NuQS Collaboration

# Gluon Digitization via Character Expansion for Quantum Computers

Yao Ji<EMAIL_ADDRESS>Physik Department T31, James-Franck-Straße 1, Technische
Universität München, D-85748 Garching, Germany  Henry Lamm<EMAIL_ADDRESS>Fermi National Accelerator Laboratory, Batavia, Illinois, 60510, USA  Shuchen
Zhu<EMAIL_ADDRESS>Department of Computer Science, Georgetown
University, Washington, DC 20057, USA

###### Abstract

Efficient digitization is required for quantum simulations of gauge theories.
Schemes based on discrete subgroups use a smaller, fixed number of qubits at
the cost of systematic errors. We systematize this approach by deriving the
single plaquette action through matching the continuous group action to that
of a discrete one via group character expansions modulo the field fluctuation
contributions. We accompany this scheme by simulations of pure gauge over the
largest discrete crystal-like subgroup of $SU(3)$ up to the fifth-order in the
coupling constant.

††preprint: FERMILAB-PUB-21-496-T, TUM-HEP-1368/21

##  I Introduction

Quantum computing has the potential to dramatically advance our understanding
of quantum field theory [1, 2, 3] – especially for the dynamics of QCD [4, 5,
6]. Performing such simulations will require a large, albeit at present
unknown, scale of the quantum computers in terms of memory and circuit depths
[7]. This scale is certainly beyond the current, noisy devices. Thus, it is
therefore essential to explore the possibilities of reducing the requirement
of quantum resources so that any near-future field theory simulations on a
quantum computer become feasible. Such investigations could not only help to
determine the advantages of various quantum algorithms in different
circumstances, but also be beneficial in the future when quantum computing
becomes commonplace by providing general frameworks for cost-effective QCD
simulations.

For real-time QCD simulations, large quantum resources are allocated to
digitize the gluon fields due to their bosonic nature and thus unbounded
Hilbert space. Many approaches exist with different and presently poorly-
understood costs. Prominent proposals for digitization [8] can be broadly
classified into: Casimir dynamics [9, 10, 11, 12, 13, 14, 15, 16] potentially
with auxiliary fields [17], conformal truncation [18], discrete groups [19,
20, 21, 22, 23], dual variables [24, 25, 26, 27, 28], light-front kinematics
[5, 29], loop-string-hadron formulation [30, 31, 32], meshes and subsets [19,
33], quantum link models [34, 35, 36, 37], and qubit regularization [38, 39,
40]. Each approximation reduces symmetries – either explicitly or through
finite-truncations [11]. Thus, one must proceed with caution as the regulated
theory may not have the original theory as its continuum limit [41, 42, 43,
44, 45, 46]. In this spirit, a number of recent works have attempted to
quantify the truncation needed to achieve fixed accuracy for a lattice
simulation [32, 47, 7, 48].

Studies of the quantum simulations of lattice field theories typically use the
Hamiltonian formalism, with the Kogut-Susskind Hamiltonian [49] being the most
investigated for gauge theories. In this work, we will instead work in the
action formalism. This allows us to construct modified actions that can be
studied nonperturbatively on classical computers today. For the eventual
simulation on quantum computers, one can derive the modified Hamiltonian
straightforwardly via the transfer matrix [50, 51, 52].

One strategy pursued in the early days of lattice QCD involved approximating
the gauge group $SU(3)$ by its largest crystal-like subgroup $\mathbb{V}$ [53,
54], thus maintaining a remnant of the gauge symmetry of the parent group.
Depending on the action chosen, the accuracy of the approximation varies
greatly. The first studies considered merely replacing the continuous group
$G$ by its discrete subgroup $H$ in the Wilson gauge action. The viability of
this approximation was studied in detail for $U(1)$ [55, 56], and $SU(N)$ [57,
58, 59], with the inclusion of fermions [60, 61]. These studies met with mixed
success. In order to improve the agreement between lattice observables for $H$
and $G$, modified actions in $H$ were considered, which generally lead to
vastly improved results [20, 62]. The drawback to these ad-hoc actions is that
they were found empirically through substantial classical simulations, and
thus a theoretical and systematic understanding of them is lacking.

The resulting discrepancy from such approximations can be analyzed
systematically through expansions of various parameters in the same spirit of
the modern effective field theory (EFT) approach with a matching procedure. So
far, two parameters have been employed for two independent matching procedures
with the most obvious choice being the coupling, commonly denoted as $\beta$
(see Eq. (2)), and the other one being the $\mathbb{V}$ group characters. The
former option was exploited recently in matching ${\cal O}(\beta^{3})$
contributions between $SU(3)$ and $\mathbb{V}$ through a group decimation
technique [22]. These systematic results gave insight into how the Wilson
action should be modified when replacing $G$ by $H$. In this paper, we examine
the latter possibility, namely, matching the character contributions between
$SU(3)$ and $\mathbb{V}$.

The paper is organized as follows. In Sec. II, we summarize the group
properties of $\mathbb{V}$, the largest crystallike subgroup of $SU$(3). This
is followed by Sec. III, where we demonstrate the matching via a systematic
character expansion. Sec. IV is reserved for numerical results with analysis
of the viability of character expansion. The paper is finally concluded in
Sec. V.

##  II Characters of $\mathbb{V}$

The 1080 elements of the $SU(3)$ subgroup $\mathbb{V}$ can be classified into
$17$ conjugacy classes [63], generated by its total 17 independent characters
for which we denote as $\chi_{r}^{\prime}$ with $r=1,2,\cdots,17$. These 17
characters are linearly related to a subset of $SU(3)$ characters
$\chi_{(\lambda,\mu)}$ which are organized by two non-negative integers111We
note that in Ref. [22], another notation was adopted for general SU(N)
characters labeled by $N$ integers
$\\{\lambda_{1},\lambda_{2},\cdots,\lambda_{N}\\}$ with
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}$. For the special case of
SU(3), which is our sole interest throughout this paper, all characters can be
denoted by only two non-negative integers $(\lambda,\mu)$ [64]. We employ the
$(\lambda,\mu)$ convention hereafter. $\lambda$ and $\mu$,

$\displaystyle\chi^{\prime}_{r}$
$\displaystyle=\sum_{(\lambda,\mu)}m_{r,(\lambda,\mu)}\,\chi_{(\lambda,\mu)}\,,$
(1)

where $m_{r,(\lambda,\mu)}$ are a set of integers systematically obtainable by
matching the character definitions of the two groups. The explicit linear
relations between $\chi_{r}^{\prime}$ and $\chi_{(\lambda,\mu)}$ are given in
Table 1 for $r=1,2,\cdots,9,12,\cdots,17$.

Table 1: Character expansion of $\mathbb{V}$ characters in terms of $SU(3)$ characters, and in trace representation. $\chi^{\prime}_{r}$ of $\mathbb{V}$ | $\sum_{(\lambda,\mu)}m_{r,(\lambda,\mu)}\,\chi_{(\lambda,\mu)}$ | Trace Representation
---|---|---
$\chi^{\prime}_{1}(u)$ | $\chi_{(0,0)}(u)$ | $1$
$\chi^{\prime}_{2}(u)$ | $\chi_{(1,0)}(u)$ | $\mathop{\rm tr}\nolimits(u)$
$\chi^{\prime}_{3}(u)$ | $\chi_{(0,1)}(u)$ | $\mathop{\rm tr}\nolimits(u^{\dagger})$
$\chi^{\prime}_{4}(u)$ | $\chi_{(1,1)}(u)$ | $\mathop{\rm tr}\nolimits(u)\mathop{\rm tr}\nolimits(u^{\dagger})-1$
$\chi^{\prime}_{5}(u)$ | $\chi_{(2,0)}(u)$ | $\frac{1}{2}\left(\mathop{\rm tr}\nolimits^{2}(u)+\mathop{\rm tr}\nolimits(u^{2})\right)$
$\chi^{\prime}_{6}(u)$ | $\chi_{(0,2)}(u)$ | $\frac{1}{2}\left(\mathop{\rm tr}\nolimits^{2}(u^{\dagger})+\mathop{\rm tr}\nolimits(u^{\dagger 2})\right)$
$\chi^{\prime}_{7}(u)$ | $\chi_{(2,1)}(u)$ | $\frac{1}{2}\mathop{\rm tr}\nolimits(u^{\dagger})\left(\mathop{\rm tr}\nolimits^{2}(u)+\mathop{\rm tr}\nolimits(u^{2})\right)-\mathop{\rm tr}\nolimits(u)$
$\chi^{\prime}_{8}(u)$ | $\chi_{(1,2)}(u)$ | $\frac{1}{2}\mathop{\rm tr}\nolimits(u)\left(\mathop{\rm tr}\nolimits^{2}(u^{\dagger})+\mathop{\rm tr}\nolimits(u^{\dagger 2})\right)-\mathop{\rm tr}\nolimits(u^{\dagger})$
$\chi^{\prime}_{9}(u)$ | $\chi_{(3,0)}(u)$ | $\frac{1}{6}\left(\mathop{\rm tr}\nolimits^{3}(u)+2\mathop{\rm tr}\nolimits(u^{3})+3\mathop{\rm tr}\nolimits(u)\mathop{\rm tr}\nolimits(u^{2})\right)$
$\chi^{\prime}_{10}(u)+\chi^{\prime}_{11}(u)$ |  | $\chi_{(1,1)}(u)+\chi_{(2,2)}(u)$
---
$+\chi_{(3,0)}(u)-\chi_{(4,1)}(u)$
| $-1+\frac{1}{4}\left(\mathop{\rm tr}\nolimits^{2}(u)\mathop{\rm
tr}\nolimits^{2}(u^{\dagger})+\mathop{\rm tr}\nolimits^{2}(u)\mathop{\rm
tr}\nolimits(u^{\dagger 2})+\mathop{\rm
tr}\nolimits^{2}(u^{\dagger})\mathop{\rm tr}\nolimits(u^{2})+\mathop{\rm
tr}\nolimits(u^{\dagger 2})\mathop{\rm tr}\nolimits(u^{2})\right)$
---
$-\frac{1}{24}\mathop{\rm tr}\nolimits(u^{\dagger})\left(\mathop{\rm
tr}\nolimits^{4}(u)+6\mathop{\rm tr}\nolimits^{2}(u)\mathop{\rm
tr}\nolimits(u^{2})+3\mathop{\rm tr}\nolimits^{2}(u^{2})+8\mathop{\rm
tr}\nolimits(u)\mathop{\rm tr}\nolimits(u^{3})+6\mathop{\rm
tr}\nolimits(u^{4})\right)$
$+\frac{1}{3}\left(\mathop{\rm tr}\nolimits^{3}(u)+3\mathop{\rm
tr}\nolimits(u)\mathop{\rm tr}\nolimits(u^{2})+2\mathop{\rm
tr}\nolimits(u^{3})\right)$
$\chi^{\prime}_{12}(u)$ |  | $\chi_{(3,0)}(u)+\chi_{(2,2)}(u)$
---
$+\chi_{(4,1)}(u)-\chi_{(3,3)}(u)$
| $-\mathop{\rm tr}\nolimits(u)\mathop{\rm
tr}\nolimits(u^{\dagger})-\frac{1}{4}\left(\mathop{\rm
tr}\nolimits^{2}(u^{\dagger})+\mathop{\rm tr}\nolimits(u^{\dagger
2})\right)\left(\mathop{\rm tr}\nolimits^{2}(u)+\mathop{\rm
tr}\nolimits(u^{2})\right)$
---
$+\frac{1}{4}\left(\mathop{\rm tr}\nolimits^{2}(u)\mathop{\rm
tr}\nolimits^{2}(u^{\dagger})+\mathop{\rm tr}\nolimits^{2}(u)\mathop{\rm
tr}\nolimits(u^{\dagger 2})+\mathop{\rm
tr}\nolimits^{2}(u^{\dagger})\mathop{\rm tr}\nolimits(u^{2})+\mathop{\rm
tr}\nolimits(u^{2})\mathop{\rm tr}\nolimits(u^{\dagger 2})\right)$
$+\frac{1}{36}\left(\mathop{\rm tr}\nolimits^{3}(u^{\dagger})+3\mathop{\rm
tr}\nolimits(u^{\dagger})\mathop{\rm tr}\nolimits(u^{\dagger 2})+2\mathop{\rm
tr}\nolimits(u^{\dagger 3})\right)\left(\mathop{\rm
tr}\nolimits^{3}(u)+3\mathop{\rm tr}\nolimits(u)\mathop{\rm
tr}\nolimits(u^{2})+2\mathop{\rm tr}\nolimits(u^{3})\right)$
$+\frac{1}{24}\mathop{\rm tr}\nolimits(u^{\dagger})\left(\mathop{\rm
tr}\nolimits^{2}(u)+6\mathop{\rm tr}\nolimits^{2}(u)\mathop{\rm
tr}\nolimits(u^{2})+3\mathop{\rm tr}\nolimits^{2}(u^{2})+8\mathop{\rm
tr}\nolimits(u)\mathop{\rm tr}\nolimits(u^{3})+6\mathop{\rm
tr}\nolimits(u^{4})\right)$
$\chi^{\prime}_{13}(u)$ |  | $-\chi_{(1,1)}(u)-2\chi_{(3,0)}(u)$
---
$-\chi_{(2,2)}(u)+\chi_{(3,3)}(u)$
| $1-\frac{1}{4}\left(\mathop{\rm tr}\nolimits^{2}(u^{\dagger})+\mathop{\rm
tr}\nolimits(u^{\dagger 2})\right)\left(\mathop{\rm
tr}\nolimits^{2}(u)+\mathop{\rm
tr}\nolimits(u^{2})\right)+\frac{1}{3}\left(-\mathop{\rm
tr}\nolimits^{3}(u)-3\mathop{\rm tr}\nolimits(u)\mathop{\rm
tr}\nolimits(u^{2})-2\mathop{\rm tr}\nolimits(u^{3})\right)$
---
$+\frac{1}{4}\left(-\mathop{\rm tr}\nolimits^{2}(u)\mathop{\rm
tr}\nolimits^{2}(u^{\dagger})-\mathop{\rm tr}\nolimits^{2}(u)\mathop{\rm
tr}\nolimits(u^{\dagger 2})-\mathop{\rm
tr}\nolimits^{2}(u^{\dagger})\mathop{\rm tr}\nolimits(u^{2})-\mathop{\rm
tr}\nolimits(u^{\dagger 2})\mathop{\rm tr}\nolimits(u^{2})\right)$
$+\frac{1}{36}\left(\mathop{\rm tr}\nolimits^{3}(u^{\dagger})+3\mathop{\rm
tr}\nolimits(u^{\dagger})\mathop{\rm tr}\nolimits(u^{\dagger 2})+2\mathop{\rm
tr}\nolimits(u^{\dagger 3})\right)\left(\mathop{\rm
tr}\nolimits^{3}(u)+3\mathop{\rm tr}\nolimits(u)\mathop{\rm
tr}\nolimits(u^{2})+2\mathop{\rm tr}\nolimits(u^{3})\right)$
$\chi^{\prime}_{14}(u)$ | $\chi_{(3,1)}(u)-\chi_{(1,2)}(u)$ |  | $\frac{1}{6}\left(\mathop{\rm tr}\nolimits^{3}(u)+2\mathop{\rm tr}\nolimits(u^{3})+3\mathop{\rm tr}\nolimits(u^{2})\mathop{\rm tr}\nolimits(u)\right)\mathop{\rm tr}\nolimits(u^{\dagger})-\frac{1}{2}\left(\mathop{\rm tr}\nolimits^{2}(u)+\mathop{\rm tr}\nolimits(u^{2})\right)$
---
$-\frac{1}{2}\mathop{\rm tr}\nolimits(u)\left(\mathop{\rm
tr}\nolimits^{2}(u^{\dagger})+\mathop{\rm tr}\nolimits(u^{\dagger
2})\right)+\mathop{\rm tr}\nolimits(u^{\dagger})$
$\chi^{\prime}_{15}(u)$ | $\chi_{(1,3)}(u)-\chi_{(2,1)}(u)$ | $(\chi^{\prime}_{14}(u))^{*}$
$\chi^{\prime}_{16}(u)$ |  | $\chi_{(3,2)}(u)-\chi_{(2,1)}(u)$
---
$-\chi_{(1,3)}(u)$
| $-\frac{1}{2}\mathop{\rm tr}\nolimits(u^{\dagger})(\mathop{\rm
tr}\nolimits^{2}(u)+\mathop{\rm tr}\nolimits(u^{2}))+\frac{1}{12}(\mathop{\rm
tr}\nolimits^{2}(u^{\dagger})+\mathop{\rm
tr}\nolimits((u^{\dagger})^{2}))(\mathop{\rm tr}\nolimits^{3}(u)+3\mathop{\rm
tr}\nolimits(u)\mathop{\rm tr}\nolimits(u^{2})+2\mathop{\rm
tr}\nolimits(u^{3}))$
---
$\quad-\frac{1}{2}\mathop{\rm tr}\nolimits(u^{\dagger})\left(\mathop{\rm
tr}\nolimits^{2}(u)+\mathop{\rm tr}\nolimits(u^{2})\right)+\mathop{\rm
tr}\nolimits(u)-\frac{1}{6}\left(\mathop{\rm
tr}\nolimits^{3}(u^{\dagger})+2\mathop{\rm tr}\nolimits(u^{\dagger
3})+3\mathop{\rm tr}\nolimits(u^{\dagger 2})\mathop{\rm
tr}\nolimits(u^{\dagger})\right)\mathop{\rm tr}\nolimits(u)$
$+\frac{1}{2}\left(\mathop{\rm tr}\nolimits^{2}(u^{\dagger})+\mathop{\rm
tr}\nolimits(u^{\dagger 2})\right)$
$\chi^{\prime}_{17}(u)$ |  | $\chi_{(2,3)}(u)-\chi_{(1,2)}(u)$
---
$-\chi_{(3,1)}(u)$
$(\chi^{\prime}_{16}(u))^{*}$

It is worth nothing that although $\chi^{\prime}_{10}+\chi^{\prime}_{11}$ is
expressible in terms of $\chi_{(\lambda,\mu)}$ in a simple fashion as given in
Table 1, individual expressions for $\chi^{\prime}_{10}$ and
$\chi^{\prime}_{11}$ in terms of $SU(3)$ characters are rather lengthy. We
note that, in practice, $\chi^{\prime}_{10}$ and $\chi^{\prime}_{11}$ always
appear together as $\chi^{\prime}_{10}+\chi^{\prime}_{11}$ and thus the
absence of their expressions individually leads to no issues in our
derivations.

On the other hand, in order to obtain the action of $\chi^{\prime}_{10}$ and
$\chi^{\prime}_{11}$ on all $\mathbb{V}$ elements, we exploit the orthonormal
condition of the character representation, allowing us to fix the last
elements in the character table in Table 2. We point out that the conjugacy
class $C_{8}$ and $C_{9}$, both of which are composed of traceless
$\mathbb{V}$ elements, are distinguishable only through these two characters
$\chi^{\prime}_{10}$ and $\chi^{\prime}_{11}$, as can be read from Table 2.

Table 2: Character table for $\mathbb{V}$ [63]. $\mu_{1}=(1-\sqrt{5})/2$; $\mu_{2}=(1+\sqrt{5})/2$, $\omega=(1+i\sqrt{3})/2$, $\omega^{*}=(1-i\sqrt{3})/2$. The integer in the second row indicates the number of elements in each class, and the letter behind denotes the cycle induced by the class elements. class: | $C_{1}$ | $C_{2}$ | $C_{3}$ | $C_{4}$ | $C_{5}$ | $C_{6}$ | $C_{7}$ | $C_{8}$ | $C_{9}$ | $C_{10}$ | $C_{11}$ | $C_{12}$ | $C_{13}$ | $C_{14}$ | $C_{15}$ | $C_{16}$ | $C_{17}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| $1E$ | $72c_{5}$ | $90c_{4}$ | $45c_{6}$ | $45c^{\prime}_{6}$ | $72c_{15}$ | $72c^{\prime}_{15}$ | $120c_{3}$ | $120c^{\prime}_{3}$ | $90c_{12}$ | $90c^{\prime}_{12}$ | $72c^{\prime}_{5}$ | $72c_{15}$ | $72c^{\prime}_{15}$ | $45c_{2}$ | $1c^{\prime\prime}_{3}$ | $1c^{\prime\prime\prime}_{3}$
$\chi^{\prime}_{1}(u)$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$\chi^{\prime}_{2}(u)$ | $3$ | $\mu_{2}$ | $1$ | $\omega$ | $\omega^{*}$ | $-\mu_{1}\omega$ | $-\mu_{1}\omega^{*}$ | $0$ | $0$ | -$\omega^{*}$ | $-\omega$ | $\mu_{1}$ | $-\mu_{2}\omega^{*}$ | $-\mu_{2}\omega$ | $-1$ | $-3\omega^{*}$ | $-3\omega$
$\chi^{\prime}_{3}(u)$ | $3$ | $\mu_{2}$ | $1$ | $\omega^{*}$ | $\omega$ | $-\mu_{1}\omega^{*}$ | $-\mu_{1}\omega$ | $0$ | $0$ | $-\omega$ | $-\omega^{*}$ | $\mu_{1}$ | $-\mu_{2}\omega$ | $-\mu_{2}\omega^{*}$ | $-1$ | $-3\omega$ | $-3\omega^{*}$
$\chi^{\prime}_{4}(u)$ | $8$ | $\mu_{2}$ | $0$ | $0$ | $0$ | $\mu_{1}$ | $\mu_{1}$ | $-1$ | $-1$ | $0$ | $0$ | $\mu_{1}$ | $\mu_{2}$ | $\mu_{2}$ | $0$ | $8$ | $8$
$\chi^{\prime}_{5}(u)$ | $6$ | $1$ | $0$ | $-2\omega^{*}$ | $-2\omega$ | $-\omega^{*}$ | $-\omega$ | $0$ | $0$ | $0$ | $0$ | $1$ | $-\omega$ | $-\omega^{*}$ | $2$ | $-6\omega$ | $-6\omega^{*}$
$\chi^{\prime}_{6}(u)$ | $1$ | $1$ | $0$ | $-2\omega$ | $-2\omega^{*}$ | $-\omega$ | $-\omega^{*}$ | $0$ | $0$ | $0$ | $0$ | $1$ | $-\omega^{*}$ | $-\omega$ | $2$ | $-6\omega^{*}$ | $-6\omega$
$\chi^{\prime}_{7}(u)$ | $15$ | $0$ | $-1$ | $\omega$ | $\omega^{*}$ | $0$ | $0$ | $0$ | $0$ | $\omega^{*}$ | $\omega$ | $0$ | $0$ | $0$ | $-1$ | $-15\omega^{*}$ | $-15\omega$
$\chi^{\prime}_{8}(u)$ | $15$ | $0$ | $-1$ | $\omega^{*}$ | $\omega$ | $0$ | $0$ | $0$ | $0$ | $\omega$ | $\omega^{*}$ | $0$ | $0$ | $0$ | $-1$ | $-15\omega$ | $-15\omega^{*}$
$\chi^{\prime}_{9}(u)$ | $10$ | $0$ | $0$ | $-2$ | $-2$ | $0$ | $0$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-2$ | $10$ | $10$
$\chi^{\prime}_{10}(u)$ | $5$ | $0$ | $-1$ | $1$ | $1$ | $0$ | $0$ | $-1$ | $2$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $1$ | $5$ | $5$
$\chi^{\prime}_{11}(u)$ | $5$ | $0$ | $-1$ | $1$ | $1$ | $0$ | $0$ | $2$ | $-1$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $1$ | $5$ | $5$
$\chi^{\prime}_{12}(u)$ | $8$ | $\mu_{1}$ | $0$ | $0$ | $0$ | $\mu_{2}$ | $\mu_{2}$ | $-1$ | $-1$ | $0$ | $0$ | $\mu_{2}$ | $\mu_{1}$ | $\mu_{1}$ | $0$ | $8$ | $8$
$\chi^{\prime}_{13}(u)$ | $9$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $0$ | $0$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $1$ | $9$ | $9$
$\chi^{\prime}_{14}(u)$ | $9$ | $-1$ | $1$ | $-\omega^{*}$ | $-\omega$ | $\omega^{*}$ | $\omega$ | $0$ | $0$ | $-\omega$ | $-\omega^{*}$ | $-1$ | $\omega$ | $\omega^{*}$ | $1$ | $-9\omega$ | $-9\omega^{*}$
$\chi^{\prime}_{15}(u)$ | $9$ | $-1$ | $1$ | $-\omega$ | $-\omega^{*}$ | $\omega$ | $\omega^{*}$ | $0$ | $0$ | $-\omega^{*}$ | $-\omega$ | $-1$ | $\omega^{*}$ | $\omega$ | $1$ | $-9\omega^{*}$ | $-9\omega$
$\chi^{\prime}_{16}(u)$ | $3$ | $\mu_{1}$ | $1$ | $\omega$ | $\omega^{*}$ | $-\mu_{2}\omega$ | $-\mu_{2}\omega^{*}$ | $0$ | $0$ | $-\omega^{*}$ | $-\omega$ | $\mu_{2}$ | $-\mu_{1}\omega^{*}$ | $-\mu_{1}\omega$ | $-1$ | $-3\omega^{*}$ | $-3\omega$
$\chi^{\prime}_{17}(u)$ | $3$ | $\mu_{1}$ | $1$ | $\omega^{*}$ | $\omega$ | $-\mu_{2}\omega^{*}$ | $-\mu_{2}\omega$ | $0$ | $0$ | $-\omega$ | $-\omega^{*}$ | $\mu_{2}$ | $-\mu_{1}\omega$ | $-\mu_{1}\omega^{*}$ | $-1$ | $-3\omega$ | $-3\omega^{*}$

##  III Character expansion of Wilson action

In this section, we first carry out the character expansion for the Boltzmann
weight using the Wilson action to sufficiently high orders in SU(3). The
domain of the resulting expansion is then reduced from SU(3) to $\mathbb{V}$
with the help of Eq. (1) of which the lowest-weight characters are found in
Table 1. Clearly, this step lowers the theory complexity at the cost of
approximation errors which are quantifiable through theoretical and/or
numerical means (e.g., [22]). Finally, we derive the effective action
$\widetilde{S}(u)$ over $\mathbb{V}$ by matching $\mathop{\rm
e}\nolimits^{-\widetilde{S}(u)}$ onto the corresponding character expansion.

We start with the pure gauge Wilson action,

$\displaystyle S(U)=-\sum_{p}\frac{\beta}{N}\mathop{\rm
Re}\nolimits\mathop{\rm Tr}\nolimits(U_{p})\,,\qquad U\in SU(3)\,,$ (2)

where the summation runs over all plaquettes. $U_{p}$ denotes a product of
gauge links in the adjoint representation of SU(3). The resulting Boltzmann
weight, $\mathop{\rm e}\nolimits^{-S(U)}$, can be decomposed into the
orthonormal $SU(3)$ characters basis,

$\displaystyle\mathop{\rm e}\nolimits^{-S(U)}$
$\displaystyle=\sum_{(\lambda,\mu)}\beta_{(\lambda,\mu)}\chi_{(\lambda,\mu)}(U)\,,$
(3)

giving rise to a series of character coupling constants
$\beta_{(\lambda,\mu)}$. The resulting partition function central to the
classical lattice calculation can therefore be presented as,

$\displaystyle Z\equiv\int DU\,\mathop{\rm
e}\nolimits^{-S(U)}=\sum_{(\lambda,\mu)}\int
DU\beta_{(\lambda,\mu)}\chi_{(\lambda,\mu)}(U)\,.$ (4)

For the Wilson action defined in Eq. (2), the explicit form of
$\beta_{(\lambda,\mu)}$ can be written as an infinite sum in terms of Bessel
functions [64],

$\displaystyle\beta_{(\lambda,\mu)}(\beta)$
$\displaystyle=\sum_{n=-\infty}^{\infty}\det\begin{pmatrix}I_{\rho+n}(\frac{\beta}{3})&I_{\sigma-1+n}(\frac{\beta}{3})&I_{n-2}(\frac{\beta}{3})\\\
I_{\rho+1+n}(\frac{\beta}{3})&I_{\sigma+n}(\frac{\beta}{3})&I_{n-1}(\frac{\beta}{3})\\\
I_{\rho+2+n}(\frac{\beta}{3})&I_{\sigma+1+n}(\frac{\beta}{3})&I_{n}(\frac{\beta}{3})\end{pmatrix},$
(5)

where $\rho=\lambda+\mu$, $\sigma=\mu$, and $I_{k}$ is the Bessel function of
the first kind [64]. Explicit expressions of $\beta_{(\lambda,\mu)}$ that are
relevant for our current study are collected in Table 3 in the form of Taylor
series of $\beta$. In practice, we found taking $|n|\leq 40$ in Eq. (III) is
sufficient numerically.

Table 3:  Series expansions of $\beta_{r}$, $x\equiv\beta/6$ up to $\mathcal{O}(x^{8})$. $(\lambda,\mu)$ | $\beta_{(\lambda,\mu)}$
---|---
$(0,0)$ | $1-x^{2}+\frac{x^{4}}{2}-\frac{11x^{6}}{72}+\frac{91x^{8}}{2880}+\cdots$
$(1,0)$ | $x+\frac{x^{2}}{2}-x^{3}-\frac{3x^{4}}{8}+\frac{11x^{5}}{24}+\frac{11x^{6}}{80}-\frac{91x^{7}}{720}-\frac{91x^{8}}{2880}+\cdots$
$(1,1)$ | $-x^{2}+x^{4}-\frac{77x^{6}}{180}+\frac{13x^{8}}{120}+\cdots$
$(2,0)$ | $\frac{x^{2}}{2}+\frac{x^{3}}{2}-\frac{5x^{4}}{12}-\frac{11x^{5}}{30}+\frac{x^{6}}{6}+\frac{56x^{7}}{365}-\frac{59x^{8}}{1440}+\cdots$
$(2,1)$ | $-\frac{x^{3}}{2}-\frac{5x^{4}}{24}+\frac{11x^{5}}{24}+\frac{7x^{6}}{48}-\frac{13x^{7}}{72}-\frac{3x^{8}}{64}+\cdots$
$(3,0)$ | $\frac{x^{3}}{6}+\frac{x^{4}}{4}-\frac{x^{5}}{12}-\frac{25x^{6}}{144}+\frac{x^{7}}{48}+\frac{113x^{8}}{2016}+\cdots$
$(2,2)$ | $\frac{x^{4}}{4}-\frac{9x^{6}}{40}+\frac{27x^{8}}{320}-\frac{187x^{10}}{10080}+\ldots$
$(3,1)$ | $-\frac{x^{4}}{6}-\frac{2x^{5}}{15}+\frac{x^{6}}{8}+\frac{4x^{7}}{45}-\frac{3x^{8}}{70}+\cdots$
$(4,0)$ | $\frac{x^{4}}{24}+\frac{x^{5}}{12}-\frac{53x^{7}}{1008}-\frac{19x^{8}}{2880}+\cdots$
$(3,2)$ | $\frac{x^{5}}{12}+\frac{7x^{6}}{240}-\frac{49x^{7}}{720}-\frac{3x^{8}}{160}+\cdots$
$(4,1)$ | $-\frac{x^{5}}{24}-\frac{7x^{6}}{144}+\frac{x^{7}}{48}+\frac{29x^{8}}{960}+\cdots$
$(5,0)$ | $\frac{x^{5}}{120}+\frac{x^{6}}{48}+\frac{x^{7}}{180}-\frac{13x^{8}}{1152}+\cdots$
$(3,3)$ | $-\frac{x^{6}}{36}+\frac{x^{8}}{45}+\cdots$
$(4,2)$ | $\frac{x^{6}}{48}+\frac{x^{7}}{72}-\frac{71x^{8}}{5040}+\cdots$
$(5,1)$ | $-\frac{x^{6}}{120}-\frac{4x^{7}}{315}+\frac{x^{8}}{720}+\cdots$
$(6,0)$ | $\frac{x^{6}}{720}+\frac{x^{7}}{240}+\frac{x^{8}}{480}+\cdots$

We are now ready to extract the effective action over $\mathbb{V}$, the finite
discrete crystal subgroup of SU(3). The starting point of our procedure is to
substitute $U\in SU(3)$ on the R.H.S. of Eq. (3) by $u\in\mathbb{V}$,
effectively reducing the domain of the character expansion. The $SU(3)$
characters $\chi_{(\lambda,\mu)}(U)$ are then mapped to their $\mathbb{V}$
counterparts $\chi^{\prime}_{r}(u)$ using either Eq. (1), or Table 2 by
solving a system of 17 linear equations. Subsequently, the SU(3) character
couplings $\beta_{(\lambda,\mu)}$ are also rearranged into $\mathbb{V}$
character couplings $\beta^{\prime}_{r}$. Detailed relations between
$\beta^{\prime}_{r}$ and $\beta_{(\lambda,\mu)}$ are collected in Table 4. The
resulting $\mathbb{V}$ character expansion is finally matched onto the
$\mathbb{V}$ effective action $\widetilde{S}(u)$ as follows,

$\displaystyle
e^{-\widetilde{S}(u)}=\sum_{r=1}^{17}\beta^{\prime}_{r}\chi^{\prime}_{r}(u)\,,\qquad
u\in\mathbb{V}\,.$ (6)

We emphasize that once the order of truncation for the SU(3) character
expansion is determined, the R.H.S of Eq. (6) is completely fixed through Eq.
(1) (or Table 1). The relations between $\beta^{\prime}_{r}$ and
$\beta_{(\lambda,\mu)}$ given in Table 4 provide a complete basis for SU(3)
character $\chi_{(\lambda,\mu)}$ expansion up to $0\leq\lambda+\mu\leq 6$. It
is also important to remark that here we have systematically neglected the
contribution of $SU(3)$ elements which are absent in $\mathbb{V}$, effectively
providing us with a “leading-order” approximation. Such approximations are
improvable systematically by parametrizing $U=\epsilon u$ [22] yielding,

$\displaystyle Z$ $\displaystyle=\sum_{u\in\mathbb{V}}\sum_{(\lambda,\mu)}\int
D\epsilon\,\beta_{(\lambda,\mu)}\chi_{(\lambda,\mu)}(\epsilon u)$
$\displaystyle=\sum_{u\in\mathbb{V}}\sum_{(\lambda,\mu)}\beta_{(\lambda,\mu)}\,F(V_{(\lambda,\mu)},\chi_{(\lambda,\mu)}(u))\,,$
(7)

where

$\displaystyle\int D\epsilon\,\chi_{(\lambda,\mu)}(\epsilon
u)=F(V_{(\lambda,\mu)},\chi_{(\lambda,\mu)}(u))=\widetilde{F}(V_{r}^{\prime},\chi_{r}^{\prime}(u))\,.$
(8)

We have schematically applied relations in Table. 1 or 2 at the second
equality in Eq. (8), and $\displaystyle
V_{(\lambda,\mu)}=\frac{1}{d_{(\lambda,\mu)}}\int_{\Omega}D\epsilon\mathop{\rm
Re}\nolimits{\chi_{(\lambda,\mu)}}(\epsilon)$, where $d_{(\lambda,\mu)}$ is
the dimension of the character $\chi_{(\lambda,\mu)}$ as defined in [64]. The
function $F(V_{(\lambda,\mu)},\chi_{(\lambda,\mu)})$, which is linear in
$\chi_{(\lambda,\mu)}$, is obtainable through systematic albeit tedious
algebraic manipulations as explicitly demonstrated in [22]. At higher orders,
however, it is more advantageous to derive the $\chi_{r}^{\prime}(u)$
dependence of $\widetilde{F}(V_{r}^{\prime},\chi_{r}^{\prime}(u))$ numerically
by solving a system of 17 linear equations, generated by $u\in\mathbb{V}$ from
17 different conjugacy classes. The domain of Eq. (7) hence reduces to
$\mathbb{V}$ leading to

$\displaystyle\chi_{(\lambda,\mu)}=\sum_{r=1}^{17}c_{(\lambda,\mu)}^{r}\chi^{\prime}_{r}\,.$
(9)

Table 4: $\beta^{\prime}_{r}$ as linear combinations of $\beta_{(\lambda,\mu)}$. $\beta^{\prime}_{1}$ | $\beta_{(0,0)}+2\beta_{(6,0)}$
---|---
$\beta^{\prime}_{2}$ | $\beta_{(1,0)}+\beta_{(5,0)}+\beta_{(5,1)}$
$\beta^{\prime}_{3}$ | $\beta^{\prime}_{2}$
$\beta^{\prime}_{4}$ | $\beta_{(1,1)}+2\beta_{(4,1)}+\beta_{(3,3)}+2\beta_{(6,0)}$
$\beta^{\prime}_{5}$ | $\beta_{(2,0)}+\beta_{(4,0)}+2\beta_{(4,2)}+\beta_{(5,1)}$
$\beta^{\prime}_{6}$ | $\beta^{\prime}_{5}$
$\beta^{\prime}_{7}$ | $\beta_{(2,1)}+\beta_{(3,1)}+2\beta_{(3,2)}+\beta_{(5,0)}+2\beta_{(4,2)}+2\beta_{(5,1)}$
$\beta^{\prime}_{8}$ | $\beta^{\prime}_{7}$
$\beta^{\prime}_{9}$ | $2\beta_{(3,0)}+2\beta_{(4,1)}+2\beta_{(3,3)}$
$\beta^{\prime}_{10}$ | $\beta_{(2,2)}+\beta_{(3,3)}+2\beta_{(6,0)}$
$\beta^{\prime}_{11}$ | $\beta^{\prime}_{10}$
$\beta^{\prime}_{12}$ | $\beta_{(2,2)}+2\beta_{(4,1)}+\beta_{(3,3)}$
$\beta^{\prime}_{13}$ | $\beta_{(2,2)}+2\beta_{(4,1)}+2\beta_{(3,3)}+2\beta_{(6,0)}$
$\beta^{\prime}_{14}$ | $\beta_{(3,1)}+\beta_{(4,0)}+\beta_{(3,2)}+2\beta_{(4,2)}+\beta_{(5,1)}$
$\beta^{\prime}_{15}$ | $\beta^{\prime}_{14}$
$\beta^{\prime}_{16}$ | $\beta_{(3,2)}+\beta_{(5,0)}$
$\beta^{\prime}_{17}$ | $\beta^{\prime}_{16}$

With the determined values of $\beta^{\prime}_{i}$, we can now reexponentiate
to obtain the effective action $\widetilde{S}(u)$ in Eq. (6). Rewriting the
effective action $\widetilde{S}(u)$ as an expansion in terms of
$\chi_{r}^{\prime}$, we obtain the following matching formula from which the
coefficients $\gamma_{i}$ are fixed to the targeted order222Both
$\beta^{\prime}_{i}$ and $\gamma_{i}$ are $\beta$-dependent. in $\beta$,

$\displaystyle\exp(-\widetilde{S}(u))=\exp{\left(-\sum_{i=1}^{17}\gamma_{i}\chi^{\prime}_{i}\right)}$
$\displaystyle=\sum_{i=1}^{17}\beta^{\prime}_{i}\chi^{\prime}_{i}\,,$ (10)

which implies

$\displaystyle-\sum_{i=1}^{17}\gamma_{i}\chi^{\prime}_{i}$
$\displaystyle=\log\left(1+\sum_{i=1}^{17}\beta^{\prime}_{i}\chi^{\prime}_{i}-1\right)\equiv\log\left(1+z\right).$
(11)

Coefficients $\gamma_{i}$ are then extracted by expanding the R.H.S. with
respect to $z$ and matching onto the L.H.S. in terms of the 17 characters, the
completeness of $\chi_{r}^{\prime}$ as a character representation of
$\mathbb{V}$ validates

$\displaystyle\chi^{\prime}_{i}\chi^{\prime}_{j}$
$\displaystyle=\sum_{k=1}^{17}c_{ijk}\chi^{\prime}_{k}\,,$ (12)

with $c_{ijk}$ being integers easily calculable from Table 2 by solving a
system of 17 linear equations with 17 unknowns.

Substituting Eq. (12) into Eq. (11) then yields all the coefficients
$\gamma_{i}$ as polynomials of the strong coupling $\beta$. Explicit
expressions with parametrization invariance, a general principle for
constructing effective actions which we explain below, are summarized in Table
5.

Finally, it is important to notice that the original Wilson action $S(U)$ is
invariant under reparameterization $\\{\beta,U\\}\mapsto\\{\beta/c,cU\\}$ with
$c\neq 0$ being an arbitrary constant. This property, which we call
reparemeterization invariance (RI), is no longer present when the character
expansion is truncated at finite orders. We explictly reintroduce it to remove
finite order terms in the effective action $\widetilde{S}(u)$ that are
destined to sum up to zero. This not only gives us a much simpler expression
for the effective action but also helps reduce the computational cost in
lattice simulations and should improve agreement with the $SU(3)$ action.

Table 5: $\gamma_{i}$ computed in terms of $\beta$ to $\mathcal{O}(\beta^{5})$ with reparameterization invariance. $\gamma_{1}$ |  | $-\frac{1}{18}\beta^{2}-0.00308642\beta^{4}$
---
$\gamma_{2}$ |  | $\frac{1}{6}\beta+0.00925926\beta^{3}+0.00174683\beta^{5}$
---
$\gamma_{3}$ |  | $\gamma_{2}$
---
$\gamma_{4}$ |  | $-\frac{1}{18}\beta^{2}-0.00308642\beta^{4}-0.000364369\beta^{5}$
---
$\gamma_{5}$ |  | $-0.000257202\beta^{4}$
---
$\gamma_{6}$ |  | $\gamma_{5}$
---
$\gamma_{7}$ |  | $0.00925926\beta^{3}-0.000257202\beta^{4}+0.00524049\beta^{5}$
---
$\gamma_{8}$ |  | $\gamma_{7}$
---
$\gamma_{9}$ |  | $-0.000728738\beta^{5}$
---
$\gamma_{10}$ |  | $-0.00308642\beta^{4}$
---
$\gamma_{11}$ |  | $\gamma_{10}$
---
$\gamma_{12}$ |  | $-0.00308642\beta^{4}-0.000364369\beta^{5}$
---
$\gamma_{13}$ |  | $\gamma_{12}$
---
$\gamma_{14}$ |  | $-0.000257202\beta^{4}+0.00174683\beta^{5}$
---
$\gamma_{15}$ |  | $\gamma_{14}$
---
$\gamma_{16}$ |  | $0.00174683\beta^{5}$
---
$\gamma_{17}$ |  | $\gamma_{16}$
---

The effective action we have derived from the character expansion in terms of
the $\gamma_{i}$ in Table 5 can be compared to the group decimated action,
$S_{GD}$, defined by the couplings in Table III of [22]. $S_{GD}$ was only
calculated to $\mathcal{O}(\beta^{3})$ and thus the scaling behavior for
certain $\chi^{\prime}_{r}$ are unknown. In contrast, using the character
expansion we can determine the leading power of $\beta$ for all
$\chi^{\prime}_{r}$.

Between the two schemes, there are different $\beta$ dependences. In
particular, $\chi^{\prime}_{1}$ in the character expansion contains only odd
powers of $\beta$ while $S_{GD}$ generates all new terms at all orders. When
comparing the two ad-hoc modified $\mathbb{V}$ actions which individually
added $\chi^{\prime}_{4}$ [65] and $\chi^{\prime}_{5}$ [20, 62], the group
decimation procedure suggested no preference – they both are generated at
$\mathcal{O}(\beta^{2})$. Within the character expansion, only
$\chi^{\prime}_{4}$ appears at $\mathcal{O}(\beta^{2})$. $\chi^{\prime}_{5}$
only arises at $\mathcal{O}(\beta^{4})$.

##  IV Numerical Results

In order to gauge the effectiveness of our approximations, the effective
$\mathbb{V}$ action induced by character expansion was simulated at each order
in $\beta$. For these computations 102 configurations separated by 102 sweeps
were collected on a $4^{4}$ lattice. The average energy per plaquette $\langle
E_{0}\rangle$ versus $\beta$ is plotted in Fig. 1.

Figure 1: Average energy per plaquette, $\langle E_{0}\rangle=1-\mathop{\rm
Re}\nolimits\langle\mathop{\rm Tr}\nolimits u_{p}\rangle/3$, vs $\beta$ on a
$4^{4}$ lattice for $\mathbb{V}$ action with corrections of: ( )
$\mathcal{O}(\beta^{1})$, ( ) $\mathcal{O}(\beta^{2})$, ( )
$\mathcal{O}(\beta^{3})$, ( ) $\mathcal{O}(\beta^{4})$, and ( )
$\mathcal{O}(\beta^{5})$. The black line is the $SU(3)$ result.

If the effective action behaved as the continuous group action, $\langle
E_{0}\rangle$ would be observed to be monotonic in $\beta$. We observe that
including $\mathcal{O}(\beta^{2})$ or higher terms that instead of the simple
freeze-out story of discrete groups, after a large drop in $\langle
E_{0}\rangle$ the expectation value begins to rise again before asymptoting to
a fixed value. Comparing the actions at increasing orders in $\beta$, the
convergence to the continuous group results is slow and alternating. Despite
this, higher-order effective action seems to correspond to a wider range of
$\langle E_{0}\rangle$ being accessible. For the $\mathcal{O}(\beta^{5})$
action, it was found to be possible to obtain $0.2\leq\langle E_{0}\rangle\leq
0.4$. This naively suggests that the action might allow for smaller lattice
spacing than the unmodified one - including physics within the scaling regime.

It should be noted that the behavior observed by this expansion is
qualitatively different to that of the group decimation procedure of [22]. In
that work, the effective actions obtained at various orders of $\beta$ were
found to lack the first-order phase-transition behavior of a discrete group,
but the range of $\langle E_{0}\rangle$ was limited to much larger values of
$0.5-1$ in the small $\beta$ region.

##  V Conclusion

In this work, we used the character expansion to develop a systematic method
for improving lattice actions that replace continuous gauge group $SU(3)$ by
its discrete subgroup $\mathbb{V}$. Moreover, this method can be generalized
to any continuous gauge group $SU(N)$. We also spell out a new principle
called reparameterization invariance as a guideline for constructing effective
actions, allowing to reduce the cost of computation and match the original
action more closely at the same time. This is the ongoing effort toward
developing efficient digitization schemes on quantum computers.

We computed to $\mathcal{O}(\beta^{5})$ for the single-plaquette $\mathbb{V}$
action as an approximation of the Wilson action of $SU(3)$. These higher-order
terms suggest a different scaling with $\beta$ compared to previous expansions
based on group decimation [22]. The higher-order (${\cal O}(\beta^{2})$)
result in this work does not take into account the contribution of quantum
fluctuations around $\mathbb{V}$ elements. So as expected it deviates from the
$SU(3)$ result with a hard truncation in the character expansion. It would be
beneficial to include the quantum fluctuation through Table 1 in future work
to achieve better approximations.

Moreover, while the group decimation procedure was observed to suggest the
necessity of introducing two characters at $\mathcal{O}(\beta^{2})$, the
character expansion only require the adjoint character to be present.

The most immediate next steps are to compute more Euclidean observables (i.e.
Wilson flow parameter and pseudocritical temperature and etc.) from the
character expansion action to determine the lattice spacings achievable and to
compare them to [20, 62]. Given the large corrections order by order for
$\mathbb{V}$ and the interesting even versus odd behavior, additional work
should be devoted to computing the higher order contributions and corrections
induced by fluctuations around $\mathbb{V}$ elements - potentially via some
resummation technique.

Another obvious step in studying the feasibility of this procedure is to
derive the modified Hamiltonian, and explicitly construct the primitive
quantum gates à la [66, 67]. Together with classical lattice results, this
would allow for complete resource counts for extracting continuum physics.

###### Acknowledgements.

The authors would like to thank Justin Thaler for helpful comments on this
work. Y.J. is grateful for the support of Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) grant SFB TR 110/2. H.L is 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. S.Z. is supported by the National
Science Foundation CAREER award (grant CCF-1845125).

## References

  * Feynman [1982] R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).
  * Jordan  _et al._ [2017] S. P. Jordan, H. Krovi, K. S. M. Lee, and J. Preskill, (2017), arXiv:1703.00454 [quant-ph] .
  * Bañuls  _et al._ [2020] M. C. Bañuls _et al._ , Eur. Phys. J. D 74, 165 (2020), arXiv:1911.00003 [quant-ph] .
  * Lamm  _et al._ [2019a] H. Lamm, S. Lawrence, and Y. Yamauchi (NuQS), (2019a), arXiv:1908.10439 [hep-lat] .
  * Kreshchuk  _et al._ [2020a] M. Kreshchuk, W. M. Kirby, G. Goldstein, H. Beauchemin, and P. J. Love, “Quantum Simulation of Quantum Field Theory in the Light-Front Formulation,” (2020a), arXiv:2002.04016 [quant-ph] .
  * Echevarria  _et al._ [2020] M. Echevarria, I. Egusquiza, E. Rico, and G. Schnell, “Quantum Simulation of Light-Front Parton Correlators,” (2020), arXiv:2011.01275 [quant-ph] .
  * Kan and Nam [2021] A. Kan and Y. Nam, (2021), arXiv:2107.12769 [quant-ph] .
  * Gustafson  _et al._ [2020] E. Gustafson, H. Kawai, H. Lamm, I. Raychowdhury, H. Singh, J. Stryker, and J. Unmuth-Yockey, Snowmass 2021 LOI TF10-97 (2020).
  * Zohar  _et al._ [2012] E. Zohar, J. I. Cirac, and B. Reznik, Phys. Rev. Lett. 109, 125302 (2012), arXiv:1204.6574 [quant-ph] .
  * Zohar  _et al._ [2013a] E. Zohar, J. I. Cirac, and B. Reznik, Phys. Rev. Lett. 110, 125304 (2013a), arXiv:1211.2241 [quant-ph] .
  * Zohar  _et al._ [2013b] E. Zohar, J. I. Cirac, and B. Reznik, Phys. Rev. A88, 023617 (2013b), arXiv:1303.5040 [quant-ph] .
  * Zohar and Burrello [2015] E. Zohar and M. Burrello, Phys. Rev. D91, 054506 (2015), arXiv:1409.3085 [quant-ph] .
  * Zohar  _et al._ [2016] E. Zohar, J. I. Cirac, and B. Reznik, Rept. Prog. Phys. 79, 014401 (2016), arXiv:1503.02312 [quant-ph] .
  * Zohar  _et al._ [2017] E. Zohar, A. Farace, B. Reznik, and J. I. Cirac, Phys. Rev. A95, 023604 (2017), arXiv:1607.08121 [quant-ph] .
  * Klco  _et al._ [2020] N. Klco, J. R. Stryker, and M. J. Savage, Phys. Rev. D 101, 074512 (2020), arXiv:1908.06935 [quant-ph] .
  * Ciavarella  _et al._ [2021] A. Ciavarella, N. Klco, and M. J. Savage, “A Trailhead for Quantum Simulation of SU(3) Yang-Mills Lattice Gauge Theory in the Local Multiplet Basis,” (2021), arXiv:2101.10227 [quant-ph] .
  * Bender  _et al._ [2018] J. Bender, E. Zohar, A. Farace, and J. I. Cirac, New J. Phys. 20, 093001 (2018), arXiv:1804.02082 [quant-ph] .
  * Liu and Xin [2020] J. Liu and Y. Xin, “Quantum simulation of quantum field theories as quantum chemistry,” (2020), arXiv:2004.13234 [hep-th] .
  * Hackett  _et al._ [2019] D. C. Hackett, K. Howe, C. Hughes, W. Jay, E. T. Neil, and J. N. Simone, Phys. Rev. A 99, 062341 (2019), arXiv:1811.03629 [quant-ph] .
  * Alexandru  _et al._ [2019] A. Alexandru, P. F. Bedaque, S. Harmalkar, H. Lamm, S. Lawrence, and N. C. Warrington (NuQS), Phys.Rev.D 100, 114501 (2019), arXiv:1906.11213 [hep-lat] .
  * Yamamoto [2021] A. Yamamoto, PTEP 2021, 013B06 (2021), arXiv:2008.11395 [hep-lat] .
  * Ji  _et al._ [2020] Y. Ji, H. Lamm, and S. Zhu (NuQS), Phys. Rev. D 102, 114513 (2020), arXiv:2005.14221 [hep-lat] .
  * Haase  _et al._ [2020] J. F. Haase, L. Dellantonio, A. Celi, D. Paulson, A. Kan, K. Jansen, and C. A. Muschik, “A resource efficient approach for quantum and classical simulations of gauge theories in particle physics,” (2020), arXiv:2006.14160 [quant-ph] .
  * Bazavov  _et al._ [2019] A. Bazavov, S. Catterall, R. G. Jha, and J. Unmuth-Yockey, Phys. Rev. D 99, 114507 (2019).
  * Bazavov  _et al._ [2015] A. Bazavov, Y. Meurice, S.-W. Tsai, J. Unmuth-Yockey, and J. Zhang, Phys. Rev. D92, 076003 (2015), arXiv:1503.08354 [hep-lat] .
  * Zhang  _et al._ [2018] J. Zhang, J. Unmuth-Yockey, J. Zeiher, A. Bazavov, S. W. Tsai, and Y. Meurice, Phys. Rev. Lett. 121, 223201 (2018), arXiv:1803.11166 [hep-lat] .
  * Unmuth-Yockey  _et al._ [2018] J. Unmuth-Yockey, J. Zhang, A. Bazavov, Y. Meurice, and S.-W. Tsai, Phys. Rev. D98, 094511 (2018), arXiv:1807.09186 [hep-lat] .
  * Unmuth-Yockey [2019] J. F. Unmuth-Yockey, Phys. Rev. D 99, 074502 (2019), arXiv:1811.05884 [hep-lat] .
  * Kreshchuk  _et al._ [2020b] M. Kreshchuk, S. Jia, W. M. Kirby, G. Goldstein, J. P. Vary, and P. J. Love, “Simulating Hadronic Physics on NISQ devices using Basis Light-Front Quantization,” (2020b), arXiv:2011.13443 [quant-ph] .
  * Raychowdhury and Stryker [2018] I. Raychowdhury and J. R. Stryker, (2018), arXiv:1812.07554 [hep-lat] .
  * Raychowdhury and Stryker [2020] I. Raychowdhury and J. R. Stryker, Phys. Rev. D 101, 114502 (2020), arXiv:1912.06133 [hep-lat] .
  * Davoudi  _et al._ [2020] Z. Davoudi, I. Raychowdhury, and A. Shaw, “Search for Efficient Formulations for Hamiltonian Simulation of non-Abelian Lattice Gauge Theories,” (2020), arXiv:2009.11802 [hep-lat] .
  * Hartung  _et al._ [2022] T. Hartung, T. Jakobs, K. Jansen, J. Ostmeyer, and C. Urbach, (2022), arXiv:2201.09625 [hep-lat] .
  * Wiese [2014] U.-J. Wiese,  _Proceedings, 24th International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions (Quark Matter 2014): Darmstadt, Germany, May 19-24, 2014_ , Nucl. Phys. A931, 246 (2014), arXiv:1409.7414 [hep-th] .
  * Luo  _et al._ [2019] D. Luo, J. Shen, M. Highman, B. K. Clark, B. DeMarco, A. X. El-Khadra, and B. Gadway, “A Framework for Simulating Gauge Theories with Dipolar Spin Systems,” (2019), arXiv:1912.11488 [quant-ph] .
  * Brower  _et al._ [2019] R. C. Brower, D. Berenstein, and H. Kawai, PoS LATTICE2019, 112 (2019), arXiv:2002.10028 [hep-lat] .
  * Mathis  _et al._ [2020] S. V. Mathis, G. Mazzola, and I. Tavernelli, Phys. Rev. D 102, 094501 (2020), arXiv:2005.10271 [quant-ph] .
  * Singh [2019] H. Singh, “Qubit $O(N)$ nonlinear sigma models,” (2019), arXiv:1911.12353 [hep-lat] .
  * Singh and Chandrasekharan [2019] H. Singh and S. Chandrasekharan, Phys. Rev. D 100, 054505 (2019), arXiv:1905.13204 [hep-lat] .
  * Buser  _et al._ [2020] A. J. Buser, T. Bhattacharya, L. Cincio, and R. Gupta, “Quantum simulation of the qubit-regularized O(3)-sigma model,” (2020), arXiv:2006.15746 [quant-ph] .
  * Hasenfratz and Niedermayer [2001a] P. Hasenfratz and F. Niedermayer,  _Proceedings, 2001 Europhysics Conference on High Energy Physics (EPS-HEP 2001): Budapest, Hungary, July 12-18, 2001_ , PoS HEP2001, 229 (2001a), arXiv:hep-lat/0112003 [hep-lat] .
  * Caracciolo  _et al._ [2001a] S. Caracciolo, A. Montanari, and A. Pelissetto, Phys. Lett. B513, 223 (2001a), arXiv:hep-lat/0103017 [hep-lat] .
  * Hasenfratz and Niedermayer [2001b] P. Hasenfratz and F. Niedermayer,  _Lattice field theory. Proceedings, 18th International Symposium, Lattice 2000, Bangalore, India, August 17-22, 2000_ , Nucl. Phys. Proc. Suppl. 94, 575 (2001b), arXiv:hep-lat/0011056 [hep-lat] .
  * Patrascioiu and Seiler [1998] A. Patrascioiu and E. Seiler, Phys. Rev. E 57, 111 (1998).
  * Krcmar  _et al._ [2016] R. Krcmar, A. Gendiar, and T. Nishino, Phys. Rev. E 94, 022134 (2016).
  * Caracciolo  _et al._ [2001b] S. Caracciolo, A. Montanari, and A. Pelissetto, Physics Letters B 513, 223 (2001b).
  * Shaw  _et al._ [2020] A. F. Shaw, P. Lougovski, J. R. Stryker, and N. Wiebe, Quantum 4, 306 (2020), arXiv:2002.11146 [quant-ph] .
  * Tong  _et al._ [2021] Y. Tong, V. V. Albert, J. R. McClean, J. Preskill, and Y. Su, (2021), arXiv:2110.06942 [quant-ph] .
  * Kogut and Susskind [1975] J. Kogut and L. Susskind, Phys. Rev. D 11, 395 (1975).
  * Creutz [1985] M. Creutz,  _Quarks, gluons and lattices_ , Cambridge Monographs on Mathematical Physics (Cambridge Univ. Press, Cambridge, UK, 1985).
  * Carena  _et al._ [2021] M. Carena, H. Lamm, Y.-Y. Li, and W. Liu, “Lattice Renormalization of Quantum Simulations,” (2021), arXiv:2107.01166 [hep-lat] .
  * Luo  _et al._ [1999] X.-Q. Luo, S.-H. Guo, H. Kroger, and D. Schutte, Phys. Rev. D 59, 034503 (1999), arXiv:hep-lat/9804029 .
  * Flyvbjerg [1984a] H. Flyvbjerg, Nucl. Phys. B243, 350 (1984a).
  * Flyvbjerg [1984b] H. Flyvbjerg, Nucl. Phys. B240, 481 (1984b).
  * Creutz  _et al._ [1979] M. Creutz, L. Jacobs, and C. Rebbi, Phys. Rev. D20, 1915 (1979).
  * Creutz and Okawa [1983] M. Creutz and M. Okawa, Nucl. Phys. B220, 149 (1983).
  * Bhanot and Rebbi [1981] G. Bhanot and C. Rebbi, Phys. Rev. D24, 3319 (1981).
  * Petcher and Weingarten [1980] D. Petcher and D. H. Weingarten, Phys. Rev. D22, 2465 (1980).
  * Bhanot [1982] G. Bhanot, Phys. Lett. 108B, 337 (1982).
  * Weingarten and Petcher [1981] D. H. Weingarten and D. N. Petcher, Phys. Lett. 99B, 333 (1981).
  * Weingarten [1982] D. Weingarten, Phys. Lett. 109B, 57 (1982), [,631(1981)].
  * Alexandru  _et al._ [2021] A. Alexandru, P. F. Bedaque, R. Brett, and H. Lamm, (2021), arXiv:2112.08482 [hep-lat] .
  * Flyvbjerg [1985] H. Flyvbjerg, J. Math. Phys. 26, 2985 (1985).
  * Drouffe and Zuber [1983] J.-M. Drouffe and J.-B. Zuber, Phys. Rept. 102, 1 (1983).
  * Lamm  _et al._ [2022] H. Lamm _et al._ , in preparation (2022).
  * Lamm  _et al._ [2019b] H. Lamm, S. Lawrence, and Y. Yamauchi (NuQS), Phys. Rev. D100, 034518 (2019b), arXiv:1903.08807 [hep-lat] .
  * Alam  _et al._ [2021] M. S. Alam, S. Hadfield, H. Lamm, and A. C. Y. Li, (2021), arXiv:2108.13305 [quant-ph] .