hadamard
gate. This group only has $2$ elements since $H_{1}=H_{1}^{-1}$ and
$H_{1}^{2}=I$. Distinctly, the Hadamard gate stabilizes no element of $S_{n}$,
and thus each state in $S_{1}$ is stabilized by only the identity
$(H_{1}^{2})$. Each ${|{\Psi}\rangle}\in S_{1}$ subsequently has an orbit of
length $2$, as shown in Table 2, which results in the decomposition of the
reachability graph into pairs of states (Figure 3), equivalent up to a change
of basis.

In the remainder of the paper, we will often consider splitting the
reachability graph into subgraphs constructed from a subset of the Clifford
gates. Ultimately, the underlying structure behind this graphical
representation is precisely the partitioning of the stabilizer set by orbit
length.

As the alert reader will have realized from our notation, we can extend the
definitions of the Pauli group, stabilizer states, and Clifford gates to more
than one qubit. The Pauli group $\Pi_{n}$ on $n$ qubits consists of “Pauli
strings” acting on each of the qubits, and is generated by111Note that writing
a Pauli string requires not just a factorization of the $2^{n}$ dimensional
Hilbert space into tensor factors representing qubits (each of which has a
specified basis $\\{0,1\\}$), but a particular _ordering_ of the qubits from
$1$ to $n$: we write ${|{a_{1}\ldots
a_{n}}\rangle}\equiv{|{a_{1}}\rangle}_{1}\otimes\ldots{|{a_{n}}\rangle}_{n}$.
It should be clear that the set of length-one Pauli strings as a whole are the
independent of choice of ordering, and hence so is the Pauli group $\Pi_{n}$
they generate. This will also be the case for the Clifford group $C_{n}$ and
the set of all Clifford gates. However, individual gates will of course depend
on the choice of ordering. We will often consider a subset of the Clifford
gates which act only on the first two qubits, which again depends on the
choice of ordering, but there is an equivalent subset which acts on any two
specified qubits, so although the position of a given state within the graphs
we will generate depends on a choice of ordering, the overall graph structures
themselves will not. the length-one Pauli strings like
$I^{1}\otimes\ldots\otimes I^{k-1}\otimes\sigma_{Z}^{k}\otimes
I^{k+1}\otimes\ldots\otimes I^{n}$. The stabilizer states $S_{n}$ are those
states ${|{\Psi}\rangle}\in(\mathbb{C}^{2})^{\otimes n}$ with stabilizer
groups of maximal size,
$\mathrm{dim}\>G_{|{\Psi}\rangle}=2^{n}$.aaronson2004improved ;
garcia2017geometry

$\left|S_{n}\right|=2^{n}\prod_{k=0}^{n-1}(2^{n-k}+1)=2(2^{n}+1)\left|S_{n-1}\right|\approx
2^{(.5+o(1))n^{2}}.$ (10)

We have given a closed-form expression, recursion relation (with base case
$|S_{1}|=6$, as constructed explicitly above), and asymptotic expression.

The Clifford group $C_{n}$ is again the group of unitaries which normalize
$\Pi_{n}$, which contains the Hadamard and phase gates acting on each
individual qubit. However, these gates no longer suffice to generate the full
Clifford group: because the gates act only on a single qubit, they cannot
change the entanglement structure of a state. Yet not every stabilizer state
has the same entanglement structure. One subset of the two-qubit stabilizer
states consists of the tensor product of a one-qubit stabilizer state on the
first qubit and another one-qubit stabilizer state on the second qubit: these
states are, of course, product states. But, for example, the Bell state
$\frac{1}{\sqrt{2}}({|{00}\rangle}+{|{11}\rangle})$ is a two-qubit stabilizer
state, stabilized by
$\\{I^{1}I^{2},\sigma_{X}^{1}\sigma_{X}^{2},-\sigma_{Y}^{1}\sigma_{Y}^{2},\sigma_{Z}^{1}\sigma_{Z}^{2}\\}$.
Hence any set of operators generating $C_{n}$ must contain operators which map
product states to entangled states and vice versa.

One convenient gate which accomplishes this task is the $CNOT_{i,j}$ gate,
which performs a controlled $NOT$ operation on the $j$th qubit depending on
the state of the $i$th qubit:

$CNOT_{1,2}\equiv\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}\in L(\mathbb{C}^{4}).$ (11)

To check that $CNOT_{i,j}\in C_{n}$, it suffices to check its action on the
length-one Pauli strings consisting of
$\\{\sigma_{X}^{i},\sigma_{Z}^{i},\sigma_{X}^{j},\sigma_{Z}^{j}\\}$ tensored
with identities on every other qubit. A standard calculation shows that
conjugation by $CNOT_{i,j}$ maps these four strings to
$\\{\sigma_{X}^{i}\otimes\sigma_{X}^{j},\sigma_{Z}^{i},\sigma_{X}^{j},\sigma_{Z}^{i}\otimes\sigma_{Z}^{j}\\}$
tensored with identities, respectively. Hence the CNOT gates are indeed
members of the Clifford group; together with the Hadamards and phase gates,
they generate222In fact, we only need half of the CNOT gates, the $n(n-1)/2$
gates $CNOT_{i,j}$ with $i<j$, because we have the relation
$CNOT_{j,i}=H_{i}H_{j}CNOT_{i,j}H_{j}H_{i}$. Note that this expression is not
unique, because $H_{i}$ and $H_{j}$ acting on distinct qubits commute.
Following convention, we will nevertheless take the Clifford gates to include
all $n(n-1)$ CNOT gates; e.g. our two-qubit reachability graph will show the
actions of both $CNOT_{1,2}$ and $CNOT_{2,1}$. $C_{n}$, so the _$n$ -qubit
Clifford gates_ are taken to be the $n$ Hadamard gates $H_{n}$, the $n$ phase
gates $P_{n}$, and the $n(n-1)$ gates $CNOT_{i,j}$ (for $i\neq j$).

We can thus construct, for the Hilbert space of $n$ qubits $\mathbb{C}^{2n}$,
a reachability graph for the stabilizer states $S_{n}$ using the Clifford
gates which generate $C_{n}$. We will devote the rest of the paper to studying
this object, and the subgraphs formed from it by restricting to a subset of
the Clifford gates, at various qubit numbers $n>1$. Before we begin this
study, however, we will first categorize the various possible allowed entropic
structures of the stabilizer states $S_{n}$.

###  2.2 Review of the Entropy Cone

All stabilizer states themselves are pure states; that is, for a given
stabilizer state ${|{\psi}\rangle}$, the density matrix $\rho_{\psi}$ is
idempotent and thus has zero total entropy:

$\rho_{\psi}={|{\psi}\rangle}\bra{\psi},\quad\rho_{\psi}^{2}=\rho_{\psi},\quad
S(\psi)=-\tr\rho_{\psi}\log_{2}\rho_{\psi}=0.$ (12)

For this paper, we measure entropy in _bits_ : every $\log$ should be
interpreted as $\log_{2}$ throughout. As we will see shortly, this convention
results in positive integer entries for every element in the entropy vector
for all stabilizer states.

Non-trivial entropic structure arises when we consider how one subset of
qubits relates to its complement. Suppose we pick a $p$-qubit subset $I$ of
the $n$ qubits in a full stabilizer state. Then, the entanglement entropy
between the $p$-qubit subset $I$ and its $(n-p)$-qubit complement $\bar{I}$ is
given by

$\rho_{I}=\tr_{I}{|{\psi}\rangle}\bra{\psi},\quad
S_{I}=-\tr\rho_{I}\log_{2}\rho_{I}.$ (13)

Here the trace $\tr_{I}$ is taken over only the $p$ qubits in the subset $I$.
The density matrix $\rho_{I}$ is called a reduced density matrix, and since
stabilizer states are pure only their reduced density matrices have nonzero
entropy. Since the entropy for the full state is zero, we also have
$S_{I}=S_{\bar{I}}$.

For an $n$-qubit stabilizer state, there are thus $2^{n-1}-1$ entropies.
Listing all of these entropies produces the _entropy vector_ for a given
state. As an example, 2-qubit stabilizer states have full entropy vector
$\vec{S}=(S_{A},S_{O},S_{AO})$. However, since the state is pure, $S_{AO}=0$
and $S_{A}=S_{O}$. We thus write the 2-qubit entropy vector as just
$\vec{S}=(S_{A})$. Here, we have labelled our last qubit with $O$ to indicate
it acts as a purifier for the other qubits.

Similarly, for a $3$-qubit state, we write $\vec{S}=(S_{A},S_{B},S_{O})$, or
equivalently, $\vec{S}=(S_{A},S_{B},S_{AB})$. For $4$ qubits we have
$\vec{S}=(S_{A},S_{B},S_{C},S_{O},S_{AB},S_{AC},S_{AO})$. We could have
written $S_{O}=S_{ABC}$ and $S_{AO}=S_{BC}$ instead; some sources choose a
different ordering for the entropy vector accordingly.

As reviewed in Section 2.1, only CNOT gates can create or destroy entanglement
entropy. We can now refine this statement: the $CNOT_{i,j}$ gate can only
alter entropies $S_{I}$ where qubit $i\in I$ but qubit $j\in\bar{I}$, or vice
versa.

In addition to the equation $S_{I}=S_{\bar{I}}$, which holds for any pure
state, entropies for subsets of qubits also obey entropy inequalities. The
full set of entropy inequalities obeyed by a given set of states defines the
_entropy cone_ 10.1109/18.641561 ; 1193790 . The quantum entropy cone is the
largest region we will discuss; any quantum state obeys the inequalities that
define its boundaries. The Araki-Lieb inequality cmp/1103842506
$S_{IJ}+S_{I}\geq S_{J}$ and subadditivity $S_{I}+S_{J}\geq S_{IJ}$, where
$I,J$ are disjoint sets of qubits, are both examples of inequalities obeyed by
all quantum states. The full quantum cone at arbitrary qubit number is not
known, but many classes of inequalities are 681320 ; 4215134 ; Linden:2004ebt
; Schnitzer:2022exe .

Instead, we will be interested in two smaller cones: the stabilizer cone, and
the holographic entropy cone. The stabilizer cone Linden:2013kal ;
doi:10.1063/1.4818950 ; HernandezCuenca:2019wgh ; Bao:2020zgx ; Bao:2020mqq
is defined as the smallest convex cone which contains all stabilizer states.
Since all states we study are stabilizer states, they will all lie within the
stabilizer cone. Although we will not study this cone in further detail, we
will use the fact that it is larger than our next cone: the holographic
entropy cone.

As defined in Bao:2015bfa , the holographic entropy cone is the smallest
convex cone in entropy space which contains all quantum states that have a
dual representation as a classical gravity state. The Ryu-Takayanagi formula
relates the entanglement entropies for subregions of field theoretic states to
areas of extremal surfaces in their dual holographic geometries. The geometry
of these extremal surfaces constrains the allowed entropy vectors. The first
such constraint was the monogamy of mutual information333As discussed in
Bao:2015bfa ; HernandezCuenca:2019wgh , at 6 qubits (5 regions), further
entropy inequalities arise not described here. Since we limit our detailed
discussion to 5 qubits or fewer, the Araki-Lieb, subadditivity, and monogamy
inequalities are sufficient to test if a state lies within the holographic
entropy cone.  Hayden:2011ag ,

$S_{IJ}+S_{IK}+S_{JK}\geq S_{IJK}+S_{I}+S_{J}+S_{K}.$ (14)

Here again $I,\,J,\,K$ are disjoint sets of qubits. This inequality is not
obeyed by all quantum states, nor by all stabilizer states, but it is obeyed
by all states which have a dual smooth classical geometry. As a consequence,
it sets the first boundary between the stabilizer and holographic cones. As we
will review below, beginning at four qubits (or, in the holographic dual
language, three regions plus a purifier), some stabilizer states cannot have a
smooth holographic dual, because they do not lie within the holographic cone.
One of our interests in studying the reachability diagrams is to understand
what gate actions on a given state can move it from within the holographic
cone to outside of it. These gate actions then describe how to create a state
whose geometry is definitely nonclassical.

##  3 The Two-Qubit Stabilizer Graph

At two qubits, the reachability graph contains $60$ vertices, representing the
full set of 2-qubit stabilizer states. These states are connected by the six
Clifford gates: $H_{1},\,H_{2},\,P_{1},\,P_{2},\,CNOT_{1,2},$ and
$CNOT_{2,1}$. The full graph is visible in Figure 4.

As discussed in section 2.2, 2-qubit stabilizer states have a one-component
reduced entropy vector $S_{A}$. For these states, $S_{A}$ is either zero or
one,444Since we are working with qubits, we measure entropies using
$\log_{2}$. That is, the reduced density matrix of one qubit in a maximally-
entangled pair has von Neumann entropy 1. so states on the reachability graph
(represented by vertices) are either unentangled (blue) or form an maximally
entangled pair (red). Since only a CNOT gate can alter the entropy vector, the
graph has two subgraphs which are connected only by CNOT gates (pink lines).
One subgraph has all of the entangled stabilizer states, while the other has
all of the unentangled ones. At any number of qubits, removing all of the CNOT
gates breaks the full graph into subcomponents. Each subcomponent has the same
entropy vector throughout.

The graph here depicts every gate acting on each vertex. Since some of these
gate actions act trivially on particular states, the graph contains loops.
These loops thus represent gate actions which stabilize the state represented
by the vertex attached to the loop. This graph also contains degenerate gate
action, i.e. multiple edges that map one vertex to another as can be seen in
the bottom-rightmost pair connected by $H_{1}$ and $H_{2}$. Beginning in
section 3.2, we will suppress trivial loops since we are most interested in
understanding the gates that move us between states.

Figure 4: This complete 2-qubit reachability graph depicts the full map
between all stabilizer states under action of the Clifford group. Edges
depicting $H_{i}$ and $CNOT_{i,j}$ are undirected since they are each their
own inverse. $P$ is not its own inverse since $P^{4}=I$; consequently the
phase gates are represented by directed edges. The color of the edge indicates
the type of gate, and the line texture (solid vs. dashed) indicates the qubits
which the gate acts on.

While Figure 4 completely describes the connections between 2-qubit stabilizer
states via the Clifford gates, the graph’s complexity obscures some of the
important features. At higher qubit numbers, the full graphs quickly increase
in complexity; we thus relegate their complete graphs to Appendix A. In order
to further explore the structure of the 2-qubit graph, and to extend our
understanding to higher qubits, we will now explore restricted graphs which
only depict the action of various subsets of the Clifford gates.

###  3.1 Restricted Graphs

Beginning at three qubits, we will further restrict our focus to the
restricted graphs composed of the Hadamard and CNOT gates on only the first
two qubits. To help motivate why we concentrate on this gate subset, we begin
by constructing several different restricted graphs for two qubits.

We construct a restricted graph by considering only select operations of the
full Clifford group. This restriction corresponds to removing edges,
representing eliminated gate operations, from the complete reachability graph.
Each restricted graph reveals different details about the connectivity and
physics of the stabilizer states.

####  3.1.1 Two-Qubit Hadamard

We begin by considering only the two Hadamard gates on two qubits, $H_{1}$ and
$H_{2}$, as in Figure 5. The Hadamard gate $H_{i}$ enacts a basis change on
the $i$th qubit. Consequently, Hadamards on different qubits commute. Since
each Hadamard gate also satisfies $H_{i}^{2}=\mathbbm{1}$, each subgraph can
have at most four different states. Indeed, the majority of the 2-qubit states
organize themselves into squares, consisting of a starting state
${|{\psi}\rangle}$ and the states $H_{1}{|{\psi}\rangle}$,
$H_{2}{|{\psi}\rangle}$, and
$H_{1}H_{2}{|{\psi}\rangle}=H_{2}H_{1}{|{\psi}\rangle}$. Additionally, since
Hadamard gates cannot change entanglement, each square has the same entropy
vector throughout.

Figure 5: The $14$ squares and $2$ connected pair subgraphs in the graph
restricted to $H_{1}$ and $H_{2}$ at two qubits. Since $[H_{1},H_{2}]=0$ and
$H_{1}^{2}=H_{2}^{2}=\mathbbm{1}$, these subgraphs are the only allowed
shapes. The four states which connect in pairs rather than squares are given
in Equation (15).

The four states arranged in pairs in the lower right of Figure 5 are worth
further note. They are

$\displaystyle{|{00}\rangle}+{|{11}\rangle},\qquad{|{00}\rangle}+{|{01}\rangle}+{|{10}\rangle}-{|{11}\rangle}$
$\displaystyle=H_{1}\left({|{00}\rangle}+{|{11}\rangle}\right)=H_{2}\left({|{00}\rangle}+{|{11}\rangle}\right);$
$\displaystyle{|{01}\rangle}-{|{10}\rangle},\qquad{|{00}\rangle}-{|{01}\rangle}-{|{10}\rangle}-{|{11}\rangle}$
$\displaystyle=H_{1}\left({|{01}\rangle}-{|{10}\rangle}\right)=H_{2}\left({|{01}\rangle}-{|{10}\rangle}\right).$
(15)

Since $H_{1}$ and $H_{2}$ produce the same action on each of these states,
their subgraphs thus form degenerate pairs instead of full squares. That is,
the four states given in (15) are eigenstates of $H_{1}\otimes H_{2}$; the two
states given in the first line have eigenvalue $+1$, while the two states on
the second line have eigenvalue $-1$.

####  3.1.2 Two-Qubit Phase and Hadamard

Considering both the Hadamard and phase gates yields the restricted graph
depicted in Figure 6. The right subgraph of Figure 6 contains all entangled
states. The left subgraph of this figure consists entirely of unentangled
states. This bisection of the restricted graph is required by the removal of
CNOT edges since neither Hadamard nor phase can alter the entanglement
entropies.

Figure 6: Removing the set of CNOT operations from the full reachability graph
Figure 4 reveals two disconnected subgraphs. Since only the CNOT gates can
change the entropy, each subgraph has the same entropy vector for all of its
states.

The Hadamard boxes from Figure 5 are clearly visible in the unentangled
subgraph. For the entangled state subgraph, the two degenerate pairs are
present at the upper left and lower right, while the Hadamard boxes are still
present but slightly harder to visualize. Removing the phase gates of course
reproduces the Hadamard-only Figure 5, while removing both Hadamard operations
yields the phase-only restricted graph (Figure 19 in Appendix A).

The four basis states
$({|{00}\rangle},{|{01}\rangle},{|{10}\rangle},{|{11}\rangle})$, are located
at the corners of the unentangled subgraph.555Qubits added to a system are
appended to the right of the quantum register, as described in section 2.1.
Thus, an $n$-qubit product state is represented by ${|{a_{1}\ldots
a_{n}}\rangle}\equiv{|{a_{1}}\rangle}_{1}\otimes\ldots{|{a_{n}}\rangle}_{n}$.
Since both phase gates act trivially on basis states, each corner has two
attached directed loops. For the other sixteen states on the outside of the
unentangled subgraph (four on each side), one of the phase gates is trivial
while the other makes a square (since $P_{i}^{4}=\mathbbm{1}$). As an example,
the state ${|{+0}\rangle}=H_{1}{|{00}\rangle}$ satisfies

$P_{2}{|{+0}\rangle}={|{+0}\rangle},\quad
P_{1}{|{+0}\rangle}={|{i0}\rangle},\quad
P_{1}^{4}{|{+0}\rangle}={|{+0}\rangle}.$ (16)

The remaining 16 states in the center of the unentangled graph are connected
by 4 $P_{1}$ squares and 4 $P_{2}$ squares, arising because
$P_{1}^{4}=P_{2}^{4}=\mathbbm{1}$.

In the entangled subgraph, again the 16 states in the center are connected by
4 $P_{1}$ squares and 4 $P_{2}$ squares. The remaining 8 entangled states, 4
at the top and 4 at the bottom of the entangled subgraph in Figure 6, are in
degenerate pairs. For these states, either
$P_{1}{|{\psi}\rangle}=P_{2}^{3}{|{\psi}\rangle}$ or
$P_{1}{|{\psi}\rangle}=P_{2}{|{\psi}\rangle}$. More precisely,
${|{01}\rangle}+{|{10}\rangle}$ is one of the states at the top of the figure,
so it satisfies

$P_{1}({|{01}\rangle}+{|{10}\rangle})=P_{2}^{3}({|{01}\rangle}+{|{10}\rangle})$
(17)

while ${|{00}\rangle}+{|{11}\rangle}$ is at the bottom of the figure and
satisfies

$P_{1}({|{00}\rangle}+{|{11}\rangle})=P_{2}({|{00}\rangle}+{|{11}\rangle}.$
(18)

Consequently, the phase squares degenerate to connected pairs on all eight of
these states.

We can also see the single-qubit reachability diagram reflected here. In the
unentangled graph, removing $H_{2}$ results in 6 copies of the one-qubit
diagram in Figure 2, arranged vertically in Figure 6. The leftmost copy
results from tensoring the six stabilizer states on the first qubit
${|{0}\rangle},\,{|{1}\rangle},\,{|{\pm}\rangle},\,{|{\pm i}\rangle}$ with the
state ${|{0}\rangle}$ on the second qubit. Similarly the rightmost copy
includes the states
${|{01}\rangle},\,{|{11}\rangle},\,{|{\pm,1}\rangle},\,{|{\pm i,1}\rangle}$.
The four copies in the middle, still connected by the phase gate $P_{2}$, are
constructed similarly except with ${|{\pm}\rangle},\,{|{\pm i}\rangle}$ for
the second qubit. This tensoring is our first example of a lift, in this case
from a one-qubit structure to a two-qubit structure.

In general, a lift of a $k$-qubit state ${|{\psi}\rangle}$ to $n$ qubits is a
quantum channel which maps ${|{\psi}\rangle}$ into $\mathbb{C}^{2n}$ by first
tensoring on a $(n-k)$-qubit state ${|{\phi}\rangle}$ and then applying an
operator $\mathcal{O}$:

${|{\psi}\rangle}\rightarrow\mathcal{O}\left({|{\Psi}\rangle}\otimes{|{\phi}\rangle}\right):\mathcal{O}\in
L(\mathbb{C}^{2n}),\>{|{\phi}\rangle}\in\mathbb{C}^{2(n-k)}.$ (19)

We will always have in mind lifts which take stabilizer states to stabilizer
states, so we will take ${|{\phi}\rangle}$ to be an $(n-k)$-qubit stabilizer
state and $\mathcal{O}$ to be a product of $n$-qubit Clifford gates. Given
these restrictions, all lifts from $\mathbb{C}^{2k}$ to $\mathbb{C}^{2n}$ are
on the same footing, and indeed lifts of ${|{\psi}\rangle}$ also successfully
lift any other $k$-qubit stabilizer state. What makes a particular lift useful
is that it preserves some of the structure seen at $k$ qubits when we go to
$n$ qubits. In the example above, the lift given by
${|{\phi}\rangle}={|{0}\rangle}$, $\mathcal{O}=I$ mapped all six one-qubit
stabilizer states to six two-qubit stabilizer states, preserving their
arrangement in the one-qubit reachability graph.

The entangled subgraph is not composed of tensor products of one-qubit
stabilizer states, so it is unsurprising that the one-qubit reachability
diagram has become more complicated. We still see four almost-copies of the
one-qubit diagram, except the Hadamard gate which connects ${|{\pm i}\rangle}$
in Figure 2 instead connects the four almost-copies to each other. Last,
removing $H_{1}$ instead of $H_{2}$ results in exactly the same structures; we
have just chosen to arrange the entangled subgraph to prioritize the $H_{1}$
structure.

####  3.1.3 Two-Qubit Phase and CNOT

Considering only operations of the subgroup generated by phase and CNOT, we
observe a reduced graph composed of five disconnected substructures (Figure
7). Each substructure is inherited from a combination of phase-only and CNOT-
only restricted graphs (Figures 19 and 20 in Appendix A).

Figure 7: Subgraph of 2-qubit complete reachability graph restricted the
subgroup generated by CNOT and phase operations. Included structures are
various attachments of the simpler structures found in the only-phase and
only-CNOT restricted graphs in Figures 19 and 20 in Appendix A.

Again, only CNOT gates can alter the entropy, so states with different
entanglement are connected only via CNOT gates. However, only some CNOT gates
modify the entropy. For example,

$CNOT_{1,2}{|{00}\rangle}=CNOT_{2,1}{|{00}\rangle}={|{00}\rangle}.$ (20)

Since $P_{1}$ and $P_{2}$ also stabilize ${|{00}\rangle}$, this state is
represented by the isolated vertex in Figure 7. The CNOT gates permute the
remaining basis states ${|{01}\rangle},\,{|{10}\rangle},\,{|{11}\rangle}$
among each other since CNOT can only flip a bit, not introduce a
superposition. As with ${|{00}\rangle}$, phase acts trivially on all the
remaining basis states.

The upper right subgraph in Figure 7 consists of $8$ unentangled states and
$4$ entangled states. The unentangled states arrange into a $P_{1}^{4}$ cycle
to the left, and a $P_{2}^{4}$ cycle to the right. For the four central
entangled states, $P_{1}{|{\psi}\rangle}=P_{2}{|{\psi}\rangle}$, so they are
linked in one phase cycle. The entangled states and unentangled states are
necessarily connected by CNOT gates. In this subgraph, all CNOT gates either
act trivially or move between entangled and unentangled states. The CNOT gates
alone connect the states into 4 lines of 3 states each.

The bottom left subgraph is similar, except the central four entangled states
satisfy $P_{1}{|{\psi}\rangle}=P_{2}^{-1}{|{\psi}\rangle}$ instead. The three
top states, and the three bottom states, both have either trivial CNOT action
or CNOT moves between an entangled and unentangled states. For the middle
states, however, both CNOT gates act nontrivially on each state, producing a
hexagon. Explicitly, we have the cycle

${|{1i}\rangle}=CNOT_{2,1}CNOT_{1,2}CNOT_{2,1}CNOT_{1,2}CNOT_{2,1}CNOT_{1,2}{|{1i}\rangle},$
(21)

where only two of the states along the way are entangled.

Finally the largest structure, located in the upper left of Figure 7, contains
only states stabilized by neither $P_{1}$ nor $P_{2}$. Again the CNOT gates
are the only connections between the entangled and unentangled states.
Considering only these CNOT gates, this subgraph contains the remaining $2$
hexagonal cycles from the CNOT-only graph (Appendix A, Figure 20), as well as
$6$ 3-state lines of paired CNOT edges, and $2$ additional states invariant
under both CNOT gates.

###  3.2 Two-Qubit Hadamard and CNOT

For the remainder of this work, we will focus on restricted graphs generated
by $H_{1},\,H_{2},$ $CNOT_{1,2},$ and $CNOT_{2,1}$ (see Figure 8 for the two-
qubit version). In order to understand changes in entanglement, we need to
consider a restricted graph which includes CNOT gates since they are the only
gates which alter entropy. We specifically choose to include the Hadamard
gates (and exclude the phase gates) because in the Hadamard-CNOT restricted
graphs, each subgraph contains states with different entropy vectors.
Additionally, at two qubits, the Hadamard-CNOT graph will have only two
subgraphs; we will see echoes of these structures repeated at higher qubit
number. When we go to higher qubit number, we will continue to use only the
gate set $H_{1},\,H_{2},\,CNOT_{1,2},CNOT_{2,1}$ because all entropic
arrangements found in stabilizer states can be built from successive bipartite
entanglements.

\begin{overpic}[width=346.89731pt]{TwoQubitH1H2CNOT12CNOT21.pdf}
\put(37.0,33.0){$g_{24}$} \put(27.0,13.0){$g_{36}$}
\put(76.6,41.4){\footnotesize{$\leftarrow{|{00}\rangle}$}}
\put(54.1,42.0){\footnotesize{${|{GHZ}\rangle}\downarrow$}} \end{overpic}
Figure 8: The 2-qubit subgraph restricted to $H_{1},H_{2},CNOT_{1,2},$ and
$CNOT_{2,1}$ has two subgraphs which are connected only via phase gates.
Trivial loops have been removed in this representation of the 2-qubit
$H_{1},H_{2},CNOT_{1,2},CNOT_{2,1}$ restricted graph.

Beginning with this graph, and continuing below, we omit any gate whose action
is the identity; thus no further trivial loops will appear. Because we have
four possible gates that can act on each state, a vertex with valency $4-k$
has $k$ trivial loops. We also label the subgraph structures by the number of
vertices they contain, so e.g. we use $g_{24}$ for the $24$-vertex
substructure in Figure 8. As noted in Footnote 2 above, we have the relation

$CNOT_{2,1}=H_{1}H_{2}CNOT_{1,2}H_{2}H_{1},$ (22)

which can be checked explicitly for $2$-qubit states using the figure; hence
the presence of the $CNOT_{2,1}$ edges is completely fixed by the structure of
the other three gates.

The subgraph $g_{24}$ contains all 2-qubit stabilizer states connected to the
basis states $({|{00}\rangle},{|{01}\rangle},{|{10}\rangle},{|{11}\rangle})$
via only Hadamard and CNOT operations. As we can see from the graph, acting a
Hadamard and then a CNOT on ${|{00}\rangle}$ produces the GHZ state, which is
entangled:

$CNOT_{1,2}H_{1}{|{00}\rangle}={|{GHZ}\rangle}.$ (23)

Because the phase gate is the only Clifford gate with imaginary matrix
elements, all states in $g_{24}$ can be written as superpositions of the basis
states with purely real coefficients. States located in $g_{36}$, on the other
hand, have relative phases $\pm i$ between different basis components.
Accordingly, a phase gate is required to move between the two subgraphs. In
fact, every phase gate either acts trivially, or connects states in $g_{24}$
with states in $g_{36}$, since the CNOT and Hadamard gates are both Hermitian,
but phase is not. Thus, the only way a product of CNOTs and Hadamards can have
the same action as a non-Hermitian operator $O$ on a state is when the state
has support only on the eigenspaces of the operator $O$ with real eigenvalues.
For $O=\mathrm{phase}$, the only eigenspace with real eigenvalue has
eigenvalue one. Hence the phase gate either acts as the identity or its action
is non-Hermitian (and thus moves us between the $g_{24}$ and $g_{36}$
subgraphs). We will see echoes of this structure when comparing subgraphs in
the $H_{1},\,H_{2},\,CNOT_{1,2},\,CNOT_{2,1}$ restricted graphs at higher
qubit number.

In our analysis at higher qubits, we will also rely on Hamiltonian paths and
Hamiltonian cycles. Hamiltonian paths visit every vertex in a graph only once
(and thus do not self-intersect). Hamiltonian cycles are closed loops with the
same property. The subgraph $g_{24}$ has no Hamiltonian paths, and therefore
no Hamiltonian cycles either. Subgraph $g_{36}$ does have Hamiltonian paths
(although again no Hamiltonian cycles). One example is shown in Figure 9. The
specific circuit depicted is

$\begin{split}\mathcal{C}\equiv(H_{2},&H_{1},H_{2},CNOT_{1,2},H_{2},H_{1},H_{2},CNOT_{1,2},H_{2},CNOT_{1,2},H_{2},CNOT_{1,2},\\\
&H_{1},H_{2},H_{1},CNOT_{1,2},H_{1},CNOT_{1,2},H_{2},CNOT_{1,2},H_{2},H_{1},H_{2},CNOT_{1,2},\\\
&H_{1},CNOT_{1,2},H_{2},CNOT_{1,2},H_{2},H_{1},H_{2},CNOT_{2,1},H_{2},H_{1},H_{2}).\end{split}$
(24)

When applying this circuit to the graph, the leftmost gate acts first and the
rightmost gate last.

\begin{overpic}[width=411.93767pt]{TwoQubitHamiltonianPath.pdf}
\put(-6.5,18.0){\footnotesize{${|{-,-i}\rangle}$}}
\put(-3.0,12.0){\footnotesize{${|{i,1}\rangle}$}} \end{overpic} Figure 9:
Hamiltonian path $\mathcal{C}$ beginning on state ${|{i,1}\rangle}$ (encircled
in green) and ending on state ${|{-,-i}\rangle}$ (encircled in red).

Note that the path $\mathcal{C}$ starting on ${|{i1}\rangle}$ is not unique;
other Hamiltonian paths exist on $g_{36}$. First, $\mathcal{C}$ also traverses
a Hamiltonian path starting on ${|{-i,1}\rangle}$ instead. Another example is
$\mathcal{C^{\top}}$, which swaps qubits 1 and 2 in (24). Two further
possibilities are the inverse paths $\mathcal{C}^{-1}$ and
$(\mathcal{C}^{\top})^{-1}$, which simply apply the respective circuits in
reverse order. In fact, there exists a Hamiltonian path on $g_{36}$ starting
from $30$ (out of $36$) of its vertices.

For definiteness, we will use the Hamiltonian path $\mathcal{C}$, starting on
${|{i1}\rangle}$, in the lifting procedure introduced in Section 4.2, where we
turn to the three-qubit stabilizer graph and a detailed analysis of its
reduced graphs.

##  4 The Three-Qubit Stabilizer Graph

At three qubits there are now $1080$ stabilizer states, and accordingly the
full reachability graph, which we defer to Figure 21 in Appendix A, becomes
unwieldy; we thus proceed in this section immediately to the restricted
graphs. For three qubits, many of the reduced graphs show features similar to
two qubits. As an example, the Hadamard-only restricted graph at three qubits
contains cubes and degenerate squares instead of squares and degenerate pairs.
As before, each subgraph has only one entropy type, since Hadamard gates
cannot alter the entropy vector. Figure 10 shows the structures exhibited in
the $H_{1},\,H_{2},\,H_{3}$ restricted three-qubit graph. The phase graph at
three qubits similarly extends, as exhibited by the full phase graph
$P_{1},\,P_{2},\,P_{3}$, shown in Appendix A in Figure 22.

Figure 10: Pictured are the two unique subgraph types that occur in the
3-qubit restricted graph of only $H_{1},H_{2},$ and $H_{3}$ operations.
Degenerate pairs in the 2-qubit $H_{1},H_{2}$ restricted graph (Figure 5) are
promoted to boxes at three qubits. Boxes from the 2-qubit $H_{1},H_{2}$ graph
are promoted to cubes by the addition of the $H_{3}$ gate.

In all of these graphs, the color of the vertex indicates the entropy vector
of the associated state. As reviewed in section 2.2 the entropy vector at
three qubits is $\vec{S}=(S_{A},S_{B},S_{AB})$. There are five different
entropy vectors among the 3-qubit stabilizer states, as shown in Table 3. All
five entropy vectors lie within the holographic cone (that is, they satisfy
the inequalities required for a holographic state, as discussed in Section
2.2).

Just as in the 2-qubit case, the entropy vector can only be changed by the
action of a CNOT gate. Accordingly the phase– and Hadamard–restricted graphs
do not allow us to study changes in entropy. Instead, as discussed in Section
3.2, we are most interested in the restricted graph which considers only
$H_{1},\,H_{2},\,CNOT_{1,2},$ and $CNOT_{2,1}$, since it will allow us to
understand the changes in entropy induced by the CNOT gates on a pair of
qubits.

###  4.1 Three-Qubit CNOT+Hadamard on 1 and 2 only

We extend our analysis to three qubits, restricting the full stabilizer group
to the subgroup generated by $H_{1},\,H_{2},\,CNOT_{1,2}$, and $CNOT_{2,1}$.
We concentrate on this gate set in order to focus on the entropic structure of
qubits $1$ and $2$. The choice of qubits $1$ and $2$ is arbitrary; any pair
would do. Because all stabilizer gates act on at most two qubits, and all
entropic structure can be built from these bipartite interactions, our
analysis of qubits $1$ and $2$ is sufficient to understand reachability for
all $n$ qubits.

The full restricted graph for the gate set
$H_{1},\,H_{2},\,CNOT_{1,2},\,CNOT_{2,1}$ is shown in Figure 23 of Appendix A.
There are only four types of subgraph, shown in Figure 11, which arise in the
full restricted graph. The full graph consists of $6$ copies of $g_{24}$ and
$g_{36}$, $3$ copies of $g_{144}$, and a single copy of $g_{288}$, where as in
the previous subsection the subscript denotes the number of vertices in the
subgraph.

\begin{overpic}[width=407.60385pt]{ThreeQubitH1H2CNOT12CNOT21Subgraphs.pdf}
\put(20.0,54.0){$g_{24}$} \put(58.0,53.0){$g_{36}$} \put(17.1,42.0){$g_{144}$}
\put(56.6,43.0){$g_{288}$} \end{overpic} Figure 11: This figure depicts the
four unique subgraphs which arise in the full 3-qubit
$H_{1},H_{2},CNOT_{1,2},CNOT_{2,1}$ restricted graph, pictured in Figure 23 of
Appendix A. Each subgraph is labeled by its corresponding vertex count. The
full graph consists of $6$ copies of $g_{24}$ and $g_{36}$, $3$ copies of
$g_{144}$, and a single copy of $g_{288}$.

As mentioned above, there are only five different entropy vectors at three
qubits. In Table 3, we record the number of stabilizer states with each of
these entropy vectors, and also record the subgraph types those states appear
in.

Holographic | $\left(S_{A},S_{B},S_{AB}\right)$ | Number of States | Subgraph
---|---|---|---
Yes | ${\color[rgb]{0,0,1}\bullet}(0,0,0)$ | $216$ | $g_{24},g_{36}$
Yes | ${\color[rgb]{1,0,0}\bullet}(1,1,0)$ | $144$ | $g_{24},g_{36}$
Yes | ${\color[rgb]{.5,0,.5}\bullet}(0,1,1)$ | $144$ | $g_{144},g_{288}$
Yes | ${\color[rgb]{1,.5,0}\bullet}(1,0,1)$ | $144$ | $g_{144},g_{288}$
Yes | ${\color[rgb]{1,1,0}\bullet}(1,1,1)$ | $432$ | $g_{144},g_{288}$
Table 3: Table of 3-qubit entropy vectors. The last column lists which
subgraph types exhibit each entropy vector.

As discussed in section 3.2, phase actions on states in the
$H_{1},\,H_{2},\,CNOT_{1,2},\,CNOT_{2,1}$ restricted graph are always either
trivial, or move you to a different subgraph. However, unlike in the 2-qubit
case, in addition to the phase gates we are also not representing $H_{3}$, nor
any CNOT gate involving the third qubit. Thus, not every subgraph can be
reached by phase actions alone: applying a phase gate does not change the
entropy vector, and two of the five entropy vectors appear only in
$g_{24},g_{36}$ while the other three appear in only $g_{144},g_{288}$. We are
also not representing $H_{3}$, nor any CNOT gate involving the third qubit.

In particular, the subgraphs of type $g_{144}$ and $g_{288}$ contain the
entropy vectors $(0,1,1)$, $(1,0,1)$, and $(1,1,1)$, but the subgraphs of type
$g_{24}$ and $g_{36}$ only contain the entropy vectors $(0,0,0)$ and
$(1,1,0)$. This separation occurs because CNOT gates involving the third qubit
are required to change between the two sets of entropy vectors. As reviewed in
section 2.2, the $CNOT_{i,j}$ gate can only alter entropies $S_{I}$ where
qubit $i\in I$ but qubit $j\in\bar{I}$. Our entropy vectors are listed as
$(S_{A},S_{B},S_{AB})$, where $A$ refers to qubit 1 and $B$ refers to qubit 2.
So the $CNOT_{1,2}$ or $CNOT_{2,1}$ gates can only affect $S_{A}$ and $S_{B}$,
not $S_{AB}$. Thus, entropy vectors with different $S_{AB}$ can only show up
on separate subgraphs in our restricted graph.

As we will see in the next section, we can understand both the number of
copies of each subgraph, as well as the shape of the subgraphs themselves, by
seeing how the 2-qubit states and subgraphs can be embedded into the 3-qubit
structures.

###  4.2 Lifting States from Two Qubits to Three Qubits

We apply the lifting procedure introduced in Section 3.1.2 to lift 2-qubit
states to three qubits. States in the 2-qubit $g_{24}$ subgraph all lift to
states in the 3-qubit $g_{24}$ subgraphs by tensoring on a third qubit. For
example, starting with the state ${|{00}\rangle}$ on qubits 1 and 2, we can
tensor on ${|{0}\rangle}$ on qubit 3 to find

${|{000}\rangle}={|{00}\rangle}\otimes{|{0}\rangle}.$ (25)

There are $6$ possible states of the third qubit:
$\\{{|{0}\rangle},\,{|{1}\rangle},\,{|{\pm}\rangle},\,{|{\pm i}\rangle}\\}$.
Tensoring on all $6$ of these states, as in (25), to the $24$ states in the
2-qubit $g_{24}$ subgraph generates $144$ of the 3-qubit stabilizer states.
These $144$ states make up the $6$ copies of $g_{24}$ at three qubits (shown
in Figure 23 of Appendix A).

The next simplest set of lifts begins with states in the 2-qubit $g_{36}$
subgraph, and then tensors on a third qubit. We start with the 2-qubit state
${|{i1}\rangle}$, since the circuit $\mathcal{C}$ (24) defines a Hamiltonian
path on $g_{36}$ starting at this state (Figure 9). The process

${|{i1+}\rangle}={|{i1}\rangle}\otimes{|{+}\rangle},$ (26)

lifts ${|{i1}\rangle}$ to one of the six 3-qubit $g_{36}$ subgraphs. Since
there are again 6 1-qubit stabilizer states available for the third qubit, we
generate a further 216 states via this lift. These 216 states are all located
on one of the $6$ copies of $g_{36}$ at three qubits, completely covering
those subgraphs.

When we tensor on a third qubit, we extend the entropy vector of the original
2-qubit state. For a 2-qubit state with entropy vector $(S_{A})$, we have
$S_{A}=S_{B}$ since it is a pure state. When we tensor on the third qubit,
these two values stay the same. Additionally, the third qubit is just tensored
on, so it is not entangled; thus $S_{O}=S_{AB}=0$. Accordingly, states with
the entropy vector $(1)$ lift to states with $(1,1,0)$ and those with entropy
vector $(0)$ lift to $(0,0,0)$ under the tensoring lift of Equations (25) and
(26).

We have accounted so far for all of the states in the $g_{24}$ and $g_{36}$
subgraphs at three qubits. As shown in Table 3, these are the only states with
entropy vector $(0,0,0)$ or $(1,1,0)$. To modify entanglement and reach new
3-qubit entropy arrangements, we must act with CNOT gates involving the third
qubit, i.e. $CNOT_{1,3},\,CNOT_{3,1},\,CNOT_{2,3},$ or $CNOT_{3,2}$. For
example, acting with $CNOT_{3,1}$ on the lifted state ${|{i1+}\rangle}$ in
Equation (26) gives

$CNOT_{3,1}{|{i1+}\rangle}={|{010}\rangle}+i{|{011}\rangle}+i{|{110}\rangle}+{|{111}\rangle}.$
(27)

This procedure lifts ${|{i1}\rangle}$ from the 2-qubit $g_{36}$ subgraph to a
state on one $g_{144}$ subgraph at three qubits. Acting with $CNOT_{3,1}$ on
${|{i1+}\rangle}$ entangles qubits $1$ and $3$, resulting in a state with
entropy vector $\vec{S}=(1,0,1)$, indicated with an orange vertex in Figure
11. Similarly replacing the third qubit in ${|{i1+}\rangle}$ with
${|{-}\rangle},\,{|{i}\rangle},$ or ${|{-i}\rangle}$ instead also results in
states on the same copy of $g_{144}$. ${|{i10}\rangle}$ and ${|{i11}\rangle}$,
however, return to the same copies of $g_{36}$ they started on under the
action of $CNOT_{3,1}$.

Acting a different gate on a lifted version of ${|{i1}\rangle}$ can result in
a lifted state on a different copy of $g_{144}$. For example, the gate
$CNOT_{1,3}$ on state
${|{i10}\rangle}\equiv{|{i1}\rangle}\otimes{|{0}\rangle}$ gives

$CNOT_{1,3}{|{i10}\rangle}={|{010}\rangle}+i{|{111}\rangle},$ (28)

which resides on a different copy of $g_{144}$ than the state lifted in
Equation (27). As before, the third ket can be replaced with
${|{1}\rangle},\,{|{i}\rangle}$ or ${|{-i}\rangle}$ resulting in three other
states on the same copy of $g_{144}$, but using ${|{\pm}\rangle}$ results in
$CNOT_{1,3}$ mapping back to the same copies of $g_{36}$.

The third copy of $g_{144}$ is phase-separated from the previous two. Lifting
2-qubit starting states to this subgraph requires a phase gate. One possible
such lift to this subgraph is

$P_{3}(CNOT_{1,3}{|{i10}\rangle})={|{010}\rangle}-{|{111}\rangle}.$ (29)

Again, the third qubit can be replaced with ${|{1}\rangle},\,{|{i}\rangle}$ or
${|{-i}\rangle}$ giving three further states on the last copy of $g_{144}$.

There are two ways that we can lift the 2-qubit state ${|{i1}\rangle}$ to the
3-qubit $g_{288}$ subgraph. The first begins by tensoring on a third qubit,
then applying the appropriate CNOT gate to move the lifted state from $g_{36}$
directly to $g_{288}$. For example,

$CNOT_{3,2}{|{i1+}\rangle})={|{010}\rangle}+i{|{011}\rangle}+{|{100}\rangle}+i{|{101}\rangle}$
(30)

resides on $g_{288}$. It has entropy vector $\vec{S}=(0,1,1)$, and is
represented by a purple vertex in $g_{288}$ in Figure 11. The same procedure
works when we replace the third qubit by ${|{-}\rangle},\,{|{i}\rangle},$ or
${|{-i}\rangle}$.

The second method first lifts state ${|{i1}\rangle}$ to $g_{144}$ as in
Equation (27), entangling qubits $1$ and $3$. Applying a second CNOT gate then
entangles qubit $2$ with the other qubits. Explicitly, we have e.g.

$CNOT_{3,2}(CNOT_{3,1}{|{i1+}\rangle})={|{010}\rangle}+i{|{011}\rangle}+i{|{100}\rangle}+{|{101}\rangle}.$
(31)

This process results in a final state on $g_{288}$. Its entropy vector is
$(1,1,1)$, represented by a yellow vertex in $g_{288}$. Three further states
arise by replacing the third qubit ${|{+}\rangle}$ with
${|{-}\rangle},\,{|{i}\rangle},$ or ${|{-i}\rangle}$.

In the next subsection, we will use the lifted states described here, combined
with the 2-qubit Hamiltonian path $\mathcal{C}$, to reach the remaining states
as well as to understand the new $g_{144}$ and $g_{288}$ subgraph structures
that arise at three qubits.

###  4.3 Lifting Paths from Two Qubits to Three Qubits

At two qubits, the Hamiltonian path $\mathcal{C}$ (24) starting from the state
${|{i1}\rangle}$ covered the $g_{36}$ subgraph. Similarly, $\mathcal{C}$
starting from the lifted states described in the previous section, listed in
Table 4, cover every vertex exactly once on each of the $g_{36}$, $g_{144}$,
and $g_{288}$ subgraphs of the $H_{1},\,H_{2},\,CNOT_{1,2},\,CNOT_{2,1}$
restricted graph at three qubits.

Lift | Starting State | Subgraph
---|---|---
${|{i1}\rangle}\otimes{|{0}\rangle}$ | ${|{i10}\rangle}$ | $g^{1}_{36}$
${|{i1}\rangle}\otimes{|{1}\rangle}$ | ${|{i11}\rangle}$ | $g^{2}_{36}$
${|{i1}\rangle}\otimes{|{+}\rangle}$ | ${|{i1+}\rangle}$ | $g^{3}_{36}$
${|{i1}\rangle}\otimes{|{-}\rangle}$ | ${|{i1-}\rangle}$ | $g^{4}_{36}$
${|{i1}\rangle}\otimes{|{i}\rangle}$ | ${|{i1i}\rangle}$ | $g^{5}_{36}$
${|{i1}\rangle}\otimes{|{-i}\rangle}$ | ${|{i1-i}\rangle}$ | $g^{6}_{36}$
$CNOT_{3,1}({|{i1}\rangle}\otimes{|{+}\rangle})$ | ${|{i10}\rangle}+i{|{-i,11}\rangle}$ | $g_{144}^{1}$
$CNOT_{3,1}({|{i1}\rangle}\otimes{|{-}\rangle})$ | ${|{i10}\rangle}-i{|{-i,11}\rangle}$
$CNOT_{3,1}({|{i1}\rangle}\otimes{|{i}\rangle})$ | ${|{i10}\rangle}-{|{-i,11}\rangle}$
$CNOT_{3,1}({|{i1}\rangle}\otimes{|{-i}\rangle})$ | ${|{i10}\rangle}+{|{-i,11}\rangle}$
$CNOT_{1,3}({|{i1}\rangle}\otimes{|{0}\rangle})$ | ${|{010}\rangle}+i{|{111}\rangle}$ | $g_{144}^{2}$
$CNOT_{1,3}({|{i1}\rangle}\otimes{|{1}\rangle})$ | ${|{011}\rangle}+i{|{110}\rangle}$
$CNOT_{1,3}({|{i1}\rangle}\otimes{|{i}\rangle})$ | ${|{01i}\rangle}-{|{11,-i}\rangle}$
$CNOT_{1,3}({|{i1}\rangle}\otimes{|{-i}\rangle})$ | ${|{01,-i}\rangle}+{|{11i}\rangle}$
$P_{3}CNOT_{1,3}({|{i1}\rangle}\otimes{|{0}\rangle})$ | ${|{010}\rangle}-{|{111}\rangle}$ | $g_{144}^{3}$
$P_{3}CNOT_{1,3}({|{i1}\rangle}\otimes{|{1}\rangle})$ | ${|{011}\rangle}+{|{110}\rangle}$
$P_{3}CNOT_{1,3}({|{i1}\rangle}\otimes{|{i}\rangle})$ | ${|{01-}\rangle}-{|{11+}\rangle}$
$P_{3}CNOT_{1,3}({|{i1}\rangle}\otimes{|{-i}\rangle})$ | ${|{01+}\rangle}+{|{11-}\rangle}$
$CNOT_{3,2}({|{i1}\rangle}\otimes{|{+}\rangle})$ | ${|{i10}\rangle}+{|{i01}\rangle}$ | $g_{288}$
$CNOT_{3,2}({|{i1}\rangle}\otimes{|{-}\rangle})$ | ${|{i10}\rangle}-{|{i01}\rangle}$
$CNOT_{3,2}({|{i1}\rangle}\otimes{|{i}\rangle})$ | ${|{i10}\rangle}+i{|{i01}\rangle}$
$CNOT_{3,2}({|{i1}\rangle}\otimes{|{-i}\rangle})$ | ${|{i10}\rangle}-{|{i01}\rangle}$
$CNOT_{3,2}CNOT_{3,1}({|{i1}\rangle}\otimes{|{+}\rangle})$ | ${|{i10}\rangle}+i{|{-i,01}\rangle}$
$CNOT_{3,2}CNOT_{3,1}({|{i1}\rangle}\otimes{|{-}\rangle})$ | ${|{i10}\rangle}-i{|{-i,01}\rangle}$
$CNOT_{3,2}CNOT_{3,1}({|{i1}\rangle}\otimes{|{i}\rangle})$ | ${|{i10}\rangle}-{|{-i,01}\rangle}$
$CNOT_{3,2}CNOT_{3,1}({|{i1}\rangle}\otimes{|{-i}\rangle})$ | ${|{i10}\rangle}+{|{-i,01}\rangle}$
Table 4: Using the 26 starting states in this table, and the path
$\mathcal{C}$ (24), we cover every vertex on the $g_{36}$, $g_{144}$, and
$g_{288}$ subgraphs of the $H_{1},\,H_{2},\,CNOT_{1,2},\,CNOT_{2,1}$
restricted graph at 3 qubits. The $g_{24}$ subgraphs cannot be covered by a
single path, but all states in them can be generated from states in the
2-qubit $g_{24}$ subgraph via the lift described in (26).

For the $g_{36}$ subgraphs, the lift only involved a simple tensor product
with the third qubit. Additionally, the lift actually worked on every state in
the $g_{36}$ subgraph, so the structure of $g_{36}$ in each of the $6$ copies
is completely preserved. All lifted states from the 2-qubit $g_{36}$ subgraph
lift to the same relative position in the 3-qubit $g_{36}$ subgraphs.
Therefore, applying $\mathcal{C}$ on each lift of ${|{i1}\rangle}$ on a
3-qubit copy $g_{36}$ still builds a Hamiltonian path on each subgraph.
Accordingly, lifting $\mathcal{C}$ from two qubits to three qubits by lifting
the starting state gives a complete vertex covering of each 3-qubit copy of
$g_{36}$, just as in Figure 9. This covering of higher qubit $g_{36}$
subgraphs by the 2-qubit $g_{36}$ structure persists to arbitrary qubit
number.

Lifting $\mathcal{C}$ to the three copies of $g_{144}$ and the single
$g_{288}$ subgraph illustrates how the two-qubit subgraph $g_{36}$ is embedded
into larger subgraphs at three qubits. Let us consider a specific example,
beginning with the second $g_{144}$ subgraph, which we have termed
$g_{144}^{2}$ in Table 4. As described in Equation (29), the lifted state
$CNOT_{1,3}{|{i10}\rangle}$ is on this subgraph. Applying the circuit
$\mathcal{C}$ to this state produces a non-intersecting path on $g_{144}$,
covering $1/4$ of its states as in Figure 12.

\begin{overpic}[width=346.89731pt]{ThreeQubitLiftedHamiltonianPath.pdf}
\put(27.0,37.3){\footnotesize{$\leftarrow{|{\psi}\rangle}$}} \end{overpic}
Figure 12: Lift of Hamiltonian path $\mathcal{C}$ defined in Equation (24),
from the 2-qubit $g_{36}$ subgraph to a copy of $g_{144}$ at three qubits. The
path begins on ${|{\psi}\rangle}\equiv CNOT_{1,3}{|{i10}\rangle}$, the state
lifted in Equation (29).

Each of the lifted starting states listed under $g_{144}^{2}$ in Table 4 have
entropy vector $\vec{S}=(1,0,1)$, and are located around the central octagonal
structure of $g_{144}$. Applying $\mathcal{C}$ to each of the lifted states
defines a separate path, covering $36$ vertices each, on $g_{144}$. The union
of these $4$ lifts of $\mathcal{C}$ defines a vertex covering of $g_{144}$,
displayed in Figure 13. The other two $g_{144}$ subgraphs can be covered
similarly, by four copies of $\mathcal{C}$ starting on each of the lifted
starting states listed in the table.

\begin{overpic}[width=346.89731pt]{HamiltonianPathCoveringG144.pdf}
\put(84.0,60.5){\footnotesize{$CNOT_{1,3}{|{i,1,0}\rangle}$}}
\put(84.0,54.0){\footnotesize{$CNOT_{1,3}{|{i,1,1}\rangle}$}}
\put(84.0,47.5){\footnotesize{$CNOT_{1,3}{|{i,1,i}\rangle}$}}
\put(84.0,40.8){\footnotesize{$CNOT_{1,3}{|{i,1,-i}\rangle}$}} \end{overpic}
Figure 13: Four copies of Hamiltonian path $\mathcal{C}$ that give a vertex
cover of $g_{144}^{2}$. Each path is indicated by the lifted starting state
where it begins.

A covering of the $g_{288}$ subgraph can also be constructed from lifts of
$\mathcal{C}$. First, we lift the 2-qubit $g_{36}$ state ${|{i1}\rangle}$ via
Equations (30) and (31), as listed in Table 4. We then apply $\mathcal{C}$ on
these $8$ lifted states, covering every vertex of $g_{288}$ once.

We have now reached all of the 3-qubit states, and we have covered the six
subgraphs $g_{36}$, three subgraphs $g_{144}$, and single subgraph $g_{288}$
with lifted versions of the 2-qubit Hamiltonian path $g_{36}$.

Just as we described at the end of section 3.2 for the 2-qubit $g_{36}$
subgraph, these coverings are not unique. We could have lifted the 2-qubit
state ${|{-i,1}\rangle}$ instead, since $\mathcal{C}$ is also a Hamiltonian
path from that starting point. Alternatively, we could have used the path
$\mathcal{C}^{\top}$ or $\mathcal{C}^{-1}$ instead, or any other Hamiltonian
path from the 2-qubit $g_{36}$ subgraph.

Sticking to just $\mathcal{C}$, $\mathcal{C}^{\top}$, $\mathcal{C}^{-1}$, and
$(\mathcal{C}^{\top})^{-1}$, each path has two possible starting states on the
2-qubit $g_{36}$ subgraph. Lifting these eight states to four starting states
each on $g_{144}^{2}$, we hit all 32 purple and orange states in the central
octagon in Figure 13. The same statement works for the other $g_{144}$
subgraphs, so every central octagon state is a lifted starting state for some
2-qubit Hamiltonian path, which together with 3 other states, fully covers the
$g_{144}$ subgraph. A similar situation arises for the $g_{288}$ subgraph.

Rather than exploring these other possible coverings, we will instead move on
to further structures which first arise at three qubits: Hamiltonian cycles
and Eulerian paths.

###  4.4 Hamiltonian cycles and Eulerian paths

At one qubit, the complete reachability diagram did not have a Hamiltonian
path. At two qubits, the $H_{1},\,H_{2},\,CNOT_{1,2},\,CNOT_{2,1}$ restricted
subgraph $g_{24}$ also did not have a Hamiltonian path, but $g_{36}$ did.
Beginning at three qubits, we are able to construct Hamiltonian cycles as
well, on subgraphs $g_{144}$ and $g_{288}$. Subgraph $g_{288}$ will
additionally have an Eulerian cycle, which traverses every path exactly once
(instead of visiting every vertex once).

Hamiltonian cycles cannot be built on the $g_{24}$ and $g_{36}$ subgraphs. For
$g_{24}$ this check is simple: it contains a vertex whose removal disconnects
the graph, known as a cut-vertex; the $GHZ$ state in Figure 8 is one example.
Subgraph $g_{36}$ contains no cut-vertex; however, it can be verified to have
no Hamiltonian cycle by exhaustive search.666Any graph containing a
Hamiltonian cycle must satisfy all of the following conditions: each vertex of
degree $2$ must be included in the Hamiltonian cycle, once two edges incident
to a vertex are included in the Hamiltonian cycle all other edges incident to
that vertex must be removed from consideration, and the Hamiltonian cycle must
contain no proper subcycles. These criteria are only sufficient to eliminate a
graph’s candidacy for having a Hamiltonian cycle. In general, the “Hamiltonian
path problem” of proving that a given graph does admit a Hamiltonian path or
cycle is NP-complete.

For the $g_{144}$ subgraph, the four copies of $\mathcal{C}$ in each graph do
not directly connect into a Hamiltonian cycle simply by connecting their ends.
However, other Hamiltonian paths exist on the 2-qubit $g_{36}$ subgraph, and
some of these paths, when their starting states are lifted to four states on
$g_{144}$, can be directly connected into one large loop that builds a
Hamiltonian cycle.

For the $g_{288}$ subgraph, we have three coverings. First, as in the previous
section, eight copies of $\cal{C}$ (or any other Hamiltonian path from
$g_{36}$ ) completely cover the graph. Next, the Hamiltonian cycle on
$g_{144}$ can be mapped to a pair of cycles which each cover half of
$g_{288}$, as shown in Figure 14. And last, a new Hamiltonian cycle exists on
$g_{288}$ itself.

Figure 14: The Hamiltonian cycle on $g_{144}$ can be mapped to a cycle on
$g_{288}$ through a CNOT operation connecting the two subgraphs. Two copies of
$g_{144}$ can be used to cover $g_{288}$ completely.

We also note that Eulerian cycles, which traverse every edge, first exist on
$g_{288}$. The cycle is of length 576, and it must exist because all vertices
have an even number of attached edges (namely, four). Stated another way, all
four gates in the restricted set $H_{1},\,H_{2},\,CNOT_{1,2},\,CNOT_{2,1}$ act
nontrivially and differently from each other, on every state in the subgraph.
In particular, this means that $g_{288}$ has no trivial loops.

Hamiltonian cycles can be embedded into to higher-qubit subgraphs in almost
exactly the same way as Hamiltonian paths; for cycles we have to pick an
arbitrary starting state to lift. Each Hamiltonian cycle embeds into every
subgraph of greater or equal vertex number. We will also see that embeddings
of Hamiltonian paths and cycles will continue to cover subgraphs at higher
qubit number.

##  5 Towards the $n$-qubit Stabilizer Graph

We will now generalize our discussion to the cases of four and five qubits.
Two particularly important features arise first at four qubits. First, the
last new subgraph shape in the $H_{1},\,H_{2},\,CNOT_{1,2},\,CNOT_{2,1}$
restricted graph appears. Second, some entropy vectors of stabilizer states
will disobey monogamy of mutual information (14). These states thus lie
outside the holographic entropy cone; they cannot have a classical
gravitational representation.

###  5.1 Four Qubits

In order to explore four qubits, we have explicitly generated all of the 36720
4-qubit stabilizer states. The explicit form of these states is available in
the GitHub repository github  along with associated reachability diagrams,
entropy vector data, and a Mathematica package for simulating stabilizer
circuits, generating sets of states, and constructing reachability diagrams.

Examining the $H_{1},\,H_{2},\,CNOT_{1,2},\,CNOT_{2,1}$ restricted graphs, we
find copies of the $g_{24}$ and $g_{36}$ subgraphs we have seen before. We
also see again the $g_{144}$ subgraph with 3 different entropy vectors
hereafter referred to as $g_{144}(3)$, and the $g_{288}$ with 3 entropy
vectors, now termed $g_{288}(3)$. We specify the number of entropy vectors
because we also find $g_{144}(4)$ and $g_{288}(4)$, as in Figures 15 and 16.

Figure 15: At four qubits, the $g_{144}$ subgraph can have $4$ different
entropy vectors among the vertices. The cyan vertices denote states with
entropy vector $\vec{S}=(1,1,1,1,1,1,1)$, an entropic structure that violates
monogamy of mutual information (14). These 4-qubit stabilizer states, located
on 4-qubit subgraphs $g_{144}$ and $g_{288}$, are the first instances of non-
holographic states. Figure 16: At four qubits, we witness a variation to the
$g_{288}$ subgraph first seen at three qubits. The structure is isomorphic to
its 3-qubit counterpart; however, the number of entropy vectors present has
increased to four, similar to what occurred in Figure 15.

We also obtain our last new structure: $g_{1152}$. At four qubits we find this
subgraph with either two or four different entropy vectors, as in Figures 17
and 18.

Figure 17: One of the two $g_{1152}$ occurring at four qubits. This subgraph
is isomorphic to the $g_{1152}$ subgraph in Figure 18, but with only $2$
different entropy vectors among all the states in the subgraph. Figure 18:
Another copy of the $g_{1152}$ subgraph introduced at $4$ qubits. States
located on this subgraph exhibit $4$ different entropy vectors, in contrast to
the $g_{1152}$ subgraph in Figure 17.

As in the 3-qubit case, lifting the $g_{24}$ and $g_{36}$ subgraphs from two
qubits to four qubits proceeds in a simple manner. To lift the 2-qubit
$g_{24}$ states to the 4-qubit $g_{24}$ subgraphs, we proceed as in (25). That
is, we start with a state in the 2-qubit $g_{24}$ subgraph, and then tensor on
any other 2-qubit state. Thus, both ${|{0000}\rangle}$ and ${|{00i1}\rangle}$
appear in copies of $g_{24}$ at four qubits.

Similarly for $g_{36}$, we again tensor on 2-qubit states instead of the
1-qubit state added in (26). Thus ${|{i100}\rangle}$ and ${|{i1i1}\rangle}$
are in copies of $g_{24}$ at four qubits. Since there are 60 2-qubit
stabilizer states, we get 60 copies each of $g_{24}$ and $g_{36}$. For
$g_{36}$ the Hamiltonian paths transfer just as before.

These Hamiltonian paths can be embedded in $g_{144}$ and $g_{288}$ by acting
with the CNOT gates that involve qubits 3 and/or 4, or with phase gates when
necessary. The new structure, $g_{1152}$, behaves similarly, except 32 copies
of $\mathcal{C}$ are needed for a full covering.

Not only the Hamiltonian paths lift; recall that Hamiltonian cycles exist for
$g_{144}$ and $g_{288}$. These cycles embed as well. Choosing some starting
state at 3 qubits, we tensor on a 1-qubit stabilizer state, act with a CNOT to
entangle it and arrive on one of the ten copies of $g_{1152}$, and then enact
the cycle in the same gate order as the Hamiltonian cycle at three qubits. In
order to cover one full $g_{1152}$ subgraph, four such cycles are needed.

Lastly, the $g_{1152}$ structure, like the $g_{288}$ subgraph, contains an
Eulerian cycle of length 2304. Just as for $g_{288}$, as discussed at the end
of section 4.4, this cycle exists because all four of the gates
$H_{1},\,H_{2},\,CNOT_{1,2},\,CNOT_{2,1}$ act nontrivially and differently
from each other, on every state in the subgraph; no degeneracies or loops
remain.

####  5.1.1 Entropy Vector Analysis

In Table 5, we list the 18 different entropy vectors which stabilizer states
exhibit at four qubits. In the table, the vectors are grouped together if they
appear in the same subgraphs in the $H_{1},\,H_{2},\,CNOT_{1,2},\,CNOT_{2,1}$
restricted graph. These groupings arise because $S_{C},\,S_{O},$ and $S_{AB}$
cannot be altered by the Hadamards and CNOTs on qubits 1 and 2 only.

Holographic | $\left(S_{A},S_{B},S_{C},S_{O},S_{AB},S_{AC},S_{AO}\right)$ | Number of States | Subgraph
---|---|---|---
Yes | $(0,0,0,0,0,0,0)$ | $1296$ |  $g_{24}$, $g_{36}$
Yes | $(1,1,0,0,0,1,1)$ | $864$ |  $g_{24}$, $g_{36}$
Yes | $(0,0,1,1,0,1,1)$ | $864$ |  $g_{24}$, $g_{36}$
Yes | $(1,1,1,1,0,2,2)$ | $576$ |  $g_{24}$, $g_{36}$
Yes | $(0,1,1,0,1,1,0)$ | $864$ | $g_{144},g_{288}$
Yes | $(1,0,1,0,1,0,1)$ | $864$ | $g_{144},g_{288}$
Yes | $(1,1,1,0,1,1,1)$ | $2592$ | $g_{144},g_{288}$
Yes | $(0,1,0,1,1,0,1)$ | $864$ | $g_{144},g_{288}$
Yes | $(1,0,0,1,1,1,0)$ | $864$ | $g_{144},g_{288}$
Yes | $(1,1,0,1,1,1,1)$ | $2592$ | $g_{144},g_{288}$
Yes | ${\color[rgb]{1,.5,0}\bullet}(1,0,1,1,1,1,1)$ | $2592$ | $g_{144},g_{288}$
Yes | ${\color[rgb]{.5,0,.5}\bullet}(0,1,1,1,1,1,1)$ | $2592$ | $g_{144},g_{288}$
Yes | ${\color[rgb]{1,1,0}\bullet}(1,1,1,1,1,2,2)$ | $5184$ | $g_{144},g_{288}$
No | ${\color[rgb]{0,1,1}\bullet}(1,1,1,1,1,1,1)$ | $2592$ | $g_{144},g_{288}$
Yes | ${\color[rgb]{.75,1,0}\bullet}(1,1,1,1,2,1,2)$ | $5184$ |  $g_{1152}$ (2,4)
Yes | ${\color[rgb]{0,0,0}\bullet}(1,1,1,1,2,2,1)$ | $5184$ |  $g_{1152}$ (2,4)
Yes | ${\color[rgb]{0,.5,.5}\bullet}(1,1,1,1,2,0,2)$ | $576$ |  $g_{1152}$ (4)
Yes | ${\color[rgb]{.75,0,.25}\bullet}(1,1,1,1,2,2,0)$ | $576$ |  $g_{1152}$ (4)
Table 5: Entropy vectors for the set of four-qubit stabilizer states. One
entropy vector disobeys the holographic inequalities. Graph $g_{1152}$ comes
in two varieties: one with only two different entropy vectors, and one with
four. The last two entropy vectors only appear on subgraphs with four
different entropy vectors, while the previous two appear on subgraphs with
either two or four entropy vectors.

Importantly, notice that our first non-holographic state, which disobeys the
holographic entropy cone relation (14), arises here. CNOT gates are the only
actions which could move a state from within the entropy cone to outside of
it, so by studying the restricted graphs where this non-holographic entropy
vector lives, we can define how to reach such states. In particular, notice
that in $g_{144}(4)$, shown in Figure 14, the non-holographic states divide
the subgraph into multiple parts; removing them entirely would separate the
inner octagonal structure from the outside, and divide the outer ring into
several pieces.

###  5.2 Five Qubits and Beyond

At five qubits, no new structures arise. The subgraph types may begin to have
more entropy vectors on a given subgraph, and each subgraph has more copies,
but the shapes themselves do not increase in size. Again, we have generated
the full set of stabilizer states as well as the complete reachability
diagram, and this data is fully accessible at github . The full five qubit
list of entropy vectors is in Table 7 of the appendix.

Again, we note 12 entropy vectors arise which disobey the holographic entropy
cone relations; as usual, examining the gate actions which move to and from
those entropy vectors will help us to learn how states move in and out of the
holographic entropy cone.

For now, we note that the $g_{24}$ and $g_{36}$ subgraphs each have 1080
copies, as expected since they are built by tensoring 2-qubit stabilizer
states with 3-qubit stabilizer states; there are 1080 3-qubit stabilizer
states. We also note that the number of $g_{1152}$ copies scales faster than
for the smaller index number graphs, as shown in Table 6.

Qubit # | $g_{24}$ | $g_{36}$ | $g_{144}$ | $g_{288}$ | $g_{1152}$
---|---|---|---|---|---
Two | $1$ | $1$ | - | - | -
Three | $6$ | $6$ | $3$ | $1$ | -
Four | $60$ | $60$ | $90$ | $30$ | $10$
Five | $1080$ | $1080$ | $3780$ | $1260$ | $1260$
Table 6: The number of occurrences of each subgraph structure at two, three,
four, and five qubits. The number of copies of $g_{1152}$ exhibits the most
dramatic growth with qubit number.

We believe no further new structures arise at higher qubits beyond the
$g_{1152}$ first found at four qubits.777 This claim has since been verified
explicitly by directly computing the full set of unique Clifford group
elements generated by $H_{1},H_{2},CNOT_{1,2},$ and $CNOT_{2,1}$. A simple
justification can be observed by allowing each leg of the $g_{1152}$ subgraph
to represent a unique operation generated by $H_{1},H_{2},CNOT_{1,2}$ and
$CNOT_{2,1}$. The proof of this result will be presented in forthcoming work.
However, individual subgraphs will start to contain larger numbers of
different entropy vectors, just as we found when increasing to four and then
five qubits.

##  6 Discussion

Our goal in the body of the paper has been to understand the $n$-qubit
stabilizer states by first constructing their reachability graphs, then
passing to the restricted graphs formed using only the Hadamards and CNOTs on
two qubits. We were able to understand how the simpler structures at low qubit
number ($n=1$ and $n=2$) assemble, via lifts, to build the more complicated
structures observed at higher qubit number ($n=3$ and $n=4$). We have argued
that the four graph structures shown already in the Introduction are “all
there is:” passing to higher qubit number merely proliferates larger numbers
of the four basic structures, perhaps with more complicated entropic
arrangements. We noted the presence of non-holographic states beginning at
four qubits, and their confinement to particular groups of subgraphs, as
demanded by their having entropy vectors that violate holographic
inequalities. In this Discussion, we collect some other ways of thinking about
the reachability graphs we have constructed, along with potential
generalizations and questions for further research.

We first note that, to our knowledge, already at two qubits this is the first
time that the complete, explicit reachability graphs have been presented in
the literature. We have constructed the full reachability graphs up through
five qubits, and have made them available in our GitHub repository github .

Having the explicit form of the graphs allows for direct computation of some
interesting quantities. For example, the minimum distance on the graph from
one state to another is, explicitly, the gate complexity of the stabilizer
circuit which maps the first state to the second. The state with largest
minimum distance relative to a reference state thus is the “most complicated”
state by this measure. We have explicitly computed this minimum distance
relative to the all-zero state for $n=1\ldots 4$, and find that it is circuit
distance $4,7,10,13$, respectively. Since the asymptotic complexity of
$n$-qubit stabilizer states is known to go as $n^{2}/\mathrm{log}(n)$
aaronson2004improved , this is perhaps a surprising result—it indicates that
at small, finite qubit number the main contribution to the complexity is not
the CNOT gate applications which lead to the asymptotic result. Of course, we
note that, even with the explicit reachability graph, finding a state with
maximal relative distance is a highly nontrivial problem requiring an
exhaustive numerical search.

We recall also that the group-theoretic identity in (8) equates the number of
elements in a given subgraph to the orbit length of an element in that
subgraph under the action of the subgroup of the Clifford group generated by
the subset of the Clifford gates we are considering (see Table 2). Hence we
can interpret $24,36,144,288,1152$ as the orbit lengths of various stabilizer
states under the action of the subgroup generated by
$\\{H_{1},H_{2},CNOT_{1,2},CNOT_{2,1}\\}$. It would be interesting to find
these orbits directly using a group-theoretic approach, and confirm or
disprove our conjecture that no additional structures appear beyond
$g_{1152}$.

We emphasize that we chose to focus primarily on Clifford gates acting on the
first two qubits because the stabilizer states exhibit only bipartite
entanglement, constructed explicitly through CNOT gate applications. Hence our
restricted graphs are sufficient to analyze how the entropy vector of a state
changes as it is acted upon by various stabilizer circuits. Of course, adding
additional gates to our restricted gate will result in more complicated
connected subgraphs, culminating in the full reachability graph at each qubit
number.

In the holographic literature, we typically think of each subsystem as some
portion of a spatial boundary, with the exception of an additional “purifier”
subsystem which does not necessarily have an interpretation as a spatial
subsystem. The holographic interpretation of our results will therefore differ
depending on whether we take the purifier to be one of the first two qubits,
or instead some other qubit $>=$ 3\. In the body of the paper we have taken
the purifying system $O$ to be the last qubit.

In what ways can our reachability graph be generalized? First consider staying
within the setting of $n$-qubit states. The existence of a graph structure is
most useful when we have a gate set that can be used to pick out only a finite
number of states, rather than a continuous subspace of $\mathbb{C}^{2n}$ or
the entire Hilbert space itself. Hence we do not expect choosing a
modification of the Clifford gates which yields a universal gate set to
provide interesting results. We could instead consider the stabilizers of some
other finite group of operators besides the Pauli group. It is an interesting
question whether there are any such finite groups with a compelling physical
interpretation. We could also consider allowing a small number of discrete
applications of non-Clifford gates, i.e. allowing a small amount of “magic”
bravyi2005universal ; White:2020zoz  as a resource. Or, we could consider
restricting ourselves to stabilizer states, but allowing some applications of
operators which are in the Clifford group but not Clifford generators
themselves, which could be used to “fast-forward” a circuit. Finally, it would
be very interesting to understand whether there is a set of gates that produce
a discrete set of states but allow, for example, for tripartite entanglement.

In more general Hilbert spaces which are not isomorphic to tensor products of
qubits, we expect that a similar story should hold with a different group than
the Pauli group. There are a number of approaches in the literature to
generalizing Pauli matrices to larger discrete systems. One approach, well-
suited to qudit systems, is to use Gell-Mann matrices and their
generalizations. Another, quite different, approach better suited to
approaching the continuum limit of a lattice system is to use the “clock” and
“shift” matrices that generate a “generalized Clifford algebra”
Jagannathan:2010sb ; Pollack:2018yum .

Finally, we recall our motivation in the Introduction, where we described both
entanglement entropies in a factorized system, and stabilizer states relative
to a preferred group of algebra, as two different ways of imposing structure
on the set of states in Hilbert space. We can further generalize this picture
by noting that the von Neumann entropy of a reduced density matrix of a
subsystem can be identified with the _algebraic_ entropy associated with the
algebra of operators which act as the identity everywhere outside that
subsystem. Hence entropies are defined relative to an _algebra_ of operators
while stabilizer states are defined relative to a _group_ of operators. In
particular, recall that the algebra generated by the Pauli operators acting on
a qubit is in fact the full algebra of linear operators on that qubit.

When we consider more general von Neumann algebras, instead of the Hilbert
space decomposing as a product of tensor factors, we get a more general
_Wedderburn decomposition_ wedderburn1934lectures ; Harlow:2016vwg ;
Kabernik:2019jko , which decomposes the Hilbert space as a _sum_ of products
of tensor factors, where each term in the sum represents a superselection
sector which has its own associated entropy. Thus we can envision a very
abstract version of our picture in which we fix a _set_ of operators, and then
find stabilizer states by using this set to generate a _group_ of operators
and entropies by using the same set to generate an _algebra_ of operators. It
would be interesting to see how far this picture can be taken in generic
cases. A first step would be to understand how to generate not just a single
entropy for a given reduced density matrix on a state, but rather a larger
entropy vector, which would pick out some _sequence_ of algebras, the
equivalent of the operators acting on the first, second, and in general
$n^{th}$ qubit.

##### Data Repository

The full set of states, reachability diagrams, and entropy vectors for
stabilizer states at $n\leq 5$ qubits can be accessed via the GitHub
repository github . The repository additionally includes Mathematica notebooks
used to generate the data for this paper, as well as a Stabilizer State
package designed to generate reachability graphs and analyze stabilizer state
structure.

###### Acknowledgements.

The authors thank Scott Aaronson, Ning Bao, Charles Cao, and Sergio Hernandez-
Cuenca for helpful discussions. J.P. is supported by the Simons Foundation
through _It from Qubit: Simons Collaboration on Quantum Fields, Gravity, and
Information_. C.K. and W.M. are supported by the U.S. Department of Energy
under grant number DE-SC0019470 and by the Heising-Simons Foundation
“Observational Signatures of Quantum Gravity” collaboration grant 2021-2818.

##  Appendix A Additional Graphs

Contained below are some graphs not featured in the body of this paper. Futher
graphs are available in the repository github .

###  A.1 Two Qubit Graphs

The 2-qubit complete reachability graph and some 2-qubit restricted graphs
were discussed in section 3. Here we compile additional restricted graphs to
further illustrate the relation between states under action $C_{2}$ subgroup
action. Figure 19 restricts to only phase operations, while Figure 20 shows
the restricted graph for only CNOT operation.

####  A.1.1 Two Qubit $P_{1},P_{2}$ Restricted Graph

Figure 19 shows the 2-qubit graph restricted to only phase operations. The
graph consists of $5$ distinct and disconnected substructures.

Figure 19: The 2-qubit $P_{1},P_{2}$ restricted graph contains $5$ unique
substructures. The $4$ isolated points are states on which both $P_{1}$ and
$P_{2}$ act trivially. The box-like structures come in two varieties. The box
of unentangled states has a trivial loop at each corner, while the boxes of
entangled states witness degenerate action instead. There exist two largest
structures of states (top-left) on which both phase gates act non-trivially
and non-degenerately.

####  A.1.2 Two Qubit $CNOT_{1,2},CNOT_{2,1}$ Restricted Graph

Figure 20 shows all interactions between 2-qubit states under only CNOT
operations. The CNOT gate can modify entropy structure, therefore we witness
the first occurrences of states with different entropy vectors lying in the
same substructures. Alternating action of $CNOT_{1,2}$ and $CNOT_{2,1}$ has a
maximum cycle of $6$, seen in the hexagonal structures top-left.

Figure 20: The subgroup generated by $CNOT_{1,2}$ and $CNOT_{2,1}$ partitions
the set of 2-qubit states into $3$ graph substructures. The isolated points
are states on which both CNOT gates act trivially. The linear triplets are
built of one state, on which both CNOT gates act non-trivially, connected to
two states on which opposing one CNOT gate acts trivially. These triplets can
occur with states of similar or different entropic structure. The largest
structure is a hexagon which illustrates the maximum cycle of the subgroup
generated by $CNOT_{1,2}$ and $CNOT_{2,1}$.

###  A.2 Three Qubit Graphs

The 3-qubit reachability diagrams were discussed in section 4 with a focus on
the $H_{1},H_{2},CNOT_{1,2},CNOT_{2,1}$ restricted graph (Figure 23). Here we
provide additional graphs of potential interest, including the complete
reachability graph on three qubits (Figure 21). Figure 22 displays the action
of all phase gates on three qubits, generalizing the cycles and structures
seen at two qubits (Figure 19) to three qubits.

####  A.2.1 Three Qubit Complete Reachability Graph

Figure 21 displays the full reachability graph for three qubits (trivial loops
removed). This graph contains all non-trivial information about 3-qubit
interaction under operations of the Clifford group ($C_{3}$).

Figure 21: Complete reachability diagram on three qubits with trivial loops
removed. The Hadamard and phase gates act individually on all three qubits,
while the CNOT gate acts on any pair of qubits. Line texture indicates the
particular action, e.g. a solid line for $H_{1}$ and medium dashed line for
$H_{2}$.

####  A.2.2 Three Qubit $P_{1},P_{2},P_{3}$ Restricted Graph Subgraphs

In Figure 22 is the 3-qubit graph restricted to only phase gates. The addition
of $P_{3}$ to the generating set allows for longer cycles, resulting in more
complex structures than were witness at lower qubit number.

Figure 22: There are $11$ unique subgraphs whose copies build the 3-qubit
$P_{1},P_{2},P_{3}$ restricted graph. These subgraphs segregate according to
which sequences of $P_{1},P_{2},$ and $P_{3}$ are equivalent.

####  A.2.3 Three Qubit $H_{1},H_{2},CNOT_{1,2},CNOT_{2,1}$ Restricted Graph

Figure 23 shows the 3-qubit Three-qubit restricted graph displaying only
Hadamard and CNOT operations on the first two qubits of in the system. The
graph has repeated copies of structures found at two qubits, as well as the
addition of two new subgraphs $g_{144}$ and $g_{288}$.

Figure 23: The graph contains $6$ copies of $g_{24}$ and $g_{36}$, $3$ copies
of $g_{144}$, and a single copy of $g_{288}$. The $3$ $g_{144}$ subgraphs are
isomorphic, but were generated with slightly different layouts by the software
used to build this graph.

##  Appendix B Five-Qubit Entropy Vectors

Below we present the complete set of entropy vectors for the 5-qubit
stabilizer set. There are $2423520$ stabilizer states at five qubits, with
$93$ different entropic arrangements. There are $16$ of these $93$ entropy
vectors which violate the monogamy of mutual information (Equation (14)), and
therefore correspond to non-holographic states.

Holographic | Entropy Vector | Subgraph
---|---|---
Yes | $(0,0,1,0,1,0,1,0,1,1,0,1,1,0,1)$ |  $g_{24}$, $g_{36}$
Yes | $(1,1,1,0,1,0,2,1,2,2,1,2,1,0,1)$ |  $g_{24}$, $g_{36}$
Yes | $(0,0,1,1,0,0,1,1,0,1,1,0,0,1,1)$ |  $g_{24}$, $g_{36}$
Yes | $(1,1,1,1,0,0,2,2,1,2,2,1,0,1,1)$ |  $g_{24}$, $g_{36}$
Yes | $(0,0,0,1,1,0,0,1,1,0,1,1,1,1,0)$ |  $g_{24}$, $g_{36}$
Yes | $(1,1,0,1,1,0,1,2,2,1,2,2,1,1,0)$ |  $g_{24}$, $g_{36}$
Yes | $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)$ |  $g_{24}$, $g_{36}$
Yes | $(1,1,0,0,0,0,1,1,1,1,1,1,0,0,0)$ |  $g_{24}$, $g_{36}$
Yes | $(0,0,1,1,1,0,1,1,1,1,1,1,1,1,1)$ |  $g_{24}$, $g_{36}$
Yes | $(1,1,1,1,1,0,2,2,2,2,2,2,1,1,1)$ |  $g_{24}$, $g_{36}$
Yes | $(0,1,1,1,1,1,1,1,1,0,2,2,2,2,0)$ | $g_{144},g_{288}$
Yes | $(1,0,1,1,1,1,0,2,2,1,1,1,2,2,0),$ | $g_{144},g_{288}$
Yes | $(1,1,1,1,1,1,1,2,2,1,2,2,2,2,0)$ | $g_{144},g_{288}$
Yes | $(0,1,1,1,1,1,1,1,1,2,2,0,0,2,2)$ | $g_{144},g_{288}$
Yes | $(1,0,1,1,1,1,2,2,0,1,1,1,0,2,2)$ | $g_{144},g_{288}$
Yes | $(1,1,1,1,1,1,2,2,1,2,2,1,0,2,2)$ | $g_{144},g_{288}$
Yes | $(0,1,0,1,0,1,0,1,0,1,0,1,1,0,1)$ | $g_{144},g_{288}$
Yes | $(1,0,0,1,0,1,1,0,1,0,1,0,1,0,1)$ | $g_{144},g_{288}$
Yes | $(1,1,0,1,0,1,1,1,1,1,1,1,1,0,1)$ | $g_{144},g_{288}$
Yes | $(0,1,0,0,1,1,0,0,1,1,1,0,0,1,1)$ | $g_{144},g_{288}$
Yes | $(1,0,0,0,1,1,1,1,0,0,0,1,0,1,1)$ | $g_{144},g_{288}$
Yes | $(1,1,0,0,1,1,1,1,1,1,1,1,0,1,1)$ | $g_{144},g_{288}$
Yes | $(0,1,1,0,0,1,1,0,0,0,1,1,1,1,0)$ | $g_{144},g_{288}$
Yes | $(1,0,1,0,0,1,0,1,1,1,0,0,1,1,0)$ | $g_{144},g_{288}$
Yes | $(1,1,1,0,0,1,1,1,1,1,1,1,1,1,0)$ | $g_{144},g_{288}$
Table 7: At five qubits, there are $93$ stabilizer state entropy vectors
(listed here and on the next two pages). Of these, $16$ correspond to non-
holographic states. All non-holographic states are located on subgraphs with
$4$ different entropy vectors.

Holographic | Entropy Vector | Subgraph
---|---|---
Yes | $(0,1,1,1,1,1,1,1,1,2,0,2,2,0,2)$ | $g_{144},g_{288}$
Yes | $(1,0,1,1,1,1,2,0,2,1,1,1,2,0,2)$ | $g_{144},g_{288}$
Yes | $(1,1,1,1,1,1,2,1,2,2,1,2,2,0,2)$ | $g_{144},g_{288}$
No | $(0,1,1,1,1,1,1,1,1,1,1,1,1,1,1)$ | $g_{144},g_{288}$
No | $(1,0,1,1,1,1,1,1,1,1,1,1,1,1,1)$ | $g_{144},g_{288}$
No | $(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)$ | $g_{144},g_{288}$
No | $(1,1,1,1,1,1,2,2,2,2,2,2,1,1,1)$ | $g_{144},g_{288}$
Yes | $(0,1,1,1,1,1,1,1,1,2,2,1,1,2,2)$ | $g_{144},g_{288}$
Yes | $(1,0,1,1,1,1,2,2,1,1,1,1,1,2,2)$ | $g_{144},g_{288}$
No | $(1,1,1,1,1,1,2,2,1,2,2,1,1,2,2)$ | $g_{144},g_{288}$
Yes | $(1,1,1,1,1,1,2,2,2,2,2,2,1,2,2)$ | $g_{144},g_{288}$
Yes | $(0,1,1,1,0,1,1,1,0,1,1,1,1,1,1)$ | $g_{144},g_{288}$
Yes | $(1,0,1,1,0,1,1,1,1,1,1,0,1,1,1)$ | $g_{144},g_{288}$
No | $(1,1,1,1,0,1,1,1,1,1,1,1,1,1,1)$ | $g_{144},g_{288}$
Yes | $(1,1,1,1,0,1,2,2,1,2,2,1,1,1,1)$ | $g_{144},g_{288}$
Yes | $(0,1,0,1,1,1,0,1,1,1,1,1,1,1,1)$ | $g_{144},g_{288}$
Yes | $(1,0,0,1,1,1,1,1,1,0,1,1,1,1,1)$ | $g_{144},g_{288}$
No | $(1,1,0,1,1,1,1,1,1,1,1,1,1,1,1)$ | $g_{144},g_{288}$
Yes | $(1,1,0,1,1,1,1,2,2,1,2,2,1,1,1)$ | $g_{144},g_{288}$
Yes | $(0,1,1,1,1,1,1,1,1,1,2,2,2,2,1)$ | $g_{144},g_{288}$
Yes | $(1,0,1,1,1,1,1,2,2,1,1,1,2,2,1)$ | $g_{144},g_{288}$
No | $(1,1,1,1,1,1,1,2,2,1,2,2,2,2,1)$ | $g_{144},g_{288}$
Yes | $(1,1,1,1,1,1,2,2,2,2,2,2,2,2,1)$ | $g_{144},g_{288}$
Yes | $(0,1,1,0,1,1,1,0,1,1,1,1,1,1,1)$ | $g_{144},g_{288}$
Yes | $(1,0,1,0,1,1,1,1,1,1,0,1,1,1,1)$ | $g_{144},g_{288}$
No | $(1,1,1,0,1,1,1,1,1,1,1,1,1,1,1)$ | $g_{144},g_{288}$
Yes | $(1,1,1,0,1,1,2,1,2,2,1,2,1,1,1)$ | $g_{144},g_{288}$
Yes | $(0,1,1,1,1,1,1,1,1,2,1,2,2,1,2)$ | $g_{144},g_{288}$
Yes | $(1,0,1,1,1,1,2,1,2,1,1,1,2,1,2)$ | $g_{144},g_{288}$
No | $(1,1,1,1,1,1,2,1,2,2,1,2,2,1,2)$ | $g_{144},g_{288}$
Yes | $(1,1,1,1,1,1,2,2,2,2,2,2,2,1,2)$ | $g_{144},g_{288}$
Yes | $(1,1,1,0,1,2,2,1,1,1,1,2,1,2,1)$ |  $g_{1152}$ (2,4)
Yes | $(1,1,1,0,1,2,1,1,2,2,1,1,1,2,1)$ |  $g_{1152}$ (2,4)
Yes | $(1,1,1,0,1,2,0,1,2,2,1,0,1,2,1)$ |  $g_{1152}$ (4)
Yes | $(1,1,1,0,1,2,2,1,0,0,1,2,1,2,1)$ |  $g_{1152}$ (4)

Holographic | Entropy Vector | Subgraph
---|---|---
Yes | $(1,1,0,1,1,2,1,2,1,1,1,2,1,1,2)$ |  $g_{1152}$ (2,4)
Yes | $(1,1,0,1,1,2,1,1,2,1,2,1,1,1,2)$ |  $g_{1152}$ (2,4)
Yes | $(1,1,0,1,1,2,1,0,2,1,2,0,1,1,2)$ |  $g_{1152}$ (4)
Yes | $(1,1,0,1,1,2,1,2,0,1,0,2,1,1,2)$ |  $g_{1152}$ (4)
Yes | $(1,1,1,1,0,2,2,1,1,1,2,1,2,1,1)$ |  $g_{1152}$ (2,4)
Yes | $(1,1,1,1,0,2,1,2,1,2,1,1,2,1,1)$ |  $g_{1152}$ (2,4)
Yes | $(1,1,1,1,0,2,0,2,1,2,0,1,2,1,1)$ |  $g_{1152}$ (4)
Yes | $(1,1,1,1,0,2,2,0,1,0,2,1,2,1,1)$ |  $g_{1152}$ (4)
No | $(1,1,1,1,1,2,1,2,2,2,1,1,2,2,1)$ |  $g_{1152}$ (4,6)
No | $(1,1,1,1,1,2,2,1,1,1,2,2,2,2,1)$ |  $g_{1152}$ (4,6)
Yes | $(1,1,1,1,1,2,1,2,2,2,2,2,2,2,1)$ |  $g_{1152}$ (4,6)
Yes | $(1,1,1,1,1,2,2,2,2,1,2,2,2,2,1)$ |  $g_{1152}$ (4,6)
Yes | $(1,1,1,1,1,2,0,2,2,2,1,1,2,2,1)$ |  $g_{1152}$ (6)
Yes | $(1,1,1,1,1,2,2,1,1,0,2,2,2,2,1)$ |  $g_{1152}$ (6)
No | $(1,1,1,1,1,2,1,1,2,2,2,1,1,2,2)$ |  $g_{1152}$ (4,6)
No | $(1,1,1,1,1,2,2,2,1,1,1,2,1,2,2)$ |  $g_{1152}$ (4,6)
Yes | $(1,1,1,1,1,2,2,2,1,2,2,2,1,2,2)$ |  $g_{1152}$ (4,6)
Yes | $(1,1,1,1,1,2,2,2,2,2,2,1,1,2,2)$ |  $g_{1152}$ (4,6)
Yes | $(1,1,1,1,1,2,2,2,0,1,1,2,1,2,2)$ |  $g_{1152}$ (6)
Yes | $(1,1,1,1,1,2,1,1,2,2,2,0,1,2,2)$ |  $g_{1152}$ (6)
No | $(1,1,1,1,1,2,1,2,1,2,1,2,2,1,2)$ |  $g_{1152}$ (4,6)
No | $(1,1,1,1,1,2,2,1,2,1,2,1,2,1,2)$ |  $g_{1152}$ (4,6)
Yes | $(1,1,1,1,1,2,2,1,2,2,2,2,2,1,2)$ |  $g_{1152}$ (4,6)
Yes | $(1,1,1,1,1,2,2,2,2,2,1,2,2,1,2)$ |  $g_{1152}$ (4,6)
Yes | $(1,1,1,1,1,2,1,2,1,2,0,2,2,1,2)$ |  $g_{1152}$ (6)
Yes | $(1,1,1,1,1,2,2,0,2,1,2,1,2,1,2)$ |  $g_{1152}$ (6)
Yes | $(1,1,1,1,1,2,1,2,2,2,1,2,2,2,2)$ |  $g_{1152}$ (7)
Yes | $(1,1,1,1,1,2,1,2,2,2,2,1,2,2,2)$ |  $g_{1152}$ (7)
Yes | $(1,1,1,1,1,2,2,1,2,1,2,2,2,2,2)$ |  $g_{1152}$ (7)
Yes | $(1,1,1,1,1,2,2,1,2,2,2,1,2,2,2)$ |  $g_{1152}$ (7)
Yes | $(1,1,1,1,1,2,2,2,1,1,2,2,2,2,2)$ |  $g_{1152}$ (7)
Yes | $(1,1,1,1,1,2,2,2,1,2,1,2,2,2,2)$ |  $g_{1152}$ (7)
Yes | $(1,1,1,1,1,2,2,2,2,2,2,2,2,2,2)$ |  $g_{1152}$ (7)

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