# Zeno Effect Suppression of Gauge Drift in Quantum Simulations

Carter Ball<EMAIL_ADDRESS>Department of Physics, University of Maryland,
College Park, MD 20742, USA  Thomas D. Cohen<EMAIL_ADDRESS>Department of
Physics, University of Maryland, College Park, MD 20742, USA

###### Abstract

Quantum simulation of lattice gauge theories is a promising tool for the study
of many complicated problems including ones with real-time dynamics. For gauge
theories, however, there is a major challenge in maintaining gauge invariance
during time evolution. Such theories have a full Hilbert space that is larger
than the physical space—the set of states which are gauge invariant or
equivalently respect the Gauss law. While an exact implementation of
Hamiltonian dynamics starting in the physical Hilbert space will keep the
system in the physical space, various types of errors will inevitably produce
components outside of it. This work proposes a method of suppressing this
gauge drift via the Zeno effect. As in the standard picture of the Zeno
effect, our method relies on frequent projection onto the physical subspace.
Additionally, a technique is discussed to reduce the speed of the gauge drift,
which helps to reduce the required frequency of projections. We demonstrate
our method on a $\mathbb{Z}_{2}$ gauge theory toy model.

††preprint: APS/123-QED

##  I Introduction

Quantum computers promise to be vital tools for the simulation of many
fundamental yet complicated problems [1, 2, 3, 4, 5] from a range of
subfields, including particle physics, nuclear physics, and condensed matter.
The simulation and study of complex systems often requires much more computing
power than classical computers can provide [6, 7]; thus, there is a present
need to map these systems onto quantum devices. Analog quantum devices
manufacture a system that can imitate, at least in some regime, the desired
theory while digital quantum devices map a theory onto an array of qubits and
control it via quantum gates [8]. Much research has gone into studying these
quantum devices and how they can be used for quantum simulation [9].

Like many complicated theories, lattice gauge theories can, in principle,
benefit enormously from efficient and reliable quantum simulation [10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. These theories typically
place matter degrees of freedom on the sites of the lattice while gauge field
operators are placed on the links; it is typical to work in a Hamiltonian
formulation with discretized space but continuous time [26, 27, 28]. This is
usually done in the temporal gauge [29] (which sets the time component of
gauge fields to zero) and by taking the continuous time limit. Note that the
temporal gauge does not fully exhaust the gauge freedom, leaving a residual
gauge freedom for the spatial components. Transformations of this residual
gauge are generated by Gauss law operators (in analogy to classical
electrodynamics). Physical states are then defined as those that are invariant
under all residual gauge transformations.

Much work has gone into the study of both Abelian and non-Abelian lattice
gauge theories. Notably, maintaining non-Abelian gauge invariance is
particularly difficult. When it comes to analog quantum simulation, many
methods have been developed to address this gauge invariance problem,
including constructing a system that makes gauge invaraince a consequence of
internal symmetry such as angular momentum [30, 31]. Another analog simulation
method is to impose an energy penalty on the unphysical states [32, 33, 16,
34, 35, 36, 37, 38, 39, 40], often by adding a term to the Hamiltonian
proportional to the square of the Gauss’ law operators. Then, the lowest
energy levels of the system are the physical ones, so keeping the system in
the low energy regime results, to good approximation, in keeping the system
within the physical subspace of the much larger Hilbert space. Variations on
this method have been developed, including one where the addition to the
Hamiltonian is directly proportional to the Gauss’ law operators [41]. Another
such variation on the energy penalty method uses local pseudogenerators, which
exactly match the Gauss’ law operators in the relevant physical subspace but
not in unphysical sectors, allowing these operators to be much less complex
and thus cheaper to engineer in practice [42, 43, 44]. Interestingly, for
sufficiently large energy penalties, the protection term added to the
Hamiltonian, constructed with local pseudogenerators, can be seen as a strong
projector which constrain the dynamics of the system to quantum Zeno subspaces
within a given timescale [44, 45].

The method presented in this paper concerns digital quantum simulation.
Ideally, any method to simulate a non-Abelian gauge theory of interest, such
as quantum chromodynamics (QCD), would only simulate the gauge invariant (i.e.
physical) states. Methods of this kind have been proposed [46, 47], which
encode only physical degrees of freedom. While these methods don’t have to
deal with unphysical states, they are fundamentally more challenging. Thus,
many methods simulate all states in the Hilbert space of the system, and must
contend with keeping the system within the exponentially smaller physical
subspace. Crucially, work has been done to develop methods of time evolution
that respect gauge invariance [13, 48]. Thus, ideally time evolving an initial
state that is constructed to be physical subspace would keep the system within
the physical subspace. However, quantum noise, gate errors, and various errors
from approximations can lead to the system picking up contributions from
unphysical states. Previous work has addressed this problem in a variety of
ways. Perhaps the most straightforward is to simply accept the unphysical
contributions on the premise that these contributions are reduced as the time
step is decreased [15]. Another method is dynamical decoupling [49], a concept
from quantum control theory where local symmetries are maintained via periodic
driving. Also, oracles have been developed [50] to check for violations of
Gauss’ law for Abelian lattice gauge theories, with similar oracles for non-
Abelian theories under development.

Furthermore, a method of engineering classical noise to constrain the system
to the physical subspace [51] has been proposed. This method serves as a
classical analog to the quantum Zeno effect by using classical fluctuations of
a perturbing field to dissipate unwanted evolution into unphysical regions of
the Hilbert space. The method of this paper, in contrast, rests on the
standard quantum Zeno effect [52], which uses frequent projections to keep the
system within a desired space; for our purposes the desired space is of course
the physical subspace of the system’s Hilbert space.

This paper is organized as follows: section II outlines our method, which
takes inspiration from the rodeo algorithm[53]—a method of projection—and a
technique of unitary gauge drift suppression. Section III demonstrates our
method on a toy model of simulating pure $\mathbb{Z}_{2}$ gauge theory on a
2x2 lattice with periodic boundary conditions. Section IV goes into more
detail about the implementation of our method, and Section V provides
concluding thoughts and discusses the required future work.

##  II Method

The method exploits the quantum Zeno effect to maintain a close approximation
of gauge invariance. For a non-Abelian theory, the gauge invariant, or
physical, states are given by those states $\ket{\psi}$ that satisfy

$\displaystyle G^{a}_{x}\ket{\psi}=0$ (1)

for all $a$ and $x$, where $G^{a}_{x}$ is the $a$th Gauss’ law operator at
site $x$.

To leverage the Zeno effect, we must invoke a method of projection onto the
physical subspace. Our method adapts previous work on the rodeo algorithm [53,
54, 55, 56]. We also incorporate a method of suppressing coherent gauge drift,
discussed below.

###  II.1 Projection via the Rodeo Algorithm

The Rodeo algorithm was developed to prepare eigenvectors of a given
Hamiltonian. See figure 1 for the circuit diagram of the algorithm [53].

Figure 1: The circuit diagram for the rodeo algorithm [53]

To make clear how the rodeo algorithm works, we will write out how it works
for one ancillary qubit. Starting with the system in the state
$\ket{\psi_{I}}$ and the ancillary qubit initialized to $\ket{1}$, we have

$\displaystyle\begin{bmatrix}\frac{I}{\sqrt{2}}&\frac{I}{\sqrt{2}}\\\
\frac{I}{\sqrt{2}}&-\frac{I}{\sqrt{2}}\end{bmatrix}\begin{bmatrix}I&0\\\
0&Ie^{iEt}\end{bmatrix}\begin{bmatrix}I&0\\\
0&e^{-iH_{obj}t}\end{bmatrix}\begin{bmatrix}\frac{I}{\sqrt{2}}&\frac{I}{\sqrt{2}}\\\
\frac{I}{\sqrt{2}}&-\frac{I}{\sqrt{2}}\end{bmatrix}\begin{bmatrix}0\\\
\ket{\psi_{I}}\end{bmatrix}=$ (2)
$\displaystyle\begin{bmatrix}\frac{1}{2}(I-e^{-i(H_{obj}-E)t})\ket{\psi_{i}}\\\
\frac{1}{2}(I+e^{-i(H_{obj}-E)t})\ket{\psi_{i}}\end{bmatrix}$ (3)

A successful implementation of the rodeo algorithm measures every ancillary
qubit in the state $\ket{1}$ at the end. In the single ancillary qubit example
above, this results in the transformation
$\ket{\psi_{I}}\rightarrow\frac{1}{2}(I+e^{-i(H_{obj}-E)t})\ket{\psi_{i}}$
where $E$ is the energy of the target eigenstate and $H_{obj}$ is the
Hamiltonian of the system. Decomposing our initial state into energy
eigenstates $\ket{\psi_{i}}=\sum\limits_{n}c_{n}\ket{n}$, we can see that the
probability for each energy eigenstate $n$ transformations from $|c_{n}|^{2}$
before Rodeo projection to
$|c_{n}|^{2}|\frac{1}{2}(1+e^{-i(E_{n}-E)t})|^{2}=|c_{n}|^{2}\cos^{2}((E_{obj}-E)\frac{t}{2})$
afterwards. Thus, the probability for the system to be in a particular energy
eigenstate $E_{obj}$ is multiplied by a factor
$\cos^{2}((E_{obj}-E)\frac{t}{2})$, which is notably 1 if $E=E_{obj}$ and less
than 1 otherwise. This is how the rodeo algorithm works: it constructs an
approximate version of the state of energy $E$ by suppressing all other states
with a $\cos^{2}(\frac{1}{2}t\Delta E)$ factor. Stronger suppression of
contributions from unwanted energy eigenstates can be accomplished with
multiple rounds of this algorithm, as discussed in [53, 54, 55]. Note that in
order to implement a round of the rodeo algorithm, a time $t$ must be chosen.
If the times are chosen strategically for the various rounds, remarkably
strong suppression can be achieved with very modest resources [56].

We note here that the Rodeo projection will fail to project onto the state of
interest with finite probability (i.e. if any ancillary qubit is not measured
in $\ket{1}$), where failure means that the resulting state is orthogonal to
the target state. Crucially, we note that the probability of a successful
projection is proportional to the overlap of the initial state of the system
with the target state of the projection. As we plan to use this projection to
leverage the Zeno effect, which means making sufficiently many measurements so
that the overlap with the target subspace is close to unity, the rate of
failure can be made negligibly small. The chance of failure serves as the
probabilistically defined time span for which our method works; for example,
if the probability for a failed projection is one in a million, we would
expect one failure within one million successive projections.

The rodeo algorithm is designed to construct eigenstates of a system’s
Hamiltonian. For our purposes, we use the ideas of this algorithm for a
different hermitian operator, that we will call $G^{2}$:

$\displaystyle G^{2}=\sum\limits_{a,x}(G^{a}_{x})^{2}$ (4)

From the above definition of the Gauss’ law operators, we see that the
physical states are eigenstates of $G^{2}$ with eigenvalue 0. Furthermore, all
the other eigenstates will be unphysical and with eigenvalues greater than 0,
since $G^{2}$ is positive definite by construction. It is this spectrum of
$G^{2}$ that we will exploit when using the rodeo algorithm: instead of
constructing some eigenstate of a given Hamiltonian, we will construct a
physical state by constructing a ground state of $G^{2}$ via the Rodeo
algorithm. This means replacing $H_{obj}$ in 1 with $G^{2}$ as well as
removing the phase rotation gates (since we would be aiming for an eigenstate
with an eigenvalue of 0, so the required phase rotation is 0*t = 0).

There is one minor detail that we must clarify before moving on. The
effectiveness of this projection is helped if the spectrum of the operator, in
our case $G^{2}$, is discrete. That way there will be less contamination from
the eigenstates nearest to the target eigenstate. Luckily, the spectrum of
$G^{2}$ for our principal case of interest $SU(3)$ gauge theories, is in fact
discrete. This can be seen very straightforwardly using basic quantum
mechanics, though We will point out one small detail: the basic argument
implicitly relies on the fact that all states have non-negative norms. While
this is obviously true for physical states it is not necessarily true for the
full Hilbert space of QCD. However, we are working within a Hamiltonian
formulation and using the temporal gauge, which is a ghostless gauge [57],
meaning that we do in fact have non-negative norms for all states. Thus, all
components $G^{a}$ have discrete spectra, so $G^{2}$ has a discrete spectrum
as well. This means the ground state of $G^{2}$ is gapped, which is helpful
when it comes to projection.

###  II.2 Suppressing Coherent Gauge Drift

The basic idea of the Zeno effect is that if you project back to the desired
space before straying too far out of it (i.e. frequently enough), then the
system stays in the desired space with high probability. This is for a simple
reason: projections depend on the probability that the system is in the
desired state, which corresponds to the square of the magnitude of the
amplitude. Thus, if one projects when the amplitude for the system drifting
into an unwanted state is some small value $\epsilon$, the probability of
failure is $\epsilon^{2}$—a much smaller value.

Consider what this means in practice if one wishes to study a system’s time
evolution over a time, $T$, and gauge-invariance-violating errors in the
amplitude accrue at a rate $r$. Suppose one makes $N$ evenly spaced
projections (which for the sake of this analysis are assumed to be perfect)
during the time evolution. The probability of a failure—obtaining a state
outside the physical space—at each measurement is $(rT/N)^{2}$, and the
probability of a successful run for the full time evolution (i.e. the
probability of $N$ successes and $0$ failures) is given by

$P_{s}=\left(1-\frac{r^{2}T^{2}}{N^{2}}\right)^{N}\approx
1-\frac{r^{2}T^{2}}{N}$ (5)

where the approximate equality holds in the limit of large $N$ and $1-P_{s}\ll
1$. The key point is that for $N\gtrsim r^{2}T^{2}$ there is a substantial
probability that the full time evolution can occur with the system restricted
to the physical space. Note that this means that by making sufficient
measurements, the evolution can stay in the physical space even if the
evolution time is much larger than $1/r$. This a major improvement.

Of course, if one makes a finite number of projections there is a finite
probability of failure—obtaining a state outside the physical space. If that
happens, the only recourse is to stop the calculation and begin again—an
expensive prospect. Thus, there is an optimization problem in choosing $N$: if
$N$ is too large, there is the computational cost of making more projections
than are necessary, whereas if $N$ is too small than the failure rate is too
high requiring the additional cost of starting over. The optimal value depends
on the computation costs of the projection as well as the costs of running the
simulation.

With this in mind, the method has been enlarged to include an additional way
to suppress gauge drift, an approach developed in ref. [58]. we show that a
mixed approach using both quantum Zeno and the the approach of ref. [58] is
likely to be more efficient than either one separately.

The method of ref. [58] is essentially an alternative version of the energy
penalty method for suppressing coherent gauge drift. Here, we define coherent
gauge drift as a unitary process $U_{D}$ that transforms a physical state
$\ket{\psi}$ to

$\displaystyle
U_{D}\ket{\psi}=\sqrt{1-\epsilon^{2}}\ket{\psi}+\epsilon\ket{\omega}$ (6)

where $\ket{\omega}$ is a nonphysical state and $\epsilon$ characterizes the
strength of the drift. Recall that an energy penalty method constructs a
Hamiltonian $H_{G}$ with the behavior that $H_{G}\ket{\psi}=0$ for physical
states $\ket{\psi}$ and $H_{G}\ket{\omega}>0$ for unphysical states
$\ket{\omega}$. Adding $\lambda_{G}H_{G}$ for $\lambda_{G}\gg 0$ to the
Hamiltonian of the system suppresses transitions to unphysical states via an
energy penalty. The authors of [58] note that in principle, building $H_{G}$
out of the fundamental fields of the lattice gauge theory would be quite
difficult for a non-Abelian theory; however, quantum simulation via
Trotterized time steps depends upon $e^{-iH_{G}t}$ not solely $H_{G}$.
Furthermore, they point out that this operator basically performs a random
gauge transformation on the state of the system, which leaves physical states
unchanged while unphysical states take on a phase. Thus, this approach
dispenses with the difficult task of building $H_{G}$ and instead simply
implements the principal effect of including $H_{G}$ via a random gauge
transformation. In summary, the method of [58] is to implement, after every
Trotterization time step, a random gauge transformation.

Our method aims to leverage this method wholesale: frequent projections back
to the physical subspace is expensive, so we include a random gauge
transformation after each Trotterization time step. This reduces the gauge
drift per time step, allowing us to do projections to the physical subspace
less frequently while still staying close enough to the physical subspace to
leverage the quantum Zeno effect, which allows the suppression of whatever
gauge drift that survives the random gauge transformations.

###  II.3 Calibration

Our method, as outlined above, is to conduct a random gauge transformation
after every Trotterization time step as well as to conduct projections in the
form discussed above to project back to the physical subspace. One important
detail that needs to be specified is the frequency of projections. To this
end, we include a calibration phase in the method—a set of initial studies—to
best estimate the frequency of projections that constitutes a useful
compromise between the minimization of gauge drift and the minimization of the
number of projections. We note up front that in principle doing a projection
after every time step would be most effective in terms of reducing gauge drift
as the probability of failure of the projection is very low. The problem with
this is simply that it is extremely expensive computationally. In the next
section we will outline results from a toy model that show that other, less
expensive forms of the method work well. This suggests that realistic
calculations can balance the resource costs with the requirements on effective
gauge drift suppression.

We propose an approach that should be useful in situations when a simulation
needs to be done repeatedly—in that case, a simple calibration phase to
optimize the calculation is done at the outset: start with an initial trial
frequency of projections based on whatever information and intuition one has
about the system. Here, any previous knowledge about the system being
simulated should be exploited, especially concerning the speed of gauge drift.
With an initial frequency chosen, the time evolution of the system is run a
small number of times with the chosen trial frequency of projections to see
how long before the projection step fails. If it appears that failure will in
general occur before the desired length of time of the simulation, the
frequency of projections can be increased. This process can be repeated until
an acceptable balance is found.

Behind this simple calibration phase is the fact that the average gauge drift
is linearly proportional to the number of time steps between projections.
Intuitively, this also means the probability of a failed projection is
linearly proportional to the number of time steps between projections. With
these relations in mind, it is simple to tune the frequency of projections
given a relatively few trial runs.

##  III Performance

To study this we consider a small “toy” model. It is a pure $\mathbb{Z}_{2}$
gauge theory in 2 spatial dimensions [59]. We simulate this on a 2x2 lattice
with periodic boundary conditions. The purpose of using such a small
simulation is simply that it allows one to explore the method with very little
cost. The Hamiltonian for the toy model is

$\displaystyle H=-k\sum\limits_{j}\square_{j}-g\sum\limits_{i}\sigma^{x}_{i}$
(7)

where $\square_{j}$ is the $j$th plaquette operator, $\sigma^{x}_{i}$ is the
Pauli x operator on the $i$th link, and $k$ and $g$ are parameters. A
plaquette operator in this context is a product of the $\sigma^{z}$ operators
living on the four sides of a plaquette in the lattice.

In this system, the Gauss’ law operators are constructed by taking the product
of the $\sigma^{x}$ operators living on the four links emanating from a site
on the lattice. We note here that while our method in general works by
constructing an operator $G^{2}$ consisting of a sum of the square of all
Gauss’ law operators, we have to modify this approach for this toy model as
the square of a product of $\sigma^{x}$ operators from different links is
manifestly the unity operator. Instead, we use a direct sum of the four Gauss’
law operators of this system, which we will call $G_{tot}$. We also note that
for this system, we define a physical state as one satisfying

$\displaystyle G_{a}\ket{\psi}=\ket{\psi}$ (8)

for $a=1,2,3,4$ denoting the associated site. As each $G_{a}$ has a 2-level
spectrum (1 and -1), we see that $G_{tot}$ has a 3-level spectrum (-4, 0, and
4), with the physical states satisfying $G_{tot}\ket{\psi}=4\ket{\psi}$.

Our aim is to study gauge drift, so we include in our toy model an artificial
gauge drift operator $D$ with each time step. We base our constructed gauge
drift operators off of eq. (6); specifically, we connect each of the, in our
case 32, physical states of the system to a randomly selected unphysical state
in exactly the way laid out in eq. (6). All other states are unchanged. This
provides a way for the system to drift out of the physical subspace of its
Hilbert space in a unitary way, and a parameter, $\epsilon$, to control the
strength of this drift. For our simulations, we set this parameter to
$\epsilon=0.01$. Note that each time our method calls for time evolution, we
select a different set of 32 unphysical states to appear in the gauge drift
operator so as to not bias any unphysical states.

With this defined, the algorithm can be used for the toy model. Each time step
takes a state $\ket{\psi}$ to $G_{i}DU_{t}\ket{\psi}$ where $G_{i}$ is a
randomly chosen gauge transformation, $D$ is a gauge drift operator, and
$U_{t}=\exp{-iHt}$ is the standard time evolution operator. The remaining part
of the algorithm is the rodeo projection, which occurs after a set number of
time steps (determined in the calibration phase). As discussed above, a
successful projection takes a state $\ket{\psi}$ to
$\frac{1}{2}(1+\exp(-i(G_{tot}-4)t_{r}))\ket{\psi}$ where here we note that
the time used within the projection $t_{r}$ is in general different from the
time step size $t$. Recall that all physical states are eigenstates of
$G_{tot}$ with an eigenvalue of 4, so they are unaffected while unphysical
states will suffer destructive interference.

Figure 2: Comparing gauge drift suppression methods, including no suppression
methods (green), a random gauge transformation after each time step (blue),
and a random gauge transformation followed by a rodeo projection after each
time step (orange). The right plot is the same as the left plot without the
green plot so as to zoom in on the behavior of the other two.

Figure 2 demonstrates the basic performance of the method. It shows no
suppression, a method of using a random gauge transformation after each time
step, and our method of using a random gauge transformation after each time
step plus projections, in this case after every time step. It can be seen both
that the use of random gauge transformations after every time step greatly
suppresses the gauge drift as well as that frequent rodeo projections does in
fact keep the system very close to the physical subspace.

Figure 3: Comparing performance of regimes with no gauge transformations and
various frequencies of rodeo projections with regimes with gauge
transformations and various frequencies of rodeo projections

In the rest of this section we will consider the frequency of projections. To
study this, we have run simulations to test many different regimes to compare
their effectiveness. In particular, systems without any gauge transformations
after time steps with a frequency of rodeo projections ranging from after one
time step to after 10 time steps. We also considered systems with a random
gauge transformation after each time step with a frequency of gauge
projections ranging from after one time step to after 30 time steps. For each
regime, we ran our algorithm on 200 systems and calculated the average of the
farthest from physical each run reached (which is to say the lowest value of
$\expectationvalue{G_{tot}}$).

Figure 3 shows these results. It is clear that including the random gauge
transformations into the method significantly helps to reduce the required
frequency of gauge projections. In particular, it shows the following
approximate equivalences: a projection after every time step performs
similarly to a projection every two time steps when gauge transformations are
included; a rodeo projection after every two time steps performs similarly to
a gauge projection every five time steps when gauge transformations are
included; and a gauge projection after every three time steps performs
similarly to a gauge projection every ten time steps when gauge
transformations are included. This shows that including the gauge
transformations reduces the required frequency of rodeo projections by more
than half. In particular, we see that for a system that drifts out of the
physical subspace slowly, one could perform projections infrequently and still
be able to utilize the Zeno effect to great advantage.

##  IV Implementation

It is important to consider the implementation of this method, as well as
their costs. We will start with a conisderation of the implementation of the
gauge transformations, which is discussed in [58]. First of all, there is the
cost of sampling the gauge transformations, which includes a small number of
ancillary qubits; notably, this cost does not scale with the size of the
system [48, 58]. When it comes to implementing the sampling, we make use of a
common method of sampling uniformly from the Haar measure of a group $G$
wherein a small number (e.g. 2) elements of $G$, call them $g_{1}$ and
$g_{2}$, are chosen and then strings (of a chosen length) of these elements
are randomly sampled, e.g. $g_{1}g_{2}g_{2}g_{1}g_{2}$. Importantly, this
method of sampling elements of $G$ converges to the Haar measure in the limit
of long strings. For the present purpose, however, we note that we do not rely
on a truly uniform sampling of gauge transformations; thus, relatively short
strings can be used, in a similar way to the circuit described in [60].

Next, we consider the implementation of the gauge transformations, which, as
the original authors of this method noted in ref. [58], ”requires about as
many G-multiplication gates as the original quantum simulation did”. Thus, the
total cost of using gauge transformations in this method is about double what
it would be for a straightforward simulation.

Additionally, the implementation and cost of the gauge projections needs to be
considered. We have suggested that the rodeo projection method appears to be
useful. Note that previous work has shown that the rodeo algorithm can be
optimized and made far more efficient [56] via considered choices for the
parameter $t_{r}$. We suggest using this optimized approach makes sense for
our purposes as well. This work notes that the dominant cost of the rodeo
algorithm is the controlled time evolution gate (in our case it is then the
controlled U gate with $U=\exp(-iG^{2}t_{r})$); thus, they argue that the
total time to conduct a rodeo projection acts as a proxy for computational
costs and then proceeds to lay out a method for selecting the $t_{r}$s that
maximize efficiency of the algorithm given fixed computational resources [56].
We will also note that constructing $G^{2}$ will be much more difficult than
$H$ so our version of a rodeo projection suffers a much heftier cost in
construction of the required gates, but so will alternative projection
schemes.

Furthermore, the projections themselves require a number of ancillary qubits.
Notably, the projections only use one ancillary qubit at a time, so if mid-
circuit measurements are allowed then only one ancillary qubit is required, as
it can be recycled again and again. If this is not the case, then a
significant number of ancillary qubits must be used. Specifically, we estimate
one projection would need as many ancillary qubits as sites in the lattice.

##  V Conclusion

We have outlined an algorithm developed to maintain gauge invariance (or more
precisely suppress gauge drift) during the simulation of lattice gauge
theories. Our algorithm was based off of the Zeno effect, wherein we use
frequent projections (i.e. altered rodeo projections) to keep the system
within the physical subspace of its exponentially larger Hilbert space. We
also include a technique of performing gauge transformations after every time
step to help reduce the system’s gauge drift into the unphysical sector which
in turn allows for less frequent projections while still taking full advantage
of the Zeno effect. We have shown that our method is effective for a simple
small toy model of pure $\mathbb{Z}_{2}$ gauge theory. Future work aims to
test our algorithm out on larger systems, and more complicated gauge theories
such as $SU(N)$ or gauge theories that include matter.

###### Acknowledgements.

This work was supported in part by the U.S. Department of Energy, Office of
Nuclear Physics under Award Number(s) DE-SC0021143 and DE-FG02- 93ER40762.

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