# Quantum simulation of quantum mechanical system with spatial
noncommutativity

S. Hasibul Hassan Chowdhury<EMAIL_ADDRESS>Department of
Mathematics and Natural Sciences, BRAC University, Dhaka 1212, Bangladesh
Talal Ahmed Chowdhury<EMAIL_ADDRESS>Department of Physics, University of
Dhaka, Dhaka 1000, Bangladesh  Department of Physics and Astronomy, University
of Kansas, Lawrence, Kansas 66045, USA  The Abdus Salam International Centre
for Theoretical Physics, Strada Costiera 11, I-34014, Trieste, Italy  Salah
Nasri<EMAIL_ADDRESS>Department of Physics, UAE University, P.O. Box
17551, Al-Ain, United Arab Emirates  The Abdus Salam International Centre for
Theoretical Physics, Strada Costiera 11, I-34014, Trieste, Italy  Omar Ibna
Nazim<EMAIL_ADDRESS>Department of Physics, University of Dhaka,
Dhaka 1000, Bangladesh  Shaikh Saad<EMAIL_ADDRESS>Department of
Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel,
Switzerland

###### Abstract

Quantum simulation has become a promising avenue of research that allows one
to simulate and gain insight into the models of High Energy Physics whose
experimental realizations are either complicated or inaccessible with current
technology. We demonstrate the quantum simulation of such a model, a quantum
mechanical system with spatial noncommutativity, which is inspired by the
works in Noncommutative Geometry and Noncommutative Field theory for a
universal quantum computer. We use the novel group theoretical formalism to
map the Hamiltonian of such a noncommutative quantum system into the ordinary
quantum mechanical Hamiltonian and then carry out the quantum simulation using
the Trotter-Suzuki product formula. Furthermore, we distinguish the impact of
the noncommutativity parameter on the quantum simulation, especially on the
Trotter error, and point out how its sizable value affects the simulation.

##  I Introduction

Quantum simulation, the idea put forward in the pioneering works [1, 2, 3],
describes the simulation of a seemingly complicated quantum system with
another well-controlled quantum system. It has become an active field of
research [4, 5, 6] and drawing much attention from the high energy physics
community [7, 8, 9, 10, 11]. The quantum simulator can be classified broadly
into two classes: analog and digital simulators. The analog simulator involves
a relatively isolated and controlled quantum system in the laboratory setup
whose constituent degrees of freedom resemble the degrees of freedom of the
underlying quantum system one simulates. On the other hand, the digital
simulator is a digital quantum computer that operates on an array of two-level
quantum systems known as qubits, following a set of universal and elementary
operations. Besides, the attributes of both analog and digital simulators can
be integrated for a more efficient and adaptable quantum simulation of a
complex quantum system. Although the quantum simulation was first prescribed
for the many-body Hamiltonian in [12] and subsequently improved in [13, 14,
15, 16, 17, 18, 19, 20, 21, 22], its importance was quickly realized for areas
related to the High Energy Physics (HEP). For example, the possibility of
simulating physics of many degrees of freedom, especially the lattice gauge
theories using cold atoms and trapped ions, has been shown in [23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. In addition, the quantum
algorithm to simulate the quantum field theory with a universal quantum
computer has been presented in [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53, 54, 55]. Moreover, the implementation of the lattice gauge
theories in the quantum computer has been carried out in [56, 57, 58, 59, 60,
61, 62, 63, 64, 65]. Apart from that, the application of quantum simulation
and quantum computing have been found in the determination of the Parton
distribution functions [66, 67, 68, 69, 70, 71, 72], and the simulation of the
Parton shower that captures the quantum effects [73, 74, 75]. Also, the
quantum simulation has been used to improve the jet clustering algorithm [76,
77, 78, 79, 80], to simulate full medium induced Parton shower and to probe
jet quenching [81, 82, 83], and the dynamics of the heavy quark or jet in
strongly coupled quark-gluon plasma in heavy ion collision [84, 85].
Furthermore, the usage of the quantum algorithm can facilitate more robust
data analysis in HEP [86], reconstruct the particle tracks more efficiently
[87], extract physical observables from scattering processes [88, 89], and it
can be relevant for future colliders [90]. A Quantum algorithm has also been
developed to evaluate the Feynman loop integral [91], which is essential for
precision HEP calculations. The quantum simulation of the phase transition in
gauge theories has been demonstrated in [92, 85, 93] as well. As we have
entered the era of noisy intermediate-scale Quantum (NISQ) devices [94, 95],
by exploring the quantum simulation of the lower-dimensional field theories,
for example, the Schwinger model [96, 97, 98, 99] in series of works [100,
101, 102, 103, 104, 105, 106, 107, 108, 109, 110], and Gross-Neveu model [111]
in [112], one can interpolate how to address the simulation of the full $3+1$
dimensional quantum field theories in the NISQ devices and future quantum
computers. Likewise, the quantum simulation of the $O(3)$ Sigma model has been
prescribed in [113, 114, 115, 116]. Additionally, the procedure to simulate
the non-perturbative processes like Schwinger pair production in the strong
electric field in a quantum computer has been presented in [117]. Therefore,
one can see the rapid development of algorithms and devices associated with
quantum simulation for HEP-related research areas.

Another interesting feature of quantum simulation is its capability to
simulate and provide insight into the novel quantum many-body system and the
exotic phases of matter whose experimental realizations are either difficult
or inaccessible with current technology. One such example is the time crystal
[118, 119, 120], which is the phase of the system arising from the spontaneous
breaking of time-translation symmetry, and it only exists when the system is
in a complete non-equilibrium state that seemingly contradicting our usual
notion of the stability of the phase of a many-body system. Such time crystal
phase has been observed in quantum simulator [121, 122, 123, 124, 125, 126].
Besides, many-body dynamics, especially localization phenomena and the phase
transitions, are simulated in [127, 128, 129, 130]. In addition, the
demonstration of the quantum phase transition has been presented in [131, 132,
133, 134, 135, 136], and the exotic topological phases are probed in [137,
138, 139, 140] by using quantum simulators. Hence, quantum simulation can
offer ways to enhance our physical understanding of quantum systems which are
beyond the reach of the contemporary experimental setup.

In this work, we consider one other example of such a system, the quantum
mechanical system with spatial noncommutativity, generally known as
noncommutative quantum mechanics (NCQM) [141, 142], the prototype model that
is inspired by the works on the noncommutative geometry [143] and
noncommutative field theory (for reviews, see Refs. [144, 145] and references
therein). In the NCQM, the canonical commutation relations between the
respective position and momenta operators accommodate spatial as well as
momentum noncommutativity. The position noncommutativity leads to the
fuzziness of points in the spatial context, while the momentum
noncommutativity signals the presence of a background magnetic field (see
[146] for a detailed account). Although probing the signatures of spatial
noncommutativity requires access to a very high energy scale [147, 148], which
is currently not possible for any terrestrial experiments, there are
suggestions to test such noncommutativity in an artificial analogue system
using cold Rydberg atoms [149]. Besides, the indirect effects of quantized
spacetime, such as nonlocal interactions, can be simulated in an array of
atomic ensembles within an optical cavity [150], and the low-energy imprints
of some models of quantum gravity can be probed in well-controlled matter-wave
interferometry, and optomechanical systems [151]. On the contrary, the
noncommutative geometrical description can be realized in condensed matter
system, for example, in the case of Landau levels [152, 153], integer quantum
hall effect [154], fractional quantum hall effect [155] and topological
insulators [156, 157, 158]. Therefore, we are interested in simulating the
dynamics of a two-dimensional noncommutative quantum mechanical system in a
quantum computer. While implementing the NCQM in the quantum simulator, we
have used the group theoretical formalism developed in [159, 160, 161, 162],
which is based on constructing the families of unitary irreducible
representations of the kinematical symmetry group associated with the NCQM
system. We also want to emphasize that, although, in the usual picture of NCQM
where the noncommutativity between the fundamental positions and momenta are
considered, one can go beyond and address an effective description of a
quantum system whose dynamics can be captured by a suitably constructed set of
noncommutative dynamical variables. Hence, as a starter, we have carried out
the quantum simulation of a two-dimensional noncommutative quantum system so
that we can identify how it can be used to probe the additional
noncommutativity parameter(s) of a quantum mechanical system.

This article is organized as follows. In section II, we present the group
theoretical construction of the two-dimensional noncommutative quantum
mechanical system. The details of the quantum algorithm for the quantum
simulation are given in section III. We discuss the results of the quantum
simulation and its limitations in section IV and conclude in section V.

##  II Noncommutative quantum mechanical system

###  II.1 Group theoretical formalism for the NCQM

The quantum phase space associated with the commutative two-dimensional
quantum mechanical system is described by the position and momenta operators,
which are unbounded Hermitian operators acting on the associated Hilbert
space. For a two-dimensional system, the classical phase space consists of the
position coordinates $x$ and $y$ and the respective momentum coordinates
$p_{x}$ and $p_{y}$. Now the corresponding quantum phase space coordinates
comprise of the Hermitian operators $\hat{x}$, $\hat{y}$, $\hat{p}_{x}$,
$\hat{p}_{y}$ defined on $L^{2}(\mathbb{R}^{2},dx\;dy)$ and satisfying the
following commutation relations that correspond to the Weyl-Heisenberg group,

$\displaystyle[\hat{x},\hat{p}_{x}]=[\hat{y},\hat{p}_{y}]=i\hbar\mathbb{I},$
$\displaystyle[\hat{x},\hat{y}]=[\hat{p}_{x},\hat{p}_{y}]=0,$ (1)

where $\mathbb{I}$ is the identity operator on $L^{2}(\mathbb{R}^{2},dx\;dy)$.

Now consider the two-dimensional noncommutative plane immersed in a constant
magnetic field $B$ where the spatial noncommutativity is parameterized by the
constant $\theta$. In group theoretical formulation, one first starts with a
two-dimensional configuration space with coordinates $(q_{1},q_{2})$. One
subsequently forms the associated 4-dimensional phase space, the coordinates
of which can be read off from the ordered 4-tuple $(q_{1},q_{2},p_{1},p_{2})$
where $p_{i}$ are the conjugate momenta associated with $q_{i}$. The
underlying physical system’s time evolution is described by a curve
parameterized by time $t$ in the underlying phase space. The evolving physical
system follows a trajectory in phase space subjected to some initial
condition. Now the underlying system is expected to possess translational
symmetry. The translational symmetry group $T_{(Q_{1},Q_{2},P_{1},P_{2})}$
acts on the phase space point $(q_{1},q_{2},p_{1},p_{2})$ describing the
system at a particular instant in the following way:

$T_{(Q_{1},Q_{2},P_{1},P_{2})}(q_{1},q_{2},p_{1},p_{2})=(q_{1}+Q_{1},q_{2}+Q_{2},p_{1}+P_{1},p_{2}+P_{2}).$
(2)

This set of transformations form a 4 dimensional abelian Lie group, with the
group multiplication given as follows:

$T_{(Q_{1},Q_{2},P_{1},P_{2})}T_{(Q^{\prime}_{1},Q^{\prime}_{2},P^{\prime}_{1},P^{\prime}_{2})}=T_{(Q_{1}+Q^{\prime}_{1},Q_{2}+Q^{\prime}_{2},P_{1}+P^{\prime}_{1},P_{2}+P^{\prime}_{2})}.$
(3)

All of its irreducible unitary representations are one-dimensional
representations ($\mathbb{C}$ is the representation space), and the group
generators are represented by constant multiples of identity operator on
$\mathbb{C}$.

At this stage, using the formalism of [163], we centrally extend the above-
mentioned 4-dimensional translational symmetry group by $\mathbb{R}^{3}$ so
that the center of the resulting centrally extended nonabelian group is
$\mathbb{R}^{3}$. We denote this 7-dimensional real nilpotent Lie group by
$G_{\hbox{\tiny{NC}}}$ and its Lie algebra by
$\mathfrak{g}_{\hbox{\tiny{NC}}}$. It has been established as the kinematical
symmetry group of 2-dimensional noncommutative quantum mechanics in [160,
161]. If one denotes a generic element of $G_{\hbox{\tiny{NC}}}$ by
$(\Theta,\Phi,\Psi,Q_{1},Q_{2},P_{1},P_{2})$, then the corresponding group
multiplication in $G_{\hbox{\tiny{NC}}}$ reads ([160])

$\displaystyle(\Theta,\Phi,\Psi,Q_{1},Q_{2},P_{1},P_{2})(\Theta^{\prime},\Phi^{\prime},\Psi^{\prime},Q^{\prime}_{1},Q^{\prime}_{2},P^{\prime}_{1},P^{\prime}_{2})$
$\displaystyle=(\Theta+\Theta^{\prime}+\frac{\alpha}{2}(Q_{1}P^{\prime}_{1}+Q_{2}P^{\prime}_{2}-P_{1}Q^{\prime}_{1}-P_{2}Q^{\prime}_{2}),\Phi+\Phi^{\prime}+\frac{\beta}{2}(P_{1}P^{\prime}_{2}-P_{2}P^{\prime}_{1}),$
$\displaystyle\hskip
14.45377pt\Psi+\Psi^{\prime}+\frac{\gamma}{2}(Q_{1}Q^{\prime}_{2}-Q_{2}Q^{\prime}_{1}),Q_{1}+Q^{\prime}_{1},Q_{2}+Q^{\prime}_{2},P_{1}+P^{\prime}_{1},P_{2}+P^{\prime}_{2}).$

Here $(\Theta,\Phi,\Psi)\in\mathbb{R}^{3}$ which is the centre of
$G_{\hbox{\tiny{NC}}}$, and $\alpha$, $\beta$ and $\gamma$ are some fixed
constants carrying dimensions of
$(\hbox{momentum}\times\hbox{position})^{-1}$, $(\hbox{momentum})^{-2}$ and
$(\hbox{position})^{-2}$, respectively.

The dual Lie algebra $\mathfrak{g}_{\hbox{\tiny{NC}}}^{*}$ is also
7-dimensional. There is a natural action of $G_{\hbox{\tiny{NC}}}$ on
$\mathfrak{g}_{\hbox{\tiny{NC}}}^{*}$ known as coadjoint action. The orbits of
the coadjoint action are called coadjoint orbits. The 7-dimensional vector
space $\mathfrak{g}_{\hbox{\tiny{NC}}}^{*}$ is found to be foliated by
coadjoint orbits of 3 different dimensions as found in [160]: 4 dimensional, 2
dimensional, and 0 dimensional. By a theorem due to Kirillov [164], the
coadjoint orbits of $G_{\hbox{\tiny{NC}}}$ are in one-to-one correspondence
with the unitary dual $\hat{G}_{\hbox{\tiny{NC}}}$, i.e., the set of
equivalence classes of unitary irreducible representations of
$G_{\hbox{\tiny{NC}}}$. A generic element of $\hat{G}_{\hbox{\tiny{NC}}}$ is
given by an ordered triple $(\hbar,\theta,B)$ where $\theta$ and $B$ are the
spatial noncommutativity parameter and the constant magnetic field,
respectively. The noncentral generators corresponding to the group parameters
$Q_{1}$, $Q_{2}$, $P_{1}$ and $P_{2}$ are represented by the 2-parameter
$(r,s)$ family of self-adjoint operators $\hat{Q}^{s}_{1}$, $\hat{Q}^{s}_{2}$,
$\hat{\Pi}^{r,s}_{1}$, $\hat{\Pi}^{r,s}_{2}$, respectively acting on
$L^{2}(\mathbb{R}^{2},dx\;dy)$ subjected to the following set of commutation
relations:

$[\hat{Q}^{s}_{i},\hat{\Pi}^{r,s}_{j}]=i\hbar\delta_{ij}\hat{\mathbb{I}},\;[\hat{Q}^{s}_{1},\hat{Q}^{s}_{2}]=i\theta\hat{\mathbb{I}},\;[\hat{\Pi}^{r,s}_{1},\hat{\Pi}^{r,s}_{2}]=i\hbar
B\hat{\mathbb{I}}.$ (5)

Here $\hat{\mathbb{I}}$ is the identity operator on
$L^{2}(\mathbb{R}^{2},dx\;dy)$. The quantum phase space comprising the
2-parameter family of self-adjoint operators
$\\{\hat{Q}^{s}_{1},\hat{Q}^{s}_{2},\hat{\Pi}^{r,s}_{1},\hat{\Pi}^{r,s}_{2}\\}$
subject to the commutation relations Eq. (5) physically represents the
noncommutative quantum mechanical plane immersed in a constant magnetic field.
Therefore, we can see that the parameters characterizing the spatial and
momentum noncommutativity, $\theta$ and $B$, respectively, along with the
usual quantum mechanical position-momentum noncommutativity given by $\hbar$,
arise from the central extension of the associated kinematical symmetry group
of the system.

The map that expresses the noncommutative operators $\hat{Q}^{s}_{1}$,
$\hat{Q}^{s}_{2}$, $\hat{\Pi}^{r,s}_{1}$, $\hat{\Pi}^{r,s}_{2}$ in terms of
the quantum mechanical operators
$\hat{x},\,\hat{y},\,\hat{p}_{x},\,\hat{p}_{y}$ is given by,

$\displaystyle\hat{Q}^{s}_{1}$
$\displaystyle=\hat{x}-s\dfrac{\theta}{\hbar}\hat{p}_{y},\,\,\,\hat{Q}^{s}_{2}=\hat{y}+(1-s)\dfrac{\theta}{\hbar}\hat{p}_{x},$
(6) $\displaystyle\hat{\Pi}^{r,s}_{1}$
$\displaystyle=\dfrac{B\hbar(1-r)}{\hbar-B\theta
r}\hat{y}+\dfrac{[B\theta(r+s-rs)-\hbar]}{B\theta r-\hbar}\hat{p}_{x},$
$\displaystyle\hat{\Pi}^{r,s}_{2}$
$\displaystyle=-rB\hat{x}+\left[1+r(s-1)\dfrac{B\theta}{\hbar}\right]\hat{p}_{y}.$

where the 2-parameters $(r,s)$ represent the family of unitary equivalent
irreducible representations of $G_{\hbox{\tiny{NC}}}$ associated with the
fixed ordered triple $(\hbar,\theta,B)$. Now two representations labeled by
$(r,s)$ and $(r^{\prime},s^{\prime})$ are intertwined by a unitary operator
$U$ on the given Hilbert space $L^{2}(\mathbb{R}^{2},dx\;dy)$ where the group
generators also transform, using the same unitary operator $U$, as follows

$\displaystyle\hat{Q}_{1}^{s^{\prime}}=U\hat{Q}_{1}^{s}U^{-1},\,\,\,\hat{Q}_{2}^{s^{\prime}}=U\hat{Q}_{2}^{s}U^{-1},$
(7)
$\displaystyle\hat{\Pi}_{1}^{r^{\prime},s^{\prime}}=U\hat{\Pi}^{r,s}_{1}U^{-1},\,\,\hat{\Pi}_{2}^{r^{\prime},s^{\prime}}=U\hat{\Pi}^{r,s}_{2}U^{-1},$

leading to the following unitary transformation of the underlying Hamiltonian:

$\hat{H}^{r^{\prime},s^{\prime}}=U\hat{H}^{r,s}U^{-1}.$ (8)

One can consider the parameters $(r,s)$ as the gauge parameters as they do not
affect the measurable quantities, for example, the energy spectra of the
noncommutative Hamiltonian as shown in [165].

A possible generalization: One can encounter a quantum system that is
described not by the usual position and momentum Hermitian operators but by a
set of $2n$ Hermitian operators $\hat{A}_{i}$ and $\hat{B}_{i}$ associated
with dynamical variables $a_{i}$ and $b_{i}$ with $i=1,2,..,n$, respectively
which effectively characterize the system. The set of Hermitian operators
$(\hat{A}_{i},\,\hat{B}_{i})$ follow the following set of commutation
relations: $[\hat{A}_{i},\hat{A}_{j}]=i\theta_{ij}$,
$[\hat{B}_{i},\hat{B}_{j}]=i\gamma_{ij}$ and
$[\hat{A}_{i},B_{j}]=i\alpha_{ij}$ where $\theta_{ij}$ and $\gamma_{ij}$ are
the matrix entries of some real skew-symmetric $n\times n$ matrices $\theta$
and $\gamma$, respectively. Also, $\alpha_{ij}$ are the entries of the real
diagonal matrix $\alpha$. The constant real parameters $\alpha_{ij}$,
$\theta_{ij}$ and $\gamma_{ij}$ are associated with the noncommutativity of
the operators $\hat{A}_{i}$ and $\hat{B}_{j}$ for all $i,j=1,2,..n$. Following
the group theoretical formalism, first, we identify the configuration space
$\mathcal{M}$ as a smooth manifold that is locally described by the dynamical
variables $a_{i}$. One then enumerates the respective conjugate variables
$b_{i}$ so that the $2n$ local coordinates $(a_{i},b_{i})$ describe the
underlying cotangent bundle $T^{*}\mathcal{M}=:\mathcal{N}$. Here,
$\mathcal{N}$ is a real $2n$-dimensional Poisson manifold and is naturally
endowed with a symplectic structure. Besides, one can go on to define a
Hamiltonian function $H:\mathcal{N}\rightarrow\mathbb{R}$ that can be
considered the generator of one-parameter transformation on points of
$\mathcal{N}$. At this point, one tries to identify the kinematical symmetry
group $G_{\mathcal{N}}$ associated with the manifold $\mathcal{N}$ and
centrally extend it to $G^{\mathrm{ext}}_{\mathcal{N}}$ whose center is
$\mathbb{R}^{N}$ where $N$ is the number of noncommutativity parameters
associated with the initial quantum system. In this way, one constructs the
kinematical symmetry group $G^{\mathrm{ext}}_{\mathcal{N}}$ of the underlying
quantum system. Eventually, one can construct the equivalence classes of
unitary irreducible representations of $G^{\mathrm{ext}}_{\mathcal{N}}$ that
are associated with the $(2n\times 2n)$ real matrix $\Omega$ of
noncommutativity written in the block form as
$\Omega=\begin{bmatrix}\theta&\alpha\\\ -\alpha&\gamma\end{bmatrix}$.
Moreover, the generators associated with the noncentral group parameters
$a_{i}$ and $b_{i}$ are nothing but the Hermitian operators $\hat{A}_{i}$ and
$\hat{B}_{i}$ acting on some suitable Hilbert Space that follow the above-
mentioned commutation relations. Therefore, the noncommutative two-dimensional
quantum mechanical system can be an example of the generalized group
theoretical construction.

###  II.2 Noncommutative two-dimensional oscillator

We consider simulating the noncommutative two-dimensional harmonic oscillator
[166, 167, 168], one of the representatives of the noncommutative two-
dimensional quantum mechanical system. The Hamiltonian that describing a
particle of mass $m$ subjected to a constant magnetic field $B$ along the
perpendicular direction of the plane and an anisotropic harmonic potential,
$V(\hat{Q}_{1},\hat{Q}_{2})=\frac{1}{2}m\big{[}\omega_{1}^{2}\hat{Q}_{1}^{2}+\omega_{2}^{2}\hat{Q}_{2}^{2}\big{]}$,
is given as

$\hat{H}^{NC}=\frac{1}{2m}\big{[}\hat{\Pi}_{1}^{2}+\hat{\Pi}_{2}^{2}\big{]}+\frac{1}{2}m\big{[}\omega_{1}^{2}\hat{Q}_{1}^{2}+\omega_{2}^{2}\hat{Q}_{2}^{2}\big{]}.$
(9)

Using Eq. (6), we can obtain the following form of the noncommutative
Hamiltonian expressed in terms of the quantum mechanical operators;

$\displaystyle\hat{H}=\frac{1}{2M_{1}}\hat{p}_{x}^{2}+\frac{1}{2M_{2}}\hat{p}_{y}^{2}+\frac{1}{2}M_{1}\Omega_{1}^{2}\hat{x}^{2}+\frac{1}{2}M_{2}\Omega_{2}^{2}\hat{y}^{2}-l_{1}\hat{x}\hat{p}_{y}+l_{2}\hat{y}\hat{p}_{x}.$
(10)

where,

$\displaystyle M_{1}$ $\displaystyle=\left[\frac{(\hbar-(r+s-rs)\theta
B)^{2}}{m(\hbar-r\theta
B)^{2}}+\frac{m(1-s)^{2}\theta^{2}\omega_{2}^{2}}{\hbar^{2}}\right]^{-1},\,\,\,M_{2}=\left[\frac{1}{m}\left(1-\frac{(1-s)r\theta
B}{\hbar}\right)^{2}+\frac{ms^{2}\theta^{2}\omega_{1}^{2}}{\hbar^{2}}\right]^{-1}$
$\displaystyle\Omega_{1}^{2}$
$\displaystyle=\frac{m}{M_{1}}\left(\omega_{1}^{2}+\frac{B^{2}r^{2}}{m^{2}}\right),\,\,\,\Omega_{2}^{2}=\frac{m}{M_{2}}\left(\omega_{2}^{2}+\frac{B^{2}(1-r)^{2}\hbar^{2}}{m^{2}(\hbar-r\theta
B)^{2}}\right),$ $\displaystyle l_{1}$
$\displaystyle=\frac{ms\theta\omega_{1}^{2}}{\hbar}+\frac{Br}{m}\left(1-\frac{(1-s)r\theta
B}{\hbar}\right),\,\,\,l_{2}=\frac{m(1-s)\theta\omega_{2}^{2}}{\hbar}+\frac{\hbar
B(1-r)(\hbar+\theta B(rs-r-s))}{m(\hbar-r\theta B)^{2}}.$ (11)

Here, we can see that the initial parameters of the NC Hamiltonian Eq. (9),
$\\{m,\,\omega_{i},\,\hbar,\,\theta,\,B,\,r,s\\}$ enter into the parameters,
$\left(M_{i},\Omega_{i},l_{i}\right)$ given in Eq. (11) of Eq. (10). Besides,
in the case of zero magnetic field, $B=0$, the family of unitary equivalent
irreducible representations of $G_{\hbox{\tiny{NC}}}$ is only parameterized by
$s$, and are associated with the fixed ordered doublet $(\hbar,\theta)$.
Moreover, in this case, the noncommutative momentum operators,
$\hat{\Pi}_{1,2}$ become just the usual quantum mechanical momentum operators,
$\hat{p}_{x,y}$ and $r$ parameter drops out in the relations Eq. (11). In the
following sections, we address the quantum simulation of the noncommutative
isotropic two-dimensional harmonic oscillator in the zero magnetic field for
its simplicity and leave the case of the anisotropic harmonic oscillator in a
non-zero magnetic field for separate work.

##  III Quantum algorithm

To study the evolution of a quantum system first, we choose a basis of the
Hilbert Space, which is generally associated with a complete set of mutually
commuting observables. Afterward, we find a suitable expression of the time
evolution operator in that basis and evolve an initial quantum state by
applying the time evolution operator to obtain the final state. When we
measure one of the observables, the final state will collapse onto an
eigenstate of that observable with a specific probability. If the time
evolution is simulated multiple times, we obtain the probability distribution
of the final state. The qubit is the fundamental computing element of the
quantum computer, which is a two-state quantum system. So a system of $n$
qubits will represent the Hilbert Space of $2^{n}$ dimensions and constitute
the computational basis on which the quantum computer operates. For performing
the quantum simulation of a quantum system, first, we must establish a mapping
between the eigenbasis for a relevant observable and the computational basis
corresponding to $n$ qubits where $n$ is chosen in such a way that we can
establish the one-to-one mapping between the eigenbasis and computational
basis. The next step is to find the representation of the time evolution
operator on a computational basis. The time evolution operator will be
decomposed into unitary operators known as the quantum gates, which are
applied to the qubits. At this point, we are ready to execute the simulation
in a quantum computer. First, we prepare an initial quantum state, and then by
applying appropriate quantum gates to it, we evolve that state in time to
obtain the final state. Subsequently, we implement the measurement of the
final state and note the resultant computational basis state. In this way, by
repeating the simulation many times and making measurements of the final
states, we obtain the probability distributions of the computational basis
states. Thereupon, we retrieve their probability distributions and interpret
the results by using the inverse mapping from the computational basis to the
relevant eigenbasis of the chosen observable. For the quantum simulation of
the noncommutative two-dimensional isotropic harmonic oscillator, we extend
the algorithm of [169] in the case of two-dimensional quantum mechanical
systems. In the following sections, we describe the steps in detail.

###  III.1 Mapping to computational basis states

In our simulation, we discretize the two-dimensional plane into a mesh by
dividing each dimension of the plane into $N$ divisions being labeled from
$-N/2$ to $N/2-1$ with a unit interval, and each discrete point of this
$N\times N$ mesh is considered to represent a position eigenstate. Besides, we
choose $N$ to be equal to $2^{p}$ so that $p=\mathrm{log}_{2}N$ number of
qubits are needed to obtain the same number of computational states along one
direction, for example, $x$ axis, and therefore the total number of qubits to
represent $|x,y\rangle$ position eigenstate will be $2p$. The computational
basis state corresponding to the position eigenstate $|x,y\rangle$ is obtained
by associating $x=i+N/2$ and $y=j+N/2$ to $\mathrm{bin}(i+N/2)$ and
$\mathrm{bin}(j+N/2)$, respectively, for
$i,\,j=-N/2,\,-N/2+1,\,...,\,N/2-2,\,N/2-1$ where $\mathrm{bin}(a)$ denotes
binary of the decimal number $a$ in $p$ bits. Therefore, the position
eigenstate $|x,y\rangle$ is expressed as
$\lvert\mathrm{bin}(i+N/2)\rangle\otimes\lvert\mathrm{bin}(j+N/2)\rangle\equiv\lvert\mathrm{bin}(i+N/2)\mathrm{bin}(j+N/2)\rangle$.
For example, if we divide $x-y$ plane into $8\times 8$ mesh with $N=8$ so that
$p=\mathrm{log}_{2}8=3$ and accordingly the $|-4,-4\rangle$ position state is
mapped to computational basis
$\lvert\mathrm{bin}(0)\mathrm{bin}(0)\rangle\equiv\lvert 000000\rangle$,
$|-4,-3\rangle$ to $\lvert\mathrm{bin}(0)\mathrm{bin}(1)\rangle\equiv\lvert
000001\rangle$, $|0,0\rangle$ to
$\lvert\mathrm{bin}(4)\mathrm{bin}(4)\rangle\equiv\lvert 100100\rangle$ and so
on.

###  III.2 Operator representations in the computational basis

In the discrete setup, the position operators are diagonal matrices, and
corresponding momentum operators, which will also be finite-dimensional
matrices, can be obtained using the centered quantum Fourier transform [170].
The quantum mechanical position operators $\hat{x}$ and $\hat{y}$, now
represented by $N\times N$ diagonal matrices and denoted as $X$ and $Y$,
respectively, are given as,

$X=Y=\sqrt{\frac{2\pi}{N}}\begin{pmatrix}-N/2&0&.&.&.&0\\\ 0&-N/2+1&.&.&.&0\\\
.&.&.&&&.\\\ .&.&&.&&.\\\ .&.&&&.&.\\\ 0&0&.&.&.&N/2-1\\\ \end{pmatrix},$ (12)

whereas the quantum mechanical momentum operators $\hat{p}_{x}$ and
$\hat{p}_{y}$ are denoted as $P_{X}$ and $P_{Y}$, respectively, are also
represented by $N\times N$ Hermitian matrices which are obtained from the
centered Fourier transform of the position matrices $X$ and $Y$ as follows,

$P_{X}=F^{-1}XF,\,\,\,P_{Y}=F^{-1}YF,$ (13)

and the centered quantum Fourier transform is given by the $N\times N$ unitary
matrix $F$ with matrix elements,

$F_{jk}=\frac{1}{\sqrt{N}}\mathrm{exp}\left[\frac{2\pi
i}{N}jk\right],\,\,\,j,k\in[-N/2,N/2-1].$ (14)

Moreover, the $X$ and $P_{X}$ matrices act on the more significant set of $p$
qubits of the computational basis, which is dedicated to the x-axis, and leave
the less significant set of $p$ qubits for the y-axis intact. On the other
hand, the action of the $Y$ and $P_{Y}$ matrices on the computational basis is
just reversed. As an example, 6 qubits are needed for $N=8$, and the most
general computational state can be denoted as $\lvert
q_{5}q_{4}q_{3}q_{2}q_{1}q_{0}\rangle$. The first 3 qubits $q_{2}$, $q_{1}$,
and $q_{0}$ are dedicated to the y-axis, and the rest are dedicated to the
x-axis. The matrices $X$ and $P_{X}$ act on the states $\lvert
q_{5}q_{4}q_{3}\rangle$ whereas $Y$ and $P_{Y}$ act on the states $\lvert
q_{2}q_{1}q_{0}\rangle$. In addition, $X\,P_{Y}$ and $YP_{X}$ are basically
$X\otimes P_{Y}$ and $P_{X}\otimes Y$, respectively, and can be simplified as
follows,

$\displaystyle X\otimes P_{Y}$ $\displaystyle=X\otimes(F^{-1}YF)=(I\otimes
F^{-1})(X\otimes Y)(I\otimes F)=(I\otimes F^{-1}).D.(I\otimes F),$
$\displaystyle P_{X}\otimes Y$ $\displaystyle=(F^{-1}XF)\otimes
Y=(F^{-1}\otimes I)(X\otimes Y)(F\otimes I)=(F^{-1}\otimes I).D.(F\otimes I),$

where we denote, $D=X\otimes Y$.

###  III.3 Time evolution operator representation

In terms of the finite-dimensional position operators $X$, $Y$ and momentum
operators $P_{X}$ and $P_{Y}$, the quantum mechanical Hamiltonian in Eq. (10)
associated with the noncommutative Hamiltonian of Eq. (9) is expressed as

$H=\frac{1}{2M_{1}}P_{X}^{2}+\frac{1}{2M_{2}}P_{Y}^{2}+\frac{1}{2}M_{1}\Omega_{1}^{2}X^{2}+\frac{1}{2}M_{2}\Omega_{2}^{2}Y^{2}-l_{1}X\,P_{Y}+l_{2}Y\,P_{X}\,=\,H_{X}+H_{Y}+H_{XY},$
(15)

where for notational convenience, we denote

$H_{X}=\frac{1}{2M_{1}}P_{X}^{2}+\frac{1}{2}M_{1}\Omega_{1}^{2}X^{2},\,\,\,H_{Y}=\frac{1}{2M_{2}}P_{Y}^{2}+\frac{1}{2}M_{2}\Omega_{2}^{2}Y^{2},\,\,\,H_{XY}=-l_{1}XP_{Y}+l_{2}YP_{X}.$
(16)

The time evolution operator $U(t)$ corresponding to the Hamiltonian in Eq.
(15) is given by

$U(t)=e^{-iHt/\hbar}=e^{-i(H_{X}+H_{Y}+H_{XY})t/\hbar}.$ (17)

Now focusing only on the isotropic case $\omega_{1}=\omega_{2}$, for which
$M_{1}=M_{2}=M$, $\Omega_{1}=\Omega_{2}=\Omega$ and $l_{1}=l_{2}=l$, we use
the Trotter-Suzuki formula [171, 172, 173, 174, 175, 176] to approximate the
time evolution operator,

$e^{-iHt/\hbar}\approx
e^{-i\frac{P_{X}^{2}}{2M}t/\hbar}e^{-i\frac{1}{2}M\Omega^{2}X^{2}t/\hbar}e^{-i\frac{P_{Y}^{2}}{2M}t/\hbar}e^{-i\frac{1}{2}M\Omega^{2}Y^{2}t/\hbar}e^{i\,lXP_{Y}t/\hbar}e^{-i\,lYP_{X}t/\hbar}.$
(18)

We take time steps of $\delta t=t/n$ so that

$U(t)=e^{-iHt/\hbar}=(e^{-iH\delta t/\hbar})^{n}\\\
\approx(e^{-i\frac{P_{X}^{2}}{2M}\delta
t/\hbar}e^{-i\frac{1}{2}M\Omega^{2}X^{2}\delta
t/\hbar}e^{-i\frac{P_{Y}^{2}}{2M}\delta
t/\hbar}e^{-i\frac{1}{2}M\Omega^{2}Y^{2}\delta t/\hbar}e^{i\,lXP_{Y}\delta
t/\hbar}e^{-i\,lYP_{X}\delta t/\hbar})^{n}.$ (19)

For convenience we denote, $U_{X}=e^{-i\frac{1}{2}M\Omega^{2}X^{2}\delta
t/\hbar}$ and $U_{Y}=e^{-i\frac{1}{2}M\Omega^{2}Y^{2}\delta t/\hbar}$.
Besides, the unitary operators containing $P_{X}$ and $P_{Y}$, denoted as
$U_{P_{X}}$, $U_{P_{Y}}$, $U_{XP_{Y}}$ and $U_{YP_{X}}$, are further
simplified in the following,

$\displaystyle U_{P_{X}}=e^{-i\frac{P_{X}^{2}}{2M}\delta t/\hbar}$
$\displaystyle=F^{-1}e^{-i\frac{X^{2}}{2M}\delta
t/\hbar}F=F^{-1}\tilde{U}_{X}F,\,\,\,U_{P_{Y}}=e^{-i\frac{P_{Y}^{2}}{2M}\delta
t/\hbar}=F^{-1}e^{-i\frac{Y^{2}}{2M}\delta t/\hbar}F=F^{-1}\tilde{U}_{Y}F,$
(20) $\displaystyle U_{XP_{Y}}$ $\displaystyle=e^{i\,lXP_{Y}\delta
t/\hbar}=(I\otimes F^{-1}).e^{i\,lD\delta t/\hbar}.(I\otimes F)=(I\otimes
F^{-1}).U^{+}_{XY}.(I\otimes F),$ (21) $\displaystyle U_{YP_{X}}$
$\displaystyle=e^{-i\,lYP_{X}\delta t/\hbar}=(F^{-1}\otimes I).e^{-i\,lD\delta
t/\hbar}.(F\otimes I)=(F^{-1}\otimes I).U^{-}_{XY}.(F\otimes I),$ (22)

where we also denote, $\tilde{U}_{X}=e^{-i\frac{X^{2}}{2M}\delta t/\hbar}$,
$\tilde{U}_{Y}=e^{-i\frac{Y^{2}}{2M}\delta t/\hbar}$ and $U^{\pm}_{XY}=e^{\pm
i\,lD\delta t/\hbar}$. Therefore, the time evolution operator is decomposed
into discrete Fourier transform operators and diagonal matrices.

###  III.4 Execution and interpretation

To carry out the quantum simulation, we write and execute a quantum program
via the Qasm Simulator of Qiskit (version 0.34.0) [177]. Initially, all the
qubits are in $\lvert 0\rangle$ state, and we have to prepare the initial
state of our simulation. As a starting point, our initial states are the
position eigenstate, $|x,y\rangle$, which can be easily prepared by applying
NOT gates to the appropriate qubits according to the mapping given in section
III.1. After preparing the initial state, we need to implement time evolution
according to Eq. (19). Before executing, the unitary operators of the single
trotter step in the time evolution are transpiled into quantum gates via the
built-in transpilers of Qiskit, and we use a loop to implement single trotter
steps $n$ times for describing the evolution over the chosen time interval. We
obtain the final state after implementing the time evolution. The qubits in
the final state are measured to obtain counts corresponding to each
computational state. Again, by using the inverse map from computational basis
to position eigenbasis given in section III.1, we assign the number of counts
to the corresponding position eigenstate. In this way, we can find a
probability distribution on the mesh after the time evolution of the position
eigenstate $|x,y\rangle$. Also, one can easily determine the overlapping of
the final state with another position eigenstate by noting the number of
counts to the corresponding position eigenstate. Alternatively, the
overlapping between the final state and any position eigenstate can be
calculated using the CircuitStateFn class of Qiskit Terra, which is
essentially an inner product calculation between the two state vectors. In the
following sections, we discuss the quantum circuits that can be used to
implement the quantum algorithm. Moreover, all circuit diagrams in the
subsequent sections are drawn with Qiskit.

####  III.4.1 Filter Circuit to Implement Diagonal Unitary Matrices

The diagonal unitary matrices can be implemented using filter circuits. We can
implement the filter circuit using a multi-controlled NOT gate and only one
ancilla qubit instead of multiple Toffoli gates as done in [178]. Filter
circuits select a certain state and apply a phase to that state. Using a
filter circuit for each state, we can implement an operator corresponding to a
diagonal unitary matrix. In our simulation, the exponential of $X^{2}$,
$Y^{2}$, and $D$ matrices being diagonal unitary matrices are implemented in
this manner.

Figure 1: Filter Circuit for the State $\lvert 1010\rangle$

Let us demonstrate the action of a filter circuit on a state $|1010\rangle$ as
shown in Fig. 1. The multi-controlled NOT gate acting on the ancilla qubit
$q_{A}$ is activated only when all of its control qubits are in $\lvert
1\rangle$ state. Therefore, in the case of $|1010\rangle$, first, the NOT
gates invert $q_{2}$ and $q_{0}$ to $\lvert 1\rangle$ and activate the
adjacent multi-controlled NOT gate. Now the ancilla qubit $q_{A}$, initially
in state $\lvert 0\rangle$, switches to $\lvert 1\rangle$ due to the
activation of the multi-controlled NOT gate. At this point, since $q_{A}$ is
in state $\lvert 1\rangle$, the controlled $U_{1}$($\theta$) gate will be
activated. It will act on $q_{0}$ and add the phase $e^{i\theta}$ to the
circuit. After that, the rightmost multi-controlled NOT gate will switch the
ancilla qubit $q_{A}$ from $\lvert 1\rangle$ to $\lvert 0\rangle$, and
finally, the two NOT gates at the right act on $q_{2}$ and $q_{0}$ to invert
them from $\lvert 1\rangle$ to $\lvert 0\rangle$. Hence, the state $\lvert
1010\rangle$ is transformed to $e^{i\theta}\lvert 1010\rangle$ by the action
of the filter circuit. Note that this filter circuit does not alter any state
other than $\lvert 1010\rangle$. After the operation of the filter circuit,
the ancilla qubit $q_{A}$ is in $\lvert 0\rangle$ state. We can append another
filter circuit with a new combination of NOT gates to select another state. We
can also choose another value of the applied phase $\theta$ by changing the
controlled $U_{1}$ gate parameter. Consequently, to implement a diagonal
unitary matrix, we apply filter circuits for each of the states with the phase
$e^{i\theta_{i}}$ corresponding to its $i$-th diagonal entry.

####  III.4.2 Implementation of Centered Quantum Fourier Transform

As the momentum operators are defined using centered discrete Fourier
transformation as in Eq. (13), we implement it using the quantum circuit for
the quantum Fourier transform (QFT) discussed in Qiskit textbook [179] and a
permutation circuit, explained below, to make it centered. In Fig. 2, we only
present the circuit diagram for QFT acting on four qubits. Moreover, the
inverse QFT can be implemented by applying the gates in reverse order since it
is a unitary operation.

Figure 2: QFT circuit for Four Qubits

In our implementation, the centered discrete Fourier transform, defined as
$F=P^{N/2}F^{d}P^{-N/2}$, is obtained by the permutation of the elements of
the standard discrete Fourier transform $F^{d}$ which is defined as
$F^{d}_{jk}=\frac{1}{\sqrt{N}}\mathrm{exp}\left[\frac{2\pi i}{N}jk\right]$
where $j,k=0,1,...,N-1$ [169]. Therefore, we need to define a quantum circuit
that permutes the computational basis states. The matrix representations of
the permutation denoted as $P$ and its inverse $P^{-1}$ are given by

$P=\begin{pmatrix}0&1&0&.&.&.&0\\\ 0&0&1&.&.&.&0\\\ .&.&&1&&&.\\\
.&.&&&.&&.\\\ .&.&&&&.&.\\\ .&.&&&&&1\\\ 1&0&0&.&.&.&0\\\ \end{pmatrix},\hskip
60.0ptP^{-1}=\begin{pmatrix}0&0&0&.&.&.&1\\\ 1&0&0&.&.&.&0\\\ .&1&&.&&&.\\\
.&.&1&&&&.\\\ .&.&&.&&&.\\\ .&.&&&.&&\\\ 0&0&0&.&.&1&0\\\ \end{pmatrix}.\hskip
12.0pt$ (23)

Let us note how the operator $P$ maps the computational basis states
appropriately for a case of three qubits from Table 1.

$\lvert\psi\rangle$ | $P\lvert\psi\rangle$
---|---
$q_{2}$ |  $q_{1}$ |  $q_{0}$ |  $q_{2}$ |  $q_{1}$ |  $q_{0}$
$\lvert 0\rangle$ |  $\lvert 0\rangle$ |  $\lvert 0\rangle$ |  $\lvert 1\rangle$ |  $\lvert 1\rangle$ |  $\lvert 1\rangle$
$\lvert 0\rangle$ |  $\lvert 0\rangle$ |  $\lvert 1\rangle$ |  $\lvert 0\rangle$ |  $\lvert 0\rangle$ |  $\lvert 0\rangle$
$\lvert 0\rangle$ |  $\lvert 1\rangle$ |  $\lvert 0\rangle$ |  $\lvert 0\rangle$ |  $\lvert 0\rangle$ |  $\lvert 1\rangle$
$\lvert 0\rangle$ |  $\lvert 1\rangle$ |  $\lvert 1\rangle$ |  $\lvert 0\rangle$ |  $\lvert 1\rangle$ |  $\lvert 0\rangle$
$\lvert 1\rangle$ |  $\lvert 0\rangle$ |  $\lvert 0\rangle$ |  $\lvert 0\rangle$ |  $\lvert 1\rangle$ |  $\lvert 1\rangle$
$\lvert 1\rangle$ |  $\lvert 0\rangle$ |  $\lvert 1\rangle$ |  $\lvert 1\rangle$ |  $\lvert 0\rangle$ |  $\lvert 0\rangle$
$\lvert 1\rangle$ |  $\lvert 1\rangle$ |  $\lvert 0\rangle$ |  $\lvert 1\rangle$ |  $\lvert 0\rangle$ |  $\lvert 1\rangle$
$\lvert 1\rangle$ |  $\lvert 1\rangle$ |  $\lvert 1\rangle$ |  $\lvert 1\rangle$ |  $\lvert 1\rangle$ |  $\lvert 0\rangle$
Table 1: Permutation Operation on 3 qubit states

First of all, the $q_{0}$ is inverted under the operation of $P$, i.e., if it
is initially in the state $\lvert 0\rangle$, it gets mapped to $\lvert
1\rangle$ and vice versa. Now, $q_{1}$ is inverted only when $q_{0}$ is in
$\lvert 0\rangle$ initially. Moreover, $q_{2}$ is inverted only when both
$q_{1}$ and $q_{0}$ are initially in $\lvert 0\rangle$. Both of these cases
are highlighted in Table 1. Furthermore, this operation can be extended for an
arbitrary number of qubit states. In Fig. 3, we present a representative
quantum circuit that can carry out the permutation operation by applying a NOT
gate to the first qubit and successive controlled-NOT gates on the respective
qubits. Here, the NOT gate acting on the first qubit $q_{0}$ inverts the first
qubit. After that, the $q_{1}$ will be inverted due to the activation of the
CNOT gate. Finally, when $q_{0}$ and $q_{1}$ are initially in $\lvert
0\rangle$ state, the $q_{2}$ is inverted by the CCNOT gate. Accordingly, when
one has more qubits, the addition of consecutive multi-controlled NOT gates
will carry out the permutation operation $P$ on the corresponding states. We
can apply the gates in reverse to implement $P^{-1}$ since they implement
unitary operations. The centered discrete Fourier transform can be implemented
by sandwiching the QFT circuit between $N/2$ repetitions of the permutation
circuit and its inverse.

Figure 3: Permutation circuit for 3 qubits

####  III.4.3 Implementation of a single Trotter step

$U_{YP_{X}}$$U_{XP_{Y}}$$U_{P_{X}}$$U_{P_{Y}}$ Figure 4: A schematic diagram
of a single Trotter step for 4 qubits.

In Fig. 4, we illustrate the implementation of a single trotter step given in
Eq. (19) on four qubits which consists of unitary diagonal matrices as defined
in section III.3 and $F$ being the centered discrete Fourier transform matrix.
Also, an ancilla qubit is required to implement those diagonal unitary
matrices via filter circuits which is made implicit in Fig. 4. Therefore, the
time evolution of an initial multi-qubit state, say $|i\rangle$ over time $t$,
denoted by $|f\rangle$ can be determined by applying the above-mentioned
single Trotter step $n$ successive times, each for $\delta t=t/n$, so that one
obtains $|f\rangle=\left[U(\delta t)\right]^{n}|i\rangle$.

##  IV Discussion

We carry out the quantum simulation of the two-dimensional isotropic harmonic
oscillator in zero magnetic field with spatial commutativity following the
algorithm stated in section III and discern the effect of the noncommutativity
parameter $\theta$ on the simulation itself. The associated Hamiltonian is,

$H=\frac{1}{2M}\left(P_{X}^{2}+P_{Y}^{2}\right)+\frac{1}{2}M\Omega^{2}\left(X^{2}+Y^{2}\right)-l\left(X\,P_{Y}-Y\,P_{X}\right),$
(24)

where, the parameters $M$, $\Omega$ and $l$ in terms of parameters of Eq. (9)
in the isotropic limit, $\omega_{1}=\omega_{2}=\omega$ and setting111In this
work we adopt $\hbar=1$, $c=1$ and the mass is expressed in an arbitrary unit.
Also, the length $x$ and time $t$ are both considered in the inverse of the
mass unit. Besides, $m\omega\theta$ is dimensionless in this unit system.
$\hbar=1$, $B=0$, $r=0$ and $s=1/2$ in Eq. (11), are given as

$M=\frac{m}{1+\left(\frac{m\theta\omega}{2}\right)^{2}}\,,\,\,\,\Omega=\omega\sqrt{1+\left(\frac{m\theta\omega}{2}\right)^{2}},\,\,\,l=\frac{m\theta\omega^{2}}{2}.$
(25)

We take the discrete values of $x,\ y\in[-16,...,15]$ (each multiplied with
$\sqrt{2\pi/N}$ with $N=32$) for which we require 10 qubits to express a
position state $|x,y\rangle$ in terms of computational basis. Now using this
Hamiltonian, we implement the time evolution of the quantum state of the two-
dimensional harmonic oscillator, initially taken at the origin, $|0,0\rangle$
at $t=0$, and determine its probability distribution over the two-dimensional
$32\times 32$ mesh at $t=0.2$ for two values of $\theta=0,1$ as shown in Fig.
5.

$\theta=0$

$\theta=1$

$t=0$

$t=0.1$

$t=0.2$

Figure 5: Probability distribution at $t=0$, 0.1 and 0.2 for $\theta=0$ and
$\theta=1$ generated with 8192 shots. The parameter values are $N=32$,
$m=0.5$, $\hbar$=1, $\omega$=1, $\delta t=0.02$

As it is challenging to disentangle the effect of $\theta$ on the simulated
time evolution just from Fig. 5, first, by fixing the initial and final
states, we calculate the overlapping between them directly and compare it with
the result obtained from the measurement in the simulation which we present in
Fig. 6 and 7.

(a)

(b)

(c)

(d)

Figure 6: Variation of the overlapping between two position states, $|\langle
x^{\prime},y^{\prime}\rvert U(t)\lvert x,y\rangle|^{2}$ as a function of time
$t$ for noncommutativity parameter, $\theta=0$ and $1$. Here, we consider
$N=32$, $m=0.5$, $\omega=1$ and the single Trotter time-step, $\delta t=0.02$.
We see that the theoretical values of the overlapping between different
position states given by the normalized truncated propagator $|K_{tn}|^{2}$
and those given by the direct computation $|K_{sim}|^{2}$ and the measurement
(taking 8192 shots) $|K^{m}_{sim}|^{2}$ after carrying out the simulation over
time $t=0.5$ are in good agreement. Additionally, the maximum total energy
eigenvalues $n^{\mathrm{max}}_{tot}$ considered in the normalized truncated
propagators in figs. (a), (b), (c) and (d) are 72, 78, 50 and 52,
respectively. Besides, all of the insets present the zoomed variations of the
overlapping for the larger value of time,
$t\stackrel{{\scriptstyle>}}{{{}_{\sim}}}0.1$.

Also, regarding the theoretical values of the time variation of the
overlapping between different position states of the two-dimensional isotropic
oscillator, which are included in Fig. 6 and 7, we would like to point out a
few subtle issues. First of all, the position and momentum operators of the
Hamiltonian in Eq. (10) for the isotropic case are unbounded Hermitian
operators acting on the infinite-dimensional Hilbert space of square-
integrable functions, $L^{2}(\mathbb{R}^{2},dxdy)$ whereas the implementation
of the position and momenta operators in the quantum simulations are finite
dimensional matrices. As a consequence, the commutations relations between
these operators are not equal to some c-numbers. Furthermore, as shown in
[180, 181, 182], the finite-dimensional quantum mechanical systems are
intrinsically different from the ordinary quantum mechanical systems with
truncated Hilbert space. For example, the uniformly spaced eigenspectrum of
the ordinary quantum harmonic oscillator does not hold for the finite-
dimensional case. However, by considering the Hamiltonian as a difference
operator (rather than differential operators), one can construct energy
eigenfunctions of this discrete harmonic oscillator which are only defined on
discrete equidistant points of space and admit the uniformly spaced
eigenspectrum (see for example, [183, 184, 185]). But such modification of the
Hamiltonian is structurally different than the usual Hamiltonian of the
harmonic oscillator, and the eigenfunctions also depend on additional
parameters and the lattice configuration, and therefore it is not
straightforward to adopt the eigenspectrum for different lattice structures.
Because of such technicalities, we avoid such construction for finite-
dimensional quantum systems. Besides, the group theoretical formalism, which
allowed us to map the noncommutative two-dimensional quantum mechanical system
into the ordinary quantum mechanical system, presented in section II.1, cannot
be adopted for the finite-dimensional case. Furthermore, the exact propagator
for the usual two-dimensional noncommutative isotropic oscillator (see
appendix) significantly deviates from the simulation results that encompass
the finite-dimensional system. Therefore, when comparing the results of the
simulation with theory, we first calculate the overlapping between different
position states by truncating the energy eigenvalues to a maximum value in the
corresponding propagator for each case. Then we normalize it in such a way
that at $t=0$, the overlapping between two same position states for the
truncated case matches with the finite-dimensional case, which is given by
$\langle f|i\rangle=\delta_{if}$. In Fig. 6 and 7, we denote the theoretical
values of the overlapping from normalized truncated propagator as
$|K_{tn}|^{2}$. Also, after carrying out the simulation over a certain time,
the overlapping between two different position states calculated directly and
via measurements are denoted as $|K_{sim}|^{2}$ and $|K^{m}_{sim}|^{2}$,
respectively. Here, the quantity $|K_{sim}|^{2}$ is obtained by calculating
directly the inner product between the position eigenstates in Qiskit, whereas
$|K_{sim}^{m}|^{2}$ is obtained by the counts of the considered position
eigenstate appeared in the measurement of the final state with 8192 shots.
From Fig. 6, we can see that the presence of non-zero $\theta$ results in a
slightly faster variation of the overlapping between different position states
with time compared to the case of $\theta=0$ in the simulation, which is
corroborated by the theoretical predictions, because the larger the $\theta$,
the larger the frequency $\Omega$ becomes as stated in Eq. (25).

(a)

(b)

(c)

(d)

Figure 7: Variation of $|\langle x^{\prime},y^{\prime}\rvert U(t)\lvert
x,y\rangle|^{2}$ as a function of $\theta$ at time $t=0.1$.

In addition, in Fig. 6 and 7, we see fluctuations in the overlapping between
position states determined by the measurement in the simulation. In this case,
the number of counts or how many times the quantum state (in general, the
superposition of multiple states) after the simulation collapses onto the
considered final position state determines by the overlapping between these
position states. As the measurement process is inherently random, to mitigate
the associated statistical fluctuation, we can increase the sample size by
taking more shots, which is nothing but the number of times the quantum
circuit is executed before obtaining the counts. It is evident from Fig. 8
that the deviation of the values obtained by measurement from the calculated
values decreases as a higher number of shots is taken.

Figure 8: Comparison of the overlapping $\langle 0,0|U(t)|0,0\rangle|^{2}$
among the directly calculated value $|K_{sim}|^{2}$ and the values obtained by
measurement $|K_{sim}^{m}|^{2}$ with 8192 shots and $10^{5}$ shots,
respectively, as a function of $\theta$.

Furthermore, let us elucidate on the Trotter error associated with the non-
zero $\theta$. For the Hamiltonian $H=\sum_{i=1}^{\Gamma}H_{i}$ which can be
decomposed into $\Gamma$ terms, the upper bound on the error coming from the
approximation of the time evolution operator $U(t)=e^{-iHt}$ with first-order
Trotter-Suzuki product formula
$U^{(n)}(t)=\left(e^{-iH_{\Gamma}t/n}...e^{-iH_{1}t/n}\right)^{n}$, is given
by

$\left\|U(t)-U^{(n)}(t)\right\|\leq\frac{t^{2}}{n}\left\|\sum_{i<j=1}^{\Gamma}[H_{i},H_{j}]\right\|\;,$
(26)

here, $\|.\|$ is the matrix norm222Here, we consider Frobenius norm of a
matrix $A$ which is given as $\|A\|=\sqrt{\mathrm{Tr}(A^{T}.A)}$. and $n$ is
the Trotter step. As we have simulated the time evolution of the two-
dimensional isotropic harmonic oscillator with Hamiltonian Eq. (24)
(expressing it as $H_{X}=H_{1}+H_{2}$, $H_{Y}=H_{3}+H_{4}$ and
$H_{XY}=H_{5}+H_{6}$) using this first-order product formula, the relevant
terms associated with the Trotter error (for a detailed account, see Ref.
[186]) are,

$\displaystyle\|[H_{1},H_{2}]\|$
$\displaystyle\sim\|[H_{3},H_{4}]\|\sim\Omega^{2}\sim
m^{2}\omega^{4}\theta^{2}+\omega^{2},$ $\displaystyle\|[H_{1},H_{5}]\|$
$\displaystyle\sim\|[H_{3},H_{6}]\|\sim l/M\sim m^{2}\omega^{3}\theta^{4},$
$\displaystyle\|[H_{2},H_{6}]\|$ $\displaystyle\sim\|[H_{4},H_{5}]\|\sim
lM\Omega^{2}\sim m^{2}\omega^{3}\theta^{2},$ $\displaystyle\|[H_{5},H_{6}]\|$
$\displaystyle\sim l^{2}\sim m^{2}\omega^{2}\theta^{4},$ (27)

where the rightmost terms are the approximations in the large $\theta$ limit.
Thus we can see that the angular momentum term $H_{XY}=H_{5}+H_{6}$ in the
Hamiltonian induced by the non-zero $\theta$ contributes to the Trotter error
at most at the order $O(\theta^{4})$. Consequently, for larger time $t$, the
increasing value of noncommutativity parameter $\theta$ will increase the
Trotter error and degrade the quality of the quantum simulation itself. This
behavior is demonstrated in Fig. 9. As a consequence, for a reliable quantum
simulation of a quantum mechanical system with spatial noncommutativity, one
has to maintain the value of $\theta$ small so that roughly
$c\,\theta/t^{2}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}1$ where $c$ can be a
dimensionless combination of other parameters of the quantum system itself.

Figure 9: Evaluation of the overlapping, $|\langle 0,0|U(t)|0,0\rangle|^{2}$
as a function of time for three values of $\theta$, where we can see that at
larger time $t\stackrel{{\scriptstyle>}}{{{}_{\sim}}}0.5$, the value of the
overlapping for $\theta=3$ has started deviating from the cases for $\theta=0$
and 1, and it signals the degradation of the quality of the quantum simulation
itself. Here, the values of the overlapping were directly obtained by
calculating inner products between position eigenstates in Qiskit.

Furthermore, the noncommutativity parameter $\theta$ introduces the term
$H_{XY}$ in the Hamiltonian of the two-dimensional simple harmonic oscillator
that results in the non-separability of the wavefunction in $x$ and $y$
dimensions. Therefore, the subspaces spanned by the qubits dedicated to the
$x$ and $y$ dimensions experience a bipartite entanglement between them under
the time evolution which is presented in Fig. 10. Here, the bipartite
entanglement entropy is calculated as
$S=\mathrm{Tr}[\rho_{X}\mathrm{log}\rho_{X}]=\mathrm{Tr}[\rho_{Y}\mathrm{log}\rho_{Y}]$
where $\rho_{X,Y}=\mathrm{Tr}_{Y,X}(|\psi\rangle\langle\psi|)$ (where,
$|\psi\rangle$ is a general position state) are the reduced density matrices
associated with the subspaces of qubits dedicated for $x$ and $y$ dimensions,
respectively. We started from $|0,0\rangle$ position state and as expected,
for $\theta=0$ we see no entanglement entropy and for $\theta\neq 0$ its
gradual increase with time. Moreover, for larger values of $\theta$, larger
entanglement entropy is generated with time.

Figure 10: Bipartite entanglement entropy S as a function of time t for
noncommutativity parameter, $\theta=0,0.5$ and $1$. We start with the initial
state $\lvert x=0,y=0\rangle$ and calculate the bipartite entanglement entropy
between the qubits dedicated to x-axis and y-axis at each trotter step using
the built-in functions of Qiskit [177].

###  IV.1 Noisy Simulation

In this section, we analyze the effects of various types of noise, for
example, the readout error, the depolarizing error, and the thermal relaxation
error on the simulation following [187]. We take $N=8$ to work with fewer
qubits so that the circuit depth is small enough to discern the effects of
different types of noise, as a higher number of qubits degrade the simulation
quality quickly. We compare the overlapping $|\langle 0,0\rvert U(t)\lvert
0,0\rangle|^{2}$ as a function of time for the ideal case and noisy
simulation.

Readout Error

(a)

Depolarizing Error

(b)

Depolarizing Error

(c)

Thermal Relaxation Error

(d)

Figure 11: Effect of various types of error on the simulation. Here, the
overlapping $|\langle 0,0\rvert U(t)\lvert 0,0\rangle|^{2}$ is plotted for the
ideal case and the noisy simulation for $N=8$ qubits. Besides, $m=0.5$,
$\omega=1$, $\theta=1$ and $\delta t=0.02$. (a) Readout error where p is the
probability of correct output. (b) Depolarizing error introduced only on the
single-qubit gates for the depolarization parameter $\lambda=0.001$ and
$0.01$. (c) Comparison of the depolarizing error of single-qubit gates and
multi-qubit gates for $\lambda=0.001$. (d) Comparison of the thermal
relaxation error of single-qubit gates and CNOT gates for $T_{1}=50\,\mu s$,
$T_{2}=100\,\mu s$, $T_{sq}=30\,ns$, $T_{CNOT}=300\,ns$.

Firstly, we add a readout error to each qubit with the probability of correct
output being $p$. The effect of this error is to reduce the value of the
overlapping without altering the shape of the overlapping vs. time curve since
measuring each qubit returns the incorrect value with a probability $1-p$,
thus reducing the amplitude of the correct state without affecting the
dynamics, as seen in Fig. 11 (a).

Next, we study the effect of the depolarizing error. Depolarization occurs due
to bit-flip and phase-flip errors, leading to a completely mixed state. We
compare the effect of depolarization on the single-qubit gates for the values
$\lambda=0.001$ and $0.01$ where even with a slight deviation from the ideal
case, $\lambda=0$, the simulation deteriorates drastically, as shown in Fig.
11 (b). Besides, for $\lambda=0.001$, we compare the depolarization error of
single-qubit and multi-qubit gates, and we find that the error is more
significant for the multi-qubit gates compared to the single-qubit gates, as
seen in Fig. 11(c).

Lastly, we analyze the effect of thermal relaxation error on our simulation.
Thermal relaxation error is characterized by two constants, $T_{1}$ and
$T_{2}$, where $T_{1}$ refers to the time a qubit decays from an excited state
to the ground state, and $T_{2}$ is dephasing time characterizing the loss of
coherence of a quantum state. These constants limit the number of operations
performed on a qubit. We take $T_{1}$ and $T_{2}$ from a Gaussian distribution
with mean $50\,\mu s$ and $100\,\mu s$, respectively with a standard deviation
of $10\,\mu s$. We take the single-qubit execution time to be $T_{sq}=30\,ns$
and CNOT gate execution time $T_{CNOT}$ to be $300\,ns$. These values are
approximated from the values obtained from current backends offered by IBM.
Here, we only compare the thermal relaxation error of the single-qubit gates
and the CNOT gates. Besides, the multi-qubit gates can be transpiled into
single-qubit and CNOT gates. As seen in Fig. 11 (d), the thermal relaxation
error on the CNOT gates is much more prominent than the single-qubit case due
to many CNOT gates and a ten times larger execution time. Therefore, we
conclude that the circuit depth corresponding to our case of $N=32$ qubits is
too large to execute a noisy simulation in a NISQ device.

##  V Conclusion and Outlook

In this work, we carry out a noiseless quantum simulation of the two-
dimensional isotropic quantum harmonic oscillator with spatial
noncommutativity, which is a representative quantum system of NCQM in a
quantum computer setup provided by Qiskit. First, we use the group theoretical
method to map the Hamiltonian of the two-dimensional NCQM into the ordinary
quantum mechanical Hamiltonian. Afterward, we discretize the two-dimensional
space into $32\times 32$ mesh where $x,y\in[-16,...,15]$ and set up the
corresponding computational basis with $n=2\,\mathrm{log}_{2}32=10$ qubits for
the $2D$ discrete points. Next, we construct the position operator as a
diagonal matrix for each dimension and the momentum operator as the centered
discrete Fourier transform of the position matrix, respectively, both acting
on the computational basis. Now equipped with the Hamiltonian, we assemble the
quantum gate that implements the time evolution using the Trotter-Suzuki
formula. One can easily extend this framework for higher $d$-dimensional
quantum systems that have translational symmetry with a set of
$p=d\,\mathrm{log}_{2}N$ qubits since the position and momentum operators are
identical for each dimension.

As the noncommutativity parameter $\theta$ increases the overall frequency of
the harmonic oscillator, we see the discerning shift in the time variation of
the overlapping between different position eigenstates in our simulation,
presented in Fig. 6 for a smaller value of time compared to the case of
$\theta=0$. In addition, for a fixed time, the variation of the overlapping
between different position eigenstates with respect to $\theta$ is also
observed in our simulation, as shown in Fig. 7. However, the discretization of
space required to implement our quantum simulation leads to the
incompatibility of position and momentum operators acting on the finite-
dimensional Hilbert space with the usual unbounded self-adjoint position and
momentum operators of the ordinary quantum mechanical system working on the
infinite-dimensional Hilbert space or even its truncated version. As the group
theoretical formalism to map the NCQM to ordinary quantum mechanics is not
available yet for a finite-dimensional quantum mechanical system, we use the
truncated propagator to calculate the overlapping between different position
states and, as a result, acquire some error which is seen in both Fig. 6 and 7
as the small deviation between theoretical prediction and the values obtained
from the simulation. The effects of discretization can be minimized by smaller
lattice spacing, but for the computational basis, this translates into taking
a large number of qubits.

Moreover, the main obstacle to executing the circuit in real devices is its
higher circuit depth due to a large number of trotter steps. Even noisy
simulations using the noise models produce highly erroneous results, from
which we conclude that the simulation is not suitable for the current NISQ
devices. Hence, we opted for the noiseless simulation in the qasm simulator of
Qiskit. Likewise, we consider 8192 shots to produce the probability
distribution of position states with time in our simulation that can be
extended up to $10^{5}$ in Qiskit currently, but the higher number of shots,
though reducing the statistical error, increases the execution time. Moreover,
for $N=32$ qubits representing a single two-dimensional oscillator on the
noncommutative space, the algorithm we followed requires $O(10^{5})$ gates for
a single Trotter step. So if we have $n$ time steps, the number of gates scale
as $O(10^{5}n)$. Also, if we extend our scenario of a single oscillator to $L$
number of oscillators, the resource would scale as $O(10^{5}nL)$ if we neglect
the coupling between the oscillators for simplicity.

Besides, the simulation is implemented using the first-order Trotter-Suzuki
product formula for which the presence of spatial noncommutativity parameter
induces additional large errors when its value is taken larger as seen from
the estimates given in Eq. (27) and Fig. 9. Hence, the Hamiltonian simulation
via this method is limited to a range of optimal values of time and
parameters. Nevertheless, there are algorithms for Hamiltonian simulation
based on continuous and fractional queries [188], Taylor series [189], and
quantum walk [190]. Quantum walk-based algorithms have query complexities that
are better scaled than product formulas with the Hamiltonian’s time of
evolution and sparsity, whereas the fractional query model simulation has
better scaling with the allowed error in the Hamiltonian. Hamiltonian
simulation combining these two approaches to obtain optimal dependence on
sparsity, error, and evolution time is given in [191]. It will be interesting
to carry out the Hamiltonian simulation of quantum systems with additional
noncommutativity parameters using the above-mentioned algorithms and find out
the impact of these parameters on the simulation.

## Acknowledgement

TAC would like to thank the High Energy Theory group of the Department of
Physics and Astronomy at the University of Kansas for the hospitality and
support. The work of S.N is supported by the United Arab Emirates University
under UPAR Grant No. 12S093. We acknowledge the use of IBM Quantum services
for this work. The views expressed are those of the authors and do not reflect
the official policy or position of IBM or the IBM Quantum team.

##  Appendix A The isotropic two-dimensional harmonic oscillator with spatial
noncommutativity

In the isotropic limit, the Hamiltonian of the two-dimensional harmonic
oscillator containing noncommutativity parameter $\theta$, given in Eq. (10)
has $M_{1}=M_{2}=M$, $\Omega_{1}=\Omega_{2}=\Omega$ and $l_{1}=l_{2}=l$, and
it is separable in the polar coordinates ($x=r\cos\phi,\,y=r\sin\phi$) because
of the $\theta$ induced angular momentum operator,
$\hat{L}_{z}=\hat{x}\hat{p}_{y}-\hat{y}\hat{p}_{x}$. The energy eigenfunctions
can be readily found which are given by,

$\psi_{n_{r},m_{l}}(r,\phi)=\sqrt{\frac{M\Omega}{\pi\hbar}}\sqrt{\frac{n_{r}!}{(n_{r}+|m_{l}|)!}}\left(\frac{M\Omega}{\hbar}\right)^{\frac{|m_{l}|}{2}}r^{|m_{l}|}e^{-\frac{M\Omega}{2\hbar}r^{2}}L_{n_{r}}^{|m_{l}|}\left(\frac{M\Omega}{\hbar}r^{2}\right)e^{im_{l}\phi},$
(28)

with the energy eigenvalues

$E_{n_{r},m_{l}}=\hbar\Omega(2n_{r}+|m_{l}|+1)+m_{l}\,l\hbar,$ (29)

where $n_{r}=0,1,2,...$ and $m_{l}=0,\pm 1,\pm 2,...$ and the total energy
eigenvalue is $n_{tot}=2n_{r}+|m_{l}|$.

Now we identify the position eigenstate
$|x,y\rangle=|r\cos\phi,r\sin\phi\rangle\equiv|r,\phi\rangle$. The exact
propagator $\langle r^{\prime},\phi^{\prime}|e^{-iHt/\hbar}|r,\phi\rangle$
denoted by $K(r^{\prime},\phi^{\prime},r,\phi,t)\equiv
K(x^{\prime},y^{\prime},x,y,t)$ is also easily found to be,

$K(r^{\prime},\phi^{\prime},r,\phi,t)=\frac{M\Omega}{2\pi i\hbar sin(\Omega
t)}exp\left(-\frac{M\Omega}{2i\hbar}\frac{cos(\Omega t)}{sin(\Omega
t)}(r^{\prime
2}+r^{2})\right)exp\left(-i\frac{M\Omega}{\hbar}\frac{1}{sin(\Omega
t)}rr^{\prime}cos(\phi^{\prime}-\phi+lt)\right).$ (30)

As the exact propagator is inadequate to compare with the values obtained in
the quantum simulation, we consider the truncated propagator up to a maximum
value of total energy eigenvalue, $n^{\mathrm{max}}_{tot}$.

$K_{t}(x^{\prime},y^{\prime},x,y,t)\equiv
K_{t}(r^{\prime},\phi^{\prime},r,\phi,t)=\sum_{n_{r}=0}^{n_{r}^{max}}\sum_{m_{l}=-m_{l}^{max}}^{m_{l}^{max}}e^{-iE_{n_{r},m_{l}}t/\hbar}\psi^{*}_{n_{r},m_{l}}(r^{\prime},\phi^{\prime})\psi_{n_{r},m_{l}}(r,\phi).$
(31)

## References

  * [1] P. Benioff, “The computer as a physical system: A microscopic quantum mechanical hamiltonian model of computers as represented by turing machines,” Journal of statistical physics 22 no. 5, (1980) 563–591.
  * [2] Y. Manin, Computable and Uncomputable (in Russian).  Sovetskoye Radio, Moscow, 1980.
  * [3] R. P. Feynman, “Simulating Physics with Computers,” International Journal of Theoretical Physics 21 no. 6-7, (June, 1982) 467–488.
  * [4] J. I. Cirac and P. Zoller, “Goals and opportunities in quantum simulation,” Nature physics 8 no. 4, (2012) 264–266.
  * [5] I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum Simulation,” Rev. Mod. Phys. 86 (2014) 153, arXiv:1308.6253 [quant-ph].
  * [6] A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, “Practical quantum advantage in quantum simulation,” Nature 607 no. 7920, (2022) 667–676.
  * [7] C. W. Bauer et al., “Quantum Simulation for High Energy Physics,” arXiv:2204.03381 [quant-ph].
  * [8] T. S. Humble et al., “Snowmass White Paper: Quantum Computing Systems and Software for High-energy Physics Research,” in 2022 Snowmass Summer Study.  3, 2022.  arXiv:2203.07091 [quant-ph].
  * [9] S. Catterall et al., “Report of the Snowmass 2021 Theory Frontier Topical Group on Quantum Information Science,” in 2022 Snowmass Summer Study.  9, 2022.  arXiv:2209.14839 [quant-ph].
  * [10] M. S. Alam et al., “Quantum computing hardware for HEP algorithms and sensing,” in 2022 Snowmass Summer Study.  4, 2022.  arXiv:2204.08605 [quant-ph].
  * [11] P. Spentzouris, “Quantum Computing: Advancing Fundamental Physics,” Comput. Softw. Big Sci. 4 no. 1, (2020) 11.
  * [12] S. Lloyd, “Universal quantum simulators,” Science 273 no. 5278, (1996) 1073–1078.
  * [13] S. Wiesner, “Simulations of Many-Body Quantum Systems by a Quantum Computer,” arXiv e-prints (1996) quant–ph/9603028, arXiv:quant-ph/9603028 [quant-ph].
  * [14] D. S. Abrams and S. Lloyd, “Simulation of Many-Body Fermi Systems on a Universal Quantum Computer,” Phys. Rev. Lett.  79 no. 13, (1997) 2586–2589, arXiv:quant-ph/9703054 [quant-ph].
  * [15] S. Somaroo, C. H. Tseng, T. F. Havel, R. Laflamme, and D. G. Cory, “Quantum Simulations on a Quantum Computer,” Phys. Rev. Lett.  82 no. 26, (1999) 5381–5384, arXiv:quant-ph/9905045 [quant-ph].
  * [16] C. Zalka, “Simulating quantum systems on a quantum computer,” Proceedings of the Royal Society of London Series A 454 no. 1969, (1998) 313, arXiv:quant-ph/9603026 [quant-ph].
  * [17] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, “Quantum Computation by Adiabatic Evolution,” arXiv e-prints (2000) quant–ph/0001106, arXiv:quant-ph/0001106 [quant-ph].
  * [18] G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, “Quantum algorithms for fermionic simulations,” Phys. Rev. A 64 no. 2, (2001) 022319, arXiv:cond-mat/0012334 [cond-mat].
  * [19] R. Somma, G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, “Simulating physical phenomena by quantum networks,” Phys. Rev. A 65 no. 4, (2002) 042323, arXiv:quant-ph/0108146 [quant-ph].
  * [20] R. D. Somma, G. Ortiz, E. H. Knill, and J. Gubernatis, “Quantum simulations of physics problems,” in Quantum Information and Computation, E. Donkor, A. R. Pirich, and H. E. Brandt, eds., vol. 5105, pp. 96 – 103, International Society for Optics and Photonics.  SPIE, 2003.  https://doi.org/10.1117/12.487249.
  * [21] D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders, “Efficient Quantum Algorithms for Simulating Sparse Hamiltonians,” Communications in Mathematical Physics 270 no. 2, (2007) 359–371, arXiv:quant-ph/0508139 [quant-ph].
  * [22] A. M. Childs and R. Kothari, “Simulating Sparse Hamiltonians with Star Decompositions,” in Lecture Notes in Computer Science, vol. 6519, p. 94.  2011\.
  * [23] J. I. Cirac and P. Zoller, “Quantum Computations with Cold Trapped Ions,” Phys. Rev. Lett. 74 (1995) 4091–4094.
  * [24] D. F. V. James, “Quantum dynamics of cold trapped ions with application to quantum computation,” Applied Physics B: Lasers and Optics 66 no. 2, (1998) 181–190, arXiv:quant-ph/9702053 [quant-ph].
  * [25] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett.  81 no. 15, (1998) 3108–3111, arXiv:cond-mat/9805329 [cond-mat.stat-mech].
  * [26] I. H. Deutsch, G. K. Brennen, and P. S. Jessen, “Quantum Computing with Neutral Atoms in an Optical Lattice,” Fortschritte der Physik 48 no. 9-11, (2000) 925–943, arXiv:quant-ph/0003022 [quant-ph].
  * [27] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Advances in Physics 56 no. 2, (2007) 243–379, arXiv:cond-mat/0606771 [cond-mat.other].
  * [28] I. Buluta and F. Nori, “Quantum simulators,” Science 326 no. 5949, (2009) 108–111.
  * [29] J. Casanova, L. Lamata, I. L. Egusquiza, R. Gerritsma, C. F. Roos, J. J. García-Ripoll, and E. Solano, “Quantum Simulation of Quantum Field Theories in Trapped Ions,” Phys. Rev. Lett.  107 no. 26, (2011) 260501, arXiv:1107.5233 [quant-ph].
  * [30] B. P. Lanyon, C. Hempel, D. Nigg, M. Müller, R. Gerritsma, F. Zähringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, M. Hennrich, P. Zoller, R. Blatt, and C. F. Roos, “Universal Digital Quantum Simulation with Trapped Ions,” Science 334 no. 6052, (Oct., 2011) 57, arXiv:1109.1512 [quant-ph].
  * [31] I. Bloch, J. Dalibard, and S. Nascimbene, “Quantum simulations with ultracold quantum gases,” Nature Physics 8 no. 4, (2012) 267–276.
  * [32] R. Blatt and C. F. Roos, “Quantum simulations with trapped ions,” Nature Physics 8 no. 4, (2012) 277–284.
  * [33] U.-J. Wiese, “Ultracold Quantum Gases and Lattice Systems: Quantum Simulation of Lattice Gauge Theories,” Annalen Phys. 525 (2013) 777–796, arXiv:1305.1602 [quant-ph].
  * [34] E. Zohar, J. I. Cirac, and B. Reznik, “Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices,” Reports on Progress in Physics 79 no. 1, (2016) 014401, arXiv:1503.02312 [quant-ph].
  * [35] M. C. Bañuls, R. Blatt, J. Catani, A. Celi, J. I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C. A. Muschik, B. Reznik, E. Rico, L. Tagliacozzo, K. Van Acoleyen, F. Verstraete, U.-J. Wiese, M. Wingate, J. Zakrzewski, and P. Zoller, “Simulating lattice gauge theories within quantum technologies,” European Physical Journal D 74 no. 8, (2020) 165, arXiv:1911.00003 [quant-ph].
  * [36] C. Monroe, W. C. Campbell, L. M. Duan, Z. X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, “Programmable quantum simulations of spin systems with trapped ions,” Reviews of Modern Physics 93 no. 2, (2021) 025001, arXiv:1912.07845 [quant-ph].
  * [37] F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, “Tools for quantum simulation with ultracold atoms in optical lattices,” Nature Reviews Physics 2 no. 8, (2020) 411–425, arXiv:2006.06120 [cond-mat.quant-gas].
  * [38] M. Aidelsburger, L. Barbiero, A. Bermudez, T. Chanda, A. Dauphin, D. González-Cuadra, P. R. Grzybowski, S. Hands, F. Jendrzejewski, J. Jünemann, G. Juzeliūnas, V. Kasper, A. Piga, S.-J. Ran, M. Rizzi, G. Sierra, L. Tagliacozzo, E. Tirrito, T. V. Zache, J. Zakrzewski, E. Zohar, and M. Lewenstein, “Cold atoms meet lattice gauge theory,” Philosophical Transactions of the Royal Society of London Series A 380 no. 2216, (Feb., 2022) 20210064, arXiv:2106.03063 [cond-mat.quant-gas].
  * [39] S. P. Jordan, K. S. M. Lee, and J. Preskill, “Quantum Algorithms for Quantum Field Theories,” Science 336 no. 6085, (2012) 1130, arXiv:1111.3633 [quant-ph].
  * [40] S. P. Jordan, K. S. M. Lee, and J. Preskill, “Quantum Computation of Scattering in Scalar Quantum Field Theories,” arXiv e-prints (2011) arXiv:1112.4833, arXiv:1112.4833 [hep-th].
  * [41] S. P. Jordan, K. S. M. Lee, and J. Preskill, “Quantum Algorithms for Fermionic Quantum Field Theories,” arXiv e-prints (Apr., 2014) arXiv:1404.7115, arXiv:1404.7115 [hep-th].
  * [42] S. P. Jordan, H. Krovi, K. S. M. Lee, and J. Preskill, “BQP-completeness of Scattering in Scalar Quantum Field Theory,” Quantum 2 (2018) 44, arXiv:1703.00454 [quant-ph].
  * [43] J. Preskill, “Simulating quantum field theory with a quantum computer,” PoS LATTICE2018 (2018) 024, arXiv:1811.10085 [hep-lat].
  * [44] A. Hamed Moosavian and S. Jordan, “Faster Quantum Algorithm to simulate Fermionic Quantum Field Theory,” Phys. Rev. A 98 no. 1, (2018) 012332, arXiv:1711.04006 [quant-ph].
  * [45] A. Macridin, P. Spentzouris, J. Amundson, and R. Harnik, “Electron-Phonon Systems on a Universal Quantum Computer,” Phys. Rev. Lett.  121 no. 11, (2018) 110504, arXiv:1802.07347 [quant-ph].
  * [46] A. Macridin, P. Spentzouris, J. Amundson, and R. Harnik, “Digital quantum computation of fermion-boson interacting systems,” Phys. Rev. A 98 no. 4, (2018) 042312, arXiv:1805.09928 [quant-ph].
  * [47] N. Klco and M. J. Savage, “Digitization of scalar fields for quantum computing,” Phys. Rev. A 99 no. 5, (2019) 052335, arXiv:1808.10378 [quant-ph].
  * [48] D. C. Hackett, K. Howe, C. Hughes, W. Jay, E. T. Neil, and J. N. Simone, “Digitizing Gauge Fields: Lattice Monte Carlo Results for Future Quantum Computers,” Phys. Rev. A 99 no. 6, (2019) 062341, arXiv:1811.03629 [quant-ph].
  * [49] K. Yeter-Aydeniz, E. F. Dumitrescu, A. J. McCaskey, R. S. Bennink, R. C. Pooser, and G. Siopsis, “Scalar Quantum Field Theories as a Benchmark for Near-Term Quantum Computers,” Phys. Rev. A 99 no. 3, (2019) 032306, arXiv:1811.12332 [quant-ph].
  * [50] M. Kreshchuk, S. Jia, W. M. Kirby, G. Goldstein, J. P. Vary, and P. J. Love, “Light-Front Field Theory on Current Quantum Computers,” Entropy 23 no. 5, (2021) 597, arXiv:2009.07885 [quant-ph].
  * [51] M. Kreshchuk, W. M. Kirby, G. Goldstein, H. Beauchemin, and P. J. Love, “Quantum simulation of quantum field theory in the light-front formulation,” Phys. Rev. A 105 no. 3, (2022) 032418, arXiv:2002.04016 [quant-ph].
  * [52] 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,” Quantum 5 (2021) 393, arXiv:2006.14160 [quant-ph].
  * [53] J. Liu and Y. Xin, “Quantum simulation of quantum field theories as quantum chemistry,” JHEP 12 (2020) 011, arXiv:2004.13234 [hep-th].
  * [54] T. F. Stetina, A. Ciavarella, X. Li, and N. Wiebe, “Simulating Effective QED on Quantum Computers,” Quantum 6 (2022) 622, arXiv:2101.00111 [quant-ph].
  * [55] N. Klco, A. Roggero, and M. J. Savage, “Standard model physics and the digital quantum revolution: thoughts about the interface,” Rept. Prog. Phys. 85 no. 6, (2022) 064301, arXiv:2107.04769 [quant-ph].
  * [56] T. Byrnes and Y. Yamamoto, “Simulating lattice gauge theories on a quantum computer,” Phys. Rev. A 73 no. 2, (2006) 022328, arXiv:quant-ph/0510027 [quant-ph].
  * [57] C. Muschik, M. Heyl, E. Martinez, T. Monz, P. Schindler, B. Vogell, M. Dalmonte, P. Hauke, R. Blatt, and P. Zoller, “U(1) Wilson lattice gauge theories in digital quantum simulators,” New J. Phys. 19 no. 10, (2017) 103020, arXiv:1612.08653 [quant-ph].
  * [58] NuQS Collaboration, H. Lamm, S. Lawrence, and Y. Yamauchi, “General Methods for Digital Quantum Simulation of Gauge Theories,” Phys. Rev. D 100 no. 3, (2019) 034518, arXiv:1903.08807 [hep-lat].
  * [59] NuQS Collaboration, A. Alexandru, P. F. Bedaque, S. Harmalkar, H. Lamm, S. Lawrence, and N. C. Warrington, “Gluon Field Digitization for Quantum Computers,” Phys. Rev. D 100 no. 11, (2019) 114501, arXiv:1906.11213 [hep-lat].
  * [60] NuQS Collaboration, S. Harmalkar, H. Lamm, and S. Lawrence, “Quantum Simulation of Field Theories Without State Preparation,” arXiv:2001.11490 [hep-lat].
  * [61] E. J. Gustafson and H. Lamm, “Toward quantum simulations of $\mathbb{Z}_{2}$ gauge theory without state preparation,” Phys. Rev. D 103 no. 5, (2021) 054507, arXiv:2011.11677 [hep-lat].
  * [62] M. Carena, H. Lamm, Y.-Y. Li, and W. Liu, “Lattice renormalization of quantum simulations,” Phys. Rev. D 104 no. 9, (2021) 094519, arXiv:2107.01166 [hep-lat].
  * [63] M. Carena, H. Lamm, Y.-Y. Li, and W. Liu, “Improved Hamiltonians for Quantum Simulations of Gauge Theories,” Phys. Rev. Lett. 129 no. 5, (2022) 051601, arXiv:2203.02823 [hep-lat].
  * [64] A. Ciavarella, N. Klco, and M. J. Savage, “Some Conceptual Aspects of Operator Design for Quantum Simulations of Non-Abelian Lattice Gauge Theories,”  3, 2022.  arXiv:2203.11988 [quant-ph].
  * [65] J. C. Halimeh and P. Hauke, “Stabilizing Gauge Theories in Quantum Simulators: A Brief Review,”  4, 2022.  arXiv:2204.13709 [cond-mat.quant-gas].
  * [66] NuQS Collaboration, H. Lamm, S. Lawrence, and Y. Yamauchi, “Parton physics on a quantum computer,” Phys. Rev. Res. 2 no. 1, (2020) 013272, arXiv:1908.10439 [hep-lat].
  * [67] QuNu Collaboration, T. Li, X. Guo, W. K. Lai, X. Liu, E. Wang, H. Xing, D.-B. Zhang, and S.-L. Zhu, “Partonic collinear structure by quantum computing,” Phys. Rev. D 105 no. 11, (2022) L111502, arXiv:2106.03865 [hep-ph].
  * [68] A. Pérez-Salinas, J. Cruz-Martinez, A. A. Alhajri, and S. Carrazza, “Determining the proton content with a quantum computer,” Phys. Rev. D 103 no. 3, (2021) 034027, arXiv:2011.13934 [hep-ph].
  * [69] K. Bepari, S. Malik, M. Spannowsky, and S. Williams, “Towards a quantum computing algorithm for helicity amplitudes and parton showers,” Phys. Rev. D 103 no. 7, (2021) 076020, arXiv:2010.00046 [hep-ph].
  * [70] M. G. Echevarria, I. L. Egusquiza, E. Rico, and G. Schnell, “Quantum simulation of light-front parton correlators,” Phys. Rev. D 104 no. 1, (2021) 014512, arXiv:2011.01275 [quant-ph].
  * [71] 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,” Phys. Rev. A 103 no. 6, (2021) 062601, arXiv:2011.13443 [quant-ph].
  * [72] W. M. Kirby, S. Hadi, M. Kreshchuk, and P. J. Love, “Quantum simulation of second-quantized Hamiltonians in compact encoding,” Phys. Rev. A 104 no. 4, (2021) 042607, arXiv:2105.10941 [quant-ph].
  * [73] C. W. Bauer, W. A. de Jong, B. Nachman, and D. Provasoli, “Quantum Algorithm for High Energy Physics Simulations,” Phys. Rev. Lett. 126 no. 6, (2021) 062001, arXiv:1904.03196 [hep-ph].
  * [74] C. W. Bauer, M. Freytsis, and B. Nachman, “Simulating Collider Physics on Quantum Computers Using Effective Field Theories,” Phys. Rev. Lett. 127 no. 21, (2021) 212001, arXiv:2102.05044 [hep-ph].
  * [75] G. Gustafson, S. Prestel, M. Spannowsky, and S. Williams, “Collider Events on a Quantum Computer,” arXiv:2207.10694 [hep-ph].
  * [76] A. Y. Wei, P. Naik, A. W. Harrow, and J. Thaler, “Quantum Algorithms for Jet Clustering,” Phys. Rev. D 101 no. 9, (2020) 094015, arXiv:1908.08949 [hep-ph].
  * [77] D. Pires, Y. Omar, and J. a. Seixas, “Adiabatic Quantum Algorithm for Multijet Clustering in High Energy Physics,” arXiv:2012.14514 [hep-ex].
  * [78] D. Pires, P. Bargassa, J. a. Seixas, and Y. Omar, “A Digital Quantum Algorithm for Jet Clustering in High-Energy Physics,” arXiv:2101.05618 [physics.data-an].
  * [79] J. J. M. de Lejarza, L. Cieri, and G. Rodrigo, “Quantum clustering and jet reconstruction at the LHC,” Phys. Rev. D 106 no. 3, (2022) 036021, arXiv:2204.06496 [hep-ph].
  * [80] A. Delgado and J. Thaler, “Quantum Annealing for Jet Clustering with Thrust,” arXiv:2205.02814 [quant-ph].
  * [81] J. a. Barata and C. A. Salgado, “A quantum strategy to compute the jet quenching parameter $\hat{q}$,” Eur. Phys. J. C 81 no. 10, (2021) 862, arXiv:2104.04661 [hep-ph].
  * [82] J. a. Barata, X. Du, M. Li, W. Qian, and C. A. Salgado, “Medium induced jet broadening in a quantum computer,” arXiv:2208.06750 [hep-ph].
  * [83] X. Yao, “Quantum Simulation of Light-Front QCD for Jet Quenching in Nuclear Environments,” arXiv:2205.07902 [hep-ph].
  * [84] W. A. De Jong, M. Metcalf, J. Mulligan, M. Płoskoń, F. Ringer, and X. Yao, “Quantum simulation of open quantum systems in heavy-ion collisions,” Phys. Rev. D 104 no. 5, (2021) 051501, arXiv:2010.03571 [hep-ph].
  * [85] NuQS Collaboration, T. D. Cohen, H. Lamm, S. Lawrence, and Y. Yamauchi, “Quantum algorithms for transport coefficients in gauge theories,” Phys. Rev. D 104 no. 9, (2021) 094514, arXiv:2104.02024 [hep-lat].
  * [86] A. Delgado et al., “Quantum Computing for Data Analysis in High-Energy Physics,” in 2022 Snowmass Summer Study.  3, 2022.  arXiv:2203.08805 [physics.data-an].
  * [87] C. Tüysüz, F. Carminati, B. Demirköz, D. Dobos, F. Fracas, K. Novotny, K. Potamianos, S. Vallecorsa, and J.-R. Vlimant, “Particle Track Reconstruction with Quantum Algorithms,” EPJ Web Conf. 245 (2020) 09013, arXiv:2003.08126 [quant-ph].
  * [88] J. C. Martens, J. P. Ralston, and J. D. T. Takaki, “Quantum tomography for collider physics: Illustrations with lepton pair production,” Eur. Phys. J. C 78 no. 1, (2018) 5, arXiv:1707.01638 [hep-ph].
  * [89] D. E. Kharzeev, “Quantum information approach to high energy interactions,” Phil. Trans. A. Math. Phys. Eng. Sci. 380 no. 2216, (2021) 20210063, arXiv:2108.08792 [hep-ph].
  * [90] H. M. Gray and K. Terashi, “Quantum Computing Applications in Future Colliders,” Front. in Phys. 10 (2022) 864823.
  * [91] S. Ramírez-Uribe, A. E. Rentería-Olivo, G. Rodrigo, G. F. R. Sborlini, and L. Vale Silva, “Quantum algorithm for Feynman loop integrals,” JHEP 05 (2022) 100, arXiv:2105.08703 [hep-ph].
  * [92] A. M. Czajka, Z.-B. Kang, H. Ma, and F. Zhao, “Quantum simulation of chiral phase transitions,” JHEP 08 (2022) 209, arXiv:2112.03944 [hep-ph].
  * [93] Z. Davoudi, N. Mueller, and C. Powers, “Toward Quantum Computing Phase Diagrams of Gauge Theories with Thermal Pure Quantum States,” arXiv:2208.13112 [hep-lat].
  * [94] J. Preskill, “Quantum computing in the NISQ era and beyond,” Quantum 2 (2018) 79, arXiv:1801.00862 [quant-ph].
  * [95] K. Bharti et al., “Noisy intermediate-scale quantum algorithms,” Rev. Mod. Phys. 94 no. 1, (2022) 015004, arXiv:2101.08448 [quant-ph].
  * [96] J. S. Schwinger, “Gauge Invariance and Mass. 2.,” Phys. Rev. 128 (1962) 2425–2429.
  * [97] J. H. Lowenstein and J. A. Swieca, “Quantum electrodynamics in two-dimensions,” Annals Phys. 68 (1971) 172–195.
  * [98] S. R. Coleman, R. Jackiw, and L. Susskind, “Charge Shielding and Quark Confinement in the Massive Schwinger Model,” Annals Phys. 93 (1975) 267.
  * [99] S. R. Coleman, “More About the Massive Schwinger Model,” Annals Phys. 101 (1976) 239.
  * [100] P. Hauke, D. Marcos, M. Dalmonte, and P. Zoller, “Quantum simulation of a lattice Schwinger model in a chain of trapped ions,” Phys. Rev. X 3 no. 4, (2013) 041018, arXiv:1306.2162 [cond-mat.quant-gas].
  * [101] S. Kühn, J. I. Cirac, and M.-C. Bañuls, “Quantum simulation of the Schwinger model: A study of feasibility,” Phys. Rev. A 90 no. 4, (2014) 042305, arXiv:1407.4995 [quant-ph].
  * [102] E. Ercolessi, P. Facchi, G. Magnifico, S. Pascazio, and F. V. Pepe, “Phase Transitions in $Z_{n}$ Gauge Models: Towards Quantum Simulations of the Schwinger-Weyl QED,” Phys. Rev. D 98 no. 7, (2018) 074503, arXiv:1705.11047 [quant-ph].
  * [103] B. Chakraborty, M. Honda, T. Izubuchi, Y. Kikuchi, and A. Tomiya, “Classically emulated digital quantum simulation of the Schwinger model with a topological term via adiabatic state preparation,” Phys. Rev. D 105 no. 9, (2022) 094503, arXiv:2001.00485 [hep-lat].
  * [104] A. F. Shaw, P. Lougovski, J. R. Stryker, and N. Wiebe, “Quantum Algorithms for Simulating the Lattice Schwinger Model,” Quantum 4 (2020) 306, arXiv:2002.11146 [quant-ph].
  * [105] N. H. Nguyen, M. C. Tran, Y. Zhu, A. M. Green, C. H. Alderete, Z. Davoudi, and N. M. Linke, “Digital Quantum Simulation of the Schwinger Model and Symmetry Protection with Trapped Ions,” PRX Quantum 3 no. 2, (2022) 020324, arXiv:2112.14262 [quant-ph].
  * [106] M. Honda, E. Itou, Y. Kikuchi, and Y. Tanizaki, “Negative string tension of a higher-charge Schwinger model via digital quantum simulation,” PTEP 2022 no. 3, (2022) 033B01, arXiv:2110.14105 [hep-th].
  * [107] M. Honda, E. Itou, Y. Kikuchi, L. Nagano, and T. Okuda, “Classically emulated digital quantum simulation for screening and confinement in the Schwinger model with a topological term,” Phys. Rev. D 105 no. 1, (2022) 014504, arXiv:2105.03276 [hep-lat].
  * [108] W. A. de Jong, K. Lee, J. Mulligan, M. Płoskoń, F. Ringer, and X. Yao, “Quantum simulation of nonequilibrium dynamics and thermalization in the Schwinger model,” Phys. Rev. D 106 no. 5, (2022) 054508, arXiv:2106.08394 [quant-ph].
  * [109] S. Thompson and G. Siopsis, “Quantum computation of phase transition in the massive Schwinger model,” Quantum Sci. Technol. 7 no. 3, (2022) 035001, arXiv:2110.13046 [quant-ph].
  * [110] QuNu Collaboration, X.-D. Xie, X. Guo, H. Xing, Z.-Y. Xue, D.-B. Zhang, and S.-L. Zhu, “Variational thermal quantum simulation of the lattice Schwinger model,” Phys. Rev. D 106 no. 5, (2022) 054509, arXiv:2205.12767 [quant-ph].
  * [111] D. J. Gross and A. Neveu, “Dynamical symmetry breaking in asymptotically free field theories,” Phys. Rev. D 10 no. 10, (Nov., 1974) 3235–3253.
  * [112] M. Asaduzzaman, S. Catterall, G. C. Toga, Y. Meurice, and R. Sakai, “Quantum Simulation of the N flavor Gross-Neveu Model,” arXiv:2208.05906 [hep-lat].
  * [113] R. Schutzhold and S. Mostame, “Quantum simulator for the O(3) nonlinear sigma model,” JETP Lett. 82 (2005) 248–252, arXiv:quant-ph/0412176.
  * [114] NuQS Collaboration, A. Alexandru, P. F. Bedaque, H. Lamm, and S. Lawrence, “$\sigma$ Models on Quantum Computers,” Phys. Rev. Lett. 123 no. 9, (2019) 090501, arXiv:1903.06577 [hep-lat].
  * [115] H. Singh and S. Chandrasekharan, “Qubit regularization of the $O(3)$ sigma model,” Phys. Rev. D 100 no. 5, (2019) 054505, arXiv:1905.13204 [hep-lat].
  * [116] A. J. Buser, T. Bhattacharya, L. Cincio, and R. Gupta, “State preparation and measurement in a quantum simulation of the $O$(3) sigma model,” Phys. Rev. D 102 no. 11, (2020) 114514, arXiv:2006.15746 [quant-ph].
  * [117] B. Xu and W. Xue, “3+1 Dimension Schwinger Pair Production with Quantum Computers,” arXiv:2112.06863 [quant-ph].
  * [118] F. Wilczek, “Quantum Time Crystals,” Phys. Rev. Lett. 109 (2012) 160401, arXiv:1202.2539 [quant-ph].
  * [119] D. V. Else, B. Bauer, and C. Nayak, “Floquet Time Crystals,” Phys. Rev. Lett.  117 no. 9, (2016) 090402, arXiv:1603.08001 [cond-mat.dis-nn].
  * [120] V. Khemani, A. Lazarides, R. Moessner, and S. L. Sondhi, “Phase Structure of Driven Quantum Systems,” Phys. Rev. Lett.  116 no. 25, (2016) 250401, arXiv:1508.03344 [cond-mat.dis-nn].
  * [121] J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano, I. D. Potirniche, A. C. Potter, A. Vishwanath, N. Y. Yao, and C. Monroe, “Observation of a discrete time crystal,” Nature 543 no. 7644, (2017) 217–220, arXiv:1609.08684 [quant-ph].
  * [122] I. D. Potirniche, A. C. Potter, M. Schleier-Smith, A. Vishwanath, and N. Y. Yao, “Floquet Symmetry-Protected Topological Phases in Cold-Atom Systems,” Phys. Rev. Lett.  119 no. 12, (2017) 123601, arXiv:1610.07611 [cond-mat.quant-gas].
  * [123] J. Randall, C. E. Bradley, F. V. van der Gronden, A. Galicia, M. H. Abobeih, M. Markham, D. J. Twitchen, F. Machado, N. Y. Yao, and T. H. Taminiau, “Many-body–localized discrete time crystal with a programmable spin-based quantum simulator,” Science 374 no. 6574, (2021) 1474–1478, arXiv:2107.00736 [quant-ph].
  * [124] X. Mi et al., “Time-crystalline eigenstate order on a quantum processor,” Nature 601 no. 7894, (2022) 531–536, arXiv:2107.13571 [quant-ph].
  * [125] C. Ying et al., “Floquet prethermal phase protected by U(1) symmetry on a superconducting quantum processor,” Phys. Rev. A 105 no. 1, (2022) 012418, arXiv:2107.07311 [quant-ph].
  * [126] H. Xu, J. Zhang, J. Han, Z. Li, G. Xue, W. Liu, Y. Jin, and H. Yu, “Realizing discrete time crystal in an one-dimensional superconducting qubit chain,” arXiv e-prints (2021) arXiv:2108.00942, arXiv:2108.00942 [quant-ph].
  * [127] J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, “Exploring the many-body localization transition in two dimensions,” Science 352 no. 6293, (2016) 1547–1552, arXiv:1604.04178 [cond-mat.quant-gas].
  * [128] A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, “Experimental realization of a long-range antiferromagnet in the Hubbard model with ultracold atoms,” arXiv e-prints (2016) arXiv:1612.08436, arXiv:1612.08436 [cond-mat.quant-gas].
  * [129] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551 no. 7682, (2017) 579–584, arXiv:1707.04344 [quant-ph].
  * [130] J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z. X. Gong, and C. Monroe, “Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator,” Nature 551 no. 7682, (2017) 601–604, arXiv:1708.01044 [quant-ph].
  * [131] P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, A. M. Läuchli, and A. Browaeys, “Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms,” Nature 595 no. 7866, (2021) 233–238, arXiv:2012.12268 [quant-ph].
  * [132] S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuletić, and M. D. Lukin, “Quantum phases of matter on a 256-atom programmable quantum simulator,” Nature 595 no. 7866, (2021) 227–232, arXiv:2012.12281 [quant-ph].
  * [133] S. D. Bartlett, “Programming a quantum phase of matter,” Science 374 no. 6572, (2021) 1200–1201.
  * [134] K. J. Satzinger et al., “Realizing topologically ordered states on a quantum processor,” Science 374 no. 6572, (2021) 1237–1241, arXiv:2104.01180 [quant-ph].
  * [135] G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuletić, and M. D. Lukin, “Probing topological spin liquids on a programmable quantum simulator,” Science 374 no. 6572, (2021) 1242–1247, arXiv:2104.04119 [quant-ph].
  * [136] F. Mei, Q. Guo, Y.-F. Yu, L. Xiao, S.-L. Zhu, and S. Jia, “Digital Simulation of Topological Matter on Programmable Quantum Processors,” Phys. Rev. Lett.  125 no. 16, (2020) 160503, arXiv:2003.06086 [quant-ph].
  * [137] P. T. Dumitrescu, J. G. Bohnet, J. P. Gaebler, A. Hankin, D. Hayes, A. Kumar, B. Neyenhuis, R. Vasseur, and A. C. Potter, “Dynamical topological phase realized in a trapped-ion quantum simulator,” Nature 607 no. 7919, (2022) 463–467, arXiv:2107.09676 [quant-ph].
  * [138] X. Zhang, W. Jiang, J. Deng, K. Wang, J. Chen, P. Zhang, W. Ren, H. Dong, S. Xu, Y. Gao, et al., “Digital quantum simulation of floquet symmetry-protected topological phases,” Nature 607 no. 7919, (2022) 468–473.
  * [139] M. Kiczynski, S. K. Gorman, H. Geng, M. B. Donnelly, Y. Chung, Y. He, J. G. Keizer, and M. Y. Simmons, “Engineering topological states in atom-based semiconductor quantum dots,” Nature 606 no. 7915, (2022) 694–699.
  * [140] S. Ashhab, “Simulating systems of itinerant spin-carrying particles using arrays of superconducting qubits and resonators,” New Journal of Physics 16 no. 11, (Nov., 2014) 113006, arXiv:1402.7185 [quant-ph].
  * [141] V. P. Nair and A. P. Polychronakos, “Quantum mechanics on the noncommutative plane and sphere,” Phys. Lett. B 505 (2001) 267–274, arXiv:hep-th/0011172.
  * [142] J. Gamboa, M. Loewe, and J. C. Rojas, “Noncommutative quantum mechanics,” Phys. Rev. D 64 (2001) 067901, arXiv:hep-th/0010220.
  * [143] A. Connes, Noncommutative Geometry.  Academic Press, 1995.
  * [144] M. R. Douglas and N. A. Nekrasov, “Noncommutative field theory,” Rev. Mod. Phys. 73 (2001) 977–1029, arXiv:hep-th/0106048.
  * [145] R. J. Szabo, “Quantum field theory on noncommutative spaces,” Phys. Rept. 378 (2003) 207–299, arXiv:hep-th/0109162.
  * [146] F. Delduc, Q. Duret, F. Gieres, and M. Lefrancois, “Magnetic fields in noncommutative quantum mechanics,” J. Phys. Conf. Ser. 103 (2008) 012020, arXiv:0710.2239 [quant-ph].
  * [147] S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane, and T. Okamoto, “Noncommutative field theory and Lorentz violation,” Phys. Rev. Lett. 87 (2001) 141601, arXiv:hep-th/0105082.
  * [148] R. J. Szabo, “Quantum Gravity, Field Theory and Signatures of Noncommutative Spacetime,” Gen. Rel. Grav. 42 (2010) 1–29, arXiv:0906.2913 [hep-th].
  * [149] J.-z. Zhang, “Testing spatial noncommutativity via Rydberg atoms,” Phys. Rev. Lett. 93 (2004) 043002, arXiv:hep-ph/0405143.
  * [150] A. Periwal, E. S. Cooper, P. Kunkel, J. F. Wienand, E. J. Davis, and M. Schleier-Smith, “Programmable interactions and emergent geometry in an array of atom clouds,” Nature 600 no. 7890, (2021) 630–635, arXiv:2106.04070 [quant-ph]. [Erratum: Nature 603, E29 (2022)].
  * [151] D. Carney, P. C. E. Stamp, and J. M. Taylor, “Tabletop experiments for quantum gravity: a user’s manual,” Class. Quant. Grav. 36 no. 3, (2019) 034001, arXiv:1807.11494 [quant-ph].
  * [152] R. Jackiw, “Physical instances of noncommuting coordinates,” Nucl. Phys. B Proc. Suppl. 108 (2002) 30–36, arXiv:hep-th/0110057.
  * [153] R. Jackiw, “Observations on noncommuting coordinates and on fields depending on them,” Annales Henri Poincare 4S2 (2003) S913–S919, arXiv:hep-th/0212146.
  * [154] J. Bellissard, A. van Elst, and H. Schulz-Baldes, “The noncommutative geometry of the quantum Hall effect,” Journal of Mathematical Physics 35 no. 10, (Oct., 1994) 5373–5451, arXiv:cond-mat/9411052 [cond-mat].
  * [155] F. D. M. Haldane, “Geometrical Description of the Fractional Quantum Hall Effect,” Phys. Rev. Lett. 107 (2011) 116801, arXiv:1106.3375 [cond-mat.mes-hall].
  * [156] E. Prodan, “Disordered topological insulators: a non-commutative geometry perspective,” Journal of Physics A Mathematical General 44 no. 11, (2011) 113001, arXiv:1010.0595 [cond-mat.dis-nn].
  * [157] T. Neupert, L. Santos, S. Ryu, C. Chamon, and C. Mudry, “Noncommutative geometry for three-dimensional topological insulators,” Phys. Rev. B 86 no. 3, (July, 2012) 035125, arXiv:1202.5188 [cond-mat.str-el].
  * [158] H. Schulz-Baldes, “Topological insulators from the perspective of non-commutative geometry and index theory,” arXiv e-prints (2016) arXiv:1607.04013, arXiv:1607.04013 [math-ph].
  * [159] S. H. H. Chowdhury and S. T. Ali, “The Symmetry Groups of Noncommutative Quantum Mechanics and Coherent State Quantization,” J. Math. Phys. 54 (2013) 032101, arXiv:1208.3861 [math-ph].
  * [160] S. H. H. Chowdhury and S. Twareque Ali, “Triply extended group of translations of $\mathbb{R}^{4}$ as defining group of NCQM: relation to various gauges,” J. Phys. A 47 (2014) 085301, arXiv:1309.7086 [math-ph].
  * [161] S. H. H. Chowdhury and S. T. Ali, “Wigner functions for noncommutative quantum mechanics: A group representation based construction,” J. Math. Phys. 56 no. 12, (2015) 122102, arXiv:1506.06341 [math-ph].
  * [162] S. H. H. Chowdhury, “On the plethora of representations arising in noncommutative quantum mechanics and an explicit construction of noncommutative 4-tori,” J. Math. Phys. 58 no. 6, (2017) 061702, arXiv:1507.01105 [math-ph].
  * [163] V. Bargmann, “On unitary ray representations of continuous groups,” Annals of Mathematics 59 (1954) 1–46.
  * [164] A. A. Kirillov, Lectures on the Orbit Method.  American Mathematical Society, 2004.
  * [165] S. H. H. Chowdhury, T. A. Chowdhury, and M. A. U. Duha, “Gauge invariant energy spectra in 2-dimensional noncommutative quantum mechanics,” Annals Phys. 430 (2021) 168505, arXiv:2003.12662 [math-ph].
  * [166] A. Hatzinikitas and I. Smyrnakis, “The Noncommutative harmonic oscillator in more than one-dimensions,” J. Math. Phys. 43 (2002) 113–125, arXiv:hep-th/0103074.
  * [167] A. Smailagic and E. Spallucci, “New isotropic versus anisotropic phase of noncommutative 2-D harmonic oscillator,” Phys. Rev. D 65 (2002) 107701, arXiv:hep-th/0108216.
  * [168] B. Muthukumar and P. Mitra, “Noncommutative oscillators and the commutative limit,” Phys. Rev. D 66 (2002) 027701, arXiv:hep-th/0204149.
  * [169] R. D. Somma, “Quantum simulations of one dimensional quantum systems,” Quantum Information $\&$ Computation 16 no. 13-14, (Mar., 2015) 1125–1168, arXiv:1503.06319 [quant-ph].
  * [170] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition.  Cambridge University Press, 2010.
  * [171] H. F. Trotter, “On the product of semi-groups of operators,” Proceedings of the American Mathematical Society 10 no. 4, (1959) 545–551. http://www.jstor.org/stable/2033649.
  * [172] M. Suzuki, “Generalized trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems,” Communications in Mathematical Physics 51 no. 2, (1976) 183–190.
  * [173] M. Suzuki, “On the convergence of exponential operators—the zassenhaus formula, bch formula and systematic approximants,” Communications in Mathematical Physics 57 no. 3, (1977) 193–200.
  * [174] M. Suzuki, “Decomposition formulas of exponential operators and lie exponentials with some applications to quantum mechanics and statistical physics,” Journal of mathematical physics 26 no. 4, (1985) 601–612.
  * [175] M. Suzuki, “Fractal decomposition of exponential operators with applications to many-body theories and monte carlo simulations,” Physics Letters A 146 no. 6, (1990) 319–323.
  * [176] N. Hatano and M. Suzuki, “Finding Exponential Product Formulas of Higher Orders,” Lect. Notes Phys. 679 (2005) 37, arXiv:math-ph/0506007.
  * [177] M. S. Anis et al., “Qiskit: An open-source framework for quantum computing,” 2021\.
  * [178] V. K. Jain, B. K. Behera, and P. K. Panigrahi, “Quantum simulation of discretized harmonic oscillator,” Quantum Studies: Mathematics and Foundations 8 no. 4, (2021) 375–390.
  * [179] M. S. Anis et al., “Qiskit textbook.” https://qiskit.org/textbook/ch-algorithms/quantum-fourier-transform.html.
  * [180] R. Jagannathan, T. S. Santhanam, and R. Vasudevan, “Finite-Dimensional Quantum Mechanics of a Particle,” International Journal of Theoretical Physics 20 no. 10, (Oct., 1981) 755–773.
  * [181] A. Singh and S. M. Carroll, “Quantum Decimation in Hilbert Space: Coarse-Graining without Structure,” Phys. Rev. A 97 no. 3, (2018) 032111, arXiv:1709.01066 [quant-ph].
  * [182] A. Singh and S. M. Carroll, “Modeling Position and Momentum in Finite-Dimensional Hilbert Spaces via Generalized Pauli Operators,” arXiv:1806.10134 [quant-ph].
  * [183] N. Atakishiev and S. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85 (1990) 1055–1062.
  * [184] N. M. Atakishiyev, E. I. Jafarov, S. M. Nagiev, and K. B. Wolf, “Meixner oscillators,” Rev. Mex. Fis. 44 (1998) 235–244, arXiv:math-ph/9807035.
  * [185] M. Lorente, “Continuous vs. discrete models for the quantum harmonic oscillator and the hydrogen atom,” Physics Letters A 285 no. 3, (July, 2001) 119–126, arXiv:quant-ph/0401087 [quant-ph].
  * [186] A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, “Theory of trotter error with commutator scaling,” Phys. Rev. X 11 (Feb, 2021) 011020, arXiv:1912.08854 [quant-ph].
  * [187] K. Georgopoulos, C. Emary, and P. Zuliani, “Modeling and simulating the noisy behavior of near-term quantum computers,” Phys. Rev. A 104 no. 6, (2021) 062432, arXiv:2101.02109 [quant-ph].
  * [188] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, “Exponential improvement in precision for simulating sparse Hamiltonians,” arXiv e-prints (Dec., 2013) , arXiv:1312.1414 [quant-ph].
  * [189] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, “Simulating Hamiltonian Dynamics with a Truncated Taylor Series,” Phys. Rev. Lett.  114 no. 9, (Mar., 2015) 090502, arXiv:1412.4687 [quant-ph].
  * [190] D. W. Berry and A. M. Childs, “Black-box Hamiltonian simulation and unitary implementation,” arXiv e-prints (2009) arXiv:0910.4157, arXiv:0910.4157 [quant-ph].
  * [191] D. W. Berry, A. M. Childs, and R. Kothari, “Hamiltonian simulation with nearly optimal dependence on all parameters,” in 2015 IEEE 56th annual symposium on foundations of computer science, pp. 792–809, IEEE.  2015\.