# Primitive Quantum Gates for an $SU(2)$ Discrete Subgroup: $\mathbb{BT}$

Erik J. Gustafson<EMAIL_ADDRESS>Fermi National Accelerator Laboratory,
Batavia, Illinois, 60510, USA  Henry Lamm<EMAIL_ADDRESS>Fermi National
Accelerator Laboratory, Batavia, Illinois, 60510, USA  Felicity Lovelace
<EMAIL_ADDRESS>Department of Physics, University of Illinois at Chicago,
Chicago, Illinois 60607, USA  Damian Musk<EMAIL_ADDRESS>Stanford
University Online High School, Redwood City, CA 94063, USA

###### Abstract

We construct a primitive gate set for the digital quantum simulation of the
binary tetrahedral ($\mathbb{BT}$) group on two quantum architectures. This
nonabelian discrete group serves as a crude approximation to $SU(2)$ lattice
gauge theory while requiring five qubits or one quicosotetrit per gauge link.
The necessary basic primitives are the inversion gate, the group
multiplication gate, the trace gate, and the $\mathbb{BT}$ Fourier transform
over $\mathbb{BT}$. We experimentally benchmark the inversion and trace gates
on ibm_nairobi, with estimated fidelities between $14-55\%$, depending on the
input state.

††preprint: FERMILAB-PUB-22-583-SQMS-T

##  I Introduction

Simulating the dynamics of lattice field theories offers a clear potential for
quantum advantage [1, 2, 3, 4, 5, 6]. Time evolution on quantum computers
requires efficiently implementing the unitary operator $U(t)=e^{-iHt}$.
Various approximations for $U(t)$ with different tradeoffs exist [3, 7, 8, 9,
10, 11, 11, 12, 13, 14, 14, 15, 16, 17, 18], but all of them require
implementing key group theoretic operations as quantum circuits in the case of
lattice gauge theories [19]. This separation of the problem into group-
dependent primitives [20] and algorithmic design [21] has proven fruitful in
optimizing both.

For efficient digital simulations, many proposals exist on how the lattice
gauge degrees of freedom can be truncated [22, 23, 24, 25, 26, 27, 28, 29, 7,
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]. For some regulated
theories, the desired theory may not even be the true continuum limit [62, 63,
64, 65, 66, 67, 50, 51, 53, 68, 69]. Furthermore, the relative efficacy of
schemes is dimension-dependent [45, 70, 20].

One promising digitization method is the discrete subgroup approximation [7,
31, 32, 33, 56, 34, 71, 35, 72]. This method was explored early on in
Euclidean lattice field theory to reduce resources. Replacing $U(1)$ by
$\mathbb{Z}_{N}$ was considered in [73, 74]. Extensions to the crystal-like
subgroups of $SU(N)$ were made in Refs. [75, 76, 77, 31, 32, 56, 78, 79, 80],
including with fermions [81, 82]. Theoretical studies revealed that the
discrete subgroup approximation corresponds to continuous groups broken by a
Higgs mechanism [83, 84, 85, 86, 87]. On the lattice, this causes the discrete
subgroup to poorly approximate the continuous group below a _freezeout_
lattice spacing $a_{f}$ (or beyond a coupling $\beta_{f}$).

Lattice calculations are performed at fixed lattice spacing $a=a(\beta)$ which
approaches zero as $\beta\rightarrow\infty$ for asymptotically free theories.
Finite $a$ leads to discrepancies from the continuum results, but provided one
simulates in the _scaling regime_ below $a_{s}(\beta_{s})$, these errors
should be polynomial in $a$. Any approximation error from using discrete
subgroups should be tolerable provided $a_{s}\gtrsim a_{f}$ or equivalently
that $\beta_{s}\lesssim\beta_{f}$. For the $3+1d$ Wilson action, $\beta_{f}$
are known. In the case of $U(1)$ where $\beta_{s}=1$, $\mathbb{Z}_{n>5}$
satisfies $\beta_{f}>\beta_{s}$. For nonabelian gauge groups, only a few
crystal-like subgroups exist. $SU(2)$ has three: the binary tetrahedral
$\mathbb{BT}$, the binary octahedral $\mathbb{BO}$, and the binary icosahedral
$\mathbb{BI}$. The scaling regime for $SU(2)$ occurs around $\beta_{s}=2.2$.
Therefore, a value of $\beta_{f}=2.24(8)$ for $\mathbb{BT}$ is unlikely to
prove useful with just the Kogut-Susskind Hamiltonian $H_{KS}$, although
experience with $SU(3)$ suggests modified or improved Hamiltonians $H_{I}$
would prove sufficient [32, 56, 78, 79]. The other two groups, $\mathbb{BO}$
and $\mathbb{BI}$, have values far into the scaling regime:
$\beta_{f}=3.26(8)$ and $\beta_{f}=5.82(8)$, respectively [32].

Substantial work has studied the quantum simulation of abelian theories,
particularly in low dimensions. Despite this, one must remember that
nonabelian gauge theories demonstrate many behaviors unseen in abelian ones;
thus, results for $U(1)$ or $\mathbb{Z}_{N}$ may fail to represent the full
complexity of lattice gauge theories. The group of interest in this paper, the
24-element $\mathbb{BT}$, is the smallest crystal-like subgroup of a
nonabelian theory and requires 5 qubits per register. The dihedral groups,
$D_{N}$, while not crystal-like, have previously been investigated for
simulation on quantum computers [7, 19, 20, 88]. Having $2N$ elements
respectively, they require $\lceil\log_{2}(2N)\rceil$ qubits per register.
Further studies have been undertaken to understand the $\mathbb{Q}_{8}$
subgroup of $SU(2)$ [72] which requires only 3 qubits.

In the interest of studying quantum simulations on near-term devices, we
should consider both $3+1d$ and $2+1d$ theories. Using classical lattice
simulations, we have determined that in both spacetimes
$\beta_{f}\approx\beta_{s}$ for the Wilson action (See Fig. 1). Thus quantum
simulations with $\mathbb{BT}$ require an improved Hamiltonian [21] and will
be the only one considered in this work. Since $a_{f}\propto e^{-\beta_{f}}$
within the scaling regime, only a small improvement in the Hamiltonian is
needed.

Figure 1: Euclidean calculations of lattice energy density $\langle
E_{0}\rangle$ of $\mathbb{BT}$ as measured by the expectation value of the
plaquette as a function of Wilson coupling $\beta$ on $4^{d}$ lattices for
(top) $2+1d$ (bottom) $3+1d$. The shaded region indicates
$\beta\leq\beta_{s}$.

In this paper, we construct quantum circuits implementing the four primitive
gates (inversion, multiplication, trace, and Fourier) required to simulate the
$\mathbb{BT}$ theories. We will consider two possible quantum devices when
constructing our gates. The first is a qubit-based device. The second device,
motivated by the potential for bosonic quantum comptuers [89], is a $d=24$
qudit device where only one qudit is required per register. We refer to this
d=24 state qudit as a $quicosotetrit$. Although time evolution on quantum
processors is infeasible at present due to the low gate fidelities and
coherence time, we benchmark the process fidelity of the inversion and trace
gates for $\mathbb{BT}$ on the ibm_nairobi QPU to evaluate the improvements
needed for simulations on quantum processors.

This paper is organized as follows. In Sec. II, the necessary group theoretic
properties of $\mathbb{BT}$ are summarized and the digitization scheme is
presented. A review of the basic qubit and qudit gates used in this work is
found in Sec. III. Sec. IV summarized the four primitive gates required for
implementing the group operations. This is followed by quantum circuit
constructions for these gates for $\mathbb{BT}$ gauge theories: the inversion
gate in Sec. V, the multiplication gate in Sec. VI, the trace gate in Sec.
VII, and the Fourier transform gate in Sec. VIII. Benchmark results for our
$\mathbb{BT}$ inversion and trace gates are found in Sec. IX. Using these
gates, Sec. X presents a resource estimates for simulating $3+1d$ $SU(2)$. We
conclude and discuss future work in Sec. XI.

##  II Properties of $\mathbb{BT}$

The simulation of lattice gauge theories requires the definition of a register
where one can store the state of a bosonic link variable which we call a
$G-$register. In order to construct the $\mathbb{BT}-$register in term of
integers, it is necessary to construct a mapping between the 24 elements of
the group and the integers $[0,23]$. A clean way to obtain this is to write
every element of $\mathbb{BT}$ as an ordered product of four generators111The
minimal set of generators for $\mathbb{BT}$ is two, but we have been unable to
find an ordered product with less than three. The choice of three generators
is the same as the one with four generators where
$(-1)^{m}\mathbf{i}^{n}\rightarrow\mathbf{i}^{2m+n}$. Nevertheless the qubit
costs cannot go below the current formulation’s value of
$\lceil\log_{2}(24)\rceil=5$. with exponents written in terms of the binary
variables $m,n,o,p,q$:

$g=(-1)^{m}\mathbf{i}^{n}\mathbf{j}^{o}\mathbf{l}^{p+2q},$ (1)

with

$\mathbf{l}=-\frac{1}{2}(\mathbb{1}+\mathbf{i}+\mathbf{j}+\mathbf{k})$ (2)

and $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are the unit quaternions which in
the 2d irreducible representation (irrep) correspond to Pauli matrices. With
the construction of Eq. (1), the $\mathbb{BT}-$register is given by a binary
encoding of the qubits with the ordering $|qponm\rangle$. This same mapping
can be utilized for quicosotetrits . For example, using $\eta=1+i$ one element
in the real 2d irrep is

$\frac{1}{2}\begin{pmatrix}\eta&\eta^{*}\\\
-\eta&\eta^{*}\end{pmatrix}=(-1)^{1}\mathbf{i}^{1}\mathbf{j}^{1}\mathbf{l}^{0+2\times
1}\rightarrow|10111\rangle=|23\rangle$ (3)

The $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ generators anti-commute with
each other. Additional useful relations are:

$\begin{split}&\mathbf{i}^{2}=\mathbf{j}^{2}=-\mathbb{1},~{}\mathbf{l}^{3}=\mathbb{1}\\\
&\mathbf{i}\mathbf{j}=\mathbf{k},~{}\mathbf{j}\mathbf{k}=\mathbf{i},~{}\mathbf{k}\mathbf{i}=\mathbf{j},\\\
&\mathbf{l}\mathbf{i}=\mathbf{j}\mathbf{l},~{}\mathbf{l}\mathbf{j}=\mathbf{k}\mathbf{l},~{}\mathbf{l}\mathbf{k}=\mathbf{i}\mathbf{l},\\\
&\mathbf{l}^{2}\mathbf{i}=\mathbf{k}\mathbf{l}^{2},~{}\mathbf{l}^{2}\mathbf{j}=\mathbf{k}\mathbf{l}^{2},~{}\mathbf{l}^{2}\mathbf{k}=\mathbf{j}\mathbf{l}^{2}.\\\
\end{split}$ (4)

Table 1: Character Table of $\mathbb{BT}$ including an enumeration of the elements in the given class. Size | 1 | 1 | 6 | 4 | 4 | 4 | 4
---|---|---|---|---|---|---|---
Order | 1 | 2 | 4 | 6 | 6 | 3 | 3
$\rho_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1
$\rho_{2}$ | 1 | 1 | 1 | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega$
$\rho_{3}$ | 1 | 1 | 1 | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega^{2}$
$\rho_{4}$ | 2 | $-2$ | 0 | 1 | 1 | $-1$ | $-1$
$\rho_{5}$ | 2 | $-2$ | 0 | $\omega^{2}$ | $\omega$ | $-\omega^{2}$ | $-\omega$
$\rho_{6}$ | 2 | $-2$ | 0 | $\omega^{2}$ | $\omega$ | $-\omega$ | $-\omega^{2}$
$\rho_{7}$ | 3 | 3 | $-1$ | 0 | 0 | 0 | 0
$\ket{g}$ | $\ket{0}$ | $\ket{1}$ |  $\ket{2}$,$\ket{3}$ |  $\ket{9}$,$\ket{10}$ |  $\ket{17}$,$\ket{19}$ |  $\ket{8}$,$\ket{11}$ |  $\ket{16}$,$\ket{18}$
|  |  |  $\ket{4}$,$\ket{5}$ |  $\ket{12}$,$\ket{14}$ |  $\ket{21}$,$\ket{23}$ |  $\ket{13}$,$\ket{15}$ |  $\ket{20}$,$\ket{22}$
|  |  |  $\ket{6}$,$\ket{7}$ |  |  |  |

The character table (Table 1) lists important group properties; the different
irreps can be identified by the value of their character acting on each
element. An irrep’s dimension is the value of the character of $\mathbb{1}$.
There are three $1d$ irreps, three $2d$ irreps (one real and two complex), and
one $3d$ irrep. To derive the Fourier transform, it is necessary to know a
matrix presentation of each irrep. Based on our qubit mapping, given a
presentation of $-1,\mathbf{i},\mathbf{j},$ and $\mathbf{l}$ we can construct
any element of the group from Eq. (1). With the $3$rd root of unity
$\omega=e^{2\pi i/3}$, the 1d irreps are given:

$\rho_{1}:-1=\mathbf{i}=\mathbf{j}=\mathbf{l}=1$ (5)
$\rho_{2}:-1=\mathbf{i}=\mathbf{j}=1,\;\mathbf{l}=\omega^{2}$ (6)
$\rho_{3}:-1=\mathbf{i}=\mathbf{j}=1,\;\mathbf{l}=\omega$ (7)

Now for the 2d irreps, we can use for all three irreps the same definitions:

$\begin{split}\rho_{4,5,6}:&-1=\text{diag}(-1,-1),\;\mathbf{i}=\text{diag}(i,-i),\\\
\;&\mathbf{j}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},\;\mathbf{l}=-\frac{1}{2}\begin{pmatrix}\eta&-\eta\\\
\eta^{*}&\eta^{*}\end{pmatrix}\end{split}$ (8)

then we can construct the three 2d irreps by taking:

$\rho_{4}(g)=(-1)^{m}\mathbf{i}^{n}\mathbf{j}^{o}\mathbf{l}^{p+2q}$ (9)
$\rho_{5}(g)=(-1)^{m}\mathbf{i}^{n}\mathbf{j}^{o}(\omega^{2}\mathbf{l})^{p+2q}$
(10)
$\rho_{6}(g)=(-1)^{m}\mathbf{i}^{n}\mathbf{j}^{o}(\omega\mathbf{l})^{p+2q}$
(11)

For the 3d irrep, we have

$\begin{split}\rho_{7}&:-1=\text{diag}(1,1,1),\quad\mathbf{i}=\text{diag}(-1,1,-1),\\\
&\quad\mathbf{j}=\text{diag}(1,-1,-1),\quad\mathbf{l}=\begin{pmatrix}0&1&0\\\
0&0&1\\\ 1&0&0\end{pmatrix}\end{split}$ (12)

##  III Qubit and Qudit Gates

In order to implement the group primitive gates on qubit and qudit hardware we
need a set of quantum gate operations. We begin by first enumerating the qubit
gates, followed by a discussion of the qudit gates.

The first basic qubit gates we need are the Pauli gates $p=X,Y,Z$. These can
be extended to arbitrary rotations about their respective axes
$R_{p}(\theta)=e^{i\theta p/2}$. When decomposing onto fault-tolerant devices,
the $T=\text{diag}(1,e^{i\pi/4})$ gate becomes relevant.

The first multiqubit operation we need is the SWAP operation, which swaps two
qubits:

$\text{SWAP}~{}\ket{a}\otimes\ket{b}=\ket{b}\otimes\ket{a}.$

The controlled not (CNOT) gate applies the $X$ operation on a target qubit if
the control qubit is in the state $\ket{1}$:

$\text{CNOT}\ket{a}\otimes\ket{b}=\ket{a}\otimes\ket{b\oplus a},$

where $\oplus$ indicates addition modulus 2. We also need the following
multiqubit gates: CnNOT – of which C2NOT is called the Toffoli gate – and
CSWAP (Fredkin) gates. The CnNOT gate is the further extension to the case of
where the $n$ control qubits must be in the $\ket{1}^{\otimes n}$ state. For
example, the Toffoli in terms of modular arithmetic is

$\text{C}^{2}\text{NOT}\ket{a}\otimes\ket{b}\otimes\ket{c}=\ket{a}\otimes\ket{b}\otimes\ket{c\oplus
ab}.$

The CSWAP gate swaps two qubit states if the control is in the $\ket{1}$
state:

$\begin{split}\text{CSWAP}\ket{a}\otimes\ket{b}\otimes\ket{c}=&\ket{a}\otimes\ket{b(1\oplus
a)\oplus ac}\\\ &\otimes\ket{c(1\oplus a)\oplus ab}.\\\ \end{split}$

One final qubit operation, the controlled permutation gate $C\chi$, will prove
useful to define for conciseness later. Fig. 2 constructs it in terms of CnNOT
gates.

Figure 2: The controlled permutation gate, C$\chi$.

We also need a set of gates for implementation on quicosotetrit device. In our
case, there are not specialized quicosotetrit gates but a general set of qudit
ones to consider. The single qudit gates we need are: Givens rotations, the
selective number of arbitrary photon (SNAP) [90, 91], displacement [92], and
photon blockade gates [93, 94].

Givens rotations $R_{p}^{(a,b)}(\theta)$ are generalizations of
$R_{p}(\theta)$ to qudits where rotations occur in the subspace of states
$\ket{a}$ and $\ket{b}$ while leaving all the other states untouched. We also
use the notation $p^{(a,b)}$ to indicate special Givens rotations that
correspond to the generalized Pauli gates (e.g. $X^{(3,4)}$). These gates are
useful in designing algorithms for simulating with qudits, but are difficult
to natively implement. Therefore real simulations will likely require their
decomposition.

One native gate set for single qudits in cavity QED devices is SNAP and
displacement gates [91, 92, 90]. SNAP gates can arbitrarily phase the qudit
states:

$\mathcal{S}(\vec{\theta})=\sum_{a=0}^{N-1}\ket{a}\bra{a}e^{i\theta_{a}}$ (13)

where the sum is over computational basis states $a=[0,N-1]$ of an $N-$state
qudit and $\vec{\theta}=\\{\theta_{0},\theta_{1},\ldots,\theta_{N-1}\\}$ are a
set of tunable parameters analogous to $\theta$ in $R_{Z}(\theta)$.

The displacement gate coherently changes the cavity’s photon number. In terms
of Fock-operators $\hat{a},\hat{a}^{\dagger}$, it is

$\mathcal{D}(\alpha)=e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}.$ (14)

The photon blockade operation acts as an $R_{X}(\theta)$ or $R_{Y}(\theta)$
rotation between two Fock states in a cavity by driving the system at an off-
resonant frequency while shifting the desired modes to said frequency [93,
94].

In order to obtain a universal set of gates, we require an entangling gate.
One proposal is the controlled SNAP gate which phases a target qudit based on
if a second qudit is in a state $\ket{\alpha}$ [95]:

$\mathcal{S}_{c}(\vec{\theta},\alpha)=\sum_{a=0,a\neq\alpha}^{N-1}|a\rangle\langle
a|\otimes\mathbb{1}+|\alpha\rangle\langle\alpha|\otimes\sum_{b=0}^{N-1}|b\rangle\langle
b|e^{i\theta_{b}}.$ (15)

##  IV Overview of Primitive Gates

For general gauge groups, it is possible to define any quantum circuit with
sets of primitive gates. Using this formulation confers two benefits: first,
it is possible to design algorithms in a theory- and hardware-agnostic way;
second, the circuit optimization is split into smaller, more manageable
pieces. This construction begins with defining for $G$ a $G$-register by
identifying each group element with a computational basis state $\ket{g}$,
where $g\in G$. One choice of primitive gates is: inversion
$\mathfrak{U}_{-1}$, multiplication $\mathfrak{U}_{\times}$, trace
$\mathfrak{U}_{\rm Tr}$, and Fourier transform $\mathfrak{U}_{F}$ [19].

The inversion gate, $\mathfrak{U}_{-1}$, is a single register gate which takes
a group element to its inverse:

$\mathfrak{U}_{-1}\left|g\right>=\left|g^{-1}\right>\text{.}$ (16)

The group multiplication gate acts on two $G-$registers. It takes the target
$G-$register and changes the state to the left product with the control
$G-$register:

$\mathfrak{U}_{\times}\ket{g}\ket{h}=\ket{g}\ket{gh}.$ (17)

Left multiplication is sufficient for a minimal set as right multiplication
can be implemented use two applications of $\mathfrak{U}_{-1}$ and
$\mathfrak{U}_{\times}$, albeit optimal algorithms may take advantage of an
explicit construction [21].

The trace of products of group elements appears in lattice Hamiltonians. We
can implement these terms by combining $\mathfrak{U}_{\times}$ with a single-
register trace gate:

$\mathfrak{U}_{\operatorname{Tr}}(\theta)\left|g\right>=e^{i\theta\operatorname{Re}\operatorname{Tr}g}\left|g\right>.$
(18)

The final gate required is the group Fourier transform $\mathfrak{U}_{F}$. The
Fourier transform of a finite $G$ is defined as

$\displaystyle\hat{f}(\rho)=\sqrt{\frac{d_{\rho}}{|G|}}\sum_{g\in
G}f(g)\rho(g),$ (19)

where $|G|$ is the size of the group, $d_{\rho}$ is the dimensionality of the
representation $\rho$, and $f$ is a function over $G$. The inverse transform
is given by

$\displaystyle
f(g)=\frac{1}{\sqrt{|G|}}\sum_{\rho\in\hat{G}}\sqrt{d_{\rho}}\operatorname{Tr}{(\hat{f}(\rho)\rho(g^{-1}))},$
(20)

where the dual $\hat{G}$ is the set of all irreducible representations (irrep)
of $G$. The gate that performs this acts on a single $G$-register with some
amplitudes $f(g)$ which rotate it into the Fourier basis:

$\mathfrak{U}_{F}\sum_{g\in
G}f(g)\left|g\right>=\sum_{\rho\in\hat{G}}\hat{f}(\rho)_{ij}\left|\rho,i,j\right>.$
(21)

The second sum is taken over $\rho$, the irreducible representations of $G$;
$\hat{f}$ denotes the Fourier transform of $f$. After application of the gate,
the register is denoted as a $\hat{G}$-register to indicate the change of
basis. A schematic example of this gate is show in Fig. 3

Figure 3: Example $\mathfrak{U}_{F}$ from Eq. (19) using column vectors
$\tilde{\rho}_{i,j}=\sqrt{d_{\rho}/|G|}\rho_{i,j}$ where
$\rho_{i,j}=\rho_{i}(g_{j})$. This example has three irreps with
$d_{\rho}=1,2,3$. $\mathfrak{U}_{F}$ is square since
$\sum_{\rho}d_{\rho}^{2}=|G|$

##  V Inversion Gate

Consider a $\mathbb{BT}-$register storing the group element given by
$g=(-1)^{m_{0}}\mathbf{i}^{n_{0}}\mathbf{j}^{o_{0}}\mathbf{l}^{p_{0}+2q_{0}}$.
The effect of the inversion gate on this register is to transform it to

$|g\rangle=|q_{0}p_{0}o_{0}n_{0}m_{0}\rangle\rightarrow|g^{-1}\rangle=|q_{1}p_{1}o_{1}n_{1}m_{1}\rangle.$
(22)

Using Eq. (1) and $(AB)^{-1}=B^{-1}A^{-1}$, the inverse of $g$ is

$g^{-1}=(-1)^{m_{0}+n_{0}+o_{0}}\mathbf{l}^{3-p_{0}-2q_{0}}\mathbf{j}^{o_{0}}\mathbf{i}^{n_{0}}.$

We can then put this into the normal ordering of Eq. (1) using the relations
in Eq. (4) to find:

$g^{-1}=(-1)^{m_{1}}\mathbf{i}^{n_{1}}\mathbf{j}^{o_{1}}\mathbf{l}^{p_{1}+2q_{1}}.$
(23)

where the relation between the $|g\rangle$ and $|g^{-1}\rangle$ indices are :

$\begin{split}m_{1}&=m_{0}+n_{0}+o_{0}+n_{0}\times o_{0}\\\
n_{1}&=n_{0}(1-q_{0})+o_{0}(p_{0}+q_{0})\\\
o_{1}&=o_{0}(1-p_{0})+n_{0}(p_{0}+q_{0})\\\ p_{1}&=q_{0}\\\ q_{1}&=p_{0}.\\\
\end{split}$ (24)

A qubit circuit implementation of $\mathfrak{U}_{-1}$ is shown in Fig. 4. We
can map Eq. (24) onto a quantum circuit using modular arithmetic, finding that
transforming $m_{0}$ to $m_{1}$ uses two CNOTs and a Toffoli gate. A circuit
with a CSWAP and two Toffolis is required for $n_{1}$ and $o_{1}$, while
$p_{1}$ and $q_{1}$ need one SWAP.

Figure 4: A qubit implementation of $\mathfrak{U}_{-1}$ which uses Toffoli and
CSWAP gates. These multiqubit entangling gates can be decomposed into one- and
two-qubit gates as discussed in the literature.

The quicosotetrit circuit of $\mathfrak{U}_{-1}$ is simpler, needing only 11
$X^{(a,b)}$ gates2221 $X^{(a,b)}$ of the 12 is unnecessary since
$\ket{0^{-1}}=\ket{0}$ and $\ket{1^{-1}}=\ket{1}$, as seen in Fig. 5. 24
$\mathcal{S}(\vec{\theta})$ and 25 $\mathcal{D}(\alpha)$ are sufficient to
approximate $\mathfrak{U}_{-1}$ to sub-percent infidelity. Inspecting this
circuit, the largest separation between inverses is $\ket{10}$ and $\ket{23}$.
Using only $\mathcal{S}(\vec{\theta})$ and $\mathcal{D}(\alpha)$, this could
prove noisy in terms of the necessary $\vec{\theta}$ and $\alpha$, and thus
large-separation photon blockade gates are desirable.

Figure 5: An quicosotetrit implementation of $\mathfrak{U}_{-1}$ using the
$X^{(a,b)}$ gate.

##  VI Multiplication Gate

The method to construct the $\mathfrak{U}_{\times}$ for qubits is similar to
that for $\mathfrak{U}_{-1}$. Given two $\mathbb{BT}-$registers storing $g$
and $h$:

$\displaystyle g$
$\displaystyle=(-1)^{m_{0}}\mathbf{i}^{n_{0}}\mathbf{j}^{o_{0}}\mathbf{l}^{q_{0}+2p_{0}},\,h=(-1)^{m_{1}}\mathbf{i}^{n_{1}}\mathbf{j}^{o_{1}}\mathbf{l}^{q_{1}+2p_{1}},$
(25)

we want $gh=g\times h$ and permuting $|h\rangle$ to $|gh\rangle$. Defining
$gh=(-1)^{m_{2}}i^{n_{2}}j^{o_{2}}l^{2p_{2}+q_{2}}$, we can derive via Eq.
(4):

$\displaystyle m_{2}$
$\displaystyle=o_{1}n_{0}(1-p_{1})+(n_{1}n_{0}+o_{1}o_{0})(1-q_{1})$
$\displaystyle\phantom{xxx}+n_{1}o_{0}(p_{1}+q_{1})$ $\displaystyle n_{2}$
$\displaystyle=n_{1}+n_{0}(1-q_{1})+o_{0}(p_{1}+q_{1})$ $\displaystyle o_{2}$
$\displaystyle=o_{1}+o_{0}(1-p_{1})+n_{0}(p_{1}+q_{1})$ $\displaystyle p_{2}$
$\displaystyle=p_{0}(1-q_{1})(1-p_{1})$ $\displaystyle q_{2}$
$\displaystyle=q_{0}(1-q_{1})(1-p_{1}).$ (26)

These expressions map into the qubit circuit of Fig. 6.

Figure 6: A qubit implementation of $\mathfrak{U}_{\times}$. Following
convention, filled (open) circles correspond to control on $\ket{1}$
($\ket{0}$). The C$\chi$ gate is defined in Fig. 2 and C$\chi^{-1}$ is its
inverse.

$\mathcal{U}_{\times}$ on quicosotetrits is a permutation of $\ket{h}$
controlled by $\ket{g}$ realized as

$\mathcal{U}_{\times}=\sum_{g\in G}\sum_{h\in G}|g\rangle\langle
g|\otimes|h\rangle\langle g\times h|=\sum_{g\in G}|g\rangle\langle
g|\otimes\hat{P}_{g},$

where $\hat{P}_{g}$ is a permutation matrix that depends upon $g$. This
unitary matrix can be diagonalized by one-qudit gates $V_{g}$. A quicosotetrit
circuit for $\mathfrak{U}_{\times}$ is shown in Fig. 8.

The structure of $V_{g}$ depends on the order $m$ of the element $g$ (Tab. 1)
being multiplied onto the element $h$. For a given operator $g$ the elements
$h\in G$ will break down into $24/m$ sets of $m$ elements. The elements in
each set are determined solely by the operator $g$ itself. The elements in
each set can be generated by taking $h$ and left multiplying it by $g$ until
the element $h$ is reached again. These sets of elements will provide a
presentation of a $\mathbb{Z}_{m}$ group. In this way $V_{g}$ can be rendered
into a set of $24/m$ blocks of size $m$. These blocked sections with at most
an $SU(6)$ rotation in the given subspace. The $SU(N)$ Euler angle
decompositions are provided in Appendix A along with the group cycles
generated by each $g$. Tab. 2 provides the total number of
$R_{p}^{(a,b)}(\theta)$ for each $QFT_{\mathbb{Z}_{m}}$ on the $m$-level
subspace as well as $V_{g}$.

A directed graph for the group sets for $g=-1,~{}\mathbf{l},$ and
$-\mathbf{l}$ ($\ket{1}$, $\ket{8}$, $\ket{9}$ respectively) are shown in Fig.
7. In this figure we show how multiplication of a group element $h$ on the
left by a group element $g$ will cycle through a subset of the group elements
in a directed graph. For example multiplication by $-1$ ($\ket{1}$) flip flops
elements $|2a\rangle$ and $|2a+1\rangle$. In this way the neighboring states
will have a $\mathbb{Z}_{2}$ Fourier transform applied on each pair.
Multiplication by the element $\mathbf{l}$ ($\ket{8}$) will cycle the states
$|a\rangle,~{}\ket{a+8},$ and $\ket{a+16}$ for $a\leq 7$. The cycles shown for
multiplication by $-\mathbb{l}$ ($\ket{9}$), are more complicated to write in
closed form but are shown in the right hand side of Fig. 7.

Table 2: $R_{p}^{(a,b)}(\theta)$ required for $QFT_{\mathbb{Z}_{m}}$ and $V_{g}$ for each order. $N_{\ket{g}}$ denote the number of elements with that cycle. cycle | $N_{\ket{g}}$ |  $R_{p}^{(a,b)}(\theta)$ in $QFT_{\mathbb{Z}_{m}}$ |  $R_{p}^{(a,b)}(\theta)$ in $V_{g}$
---|---|---|---
1 | 1 | 0 | 0
2 | 1 | 3 | 36
3 | 8 | 8 | 64
4 | 6 | 15 | 90
6 | 8 | 35 | 140

In this way $V^{\dagger}_{g}\hat{P}_{g}V_{g}$ will be a diagonal matrix whose
nonzero elements are phases corresponding to the eigenvalues of $\hat{P}_{g}$.
As we iterate through $g_{i}$, neighboring $V_{g_{i}}V^{\dagger}_{g_{i+1}}$
can be combine into a single qudit operation. If we use Tab. 2 as a starting
point and recognize that for the order $3$, $4$, and $6$ that $V_{g}$’s appear
in pairs of states, almost half of the $R_{p}^{(a,b)}(\theta)$ are eliminated,
leaving 2,244 to implement all $V_{g}$’s. In terms of native gates,
$\mathfrak{U}_{\times}$ needs 575 SNAP and 575 $\mathcal{D}(\alpha)$ gates in
addition to the 23 cSNAP gates for $\hat{P}_{g}$. This cost could be reduced
by pulse engineering, an active research area in bosonic quantum computers
[96, 97, 98, 99].

Figure 7: Left: Pictorial representation of the cycles for $\ket{g}$ being:
$\ket{1}$ with $m=2$, (black arrows); $\ket{8}$ with $m=3$, (orange dotted
arrows). Right: representation of the cycle for $\ket{9}$ with $m=6$,
different cycles are shown in different colors. Figure 8: A quicosotetrit
implementation of $\mathfrak{U}_{\times}$. The subscript $g_{i}$ indicates the
the ith element of the group.

##  VII Trace gate

For simulating gauge theories, $\mathfrak{U}_{\rm Tr}$ is only needed for the
fundamental representation, $\rho_{4}$. The character table (Tab. 1) provides
us with the $\operatorname{Re}\operatorname{Tr}(g_{i})$ necessary.
$\mathfrak{U}_{\rm Tr}$ can be obtained by defining a Hamiltonian, and then
exponentiating it. For our qubit-register, $H_{tr}$ for $\rho_{4}$ is

$\begin{split}H_{\rm
Tr}=&Z_{m_{0}}\bigg{(}Z_{q_{0}}\big{[}2+\big{(}1+Z_{o_{0}}\big{)}\big{(}Z_{n_{0}}+Z_{p_{0}}(1+Z_{n_{0}})\big{]}\\\
&+Z_{p_{0}}\big{[}Z_{o_{0}}+Z_{n_{0}}-1\big{]}\bigg{)}\end{split}$ (27)

were $Z_{h}$ acts on the $|h\rangle$ qubit. From this, we can decomposing
$e^{i\theta H_{tr}}$ into the linear combinations of $R_{Z}(\theta)$ gates.
The qubit-based circuit for $\mathfrak{U}_{\rm Tr}$ is shown in Fig. 9.

Figure 9: A qubit implementation of $\mathfrak{U}_{\rm Tr}$.

Implementing $\mathfrak{U}_{\rm Tr}$ on a quicosotetrit requires 9
$R^{(a,b)}_{Z}(\theta)$ gates corresponding to the 9 ($g_{i}$,$-g_{i}$) pairs
with $\operatorname{Re}\operatorname{Tr}(g_{i})\neq 0$; this gate is shown in
Fig. 10. Together, these gates can be mapped to a single SNAP gate.

Figure 10: A quicosotetrit implementation of $\mathfrak{U}_{Tr}$ using two
level $R_{Z}^{(a,b)}(\theta)$ gates.

##  VIII Fourier Transform

The standard $n$-qubit quantum Fourier transform (QFT) [100] corresponds to
the quantum version of the fast Fourier transform of $\mathbb{Z}_{2^{n}}$.
Quantum Fouirer transforms over several nonabelian groups exist in the
literature [101, 102, 103, 104, 20]. Alas, for all the crystal-like subgroups
of interest to high energy physics efficient QFT circuits are currently
unknown [105]. For the general case, there isn’t a clear algorithmic way to
construct the QFT. Therefore, we instead construct a suboptimal
$\mathfrak{U}_{F}$ from Eq. (19) using the irreps of Sec. II to obtain the
matrix in Fig. 11.

Figure 11: Matrix representation of $\mathfrak{U}_{F}$ where $\eta=1+i$ and
$\omega=e^{\frac{2\pi i}{3}}$.

Since $\mathbb{BT}$ has 24 elements, on a qubit device $\mathfrak{U}_{F}$ must
be embedded into a larger $2^{d}\times 2^{d}$ matrix. With this matrix, a
transpiler can be used to derive a circuit. Using the Qiskit transpiler,
$\mathfrak{U}_{F}$ requires 1025 CNOTs, 2139 $R_{Z}(\theta)$, and 1109
$R_{Y}(\theta)$; the Fourier gate is the most expensive qubit primitive. As
will be discussed in Sec. X, $\mathfrak{U}_{F}$ dominates the total simulation
costs and future work should be devoted to finding a $\mathbb{BT}$ QFT.

A quicosotetrit implementation follows the same generalized Euler angle
decomposition as the $V_{g}$ used for $\mathfrak{U}_{\times}$ [106]. This gate
can be implemented with 24 SNAP and 25 displacement gates to subpercent
infidelity.

##  IX Experimental Results

In this section, we discuss experimental results from running
$\mathfrak{U}_{-1}$ and $\mathfrak{U}_{\rm Tr}$ on the ibm_nairobi 7 transmon
qubit device (see Fig. 12). Transpiling Fig. 4 and Fig. 9 onto ibm_nairobi,
topology constraints require introducing additional SWAP gates (see Fig. 12).
The cost of $\mathfrak{U}_{\rm Tr}$ increases from 22 CNOTs to 39, and for
$\mathfrak{U}_{-1}$ the number of CNOTs goes from 31 to 49. The high qubit
cost of $\mathfrak{U}_{\times}$ and high gate costs of $\mathfrak{U}_{F}$
suggest they are unlikely to have reasonable fidelities and they are left for
the future.

Figure 12: An example mapping of $\ket{g}=|qponm\rangle$ onto the 7 transmon
qubit ibm_nairobi

.

We define the process fidelity $\mathcal{F}$ of $\mathfrak{U}$ on a state
$\ket{\psi}$ as

$\mathcal{F}^{|\Psi\rangle}_{\mathfrak{U}}=|\langle
0|\Psi^{{\dagger}}\mathfrak{U}^{{\dagger}}\mathfrak{U}\Psi|0\rangle|^{2}$ (28)

Without noise, the state preparation $\Psi$ and $\mathfrak{U}$ are exactly
cancelled by their complex conjugations, thus the measured result should
always be $\ket{0}^{\otimes 5}$. Determining the fidelity requires testing all
the possible states $|\Psi\rangle$, a prohibitively expensive task [107].
Therefore we consider a subset of states given by the 24 group element states
$|g\rangle$ which can be obtained by applying $X$ gates to the appropriate
qubits. For a general state, $\Psi^{{\dagger}}\mathfrak{U}^{{\dagger}}$ could
require as many CNOTs as $\mathfrak{U}\Psi$. In this case, the total circuit
cost is doubled and fidelities are reduced. In contrast, for $|g\rangle$, the
results of applying either of our gates is another $|g\rangle$, which can be
returned to $|0\rangle^{\otimes 5}$ using only $X$ gates. With this, we
compute $\mathcal{F}$ for each $\ket{g}$ for both $\mathfrak{U}_{\rm Tr}$ and
$\mathfrak{U}_{-1}$ without doubling the CNOT count. With these results we
calculate a mean value $\bar{\mathcal{F}}_{\mathfrak{U}}$.

The dominant coherent CNOT error can be mitigated through Pauli twirling [108,
109, 110, 111, 112]. This method converts coherent errors into random Pauli
channel errors and has found success in lattice applications [113, 21]. The
circuits are modified by wrapping each CNOT with a set of Pauli gates
$\\{\mathbb{1},X,Y,Z\\}$ sampled from sets which are logical equivalent to
CNOT. We ran 20 unique circuits for each gate-state pair following prior
results finding $\mathcal{O}(10)$ circuits to suffice [108]. 2000 shots were
taken for the $\mathfrak{U}_{\rm Tr}$ circuits, while 500 were gotten for
$\mathfrak{U}_{-1}$. The process fidelity for each $\ket{g}$ are shown in Fig.
13. Averaging we find that the $\bar{\mathcal{F}}^{|g\rangle}_{\rm
Tr}=55(1)\%$, while the higher CNOT count of $\mathfrak{U}_{-1}$ leads to a
lower fidelity of $\bar{\mathcal{F}}^{|g\rangle}_{-1}=37.0(8)\%$.

Figure 13: Process Fidelities, $\mathcal{F}$, for the trace and inverse gate
on IBM’s computer. The angle used was $\theta=0.7$. The averages are shown as
a shaded band.

While $\ket{g}$ are easy to implement, they are less likely to be encountered
during a simulations, since states must be gauge invariant. In the case of
$\mathfrak{U}_{-1}$, it is possible to test a gauge invariant state
$|GI\rangle=|G|^{-1/2}\sum_{g}|g\rangle$ because both $|GI\rangle$ and
$\Psi^{{\dagger}}\mathfrak{U}_{-1}^{{\dagger}}$ can be implemented with only 1
additional CNOT. For this state, we found a reduced fidelity
$\mathcal{F}^{|GI\rangle}_{-1}=14.8(12)\%$.

##  X Resource Estimates

Clearly time evolution for large lattices of $\mathbb{BT}$ is beyond the NISQ
era and we should consider fault-tolerant quantum computers. The Eastin-Knill
theorem restricts quantum error correcting codes by preventing universal sets
of gates from being implemented transversally [114]. Transversality refers to
the property that gates operating on logical qubits correspond to tensor
products on the physical qubits. For many error correcting codes such as
Calderbank-Shor-Steane (CSS) codes the Clifford gates are transversal [107,
115, 116, 117, 118] while the T gate is not. Therefore, T gates are an
important metric in fault-tolerant algorithm analysis because they require
entanglement between physical qubits [107, 119].

Table 3: Number of physical T gates and clean ancilla required to implement logical gates for (top) basic gates taken from [107] (bottom) primitive gates for $\mathbb{BT}$. Gate | T gates | Clean ancilla
---|---|---
C2NOT | 7 | 0
C3NOT | 28 | 1
CSWAP | 7 | 0
$R_{z}$ | 1.15 $\log_{2}(1/\epsilon)$ | 0
$\mathcal{U}_{-1}$ | 28 | 0
$\mathcal{U}_{\times}$ | 154 | 1
$\mathcal{U}_{Tr}$ | 12.65 $\log_{2}(1/\epsilon)$ | 0
$\mathcal{U}_{FT}$ | 1150 $\log_{2}(1/\epsilon)$ | 0

The Toffoli gate is known to require 7 T gates [107] and the CnNOT gates can
be constructed exactly using a ladder of Toffoli gates and clean ancilla
qubits333A _clean_ ancilla is a qubit initialized to $\ket{0}$. _Dirty_
ancilla indicate ones in an unknown initial state. which can be reused later
[107, 120]. Using this ladder method a $C^{n}$NOT gate can be implemented
using $4(n-1)$ Toffoli gates and $n-1$ clean ancilla qubits. Methods exist
using dirty ancilla at the cost of more T gates [121, 120]. We arrive at the
cost for the $R_{z}$ gates via [122] where these gates can be approximated to
precision $\epsilon$ with at worst $1.15\log(1/\epsilon)$ T gates using the
repeat-until-success method. Using these, we can construct fault-tolerant gate
cost estimates for $\mathbb{BT}$ (See Tab. 3).

Table 4: Number of primitive gates per link per $\delta t$ neglecting boundary effects as a function of dimension $d$ for $H_{I}$. Gate | $N[H_{I}]$
---|---
$\mathfrak{U}_{F}$ | 4
$\mathfrak{U}_{\rm Tr}$ | $\frac{3}{2}(d-1)$
$\mathfrak{U}_{-1}$ | $2+11(d-1)$
$\mathfrak{U}_{\times}$ | $4+26(d-1)$

Primitive gate costs for implementing the improved Hamiltonian, $H_{I}$, per
link per Trotter step $\delta t$ are shown in Tab. 4. Using these costs we
find that a $d$ spatial lattice simulation of $H_{I}$ for time $t=N_{t}\delta
t$ would require

$N_{T}=\left[4312d-3640+(4581.03+18.975d\log_{2}\frac{1}{\epsilon})\right]dL^{d}N_{t}$
(29)

Following [123, 124], we consider a fiducial simulation of the shear viscosity
$\eta$ on a $d=3$ lattice of $L=10$ with $N_{t}=50$, $\epsilon=10^{-8}$ and
the cost of state preparation neglected. For an $SU(2)$ simulation including
fermions, Kan and Nam estimated $3\times 10^{34}$ T gates, while neglecting
fermions allows for a more modest $3\times 10^{19}$. Here, we neglect fermions
and using $\mathbb{BT}$ to approximate $SU(2)$ requires $2.0\times 10^{10}$ T
gates for $H_{I}$. So using $\mathbb{BT}$ reduces the gate costs by 9 orders
of magnitude. The T gate density is 1 T gate per $\mathbb{BT}-$register per
clock cycle and is independent of primitive, although a QFT might increase
this. The large reduction in T gates compared to [124] comes by avoiding
quantum fixed-point arithmetic. For us, $\mathfrak{U}_{F}$ dominates the
simulations – $44\%$ of the total cost.

Compared to qubits, the field of quantum error correction for qudits is less
developed [125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137,
138, 139]. Much, but not all, of the work has focused on qutrits and relies
upon specific hardware and native gates. While this field will develop rapidly
in the coming years, we will restrict ourselves to quicosotetrit resources
estimates based on a device with native cSNAP, SNAP, and displacement gates.
The costs for each $\mathbb{BT}$ gate are shown in Tab. 5. In contrast to the
qubits, for quicosotetrit simulations the most costly gate is
$\mathfrak{U}_{\times}$ with all other gates contributing negligible amounts.
Thus, determination of the QFT is less important for quicosotetrit devices.
Summing the gates, we find the fiducial calculation of the viscosity with
3$\times 10^{3}$ quicosotetrits would require $1.9\times 10^{8}$ cSNAP and
$4.9\times 10^{9}$ SNAP and displacement gates.

Table 5: c$\mathcal{S}(\vec{\theta})$, $\mathcal{S}(\vec{\theta})$, and $\mathcal{D}(\alpha)$ gates required for $\mathbb{BT}$ (top) primitive gates (bottom) $H_{I}$ simulations per link per $\delta t$. Gate | c$\mathcal{S}(\vec{\theta})$ | $\mathcal{S}(\vec{\theta})$ | $\mathcal{D}(\alpha)$
---|---|---|---
$\mathcal{U}_{-1}$ | 0 | 24 | 25
$\mathcal{U}_{\times}$ | 23 | 575 | 575
$\mathcal{U}_{Tr}$ | 0 | 1 | 0
$\mathcal{U}_{FT}$ | 0 | 24 | 25
$e^{-iH_{I}\delta t}$ | $598d-506$ | $15215.5d-12771.5$ | $15225d-12775$

##  XI Conclusions

In this paper, we constructed the necessary primitive quantum circuits for the
simulation of $\mathbb{BT}$ – the smallest crystal-like subgroup of $SU(2)$ –
gauge theories. These circuits were constructed for both qubit and
quicosotetrit architectures and quantum resource estimates were made for the
simulation of pure $SU(2)$ shear viscosity. Compared to previous fault-
tolerant qubit estimates, we require $10^{9}$ fewer T gates by avoiding
quantum fixed point arithmetic via the discrete group approximation. While
these simulations are still far off, we performed quantum fidelity experiments
for two of the gates. Experimentally, we found the fidelity of the inversion
and trace operation to be $37.0(8)\%$ and $55(1)\%$ for classical bit string
states on the ibm_nairobi quantum processor.

Qudit-based quantum computers, like the quicosotetrit device considered here,
are known to require fewer gates, in particular entangling ones. Here we have
demonstrated an additional benefit that the construction of nonabelian group
primitives are dramatically simplified compared to the qubit case by reducing
the complex internal $G-$register logic required to preserve group structure.

Looking forward, primitive gates should be constructed for larger crystal-like
subgroups of $SU(2)$ and to the subgroups of $SU(3)$ theories. At the cost of
more qubits and larger lattice errors, a larger $SU(2)$ subgroup should allow
the possibility of using the Kogut-Susskind Hamiltonian. This would reduce
gate costs by a factor of 2 on a qubit device and a factor of 4 on a qudit one
since different primitives dominate the cost. Finally, in order to further
reduce the qubit-based simulation gate costs for all discrete subgroup
approximations, the formalism for deriving the quantum Fourier transform for
each crystal-like subgroup would be of great interest.

###### Acknowledgements.

The authors thank Doğa Kürkçüoğlu and Sophie Croll for helpful comments. EG is
supported by the U.S. Department of Energy, Office of Science, National
Quantum Information Science Research Centers, Superconducting Quantum
Materials and Systems Center (SQMS) under contract number DE-AC02-07CH11359.
HL and FL are supported by the Department of Energy through the Fermilab
QuantiSED program in the area of “Intersections of QIS and Theoretical
Particle Physics”. Fermilab is operated by Fermi Research Alliance, LLC under
contract number DE-AC02-07CH11359 with the United States Department of Energy.
We acknowledge use of the IBM Q 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 Q team.

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##  Appendix A SU(N) Euler angle decompositions

The following operators $\mathcal{U}_{2}^{(a,b)}$,
$\mathcal{U}_{3}^{(a,b,c)}$, $\mathcal{U}_{4}^{(a,b,c,d)}$, and
$\mathcal{U}_{6}^{(a,b,c,d,e,f)}$ correspond to specific $SU(N)$ rotations
that implement the $QFT_{\mathbb{Z}_{m}}$ of Tab. 2. We use the following
Euler angle decompositions where the superscripts indicate levels that are
swapped between. The $SU(2)$ Euler angle decomposition we require is built
from the well-known ZXZ rotation

$\mathcal{U}^{(a,b)}_{2}(\vec{\theta}_{II})=R^{(a,b)}_{Z}(\theta_{0})R^{(a,b)}_{X}(\theta_{1})R^{(a,b)}_{Z}(\theta_{2}).$
(30)

For the two-state rotation we need, $\vec{\theta}_{II}=[\pi/2,\pi/2,\pi/2]$.
An example of the operator $V_{g}$ with cycle $m=2$ would be for
$g=-\mathbb{1}$, which corresponds to $|1\rangle$, and is given by:

$V_{1}=\prod_{a=0}^{11}\mathcal{U}_{2}^{(2a,2a+1)}(\vec{\theta}_{II}).$ (31)

The $SU(3)$ Euler angle decomposition requires two
$\mathcal{U}^{(a,b)}_{2}(\vec{\theta})$ and two Givens rotations [140, 141,
142, 106],

$\begin{split}\mathcal{U}^{(a,b,c)}_{3}(\vec{\theta}_{III})=&\mathcal{U}^{(a,b)}_{2}(\vec{\theta_{0}})R^{(b,c)}_{X}(\theta_{1})\mathcal{U}^{(a,b)}_{2}(\vec{\theta_{2}})R_{Z}^{(b,c)}(\theta_{3}),\end{split}$
(32)

where the angles are fixed to: $\vec{\theta}_{0}=[7\pi/6,3\pi/2,\pi/2]$,
$\theta_{1}=0.608175\pi$, $\vec{\theta}_{4}=[0,-\pi/2,\pi/3]$,
$\theta_{3}=7\pi/3$. One element with $m=3$ is $g=\mathbf{l}$, corresponding
to $\ket{8}$:

$V_{8}=\prod_{a=0}^{7}\mathcal{U}_{3}^{(a,a+8,a+16)}(\vec{\theta}_{III}).$
(33)

The Euler angle decomposition of an arbitrary $SU(4)$ is given in terms of
three $\mathcal{U}^{(a,b)}_{2}(\vec{\theta})$ and six Givens rotations

$\begin{split}\mathcal{U}^{(a,b,c,d)}_{4}(\vec{\theta}_{IV})=&\,\mathcal{U}^{(a,b)}_{2}(\vec{\theta_{0}})R^{(b,c)}_{X}(\theta_{1})R^{(a,b)}_{Z}(\theta_{2})\\\
&R^{(c,d)}_{Z}(\theta_{3})\mathcal{U}^{(a,b)}_{2}(\vec{\theta_{4}})R^{(b,c)}_{X}(\theta_{5})\\\
&\mathcal{U}^{(a,b)}_{2}(\vec{\theta_{6}})R^{(b,c)}_{Z}(\theta_{7})R^{(c,d)}_{Z}(\theta_{8}),\end{split}$
(34)

where the angles required for $QFT_{\mathbb{Z}_{4}}$ are fixed to be:
$\vec{\theta}_{0}=[2\pi,\pi/2,0]$, $\theta_{1}=1.392\pi$,
$\theta_{2}=0.4511\pi$, $\theta_{3}=4\pi/3$,
$\vec{\theta}_{4}=[0.90126\pi,0.41956\pi,1.852\pi]$, $\theta_{5}=0.60817\pi$,
$\vec{\theta}_{6}=[\pi/2,\pi/4,-\pi/4]$, $\theta_{7}=-\pi/2$,
$\theta_{8}=-3\pi/4$. If we consider the example $g=\mathbf{i}$, ($\ket{2}$)
then $V_{2}$ would be:

$V_{2}=\prod_{a=0}^{5}\mathcal{U}_{4}^{(4a,4a+2,4a+1,4a+3)}(\vec{\theta}_{IV}).$
(35)

The final decomposition required for $\mathbb{BT}$ is $SU(6)$, which is given
by two $\mathcal{U}^{(a,b,c)}_{3}(\vec{\theta})$ and three
$\mathcal{U}^{(a,b)}_{2}(\vec{\theta_{0}})$:

$\displaystyle\mathcal{U}_{6}^{(a,b,c,d,e,f)}=$
$\displaystyle\mathcal{U}_{3}^{(a,b,c)}(\vec{\theta}_{III})\mathcal{U}_{3}^{(d,e,f)}(\vec{\theta}_{III})$
$\displaystyle\times\mathcal{U}_{2}^{(a,d)}(\vec{\theta}_{II})\mathcal{U}_{2}^{(b,e)}(\vec{\theta}_{II})\mathcal{U}_{2}^{(c,f)}(\vec{\theta}_{II}).$
(36)

The group element $-\mathbf{l}$ corresponding to $\ket{9}$ has order $m=6$.
The corresponding $V_{9}$ is made with four products of
$\mathcal{U}_{6}^{(a,b,c,d,e,f)}$, although it lacks the obvious structure of
the other examples shown thus far:

$\begin{split}V_{9}=&\mathcal{U}_{6}^{(0,9,16,1,8,17)}\mathcal{U}_{6}^{(2,14,20,3,15,21)}\\\
&\mathcal{U}_{6}^{(4,11,23,5,10,22)}\mathcal{U}_{6}^{(6,12,19,7,13,18)}.\end{split}$
(37)


11institutetext:

# Fast Estimation of Haemoglobin Concentration in Tissue Via Wavelet
Decomposition

Geoffrey Jones  Neil T Clancy  Xiaofei Du  Maria Robu  Simon Arridge  Daniel S
Elson  Danail Stoyanov

###### Abstract

Tissue oxygenation and perfusion can be an indicator for organ viability
during minimally invasive surgery, for example allowing real-time assessment
of tissue perfusion and oxygen saturation. Multispectral imaging is an optical
modality that can inspect tissue perfusion in wide field images without
contact. In this paper, we present a novel, fast method for using RGB images
for MSI, which while limiting the spectral resolution of the modality allows
normal laparoscopic systems to be used. We exploit the discrete Haar
decomposition to separate individual video frames into low pass and
directional coefficients and we utilise a different multispectral estimation
technique on each. The increase in speed is achieved by using fast Tikhonov
regularisation on the directional coefficients and more accurate Bayesian
estimation on the low pass component. The pipeline is implemented using a
graphics processing unit (GPU) architecture and achieves a frame rate of
approximately 15Hz. We validate the method on animal models and on human data
captured using a da Vinci stereo laparoscope.

###### Keywords:

Minimal Invasive Surgery, Intraoperative Imaging, Multispectral Imaging

##  1 Introduction

Haemoglobin concentrations in tissue are important measurements that provide
functional information and could also be used for structural characterisation
of tissue types. In minimally invasive surgery (MIS), the use of haemoglobin
concentration imaging on tissue surfaces using multispectral imaging (MSI), is
promising as an non-ionising optical imaging modality that can monitor organ
viability in transplant procedures [4], [3] or be used to detect abnormal
tissue. A major advantage of the MSI modality is that it is able to obtain
wide field of view measurements without contact thus allowing for monitoring
large areas that cannot be observed through other sensing means like oxi-meter
probes which can only give spot measurements.

Techniques for MSI compatible with MIS are typically limited for real-time
monitoring by either their capture rate [5], data processing speed [10]
limiting their use for imaging dynamic systems. Methods have been developed to
estimate tissue haemoglobin concentrations for monitoring using fast filter
wheels of minimal sets of filters that show significant promise [14]. Yet in
MIS the surgical environment makes the use specialised hardware complex
because it requires additional regulatory and sterilisation considerations to
be accounted. As such computational methods requiring minimal hardware
modification and using existing hardware are highly attractive. Computational
techniques that utilise the laparoscopic RGB video feed for tissue monitoring
at speeds greater than 15Hz, can require inflexible calibration [12], or has
difficulty dealing with scene motion [9], or requires making a trade off in
estimation accuracy of saturation (SO2) against total haemoglobin (THb) [11].
Therefore high-speed techniques for this purpose are still a major challenge.

Wavelet decompositions have been widely used in image processing to transform
data it into domain spaces where certain operations can be applied more
efficiently or components of the signal with little entropy can be omitted
[6]. In this paper, we adopt the wavelet approach and propose a framework for
processing frames of laparoscopic video by Haar decomposition, which allows us
to use separate algorithms to process the various components of the compressed
data representation. Because the Haar wavelet produces large numbers of zeros
in the transformed data set, information is concentrated in relatively few
coefficients which we can process effectively. The resulting algorithm means
we can approach RGB processing to arrive at a surrogate MSI signal though a
dual optimisation approach which seems effective in fairly smooth signals,
such as laparoscopic video data. We present preliminary results on data
captured using hardware MSI for comparison on animal models as well as results
on human tissue imaging acquired within a laboratory environment.

##  2 Method

Underpinning the technique proposed in this paper is the estimation of a
signal surrogate to MSI but obtained from RGB video frames. This relies on
knowledge of the tissue’s transmitted attenuation characteristics [2],
backscattered attenuation coefficients are derived by Monte Carlo (MC)
simulation [8], to combine attenuation due to absorption and scattering [10],
since camera and light source are on the same side of the tissue during
laparoscopic imaging. The fitting to surrogate MSI data can be performed to
satisfy the Beer-Lambert relation [5].

Once an RGB image is captured, the 2D discrete Haar transform can be used to
decompose the image into four components, three directional approximate
derivatives and a residual low pass coefficient. We exploit this fact for
efficient computation since it enables us to only perform expensive estimation
on less of the data ($1/2^{1+n}$ where $n$ is the level of decomposition). The
matrix $H$ can be expressed for an image window
$I=\\{I_{x,y},I_{x,y+1},I_{x+1,y},I_{x+1,y+1}\\}$ as:

$\text{Haar}(I)=I*H$ (1)

given that the 2d Haar matrix is formed as:

$H=0.5\cdot\begin{bmatrix}1&1&1&1\\\ 1&1&-1&-1\\\ 1&-1&-1&1\\\
1&-1&1&-1\end{bmatrix}$ (2)

As this is a just a linear transformation, if we group each pixel data in
windows of four elements we can include it in the construction of RGB data
$I_{RGB}$ from the multispectral signal $I_{\lambda}$:

$\displaystyle C\cdot I_{\lambda}\cdot H$ $\displaystyle=I_{RGB}\cdot H$ (3)
for known camera spectral sensitivity matrix $C$, the least squares solution
being: $\displaystyle I_{\lambda}$
$\displaystyle=((C^{\text{T}}C)^{-1}C^{\text{T}}\cdot I_{RGB}\cdot H)\cdot H$
(4) since $H=H^{*}$. Similarly for the Tikhonov regularise solution as used by
[11], we can apply this method to the Haar transformed data as $\displaystyle
I_{\lambda}\cdot H$
$\displaystyle=(C^{\text{T}}C+\Gamma)^{-1}C^{\text{T}}\cdot I_{RGB}\cdot H$
(5)

where $\Gamma$ is the identity matrix multiplied by a small constant $\gamma$.
Here we have used the same windowed grouping:

$\displaystyle I_{RGB}$
$\displaystyle=\\{I_{RGB,x,y},I_{RGB,x,y+1},I_{RGB,x+1,y},I_{RGB,x+1,y+1},\\}$
(6) corresponding to the multispectral window: $\displaystyle I_{\lambda}$
$\displaystyle=\\{I_{\lambda,x,y},I_{\lambda,x,y+1},I_{\lambda,x+1,y},I_{\lambda,x+1,y+1},\\}$
(7)

While this demonstrates how Tikhonov based estimation can be used on any of
the Haar coefficients, to get better accuracy it is desirable to only use this
on the sparse directional components. On the low-pass coefficients we then
utilise the Bayesian method of [10], which they show is more accurate then
Tikhonov estimation, enabling estimation of oxygen saturation. It is possible
to do so because the low pass coefficients are analogous to an over exposed
image taken at a lower resolution.

Formally we can show that the method holds by substituting in the maximisation
step of [10] with low pass transformed data. Using $H_{LP}=0.5*[1,1,1,1]$ to
represent the low pass component of the discrete Haar transform, starting from
the concentration estimation $\hat{x}$ from low pass data:

$\displaystyle\xi x$ $\displaystyle=-log(I_{\lambda}H_{LP})$ (8) where $\xi$
are the backscatter attenuation coefficients for oxy and de-oxy haemoglobin
(HbO2 and Hb) as well as a constant term. The constant term is included to
account for global illumination changes that occur equally across wavelength,
such as due to distance from the camera. We solve for $x$ using least squares
fitting, $\displaystyle\hat{x}$
$\displaystyle=(-\xi^{\prime}\xi)^{-1}\xi^{\prime}\log(I_{\lambda}H_{LP})$ (9)

These concentration estimates are then used to generate the expected value for
the multispectral data $E[I_{\lambda}]$ used to regularise then subsequent
fitting iteration, this is defined as,

$\displaystyle E[I_{\lambda}]$ $\displaystyle=e^{-\xi x}$ (10) substituting
the concentration estimate from (8) and cancelling terms the expected value
will be the multispectral data transformed by $H_{LP}$ $\displaystyle
E[I_{\lambda}]$ $\displaystyle=I_{\lambda}H_{LP}$ (11)

This can only apply to the low pass coefficients because they are all positive
and so all terms remain real throughout.

The expectation step of [10] is linear and so compatible with working on Haar
transformed data. One notable change that we make to the expectation step is
to change the prior on the expected spectrum from a value prior to a shape
prior. We do this by computing the second derivative of $E[I_{\lambda}]$ and
using that to regularise the second derivative of the estimated
$\hat{I}_{\lambda}$, this better regularises the estimation.

##  3 Experiments and Results

We conducted two experiments to validate the proposed algorithm. The first
uses RGB views generated from in vivo animal experiment MSI data sets, which
allow evaluation comparing the method to a hardware implemented gold standard.
Secondly we utilise our method to process video from a patch of tissue at the
base of the tongue, where we show that because we are able resolve at a high
frame rate we can detect the pulse rate by tracking a patch of tissue over
time.

We observed good stability of our method both on the generated RGB data and on
the in-vivo acquired RGB data. This is conformant to expectations as our
method independently combines two methods which either have a closed form
solution [11] or are empirically observed to be stable under noisy conditions
[10].

THb (g/litre) SO2 (%)

reference proposed reference proposed

Figure 1:  Top: Selected bands from a multispectral datacube and corresponding
synthesised RGB views. Bottom left: total haemoglobin. Bottom right: oxygen
saturation. Example estimation from our proposed method using RGB data
synthesised from MSI and compared with reference estimation directly from the
MSI data.

###  3.1 Comparison with Hardware Multispectral Imaging Signals

Real in vivo MSI data was used to generate synthetic RGB images using known
camera spectral sensitivity curves, calibrated from the spectral response of
cameras on a Da Vinci surgical robot [13]. The MSI data comes from experiments
in which MSI was used to monitor and evaluate organ viability, by measuring
tissue perfusion and oxygen saturation, throughout uterine transplant
procedures performed on sheep and rabbits [4]. The data was chosen as they
most closely resemble the envisioned clinical use for our method. To evaluate
computation speed and accuracy we compared against [11] and [10], however, to
make fair comparisons of speed on the same architecture we also implemented
both of these methods for GPU, using the CUDA programming language. We
performed two variants of our proposed method corresponding to single (w1) and
multilevel (w3: three levels) Haar decomposition. All methods’ accuracy are
compared against the results of estimation from directly using the MSI data
[5].

Table 1: HbO2 and Hb estimation accuracy on synthetic data compared to direct estimation from multispectral data [5]. Method | Computational | Execution time | Mean squared error
---|---|---|---
| architecture | Hz | g/litre
Bayes [10] | CPU | 0.0671 | 25
Tikhonov [11] | CPU | 0.282 | 54
Bayes [10] | CUDA | 3.54 | 25
Tikhonov [11] | CUDA | 43.5 | 54
Proposed (w1) | CUDA | 12.7 | 36
Proposed (w3) | CUDA | 14.4 | 36

As seen in Figure 1 the estimation from our method bears strong visual
similarity to the result of direct estimation from MSI data, with two notable
variations. Firstly in our method there is much less noise in the estimation,
which can be interpreted as a product of the MSI data’s band limited images
each being moderately noisy. Such noise is inherently smoothed out when
synthesising RGB data from the MSI datacube as each RGB band integrates over
many MSI bands due to wider sensitivity of the respective colour channels. On
the other hand, our RGB technique is less able to robustly estimate
haemoglobin concentrations in areas of high illumination, this can be clearly
seen in the SO2 estimation near the top. This is possibly due to the non-
linear response of the camera at extreme measurements. In addition, the
generally smoother appearance of our result means that local variations in the
MSI signal are lost due to both the synthesis of the RGB image and
subsequently due to the regularisation in our estimation approach. However,
overall our proposed method was closer to that of [10] than [11] as is shown
in Table 1.

In terms of computational performance, the timed results of haemoglobin
estimation were made using mega-pixel sized input images. Timings were
calculated including all pre-processing steps, which for our proposed method
included the initial Haar decomposition and final recomposition, this
illustrates the diminishing returns for the for multileveled (w3) variant of
our method compared to the single decomposition level version (w1) as
tabulated in Table 1. Interestingly both variations have the same error
performance albeit with minor computational variations in terms of speed.

###  3.2 Experiments on In vivo Data From a Stereo-laparoscope

(a) (b) (c)

Figure 2:  Top (a) the original RGB laparoscopic view, (b) SO2 estimation
overlaid on the laparoscopic view, (c) THbestimation overlaid on the
laparoscopic view. Bottom left: the tracked THb over time from the left and
right camera feeds separately this is nice and constant (not bleeding from my
tongue!). Bottom centre: the derivative after smoothing the THb trace for each
camera view, smoothing is required due to large amounts of high frequency
noise in the trace. (roaming high-lights on tissue surface, non-perfect
tracking) Bottom right: Frequency analysis of the change in THb over time,
fitting a polynomial curve finds the peak between between $1.2$ and $1.3$ Hz.

We captured video of the base of the tongue of an adult male using a Da Vinci
surgical robot’s stereo laparoscope as shown in Figure 2. The laparoscopic
camera was calibrated to estimate the channel sensitivity. We tracked a patch
of tissue using the method of [7] over time in both left and right cameras,
and processed the two views separately. Tracking and image warping was
necessary in order to remove residual motion artefacts due to tongue movement
allowing visualisation of the spectral and temporal variation of a selected
region within a spatio-temporally registered signal. We then compared the mean
value of total haemoglobin as estimated from each view as shown in Figure 2
and observed a strong similarity in estimation over time. While there appears
to be constant offset between the estimate from either view this could be due
to a miss calibration of either of the cameras’ spectral response curve. We
observed a large amount of high frequency noise in the time series data, which
could be due to specular highlights moving as the tissue surface deformed and
moved or to imperfect tracking which is to be expected given the difficult
conditions.

Processing the registered video signal after warping using a low pass filter
we were able to observe a periodic signal in the derivative of the time series
which was aligned in both views. We believe this to be representative of
physiological signals within the tissue due to cardiovascular activity. By
looking at the Fourier transform of the change in THb estimation the power
spectrum peaked between $1.2$ and $1.3$ Hz which corresponds to periodic
signal of between 72 and 78 cycles per minute. This resonated with the heart
rate of the subject during the experimental acquisition.

##  4 Conclusion

We have presented a hybrid method for estimating haemoglobin concentration
from RGB images. The method improves on the computational speed, with minimal
loss in accuracy. The speed improvement is sufficient to enable use in real-
time clinical applications, as it is fast enough to resolve variations in the
oxy- and de-oxygenated blood flow within tissue close the exposed surface.
During surgical procedure heart rate can become elevated to over 2 Hz [1] as
such imaging at a rate faster than 4Hz is would be required to detect this
without aliasing. The precision of the proposed method is also high enough to
evaluate the oxygen saturation within tissue, this is a major improvement on
[11]. Our results on in vivo measurements of haemoglobin concentrations using
an existing laparoscopic set up for white light RGB imaging are promising.
Meanwhile the results on data from animal transplantation experiments
illustrates the potential surrogate use to a hardware MSI approach as a
translatable method for monitoring and evaluating organs intra-operatively.
Additional controlled studies are required to validate the precise nature of
the recovered information and to explore the quantitative experimental
differences between the proposed approach and set ups that use narrow bands of
wavelengths that allow a more well constrained regression.

##  5 Acknowledgements

This work was supported by the EPSRC (EP/N013220/1, EP/N022750/1,
EP/N027078/1, NS/A000027/1, EP/P012841/1), The Wellcome Trust (WT101957,
201080/Z/16/Z) and the EU-Horizon2020 project EndoVESPA
(H2020-ICT-2015-688592).

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