# From square plaquettes to triamond lattices for SU(2) gauge theory

Ali H. Z. Kavaki<EMAIL_ADDRESS>Randy Lewis<EMAIL_ADDRESS>Department
of Physics and Astronomy, York University, Toronto, Ontario, Canada, M3J 1P3

(January 25, 2024)

###### Abstract

Lattice gauge theory should be able to address significant new scientific
questions when implemented on quantum computers. In practice, error-mitigation
techniques have already allowed encouraging progress on small lattices. In
this work we focus on a truncated version of SU(2) gauge theory, which is a
familiar non-Abelian step toward quantum chromodynamics. First, we demonstrate
effective error mitigation for imaginary time evolution on a lattice having
two square plaquettes, obtaining the ground state using an IBM quantum
computer and observing that this would have been impossible without error
mitigation. Then we propose the triamond lattice as an expedient approach to
lattice gauge theories in three spatial dimensions, deriving the Hamiltonian
and obtaining energy eigenvalues and eigenstates from a noiseless simulator
for a three-dimensional unit cell.

##  I Introduction

Quantum field theory combines quantum mechanics with special relativity and is
of widespread importance in high-energy physics, nuclear physics, and
condensed matter physics. Enforcing a spatially local symmetry leads to the
special case of a gauge theory. The standard model of particle physics is a
collection of three gauge theories describing the strong, weak and
electromagnetic forces. Because perturbation theory cannot be applied to a
large coupling constant, many aspects of the strong force are best described
by a numerical approach called lattice gauge theory.

Lattice gauge theory has been of central importance to quantum chromodynamics
for several decades and has become a precision tool for practitioners [1, 2].
It provides rigorous calculations of static properties, such as the masses and
form factors of hadrons, directly from the gauge theory of quarks and gluons.
Such calculations are crucial in the ongoing quest to understand hadron
physics, such as the tetraquarks and pentaquarks that have been discovered
experimentally in recent years [3, 4]. Lattice gauge theory calculations also
provide necessary input for experiments seeking new physics beyond the
standard model, where a famous recent example is the anomalous magnetic moment
of the muon [5].

Quantum computers offer a prospective way to calculate more than just static
properties. Traditional lattice gauge theory uses Monte Carlo methods that
would have sign problems if applied straightforwardly to situations involving
dynamics, nonzero density, or some topological interactions [6, 7]. These sign
problems can be avoided by using a Hamiltonian approach, where the
exponentially large Hilbert space can fit naturally onto a quantum computer.
Now that quantum computing hardware is becoming a reality, this powerful new
approach to lattice gauge theory is being developed by a broad community of
researchers [7, 8, 9].

With quantum chromodynamics as a motivating long-range goal, the present work
is focused on the simplest non-Abelian gauge theory. Being non-Abelian means
that gauge fields carry the relevant charge directly and can therefore
interact among themselves, like the gluons of quantum chromodynamics and in
contrast to the photons of quantum electrodynamics. Specifically our work will
consider SU(2) gauge theory in the absence of fermions, and we will truncate
the gauge field to fit on a small number of qubits. Several research groups
have already carried out exploratory studies of non-Abelian gauge theories on
quantum computing hardware [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].
There have also been many theoretical studies laying the groundwork for
anticipated implementations on larger quantum computers [22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61]. In the
present work, we address two important issues: an effective way to mitigate
errors and a practical way to extend lattices into three spatial dimensions.

Future quantum computers are expected to have smaller error rates and robust
methods for correcting those errors, but calculations in the present era face
substantial error rates and typically have insufficient resources to implement
true error correction. Instead, error mitigation methods have been devised
[62], and these can provide significant improvements for computations
performed on today’s hardware. Our study demonstrates the use of an existing
method called self-mitigation [17] but in a new context: evolution of a
quantum state through imaginary time. As is well known, after sufficiently
many steps of imaginary time the excited state contributions to a generic
initial state will become exponentially suppressed relative to the ground
state, thus providing a way to create the ground state and to determine its
eigenvalue. Our results will demonstrate that self-mitigation correctly finds
and sustains the true ground state of a two-plaquette lattice even though the
unmitigated data points are moving ever farther from the true result with each
additional time step.

With successful mitigation in hand, we then turn our attention to lattice
gauge theory in three spatial dimensions. While this can be done with a
standard cubic lattice, it is important to remember that whenever more than
three gauge links meet at a lattice site, the quantum numbers of the gauge
links themselves are insufficient to fully define the state of the gauge field
across the lattice [36, 40, 54, 58]. For example, consider four SU(2) gauge
links meeting at a site and suppose each gauge link has quantum number
$j=\frac{1}{2}$, where $j$ is notation familiar from quantized angular
momentum though here it refers to the gauge degrees of freedom. One pair of
these links could sum to $j_{\rm pair}=0$ or $1$. The other pair must have the
same sum because Gauss’s law requires the total of all four to be $j_{\rm
site}=0$. Therefore a full description of the state requires not just the $j$
values of individual links, but also the value of $j_{\rm pair}$ for some pair
selected by the user to define that site.

One way to handle this issue is to assign a sufficient number of additional
qubits at each lattice site, being careful to define a convention at each
site, thereby completing the state’s definition [36, 40, 54, 58]. In the
present work we propose an alternative based on a structure from
crystallography [63, 64, 65, 66, 67, 68]. Having only three links at each
site, our choice will avoid the need for any quantum numbers beyond the
individual link values.

Consider a lattice where each site is touched by exactly three gauge links
that have equal lengths, lie in a single plane, and are placed at equal angles
around the site, i.e. $2\pi/3$. This forms a three-dimensional lattice that
has a high degree of symmetry and needs no additional qubits beyond the links
themselves. Because of its similarity to a diamond crystal of carbon atoms but
with three links per site instead of four, it has been called the triamond
lattice [66, 67]. It is also known as the Laves lattice [64] or the $K_{4}$
lattice [65, 68]. In this work, we provide an introduction to lattice gauge
theory on the triamond framework. We derive the SU(2) Hamiltonian for a
triamond lattice and demonstrate its use on a noiseless simulator.

##  II Results

###  II.1 Imaginary time evolution on square plaquettes

Finding the ground state for a given Hamiltonian is an important ingredient of
many scientific studies. Two common approaches are variational methods and
imaginary time evolution. Variational methods rely on an ansatz chosen by the
user, and can only get as close to the true ground state as the ansatz will
allow. In contrast, imaginary time evolution always converges to the true
ground state unless the initial state is perfectly orthogonal to it.

Imaginary time evolution is used routinely in traditional lattice gauge theory
on classical computers. Our motivation to study it now on a quantum computer
is not to compete with successful classical methods, but rather to imagine
that ground-state preparation will become the first step in a larger
computation that truly does require quantum computing [69, 70].

Our computation uses a lattice comprising two side-by-side square plaquettes
that share a single gauge link, thus forming a left path, a center path, and a
right path as shown in Fig. 1. Each of the seven gauge links on this lattice
is a superposition of SU(2) representations, and Gauss’s law requires the
three on the left path to equal one another and similarly the three on the
right path must equal one another. For our calculation we truncate the basis
and retain only the two lowest states for each gauge link, $j=0$ and
$j=\frac{1}{2}$.

Figure 1: The lattice used to demonstrate self-mitigation for quantum
imaginary time evolution. The text refers to the 3 paths between sites A and B
as the left path (three links), the center path (one link), and the right path
(three links).

Any state of this system can be described by two qubits, one for the left path
and one for the right path, because Gauss’s law then determines the center
path uniquely. Such a tiny system is a valuable place to study error
mitigation because a single time step requires only a simple circuit while
more and more time steps will require more and more entangling gates. The
real-time evolution of this system was studied previously as the first example
of self-mitigation [17]. The Hamiltonian, in units of $2/g^{2}$, is

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\frac{3}{8}\left(7-3\hat{Z}_{0}-\hat{Z}_{0}\hat{Z}_{1}-3\hat{Z}_{1}\right)-\frac{x}{2}\left(3+\hat{Z}_{1}\right)\hat{X}_{0}$
(1) $\displaystyle-\frac{x}{2}\left(3+\hat{Z}_{0}\right)\hat{X}_{1}\,,$

where $x=2/g^{4}$ and $g$ is the coupling constant, i.e. the strength of the
strong interaction for this physics theory. The Pauli gates acting on qubit
$i$ are $\hat{X}_{i}$, $\hat{Y}_{i}$ and $\hat{Z}_{i}$.

Imaginary time evolution is not a unitary operation, so how can it be
implemented on a quantum computer? We use an algorithm developed by Motta et
al., called quantum imaginary time evolution (QITE) [71], which can be
described as follows. The evolution we seek is

$\left|\Psi(\tau)\right>=e^{-\tau\hat{H}}\left|\Psi(0)\right>\,,$ (2)

where $\tau$ is the real-valued Euclidean time parameter. We can define a
normalized state $\left|\psi\right>$ such that
$\left|\Psi(0)\right>=r\left|\psi\right>$ for positive $r$. The evolution can
always be expressed as

$\left|\Psi(\tau)\right>=r^{\prime}e^{-i\tau\hat{A}}\left|\psi\right>\,,$ (3)

for some unitary operator $e^{-i\tau\hat{A}}$ and positive $r^{\prime}$. An
immediate consequence is

$r^{\prime}=r\left(1-\tau\left<\psi\right|\hat{H}\left|\psi\right>\right)+O(\tau^{2})\,.$
(4)

The operator $\hat{A}$ can be determined from a state tomography of
$\left|\psi\right>$ that reflects the specific qubit connectivities of
$\hat{H}$.

For our physics example, Eq. (1) confirms that the Hamiltonian is purely real,
meaning that we can restrict ourselves to real-valued basis states as well.
Together with Eqs. (2) and (3), this realness means $\hat{A}$ must be purely
imaginary, leading to a general expression with odd powers of $\hat{Y}_{i}$
gates,

$\displaystyle\hat{A}$ $\displaystyle=$ $\displaystyle
a_{iy}\hat{Y}_{0}+a_{xy}\hat{X}_{1}\hat{Y}_{0}+a_{zy}\hat{Z}_{1}\hat{Y}_{0}$
(5)
$\displaystyle+a_{yi}\hat{Y}_{1}+a_{yx}\hat{Y}_{1}\hat{X}_{0}+a_{yz}\hat{Y}_{1}\hat{Z}_{0}\,.$

The state tomography that determines the values of the coefficients $a_{jk}$
is presented in Sec. IV.1. With those values in hand, and with $r^{\prime}$
obtained from the calculation of Eq. (4), the state at a small value of $\tau$
can be computed from Eq. (3). In particular, we use

$\displaystyle e^{-i\theta\hat{Y}_{j}}$ $\displaystyle=$ $\displaystyle
RY_{j}(2\theta)\,,$ (6) $\displaystyle e^{-i\theta\hat{X}_{j}\hat{Y}_{k}}$
$\displaystyle=$ $\displaystyle CX_{kj}RY_{k}(2\theta)CX_{kj}\,,$ (7)
$\displaystyle e^{-i\theta\hat{Z}_{j}\hat{Y}_{k}}$ $\displaystyle=$
$\displaystyle CX_{jk}RY_{k}(2\theta)CX_{jk}\,,$ (8)

where $CX_{jk}$ is a CNOT gate with qubit $j$ as control and qubit $k$ as
target.

For sufficiently small $\tau$, the operator $e^{-i\theta\hat{A}}$ can be
approximated by a product of six factors, one for each term in $\hat{A}$. For
a better approximation, that first-order Trotter expression can be replaced by
the second-order Trotter expression represented by the circuit shown in Fig.
2. To reach a larger value of $\tau$, we can perform a sequence of small steps
where the circuit is several end-to-end copies of Fig. 2. Notice that the
tomography needs to be computed separately for each step, using the state at
the previous step as input.

Figure 2: The circuit used for one second-order Trotter step of imaginary time
evolution for SU(2) gauge theory on a lattice comprising two square
plaquettes.

Because the two-qubit CNOT gate is noisier than a single-qubit gate on quantum
hardware, the gates in Fig. 2 have been ordered in a way that minimizes the
number of CNOT gates. Also, the CNOT gate at the end of one Trotter step will
cancel the CNOT gate at the beginning of the next step. As well, the very
first CNOT gate in the circuit can be omitted because the initial control
qubit is always off. Nevertheless, the open symbols in Fig. 3 show that only
the first two or three time steps approach the true ground state energy and
then subsequent time steps move further and further away. The quantum
computation is overwhelmed by hardware noise without ever reaching the correct
result.

Figure 3: Quantum imaginary time evolution, with and without self-mitigation,
for SU(2) gauge theory with $j_{\rm max}=\frac{1}{2}$ on a lattice comprising
two square plaquettes. The gauge coupling is $x=1$ and the time step is
$\Delta\tau=0.1$ in units of $2/g^{2}$. Each quantum circuit uses 50
randomized compilings with $10^{4}$ shots per compiling. All data points were
computed on ibm_lagos.

Self-mitigation is able to extract the true result from the computed data. The
original implementation [17], which was inspired by the work of Urbanek et al.
[72], has subsequently been used and extended in various ways [18, 19, 21, 56,
73, 74, 75, 76, 70, 77]. The basic idea is to run a mitigation circuit that is
identical to the physics circuit except that $\tau\to-\tau$ in the second half
of the circuit. If your circuit has an odd number of second-order Trotter
steps, use $2\tau\to 0$ in the center gate (see, for example, Fig. 2) and
insert a barrier to prevent CNOT cancellation.

The true outcome of the mitigation circuit should be identical to the initial
state, i.e. all qubits should be in the off position. Comparison of the
computed results with this known true result provides a measurement of
hardware errors. The physics circuit will have similar hardware errors because
the two circuits are identical up to the sign of $\tau$ in the latter half.

More precisely, randomized compiling (see Sec. IV.2) is used to convert CNOT
errors into incoherent noise that is well described by the depolarizing noise
model [72]. For our example of imaginary time evolution, this leads to a pair
of ratios,

$\left.\frac{\left<\hat{P}_{j}\right>_{\rm
true}}{\left<\hat{P}_{j}\right>_{\rm comp}}\right|_{\rm
physics~{}run}=\left.\frac{1}{\left<\hat{Z}_{j}\right>_{\rm comp}}\right|_{\rm
mitigation~{}run},$ (9) $\left.\frac{\left<\hat{P}_{j}\hat{P}_{j}\right>_{\rm
true}}{\left<\hat{P}_{j}\hat{P}_{k}\right>_{\rm comp}}\right|_{\rm
physics~{}run}=\left.\frac{1}{\left<\hat{Z}_{j}\hat{Z}_{k}\right>_{\rm
comp}}\right|_{\rm mitigation~{}run},$ (10)

where $\hat{P}_{j}\in\\{\hat{X}_{j},\hat{Y}_{j},\hat{Z}_{j}\\}$. These
equations give the true expectation value of a Pauli string as the ratio of
results computed by a physics run and a mitigation run. They are the
counterparts of Eq. (8) in the original paper [17] but look slightly different
because the previous work studied probabilities that equalled $\frac{1}{2}$ in
the pure noise limit whereas the present work employs expectation values of
Pauli strings that are zero for pure noise.

The filled data points of Fig. 3 were obtained by computing each term of Eq.
(1) from the ratios in Eqs. (9) and (10). The physics runs and mitigation runs
(with 50 randomized compilings for each) and the readout error mitigation runs
were all completed within a single qiskit job [78] to ensure they were
experiencing the same hardware conditions. Whereas the unmitigated data points
in Fig. 3 approach the pure noise limit of $\frac{21}{8}$, where only the
first term in Eq. (1) makes a nonzero contribution, the self-mitigated data
points in Fig. 3 remain consistent with the true ground state as $\tau$
increases. Error bars grow as the circuit gets longer, but meaningful results
are still obtained with more than 400 CNOT gates. This success provides an
incentive to move toward larger lattices, including novel three-dimensional
approaches like the triamond lattice.

###  II.2 The SU(2) Hamiltonian for a triamond lattice

Figure 4: Two views of the unit cell for a triamond lattice. Each unit cell
contains 8 sites and 12 gauge links. Figure 5: The triamond lattice having 3
unit cells along each direction.

Figs. 4 and 5 show the locations of lattice sites and gauge links on a
triamond lattice. See also our two-minute video presentation [79], which
provides views of the triamond lattice from a camera moving around it.

To introduce the triamond lattice quantitatively, we will first define six
unit vectors labeled as colors (red, green, blue, cyan, magenta, yellow) to
aid the discussion,

$\displaystyle\hat{r}$ $\displaystyle=\frac{\hat{j}-\hat{k}}{\sqrt{2}},$
$\displaystyle\hat{g}$ $\displaystyle=\frac{\hat{i}-\hat{j}}{\sqrt{2}},$
$\displaystyle\hat{b}$ $\displaystyle=\frac{\hat{k}-\hat{i}}{\sqrt{2}},$ (11)
$\displaystyle\hat{c}$ $\displaystyle=\frac{\hat{j}+\hat{k}}{\sqrt{2}},$
$\displaystyle\hat{m}$ $\displaystyle=\frac{\hat{i}+\hat{j}}{\sqrt{2}},$
$\displaystyle\hat{y}$ $\displaystyle=\frac{\hat{k}+\hat{i}}{\sqrt{2}},$ (12)

where $\hat{i}$, $\hat{j}$ and $\hat{k}$ are the standard orthonormal vectors.
Every gauge link in the triamond lattice is along one of these color
directions. As an aside, we note that the colors defined here have nothing to
do with the colors of quantum chromodynamics.

Each lattice site is touched by three gauge links that lie in a plane. The
complete lattice has just four sets of parallel planes, so we color each site
white, red, green or blue to represent which plane is at that site. The links
and sites form a color scheme that is motivated by the well-known RGB model of
colors: each white site is touched by red, green and blue links, each red site
is touched by red, magenta and yellow links, each green site is touched by
green, cyan and yellow links, and each blue site is touched by blue, cyan and
magenta links.

The angle between any two planes is $\arccos(\frac{1}{3})$, which is familiar
because this same angle lies between any pair of lines joining the center of a
cube to neighboring corners. Because the unit vector orthogonal to a plane is
only unique up to its sign, the triamond lattice is chiral. Specifically, the
mirror image of Fig. 5 is a distinct but equally valid triamond lattice.

Notice that the white sites of a triamond lattice form a body-centered cubic
(bcc) structure. It is useful to compare the options of defining a lattice
gauge theory on a triamond lattice, on a bcc lattice, or on a simple cubic
lattice. Relative to a simple cubic lattice of the same volume, a bcc lattice
has twice as many lattice sites, gauge links that are shorter by a factor of
$\sqrt{2}$, and each bcc site has 8 nearest neighbors compared to 6 for the
simple cubic lattice. This suggests that a bcc lattice could be valuable for
efficient discussions of the continuum limit, perhaps alongside simple cubic
studies. Importantly, the bcc and simple cubic lattices both have more than
three gauge links touching each lattice site, meaning that extra qubits would
be needed within each lattice site (and a convention must be chosen) to fully
define the quantum state of the lattice [36, 40, 54]. In a sense, the triamond
lattice can be viewed as a bcc lattice where those implicit qubits have been
made explicit and unambiguous by creating the red, green and blue sites.

Although the white sites do form a bcc lattice, the triamond lattice is more
economical than merely being a clarified bcc lattice. Imagine, for example,
shrinking the red, green and blue links into their white sites. The remaining
lattice would not be bcc because it would have only 6 links per unit cell (2
cyan, 2 magenta, and 2 yellow) whereas a bcc lattice would require 8 links per
unit cell.

The comparison to bcc is encouraging, but our primary motivation for proposing
the triamond lattice is the fact that it has exactly three links touching each
site, so the $j$ values of each gauge link are sufficient to define basis
states for the lattice. For planar physics, a hexagonal lattice succeeds in
having three links touching each site, and SU(2) gauge theory has recently
been implemented on a hexagonal lattice [58, 60]. Notice that the shortest
closed paths on a hexagonal lattice have six gauge links.

The shortest closed paths on a triamond lattice have 10 gauge links. We refer
to these as elementary plaquettes. Among the 10 gauge links comprising any
elementary plaquette, one of the 6 colors is absent and the other 5 colors
always occur twice. The lattice has only 6 orientations of plaquettes (the
non-red plaquette, the non-green plaquette, etc.), with copies of them
translated spatially throughout the lattice.

To perform any quantum computation on a triamond lattice, we need a
Hamiltonian. For SU(2) gauge theory, the Hamiltonian is the non-Abelian
generalization of quantum electrodynamics and is written as a sum of electric
terms and magnetic terms [22]. In the continuum limit, the Hamiltonian is

$\displaystyle H^{\rm cont}$ $\displaystyle=$ $\displaystyle H_{E}^{\rm
cont}+H_{B}^{\rm cont}\,,$ (13) $\displaystyle H_{E}^{\rm cont}$
$\displaystyle=$ $\displaystyle\frac{1}{g^{2}}\int{\rm
Tr}\left(E_{x}^{2}+E_{y}^{2}+E_{z}^{2}\right)d^{3}x\,,$ (14) $\displaystyle
H_{B}^{\rm cont}$ $\displaystyle=$ $\displaystyle\frac{1}{g^{2}}\int{\rm
Tr}\left(F_{xy}^{2}+F_{yz}^{2}+F_{zx}^{2}\right)d^{3}x\,,$ (15)

where each trace is over the SU(2) indices of the electric field $\vec{E}$ or
field strength tensor $F_{\mu\nu}$.

On a lattice, each gauge link is an element of SU(2),

$U(\vec{n},\hat{s})=e^{i\hat{s}\cdot\vec{A}(\vec{n})}\,,$ (16)

and the component of the gauge field at site $\vec{n}$ that points in
direction $\hat{s}$ is an element of the SU(2) algebra,

$A_{s}(\vec{n})=\sum_{k=1}^{3}\sigma^{k}A_{s}^{k}(\vec{n})\,.$ (17)

Our computations will be performed in the electric basis, where the magnetic
terms are best expressed as a sum of all elementary plaquettes in this lattice
and each plaquette is the trace of the ordered product of the 10 gauge links
comprising that plaquette. The somewhat tedious derivation of this lattice
Hamiltonian is outlined in Sec. IV.3. The result is

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{E}+H_{B}\,,$ (18)
$\displaystyle H_{E}$ $\displaystyle=$
$\displaystyle\frac{4\sqrt{2}a^{3}}{3g^{2}}\sum_{n={\rm links}}{\rm
Tr}\left(E_{x}^{2}(n)+E_{y}^{2}(n)+E_{z}^{2}(n)\right)\,,~{}~{}~{}~{}~{}$ (19)
$\displaystyle H_{B}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{g^{2}a}\sum_{\vec{w}={\rm
white}}\sum_{s=1}^{6}\mathcal{P}_{s}(\vec{w})\,,$ (20)

where the lattice spacing $a$ is defined to be the distance between nearest-
neighbor sites, i.e. the length of each gauge link on the lattice. The sum in
$H_{B}$ over 6 plaquettes, $\mathcal{P}_{s}(\vec{w})$, for each white lattice
site includes all plaquettes on the lattice. Specifically, there is one of
each type of plaquette (non-red, non-green, etc.) associated with each white
site, and we use the subscript $s$ to identify the color that is absent from
the plaquette.

Any eigenstate of $H_{E}$ is fully defined by providing the quantum numbers
for every gauge link on the triamond lattice, $j_{1}$, $j_{2}$, $j_{3}$,
…$j_{N}$. Our shorthand for an eigenstate is $\left|\\{j\\}\right>$. The
diagonal matrix elements of the Hamiltonian are

$\left<\\{j\\}\right|H_{E}\left|\\{j\\}\right>=\frac{4\sqrt{2}}{3g^{2}a}\sum_{n=1}^{N}j_{n}(j_{n}+1)\,,$
(21)

and the off-diagonal matrix elements from initial state $\left|\\{j\\}\right>$
to final state $\left|\\{J\\}\right>$ come from Eq. (20) with

$\displaystyle\left<\\{J\\}\right|\mathcal{P}_{s}(n)\left|\\{j\\}\right>$
$\displaystyle=$ $\displaystyle\prod_{\rm
sites}(-1)^{j_{e}+j_{f}+\frac{1}{2}+J_{b}}\sqrt{2J_{f}+1}$ (24)
$\displaystyle\sqrt{2j_{b}+1}\left\\{\begin{array}[]{ccc}j_{e}&j_{f}&j_{b}\\\
\frac{1}{2}&J_{b}&J_{f}\end{array}\right\\}\,,$

where the right-hand side has a standard $6j$ symbol and the product includes
the 10 sites in order around the plaquette. For SU(2), either direction around
the plaquette gives the same result. There are three links at each site, one
external to the plaquette (subscript $e$), one in the forward direction around
the plaquette (subscript $f$), and one in the backward direction around the
plaquette (subscript $b$). Eq. (24) is not specific to the triamond lattice
and has been obtained previously by other authors [54, 55].

###  II.3 Computations on the triamond unit cell

As shown in Fig. 4, a unit cell contains 12 gauge links. By using periodic
boundary conditions, a single unit cell becomes a viable three-dimensional
lattice having exactly three gauge links touching each site. If each SU(2)
gauge link is truncated to just two basis states, $j=0$ and $j=\frac{1}{2}$,
the unit cell accommodates $2^{12}=4096$ basis states.

A simple way to begin exploring this theory is to use the variational method,
which will find the lowest energy state among all states that couple to a
selected ansatz. Fig. 6 shows three energy eigenstates obtained from three
different ansatze, each having just one variational parameter. (We name the
parameter $\theta$ in every case, but they are three different parameters.)
Each of these examples is successful in finding a true eigenstate of the
Hamiltonian. The 12-qubit quantum circuits that produced these results can be
found in Sec. IV.4.

Figure 6: Three energy eigenvalues for one unit cell of the triamond lattice
with periodic boundary conditions. The gauge coupling is $g=1$. Each data
point was obtained by running qiskit code with $10^{4}$ shots on the ibm_qasm
noiseless simulator. For comparison, red dashed lines show the true
eigenvalues.

To gain more intuition, notice that there is no 10-sided path inside a unit
cell. This means any plaquette operator will wrap around the lattice and touch
one of the gauge links twice, thus changing that value of $j$ twice. With our
truncation to $j\in\\{0,\frac{1}{2}\\}$, this means only 8 gauge links are
changed when a plaquette operator is applied to the single-cell lattice.

For example, begin with $j=0$ on all 12 links (called the bare vacuum) and
then apply a non-red plaquette. The non-red plaquette touches one of the cyan
links twice while avoiding the other cyan link, thus producing a state with
$j=0$ for all red and cyan links but $j=\frac{1}{2}$ for the 8 other links. If
we had applied a non-cyan plaquette instead, the result would have been
exactly the same because it touches one of the red links twice. Similarly, the
non-green and non-magenta plaquettes are identical to each other ($j=0$ for
green and magenta but $j=\frac{1}{2}$ for the 8 other links), and the non-blue
and non-yellow plaquettes are identical to each other ($j=0$ for blue and
yellow but $j=\frac{1}{2}$ for the 8 other links).

In fact, any number of plaquette operators can be applied to the bare vacuum
one after the other in any order, and the final state will always be one of
these four basis states: the bare vacuum, the non-red non-cyan plaquette, the
non-green non-magenta plaquette and the non-blue non-yellow plaquette. The
part of the Hamiltonian matrix that governs these four states is

$H_{\rm
vac}=\frac{2\sqrt{2}g^{2}}{a}\left(\begin{array}[]{cccc}0&\frac{-1}{2g^{4}}&\frac{-1}{2g^{4}}&\frac{-1}{2g^{4}}\\\
\frac{-1}{2g^{4}}&8&\frac{-1}{32g^{4}}&\frac{-1}{32g^{4}}\\\
\frac{-1}{2g^{4}}&\frac{-1}{32g^{4}}&8&\frac{-1}{32g^{4}}\\\
\frac{-1}{2g^{4}}&\frac{-1}{32g^{4}}&\frac{-1}{32g^{4}}&8\end{array}\right)\,.$
(25)

Of the 4096 basis states for the periodic unit cell, only 32 obey Gauss’s law.
The requirement of Gauss’s law is that each site has either all three gauge
links with $j=0$ or exactly one gauge link with $j=0$. Because the triamond
lattice is symmetric under color interchanges, the 32$\times$32 Hamiltonian
for Gauss-compliant states can be block diagonalized into eight 4$\times$4
blocks, including the one shown explicitly in Eq. (25). Each of the 32 states
has exactly 0, 4, 6 or 8 of the gauge links with $j=\frac{1}{2}$. From top to
bottom, the three examples in Fig. 6 are dominated by a state where the number
of gauge links with $j=\frac{1}{2}$ is 6, 4 and 0 respectively. In particular,
the bottom panel of Fig. 6 is showing the true ground state of the theory,
which is built from four basis states.

Block diagonalizing the Hamiltonian is simple enough for such a small lattice,
but phenomenological applications will require lattices with more than a
single unit cell, and this is where the Hilbert space grows rapidly. A lattice
with $N$ unit cells with have $12N$ gauge links. Truncating each link to
$j\in\\{0,\frac{1}{2}\\}$ leads to $2^{12N}$ basis states, and the ground
state is built from $2^{4N-2}$ of them. Larger lattices are where the
variational method, imaginary time evolution, and other algorithms implemented
with error mitigation on quantum computers have the potential to be of great
value.

##  III Discussion

In this work, SU(2) gauge theory has been truncated to
$j\in\\{0,\frac{1}{2}\\}$ and studied on small lattices.

Self-mitigation has allowed the QITE algorithm [71] to find the ground state
of a lattice with two square plaquettes on an IBM quantum computer. As shown
in Fig. 3, the computation without self-mitigation did not attain the correct
ground state. The basic effect of self-mitigation is a rescaling of
expectation values, where raw results from the physics circuit are divided by
results from a mitigation circuit according to Eqs. (9) and (10). The
Hamiltonian’s expectation value is a linear combination of the individual
rescaled expectation values, leading to the large improvement observed in Fig.
3 where, from an initial value of zero, the self-mitigated and unmitigated
results ultimately move in opposite directions. The unmitigated result is
approaching the pure noise limit while the self-mitigated result finds the
true value quickly and remains consistent with it.

The error bars on Fig. 3 are statistical only, and self-mitigation has clearly
handled the dominant systematic error. The mitigated result at $\tau=1.3$
happens to be furthest from the true ground state, being almost four standard
deviations away, but subsequent time steps agree nicely with the true ground
state. This demonstrates a valuable feature of the QITE algorithm. Output from
one time step is needed as input for the following step, and yet the algorithm
recovers quickly from an outlier.

The application of QITE to a larger lattice will require more terms in the
expression for $\hat{A}$ in Eq. (5), leading to more expectation values that
need to be computed. However, the expression for $\hat{A}$ will generally not
require all possible Pauli strings but is instead governed by the correlation
length of the physics under consideration [71]. It will be interesting to see
future studies that use self-mitigated QITE to examine non-Abelian gauge
theories on larger lattices.

The triamond crystal offers a systematic way to define three-dimensional
lattices. Because there are only three gauge links touching each lattice site,
the quantum numbers of the links themselves are sufficient to fully define any
basis state of the lattice. This is not true for a simple cubic lattice [58].
In fact, the triamond lattice is the only highly symmetric lattice that
achieves this goal in three dimensions. To be more precise, the triamond
lattice is the only three-dimensional lattice that is strongly isotropic and
has three gauge links per site [65]. Strong isotropy refers to the fact that
rotation (or reflection) of the lattice around any site can interchange any
pair of links at that site while leaving the physical structure of the entire
lattice invariant [65]. In particular, a simple cubic lattice is not strongly
isotropic [65, 68].

Another interesting glimpse into the symmetries of the triamond lattice comes
from noticing that the six vectors comprising a triamond lattice form a
regular tetrahedron. Begin with one gauge link of each color (recall Eqs. (11)
and (12)) and translate each one spatially without any rotations. The red,
green and blue links will form the equilateral triangular base of the
tetrahedron. The cyan, magenta and yellow links rise from the corners of that
base to meet at the top of the tetrahedron. As might be expected from the
sentences preceding Eq. (25), the cyan link of the tetrahedron touches all
links except the red one, the magneta link touches all but green, and the blue
link touches all but yellow.

For eyes used to seeing cubic lattices, the triamond might seem difficult to
visualize. However, our derivation shows that the SU(2) Hamiltonian has the
familiar form of a sum over just a few plaquette orientations, in this case
six, and our quantum circuits for obtaining ground and excited states (see
Figs. 7 and 8) are quite simple due to the triamond symmetries. The triamond
lattice has only the expected parameters, namely the gauge coupling and the
lattice volume, but it provides extra gauge links inside a unit cell,
representing a smaller lattice spacing and a new trajectory for approaching
the continuum limit of a gauge theory.

Figure 7: The circuit used to obtain the bottom panel of Fig. 6. The 12 qubits
are labeled by their colors, with two copies of each color appearing in the
list. A constant angle is defined by $\phi=\arccos(1/\sqrt{3})$. Figure 8: The
circuit used to obtain the center panel of Fig. 6. The 12 qubits are labeled
by their colors, with two copies of each color appearing in the list.

The present study has explored error mitigation and the triamond lattice as
separate steps toward a common goal. Future work can combine error mitigation
with the triamond lattice to continue the progress toward a practical
implementation of lattice gauge theories on quantum computers.

##  IV Methods

###  IV.1 Tomography for the QITE algorithm

The QITE procedure [71] needs optimal values for the coefficients $a_{jk}$ in
Eq. (5). For notational convenience the subscripts can be combined into a
single subscript, so $a_{jk}\equiv a_{J}$ with $J$ running from 1 to 6. The
$a_{J}$ values can be obtained by minimizing Trotter errors, which means
minimizing the difference between

$\left|\Delta_{0}\right>\equiv e^{-i\tau\hat{A}}\left|\psi\right>$ (26)

and

$\left|\Delta\right>\equiv-i\tau\hat{A}\left|\psi\right>\,.$ (27)

It is more convenient to work with a scalar function rather than states, so we
actually minimize

$\left<\Delta_{0}-\Delta|\Delta_{0}-\Delta\right>\equiv\left<\Delta_{0}|\Delta_{0}\right>+a_{J}b_{J}+a_{J}S_{JK}a_{K}\,,$
(28)

where the 6-component vector $b$ and the 6$\times$6 matrix $S$ are real-
valued. Minimization results in

$a=-(S+S^{T})^{-1}b\,,$ (29)

with

$\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle-i\left<\psi\right|\left[\hat{H},\hat{Y}_{0}\right]\left|\psi\right>,$
(30) $\displaystyle b_{2}$ $\displaystyle=$
$\displaystyle-i\left<\psi\right|\left[\hat{H},\hat{X}_{1}\hat{Y}_{0}\right]\left|\psi\right>,$
(31) $\displaystyle b_{3}$ $\displaystyle=$
$\displaystyle-i\left<\psi\right|\left[\hat{H},\hat{Z}_{1}\hat{Y}_{0}\right]\left|\psi\right>,$
(32) $\displaystyle b_{4}$ $\displaystyle=$
$\displaystyle-i\left<\psi\right|\left[\hat{H},\hat{Y}_{1}\right]\left|\psi\right>,$
(33) $\displaystyle b_{5}$ $\displaystyle=$
$\displaystyle-i\left<\psi\right|\left[\hat{H},\hat{Y}_{1}\hat{X}_{0}\right]\left|\psi\right>,$
(34) $\displaystyle b_{6}$ $\displaystyle=$
$\displaystyle-i\left<\psi\right|\left[\hat{H},\hat{Y}_{1}\hat{Z}_{0}\right]\left|\psi\right>,$
(35)

and the only nonzero entries in $S+S^{T}$ are

$\displaystyle(S+S^{T})_{11}=(S+S^{T})_{22}$ $\displaystyle=$ $\displaystyle
2,$ (36) $\displaystyle(S+S^{T})_{33}=(S+S^{T})_{44}$ $\displaystyle=$
$\displaystyle 2,$ (37) $\displaystyle(S+S^{T})_{55}=(S+S^{T})_{66}$
$\displaystyle=$ $\displaystyle 2,$ (38)
$\displaystyle(S+S^{T})_{12}=(S+S^{T})_{21}$ $\displaystyle=$ $\displaystyle
2\left<\psi\right|\hat{X}_{1}\left|\psi\right>,$ (39)
$\displaystyle(S+S^{T})_{13}=(S+S^{T})_{31}$ $\displaystyle=$ $\displaystyle
2\left<\psi\right|\hat{Z}_{1}\left|\psi\right>,$ (40)
$\displaystyle(S+S^{T})_{14}=(S+S^{T})_{41}$ $\displaystyle=$ $\displaystyle
2\left<\psi\right|\hat{Y}_{1}\hat{Y}_{0}\left|\psi\right>,$ (41)
$\displaystyle(S+S^{T})_{25}=(S+S^{T})_{52}$ $\displaystyle=$ $\displaystyle
2\left<\psi\right|\hat{Z}_{1}\hat{Z}_{0}\left|\psi\right>,$ (42)
$\displaystyle(S+S^{T})_{26}=(S+S^{T})_{62}$ $\displaystyle=$
$\displaystyle-2\left<\psi\right|\hat{Z}_{1}\hat{X}_{0}\left|\psi\right>,$
(43) $\displaystyle(S+S^{T})_{35}=(S+S^{T})_{53}$ $\displaystyle=$
$\displaystyle-2\left<\psi\right|\hat{X}_{1}\hat{Z}_{0}\left|\psi\right>,$
(44) $\displaystyle(S+S^{T})_{36}=(S+S^{T})_{63}$ $\displaystyle=$
$\displaystyle 2\left<\psi\right|\hat{X}_{1}\hat{X}_{0}\left|\psi\right>,$
(45) $\displaystyle(S+S^{T})_{45}=(S+S^{T})_{54}$ $\displaystyle=$
$\displaystyle 2\left<\psi\right|\hat{X}_{0}\left|\psi\right>,$ (46)
$\displaystyle(S+S^{T})_{46}=(S+S^{T})_{64}$ $\displaystyle=$ $\displaystyle
2\left<\psi\right|\hat{Z}_{0}\left|\psi\right>.$ (47)

The commutators defining the elements of $b$ use the Hamiltonian from Eq. (1).
The numerical computation of $b$, $S+S^{T}$ and $r^{\prime}/r$ (from Eq. (4))
requires nine expectation values. Three of them can be obtained by preparing
the state $\left|\psi\right>$ in a quantum computer and measuring each qubit,
which gives

$\displaystyle\left<\psi\right|\hat{Z}_{0}\left|\psi\right>$ $\displaystyle=$
$\displaystyle 1-2P_{0},$ (48)
$\displaystyle\left<\psi\right|\hat{Z}_{1}\left|\psi\right>$ $\displaystyle=$
$\displaystyle 1-2P_{1},$ (49)
$\displaystyle\left<\psi\right|\hat{Z}_{1}Z_{0}\left|\psi\right>$
$\displaystyle=$ $\displaystyle 1-2P_{1\oplus 0},$ (50)

where $P_{j}$ is the probability that qubit $j$ is 1, and $P_{1\oplus 0}$ is
the probability that both qubits are 1. Three other expectation values are
obtained by preparing the state
$\left|\psi^{\prime}\right>=RY_{1}(-\frac{\pi}{2})RY_{0}(-\frac{\pi}{2})\left|\psi\right>$
in a quantum computer and measuring each qubit, which gives

$\displaystyle\left<\psi\right|\hat{X}_{0}\left|\psi\right>$ $\displaystyle=$
$\displaystyle 1-2P_{0}^{\prime},$ (51)
$\displaystyle\left<\psi\right|\hat{X}_{1}\left|\psi\right>$ $\displaystyle=$
$\displaystyle 1-2P_{1}^{\prime},$ (52)
$\displaystyle\left<\psi\right|\hat{X}_{1}X_{0}\left|\psi\right>$
$\displaystyle=$ $\displaystyle 1-2P_{1\oplus 0}^{\prime}.$ (53)

The remaining three expectation values are obtained similarly: Preparing
$\left|\psi^{\prime\prime}\right>=RX_{1}(\frac{\pi}{2})RX_{0}(\frac{\pi}{2})\left|\psi\right>$
gives

$\left<\psi\right|\hat{Y}_{1}Y_{0}\left|\psi\right>=1-2P_{1\oplus
0}^{\prime\prime}.$ (54)

Preparing
$\left|\psi^{\prime\prime\prime}\right>=RY_{0}(-\frac{\pi}{2})\left|\psi\right>$
gives

$\left<\psi\right|\hat{Z}_{1}X_{0}\left|\psi\right>=1-2P_{1\oplus
0}^{\prime\prime\prime}.$ (55)

Preparing
$\left|\psi^{\prime\prime\prime\prime}\right>=RY_{1}(-\frac{\pi}{2})\left|\psi\right>$
gives

$\left<\psi\right|\hat{X}_{1}Z_{0}\left|\psi\right>=1-2P_{1\oplus
0}^{\prime\prime\prime\prime}.$ (56)

Our computations create and measure these five circuits separately for each
time step. This means each one is self-mitigated, with 50 randomized
compilings of its physics circuit and 50 of its mitigation circuit.

It should be noted that the QITE procedure does not require $S+S^{T}$ to be an
invertible matrix. For all practical purposes, $(S+S^{T})^{-1}$ can be
interpreted as the Moore-Penrose pseudo-inverse [80, 81] and this is what we
use for our calculations.

###  IV.2 Randomized compiling for CNOT gates

The conversion of CNOT errors into incoherent noise is accomplished by
randomizing the input to each CNOT gate in a circuit [72, 17]. Two random
Pauli gates are applied immediately before the CNOT gate, one to the control
qubit and the other to the target qubit. Specifically, each gate is chosen
randomly from the set $\\{I,X,Y,Z\\}$. Immediately after the CNOT gate, two
Pauli gates are applied to ensure that the combined effect of Pauli gates
would not change the circuit’s output on error-free hardware. The 16 options
are shown explicitly in Table 1.

Table 1: The 16 options for randomized compiling. Applied before the CNOT | Applied after the CNOT
---|---
control | target | control | target
I | I | I | I
I | X | I | X
I | Y | Z | Y
I | Z | Z | Z
X | I | X | X
X | X | X | I
X | Y | Y | Z
X | Z | Y | Y
Y | I | Y | X
Y | X | Y | I
Y | Y | X | Z
Y | Z | X | Y
Z | I | Z | I
Z | X | Z | X
Z | Y | I | Y
Z | Z | I | Z

###  IV.3 Deriving the triamond SU(2) Hamiltonian

This section outlines the derivation that begins with gauge links in the form
of Eq. (16) on a triamond lattice and arrives at the Hamiltonian in Eqs.
(18-20). The coefficients in Eqs. (18-20) are defined by their need to agree
with continuum SU(2) gauge theory, so our first step will be to expand the sum
over plaquettes in powers of the lattice spacing.

Consider a general gauge link, $U(\vec{w}+\vec{t},\hat{s})$, where $\vec{w}$
is a white site on the lattice. The lattice spacing $a$ enters through the
vectors $\vec{t}$ and $\hat{s}$. As a specific example, the gauge link that
begins at the nearest green site and points toward the neighboring yellow site
is

$\displaystyle U(\vec{w}+a\hat{g},\frac{\hat{k}+\hat{i}}{\sqrt{2}})$
$\displaystyle=$ $\displaystyle
I+\frac{ia}{2}\left(A_{x}(\vec{w})+A_{z}(\vec{w})\right)$ (57)
$\displaystyle+\frac{ia^{2}}{4}(\partial_{x}-\partial_{y})\left(A_{x}(\vec{w})+A_{z}(\vec{w})\right)$
$\displaystyle-\frac{a^{2}}{8}\left(A_{x}(\vec{w})+A_{z}(\vec{w})\right)^{2}$
$\displaystyle+O(a^{3})\,.$

Although not displayed here, terms at $O(a^{3})$ and $O(a^{4})$ must also be
retained. Performing 10 such expansions provides an expression for the non-
blue plaquette, which is

$\displaystyle\mathcal{P}_{\bar{b}}$ $\displaystyle=$ $\displaystyle{\rm
Tr}\big{(}U(\vec{w},\hat{g})U(\vec{w}+a\hat{g},\hat{y})U(\vec{w}+a\hat{g}+a\hat{y},\hat{m})$
(58) $\displaystyle
U(\vec{w}+a\hat{g}+a\hat{y}+a\hat{m},\hat{c})U^{\dagger}(\vec{w}+a\hat{y}+a\hat{m}+a\hat{c},\hat{g})$
$\displaystyle
U(\vec{w}+a\hat{m}+a\hat{c}+a\hat{y},\hat{r})U^{\dagger}(\vec{w}+a\hat{r}+a\hat{m}+a\hat{c},\hat{y})$
$\displaystyle
U^{\dagger}(\vec{w}+a\hat{r}+a\hat{m},\hat{c})U^{\dagger}(\vec{w}+a\hat{r},\hat{m})U^{\dagger}(\vec{w},\hat{r})\big{)}\,,$

and can be expanded to become

$\displaystyle\mathcal{P}_{\bar{b}}$ $\displaystyle=$ $\displaystyle
2-2a^{4}{\rm
Tr}\big{(}(\partial_{x}A_{y}-\partial_{y}A_{x}+\partial_{z}A_{y}-\partial_{y}A_{z})^{2}\big{)}$
(59) $\displaystyle-4ia^{4}{\rm
Tr}\big{(}(\partial_{x}A_{y}-\partial_{y}A_{x}+\partial_{z}A_{y}-\partial_{y}A_{z})$
$\displaystyle([A_{x},A_{y}]+[A_{z},A_{y}])\big{)}$ $\displaystyle+2a^{4}{\rm
Tr}\big{(}([A_{x},A_{y}]+[A_{z},A_{y}])^{2}\big{)}+O(a^{5})\,,$

where all fields are evaluated at $\vec{w}$. The expression for the non-yellow
plaquette is similar to this non-blue result and summing the two of them
removes some cross terms,

$\displaystyle\mathcal{P}_{\bar{b}}+\mathcal{P}_{\bar{y}}$ $\displaystyle=$
$\displaystyle 4-4a^{4}{\rm
Tr}\big{(}(\partial_{x}A_{y}-\partial_{y}A_{x})^{2}\big{)}$ (60)
$\displaystyle-4a^{4}{\rm
Tr}\big{(}(\partial_{z}A_{y}-\partial_{y}A_{z})^{2}\big{)}$
$\displaystyle-8ia^{4}{\rm
Tr}\big{(}(\partial_{x}A_{y}-\partial_{y}A_{x})[A_{x},A_{y}]\big{)}$
$\displaystyle-8ia^{4}{\rm
Tr}\big{(}(\partial_{z}A_{y}-\partial_{y}A_{z})[A_{z},A_{y}]\big{)}$
$\displaystyle+4a^{4}{\rm
Tr}\big{(}[A_{x},A_{y}]^{2}+[A_{z},A_{y}]^{2}\big{)}+O(a^{5}).~{}~{}~{}~{}~{}~{}$

The sum of non-green and non-magenta plaquettes can be obtained from this by
relabeling colors and $x,y,z$ subscripts. The same is true for the sum of non-
red and non-cyan. Finally, the sum of all 6 plaquettes is

$\sum_{k=1}^{6}\mathcal{P}_{k}=12-8a^{4}{\rm
Tr}\left(F_{xy}^{2}+F_{yz}^{2}+F_{zx}^{2}\right)+O(a^{5})\,,$ (61)

where the field strength tensor is
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+i[A_{\mu},A_{\nu}]$.

The continuum expression for the magnetic Hamiltonian, Eq. (15), can easily be
converted from an integral to a sum,

$H_{B}=\frac{V}{g^{2}a}\sum_{\vec{w}={\rm white}}{\rm
Tr}\big{(}F_{xy}^{2}(\vec{w})+F_{yz}^{2}(\vec{w})+F_{zx}^{2}(\vec{w})\big{)}+O(a)\,,$
(62)

where $V$ is the volume per white site. There are two white sites in each unit
cell, and the volume of a unit cell is $(2\sqrt{2}a)^{3}$, so
$V=8\sqrt{2}a^{3}$. Therefore the magnetic Hamiltonian becomes

$H_{B}=\frac{\sqrt{2}}{g^{2}a}\sum_{\vec{w}={\rm
white}}\left(12-\sum_{k=1}^{6}\mathcal{P}_{k}(\vec{w})\right)+O(a)\,.$ (63)

The constant term has no effect on dynamics so it can be dropped and we are
left with the expression given in Eq. (20).

Converting the electric Hamiltonian from the continuum to the triamond lattice
is straightforward. An important detail is that instead of the volume per
white site, $V$, we now need the volume per link which is $V/6$. Begin with
Eq. (14) and discretize to find

$\displaystyle H_{E}$ $\displaystyle=$
$\displaystyle\frac{V}{6g^{2}}\sum_{n={\rm links}}{\rm
Tr}\left(E_{x}^{2}(n)+E_{y}^{2}(n)+E_{z}^{2}(n)\right)+O(a)$ $\displaystyle=$
$\displaystyle\frac{4\sqrt{2}a^{3}}{3g^{2}}\sum_{n={\rm links}}{\rm
Tr}\left(E_{x}^{2}(n)+E_{y}^{2}(n)+E_{z}^{2}(n)\right)+O(a)$

in agreement with Eq. (19).

###  IV.4 Hamiltonian and circuits for triamond unit cell

To obtain the results in Fig. 6, we begin with the Hamiltonian of SU(2) gauge
theory truncated to $j\in\\{0,\frac{1}{2}\\}$ on a unit cell of the triamond
lattice with periodic boundary conditions. The general expression was obtained
in Sec. II.2 and here it is written in terms of Pauli operators.

The Hamiltonian is the sum of an electric part and three magnetic parts,

$H=H_{E}+H_{\bar{r}\bar{c}}+H_{\bar{g}\bar{m}}+H_{\bar{b}\bar{y}}\,.$ (66)

The electric part is a sum of contributions from all gauge links as in Eq.
(21),

$H_{E}=\frac{2\sqrt{2}}{g^{2}a}\sum_{k=1}^{12}\frac{1-Z_{k}}{2}\,.$ (67)

The magnetic terms correspond respectively to application of the non-red non-
cyan plaquette,

$\displaystyle H_{\bar{r}\bar{c}}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{4g^{2}a}\prod_{k\notin\\{r,c\\}}X_{k}$ (68)
$\displaystyle-\frac{3\sqrt{2}}{4g^{4}a}\left(\prod_{k\in\\{r,c\\}}\frac{1-Z_{k}}{2}\right)\prod_{k\notin\\{r,c\\}}X_{k}$
$\displaystyle+\frac{3\sqrt{2}}{16g^{4}a}\left(\prod_{k\in\\{r,c\\}}\frac{1+Z_{k}}{2}\right)\prod_{k\notin\\{r,c\\}}X_{k}\,,$

application of the non-green non-magenta plaquette,

$\displaystyle H_{\bar{g}\bar{m}}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{4g^{2}a}\prod_{k\notin\\{g,m\\}}X_{k}$ (69)
$\displaystyle-\frac{3\sqrt{2}}{4g^{4}a}\left(\prod_{k\in\\{g,m\\}}\frac{1-Z_{k}}{2}\right)\prod_{k\notin\\{g,m\\}}X_{k}$
$\displaystyle+\frac{3\sqrt{2}}{16g^{4}a}\left(\prod_{k\in\\{g,m\\}}\frac{1+Z_{k}}{2}\right)\prod_{k\notin\\{g,m\\}}X_{k}\,,$

and application of the non-blue non-yellow plaquette.

$\displaystyle H_{\bar{b}\bar{y}}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{4g^{2}a}\prod_{k\notin\\{b,y\\}}X_{k}$ (70)
$\displaystyle-\frac{3\sqrt{2}}{4g^{4}a}\left(\prod_{k\in\\{b,y\\}}\frac{1-Z_{k}}{2}\right)\prod_{k\notin\\{b,y\\}}X_{k}$
$\displaystyle+\frac{3\sqrt{2}}{16g^{4}a}\left(\prod_{k\in\\{b,y\\}}\frac{1+Z_{k}}{2}\right)\prod_{k\notin\\{b,y\\}}X_{k}\,.$

Notice that each magnetic term has 8 Pauli $X$ gates and a projector acting on
the other 4 qubits.

The next thing needed to obtain the results in Fig. 6 is a quantum circuit to
prepare the ansatz containing the variational parameter. Based on the
discussion surrounding Eq. 25, we anticipate that the physical ground state
will have a contribution from the bare ground state as well as a triply
degenerate contribution from three plaquette terms. Our variational parameter
$\theta$ will be used to adjust the relative strengths of these two
contributions. Our circuit is displayed in Fig. 7 and is responsible for the
bottom panel of Fig. 6. This particular ansatz is quite simple, with two
qubits at its core and extra CNOT gates adjusting the other qubits in a
straightforward way. The circuit has been designed such that it requires
minimal connectivity among the 12 qubits and is suitable, for example, to run
on IBM’s heavy hex architecture without additional swap gates. The center
panel of Fig. 6 uses the trial state obtained from the circuit in Fig. 8. The
top panel of Fig. 6 uses a trial state that differs from Fig. 8 simply by
appending a Pauli $X$ gate at the end of the circuit for each cyan, magenta
and yellow qubit.

The final step to obtain the results in Fig. 6 is to measure the expectation
value of each term in the Hamiltonian. Each term is a product of $X$ and $Z$
gates. Expectation values involving only $Z$ gates can be obtained directly
from measurements of the state itself. Expectation values involving $X$ gates
can be obtained by rotating into the computational basis with a pair of
Hadamard gates, $X=HZH$. Each data point in Fig. 6 comes from the linear
combination of output from 4 separate runs of the circuit, one for $H_{E}$,
one for $H_{\bar{r}\bar{c}}$, one for $H_{\bar{g}\bar{m}}$, and one for
$H_{\bar{b}\bar{y}}$.

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###### Acknowledgements.

The authors thank R. M. Woloshyn for suggesting that self-mitigation be
applied to QITE, W. G. Parrott for discussions about quantum imaginary time
evolution, and M. Tsang for discussions about triamond crystals. We thank all
three for reading a draft of the manuscript. We also acknowledge the use of
IBM Quantum services for this work. The views expressed are those of the
authors, and do not reflect the official policy or position of IBM or the IBM
Quantum team. This work is affiliated with the QC4HEP Working Group and
supported by Canada’s Natural Sciences and Engineering Research Council.