$i\mathcal{Q}\frac{q_{a}}{q^{2}}j_{b}(\text{\bf q})$. In this way, the stress-
stress correlations is expressed in terms of the projected force-force
correlations, encoded by the memory kernel K given by Eq. (66). We therefore
get, using the $\text{\bf A}-\text{\bf q}$ basis,

$\eta_{xyxy}=\lim_{q\to 0}\lim_{\omega\to
0}\frac{\rho_{0}}{D_{0}\beta}\left[K_{22}(q\text{\bf e}_{x},-i\omega)-\gamma
K_{21}(q\text{\bf e}_{x},-i\omega)\right].$ (197)

Using the expression of K given by the mode-coupling approximation, Eq.(75),
and the fact that within this approximation the off-diagonal terms in K are
$0$ in the $\text{\bf A}-\text{\bf q}$ basis, we have

$\begin{split}\eta_{xyxy}&=\lim_{q\to 0}\lim_{\omega\to
0}\frac{\rho_{0}}{q^{2}D_{0}\beta}K_{22}(q\text{\bf e}_{x},-i\omega)\\\
&=\lim_{q\to 0}\frac{\rho_{0}^{2}}{2q^{2}\beta}\int_{0}^{+\infty}\differential
t\int_{\text{\bf k}}[k_{y}(c(k)-c(|q\text{\bf e}_{x}-\text{\bf k}|))]^{2}\\\
&\times S(\text{\bf k},t)S(q\text{\bf e}_{x}-\text{\bf k},t)\\\
&=\frac{1}{2\beta}\int_{0}^{+\infty}\differential t\int_{\text{\bf
k}}[\hat{k}_{x}k_{y}\rho_{0}c^{\prime}(k)]^{2}\lvert S(\text{\bf
k},t)\rvert^{2}\end{split}.$ (198)

In this expression, the dependence on $\gamma$ enters through the dynamical
evolution of $S(\text{\bf k},t)$. For $\gamma=0$, rotational symmetry is
restored, and we recover the equilibrium, mode-coupling result [64]:

$\eta_{xyxy}^{\text{eq}}=\frac{1}{60\pi^{2}\beta}\int_{0}^{+\infty}\differential
kk^{4}\left[\frac{1}{S(k)}S^{\prime}(k)\right]^{2}\phi(k,t)^{2}.$ (199)

Since the relaxation of $S(\text{\bf k},t)$ is faster in the presence of
transverse forces we have $\eta_{xyxy}(\gamma)\leq\eta_{xyxy}^{\text{eq}}$.
Also, the shear viscosity diverges at the glass transition.

###  VI.5 Odd viscosity

We now address the odd viscosity. We identify the following contribution as
the odd viscosity:

$\eta_{\text{odd}}\equiv\frac{1}{2}\left(\eta_{xyxx}-\eta_{xxxy}\right).$
(200)

Physically, a nonzero $\eta_{\text{odd}}$ means that attempts at compressing
the system along the $x$-direction generate shear flows in the $(xy)$ plane.
In a nonreciprocal fashion, shear stresses applied on the $(xy)$ plane will
generate an expansion of the liquid along the $x$-direction. In equilibrium
dynamics we have $\eta_{abcd}=\eta_{cdba}$ and therefore
$\eta_{\text{odd}}=0$. We address now how this situation changes when
transverse forces are present.

Performing a computation similar to the one done in Eq. (197) gives

$\eta_{\text{odd}}=\frac{1}{2}\lim_{q\to 0}\lim_{\omega\to
0}\frac{\rho_{0}}{q^{2}D_{0}\beta}\left[(K_{21}(q\text{\bf
e}_{x},-i\omega)+K_{12}(q\text{\bf e}_{x},-i\omega))-\gamma(K_{11}(q\text{\bf
e}_{x},-i\omega)+K_{22}(q\text{\bf e}_{x},-i\omega))\right].$ (201)

Within the mode-coupling approximation, the first two terms on the right hand
side of Eq. (201) are $0$. Taking the hydrodynamic limit of the mode-coupling
expression of $K_{ii}$ given by Eq. (75), we obtain

$\begin{split}\eta_{\text{odd}}&=-\frac{\gamma}{2}\lim_{q\to 0}\lim_{\omega\to
0}\frac{\rho_{0}}{q^{2}D_{0}\beta}\left[K_{11}(q\text{\bf
e}_{x},-i\omega)+K_{22}(q\text{\bf e}_{x},-i\omega)\right]\\\
&=-\frac{\gamma}{4\beta}\int_{0}^{+\infty}\differential t\int_{\text{\bf
k}}\left[[\rho_{0}c(k)+k_{x}\hat{k}_{x}\rho_{0}c^{\prime}(k)]^{2}+[\hat{k}_{x}k_{y}\rho_{0}c^{\prime}(k)]^{2}\right]S(\text{\bf
k},t)^{2}.\end{split}$ (202)

As the glass transition is approached the odd viscosity diverges. Physically,
this means that in the glassy state no form of transport is possible, and the
transverse, as well as the longitudinal, dynamical pathways through which the
liquid relaxes, are blocked. This directly impacts all viscosity coefficients.

##  VII Lifting: sampling the Boltzmann distribution with active particles

The term lifting is used in the Markov Chain Monte Carlo literature to denote
a sampling technique used to extract configurations from a given steady state
target distribution, see [65] for a pedagogical review. In a lifting scheme,
the number of degrees of freedom of the system is extended, and transitions
involving the additional degrees of freedom are allowed. By exploiting the
extended nature of the phase space, one can perform irreversible transitions,
while preserving the target distribution in the steady state. The
nonequilibrium nature of the dynamics can be exploited to obtain a convergence
towards the equilibrium steady state which is faster than conventional
equilibrium methods, such as the Metropolis-Hastings algorithm.

Lifted schemes have been employed in a variety of contexts: in the Ising model
in mean-field [66, 67] and in one-dimension [68], in hard spheres in one [69]
and two dimensions (where it is named Event Chain Monte Carlo method [70,
71]), in lattice random walkers [72], and glassy hard disks [73]. Based on
these examples, it appears that the performance is remarkable even when the
system evolves in a not-too-strongly convex potential. This is what occurs
when the lifted degrees of freedom are coupled with the order parameter close
to a second-order phase transition.

Lifted dynamics can be described in terms run and tumble particles with a
properly tuned tumbling rate. By exploiting this connection, we now present a
model of lifted dynamics inspired by Active Brownian Particles (ABPs). While
the mode-coupling theory of the non-lifted counterpart of these active system
has been explored recently [12, 13, 14], here we focus on systems of active
particles specifically designed to sample the Boltzmann distribution, and we
compare this model with the overdamped equilibrium dynamics counterpart. The
results obtained will also make clear that the performance of lifted schemes
in supercooled liquids is similar to the one obtained through the transverse
forces approach studied in the first part of this article, thus strengthening
the connection between lifting and transverse forces.

###  VII.1 Lifted active Brownian particles (ABP)

The lifted-ABP dynamics of $N$ particles in two dimensions is given by

$\begin{cases}\dot{\text{\bf r}}_{i}&=v_{0}\bm{\widehat{\textbf{u}}}_{i}\\\
\dot{\theta}_{i}&=v_{0}\beta\text{\bf F}_{i}\cdot\text{\bf
A}\bm{\widehat{\textbf{u}}}_{i}+\sqrt{2D_{r}}\chi_{i},\end{cases}$ (203)

where $\bm{\widehat{\textbf{u}}}_{i}=(\cos{\theta_{i}},\sin{\theta_{i}})$ is a
unit vector denoting the direction of the self-propulsion speed of particle
$i$. The vector $\bm{\widehat{\textbf{u}}}_{i}$ forms an angle $\theta_{i}$
with respect to the $x$-axis. $\chi_{i}$ is a Gaussian white noise,
$\langle\chi_{i}(t)\chi_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})$.
The self-propulsion orientation is subjected to diffusion with a diffusion
constant $D_{r}$ as for standard ABPs, but is in addition subjected to a drift
that depends on the interaction force $\text{\bf
F}_{i}=-\sum_{i}\bm{\nabla}_{j\neq i}V(\text{\bf r}_{i}-\text{\bf r}_{j})$
between the particles in position space. The latter enforces the stationarity
of the Boltzmann distribution in the steady state because the matrix
$\text{\bf A}\equiv\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}$ ensures that the
drift is always perpendicular to $\bm{\widehat{\textbf{u}}}_{i}$. The drift
term makes the ABP equations of motion non-conventional, hence the ‘lifted-
ABP’ name for Eq. (203).

In the absence of interactions when $\text{\bf F}_{i}=0$, Eq. (203) describes
freely diffusing ABPs, with diffusion constant $\frac{v_{0}^{2}}{2D_{r}}$.
When interactions are added, lifted-ABPs sample the equilibrium Boltzmann
distribution of the interacting system at fixed temperature $T$, and this can
be done at various values of $\frac{v_{0}^{2}}{2D_{r}}$. This is to be
compared with the equilibrium dynamics where temperature simultaneously fixes
the Boltzmann distribution and the free diffusion constant $D_{0}=\mu_{0}T$.

The evolution of the probability distribution $\rho(\text{\bf
r}^{N},\theta^{N},t)$ of lifted-ABPs is governed by the operator
$\Omega_{v_{0}}$:

$\begin{split}\Omega_{v_{0}}\equiv&-\sum_{i}\bm{\nabla}_{i}\cdot
v_{0}\bm{\widehat{\textbf{u}}}_{i}\\\
&+\sum_{i}\partial_{\theta_{i}}\left[-v_{0}\beta\text{\bf F}_{i}\cdot\text{\bf
A}\bm{\widehat{\textbf{u}}}_{i}+D_{r}\partial_{\theta_{i}}\right],\end{split}$
(204)

so that

$\partial_{t}\rho(\text{\bf r}^{N},\theta^{N},t)=\Omega_{v_{0}}\rho(\text{\bf
r}^{N},\theta^{N},t).$ (205)

The steady state solution of Eq. (205) is a product of independent
distributions in the respective subspaces spanned by $\text{\bf r}^{N}$ and by
$\text{\bf u}^{N}$:

$\rho_{\text{ss}}=\frac{1}{(2\pi)^{N}}\rho_{\text{\tiny{B}}}(\text{\bf
r}^{N})=\frac{1}{Z}e^{-\beta\sum_{i<j}V(\text{\bf r}_{i}-\text{\bf r}_{j})}.$
(206)

with $Z\equiv(2\pi)^{N}\int d\text{\bf r}^{N}e^{-\beta\sum_{i<j}V(\text{\bf
r}_{i}-\text{\bf r}_{j})}$.

The operator $\Omega_{v_{0}}$ has the following property, which is a
reflection of the breaking of detailed balance reminiscent of Eq. (47) for
transverse forces:

$\Omega_{v_{0}}f(\text{\bf
r}^{N},\bm{\widehat{\textbf{u}}}^{N})\Biggr{\rangle}=\left(\Omega^{\dagger}_{-v_{0}}f(\text{\bf
r}^{N},\bm{\widehat{\textbf{u}}}^{N})\right)\Biggr{\rangle}.$ (207)

We are interested in the evolution of the density mode $n(\text{\bf q})$:

$n(\text{\bf q},t)\equiv\sum_{i}\text{e}^{-i\text{\bf q}\cdot\text{\bf
r}_{i}(t)}$ (208)

and of its correlation function, the dynamical structure factor $S(\text{\bf
q},t)$:

$S(\text{\bf q},t)\equiv\frac{1}{N}\left\langle n^{*}(\text{\bf q})n(\text{\bf
q},t)\right\rangle.$ (209)

The following identity holds when inserted into an average:

$\partial_{t}n(\text{\bf q},t)=\Omega^{\dagger}_{v_{0}}n(\text{\bf q},t),$
(210)

so that $n(\text{\bf q},t)=\text{e}^{\Omega^{\dagger}_{v_{0}}}n(\text{\bf
q},t)$. This, together with Eq. (207) allows us to write

$S(\text{\bf q},t)=\frac{1}{N}\left\langle n^{*}(\text{\bf
q})\text{e}^{\Omega_{-v_{0}}t}n(\text{\bf q})\right\rangle.$ (211)

Another important quantity is the time derivative of the density mode at
$t=0$:

$\partial_{t}n(\text{\bf q})=-iv_{0}q\hat{u}_{\parallel}(\text{\bf q}),$ (212)

which is the projection of the velocity $\bm{\widehat{\textbf{u}}}_{i}$ along
the wavevector q:

$\hat{u}_{\parallel}(\text{\bf
q})\equiv\sum_{i}\left(\bm{\widehat{\textbf{q}}}\cdot\bm{\widehat{\textbf{u}}}_{i}\right)\text{e}^{-i\text{\bf
q}\cdot\text{\bf r}_{i}}.$ (213)

We are also interested in the self-intermediate scattering function
$F_{s}(\text{\bf q},t)$. Here we re-write the definition (50) by putting the
time dependence entirely into $n_{i}(\text{\bf q},t)$,

$F_{s}(\text{\bf q},t)\equiv\frac{1}{N}\sum_{i=1}^{N}\left\langle
n^{*}_{i}(\text{\bf q})n_{i}(\text{\bf q},t)\right\rangle.$ (214)

We have $\partial_{t}n_{i}(\text{\bf q})=-iv_{0}qu_{\parallel,i}(\text{\bf
q})$, with

$\hat{u}_{\parallel,i}(\text{\bf
q})\equiv\bm{\widehat{\textbf{q}}}\cdot\bm{\widehat{\textbf{u}}}_{i}\text{e}^{-i\text{\bf
q}\cdot\text{\bf r}_{i}}$ (215)

encoding the Fourier transform of the longitudinal self-propulsion velocity
with respect to the wavevector q. The small-$q$ expansion of the self-
intermediate scattering function yields the mean-squared displacement

$\Delta(t)\equiv\sum_{i=1}^{N}\left\langle\left[\text{\bf r}_{i}(t)-\text{\bf
r}_{i}(0)\right]^{2}\right\rangle.$ (216)

Using the isotropy of the dynamics given by Eq. (203) we can also connect $F$
and $\Delta$:

$F_{s}(\text{\bf q},t)=1-\frac{q^{2}}{4}\Delta(t)+O(q^{3}).$ (217)

An equation for $\Delta$ is obtained at the end of the section.

####  VII.1.1 Short-time dynamics

To gain insights on the dynamics we consider a short-time expansion of
$S(\text{\bf q},t)$:

$\displaystyle S(\text{\bf q},t)$ $\displaystyle\approx
S(q)+\frac{t}{N}\left\langle n^{*}(\text{\bf q})\Omega_{-v_{0}}n(\text{\bf
q})\right\rangle$ (218) $\displaystyle+\frac{t^{2}}{2N}\left\langle
n^{*}(\text{\bf q})\Omega_{-v_{0}}\Omega_{-v_{0}}n(\text{\bf q})\right\rangle$
$\displaystyle=S(q)-iv_{0}q\left\langle n^{*}(\text{\bf
q})\hat{u}_{\parallel}(\text{\bf q})\right\rangle$
$\displaystyle-\frac{t^{2}}{2}v_{0}^{2}q^{2}\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})\hat{u}_{\parallel}(\text{\bf q})\right\rangle.$

The second term on the right hand side of Eq. (218) vanishes due to rotational
invariance. For the third term we have, using Einstein notation for the
components of the vectors $\bm{\widehat{\textbf{q}}}$ and
$\bm{\widehat{\textbf{u}}}_{i}$,

$\displaystyle\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})\hat{u}_{\parallel}(\text{\bf q})\right\rangle$
$\displaystyle=\left\langle\sum_{i,j}\hat{u}_{i,\alpha}\hat{u}_{j,\beta}\hat{q}_{\alpha}\hat{q}_{\beta}\text{e}^{-i\text{\bf
q}\cdot\left(\text{\bf r}_{i}-\text{\bf r}_{j}\right)}\right\rangle$ (219)
$\displaystyle=\sum_{i,j}\left\langle\hat{u}_{\alpha,i}\hat{u}_{\beta,j}\right\rangle\left\langle\delta_{i,j}\delta_{\alpha\beta}\hat{q}_{\alpha}\hat{q}_{\beta}\text{e}^{-i\text{\bf
q}\cdot\left(\text{\bf r}_{i}-\text{\bf r}_{j}\right)}\right\rangle$
$\displaystyle=\frac{N}{2},$

and therefore we obtain

$S(\text{\bf q},t)=S(q)-\frac{v_{0}^{2}q^{2}}{4}t^{2}+O(t^{3}).$ (220)

This yields two results. First, we see that increasing $v_{0}$ produces a
faster decay of the dynamic structure factor at short times. Moreover, in
contrast with equilibrium overdamped Brownian dynamics, the decrease is
quadratic in time (in line with the ballistic nature of the dynamics at short
times), instead of being linear. This suggests that we should study the
dynamics of $S(\text{\bf q},t)$ to second order in time, as done in the next
section.

####  VII.1.2 Projection operator formalism

We want to find an evolution equation for $S(\text{\bf q},t)$. The starting
point is to write its second time derivative:

$\displaystyle\partial_{t}^{2}S(\text{\bf q},t)$
$\displaystyle=\frac{1}{N}\left\langle n^{*}(\text{\bf
q})\Omega_{-v_{0}}\text{e}^{\Omega_{-v_{0}}t}\Omega_{-v_{0}}n(\text{\bf
q})\right\rangle$ (221)
$\displaystyle=-\frac{iv_{0}q}{N}\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})\Omega_{-v_{0}}\text{e}^{\Omega_{-v_{0}}t}n(\text{\bf q})\right\rangle.$

By taking the Laplace Transform on both sides of Eq. (221) we get

$z^{2}S(\text{\bf
q},z)-z=-\frac{iv_{0}q}{N}\left\langle\widehat{u}^{*}_{\parallel}(\text{\bf
q})\Omega_{-v_{0}}\frac{1}{z-\Omega_{-v_{0}}}n(\text{\bf q})\right\rangle.$
(222)

We now have to choose a projection operator $\mathcal{P}$ tailored to the
relevant degrees of freedom. We assume these relevant modes to be the density
field and its time derivative, which leads to the following expression for
$\mathcal{P}$:

$\mathcal{P}\equiv\frac{1}{NS(q)}n(\text{\bf q})\rangle\langle n^{*}(\text{\bf
q})+\frac{2}{N}\hat{u}_{\parallel}(\text{\bf
q})\rangle\langle\hat{u}^{*}_{\parallel}(\text{\bf q}).$ (223)

With this choice and a resolvent identity akin to Eq. (54) previously used for
transverse forces

$\begin{split}&\frac{1}{z-\Omega_{-v_{0}}}=\frac{1}{1-\mathcal{Q}\Omega_{-v_{0}}\mathcal{Q}}\\\
&+\left(1+\frac{1}{z-\mathcal{Q}\Omega_{-v_{0}}\mathcal{Q}}\Omega_{-v_{0}}\right)\mathcal{P}\frac{1}{z-\Omega_{-v_{0}}}\mathcal{P}\\\
&\times\left(1+\Omega_{-v_{0}}\frac{1}{z-\mathcal{Q}\Omega_{-v_{0}}\mathcal{Q}}\right),\end{split}$
(224)

Eq. (222) becomes

$\displaystyle z^{2}S(\text{\bf q},z)$ $\displaystyle-
zS(q)=-\frac{iv_{0}q}{N}\Biggl{[}\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})\Omega_{-v_{0}}n(\text{\bf q})\right\rangle$ (225)
$\displaystyle\times\frac{1}{NS(q)}\left\langle n^{*}(\text{\bf
q})R(z)n(\text{\bf q})\right\rangle$
$\displaystyle+\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})\Omega_{-v_{0}}\hat{u}_{\parallel}(\text{\bf
q})\right\rangle\frac{2}{N}\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})R(z)n(\text{\bf q})\right\rangle$
$\displaystyle+\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})\Omega_{-v_{0}}\mathcal{Q}R_{\mathcal{Q}}(z)\mathcal{Q}\Omega_{-v_{0}}\hat{u}_{\parallel}(\text{\bf
q})\right\rangle$
$\displaystyle\times\frac{2}{N}\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})R(z)n(\text{\bf q})\right\rangle\Biggr{]}.$

An explicit computation gives:

$\displaystyle\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})\Omega_{-v_{0}}n(\text{\bf q})\right\rangle$
$\displaystyle=-i\frac{v_{0}qN}{2},$ (226)
$\displaystyle\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})\Omega_{-v_{0}}\hat{u}_{\parallel}(\text{\bf q})\right\rangle$
$\displaystyle=-\frac{N}{2}D_{r},$
$\displaystyle-\frac{iv_{0}q}{N}\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})\frac{1}{z-\Omega_{-v_{0}}}n(\text{\bf q})\right\rangle$
$\displaystyle=zS(\text{\bf q},z)-S(q).$

Putting everything together and taking an inverse Laplace transform yields

$\begin{split}\partial^{2}_{t}S(\text{\bf
q},t)&=-\frac{v_{0}^{2}q^{2}}{2S(q)}S(\text{\bf
q},t)-D_{r}\partial_{t}S(\text{\bf q},t)\\\ &-\int_{0}^{t}\differential\tau
M(\text{\bf q},t-\tau)\partial_{\tau}S(\text{\bf q},\tau).\end{split}$ (227)

This is an evolution equation for the dynamical structure factor for the
lifted-ABP dynamics. It allows for underdamped oscillations in the presence of
a memory kernel given by

$M(\text{\bf
q},t)\equiv-\frac{2}{N}\left\langle\hat{u}^{*}_{\parallel}(\text{\bf
q})\Omega_{-v_{0}}\mathcal{Q}e^{\mathcal{Q}\Omega_{-v_{0}}\mathcal{Q}t}\mathcal{Q}\Omega_{-v_{0}}\hat{u}_{\parallel}(\text{\bf
q})\right\rangle.$ (228)

With a similar approach, one can derive an equation for $F_{s}(\text{\bf
q},t)$. In practice the step above must be repeated using the single particle
projection operator

$\mathcal{P}_{i}\equiv n_{i}(\text{\bf q})\rangle\langle n^{*}_{i}(\text{\bf
q})+2u_{\parallel,i}(\text{\bf
q})\rangle\langle\hat{u}^{*}_{\parallel,i}(\text{\bf q}),$ (229)

instead of $\mathcal{P}$ and employ particle equivalence. The result is

$\begin{split}\partial^{2}_{t}F_{s}(\text{\bf
q},t)&=-\frac{v_{0}^{2}q^{2}}{2}F_{s}(\text{\bf
q},t)-D_{r}\partial_{t}F_{s}(\text{\bf q},t)\\\
&-\int_{0}^{t}\differential\tau M_{s}(\text{\bf
q},t-\tau)\partial_{\tau}F_{s}(\text{\bf q},\tau),\end{split}$ (230)

with the self-memory kernel defined as

$M_{s}(\text{\bf
q},t)\equiv-\frac{2}{N}\sum_{i=1}^{N}\left\langle\hat{u}^{*}_{\parallel,i}(\text{\bf
q})\Omega_{-v_{0}}\mathcal{Q}\text{e}^{\mathcal{Q}\Omega_{-v_{0}}\mathcal{Q}t}\mathcal{Q}\Omega_{-v_{0}}\hat{u}_{\parallel,i}(\text{\bf
q})\right\rangle.$ (231)

All the steps performed so far are exact. To proceed, approximations are
needed to compute $M$ and $M_{s}$. In the next section, we approximate the
memory kernels using the mode-coupling scheme.

####  VII.1.3 Mode-coupling expansion

We expand the memory kernel as a product of density mode correlations. The
physical justification for this choice is that the non-trivial part in the
time evolution of $\hat{u}_{\parallel}(\text{\bf q})$ comes from the coupling
with the fluid, which involves pairwise forces between particles. We therefore
write

$\begin{split}\mathcal{Q}\Omega_{-v_{0}}\hat{u}_{\parallel}(\text{\bf
q})\Biggr{\rangle}&=\frac{1}{2}\sum_{\text{\bf k}}\frac{\left\langle
n^{*}(\text{\bf k})n^{*}(\text{\bf q}-\text{\bf
k})\mathcal{Q}\Omega_{-v_{0}}\hat{u}_{\parallel}(\text{\bf
q})\right\rangle}{N^{2}S(k)S(\lvert\text{\bf q}-\text{\bf k}\rvert)}\\\
&n(\text{\bf k})n(\text{\bf q}-\text{\bf k})\Biggr{\rangle},\end{split}$ (232)

with the factor $\frac{1}{2}$ inserted to avoid double counting of the density
modes products. We now calculate the expectation value on the right hand side
of Eq. (232) thanks to the convolution approximation $\left\langle
n^{*}(\text{\bf k})n^{*}(\text{\bf q}-\text{\bf k})n(\text{\bf
q})\right\rangle\approx NS(q)S(k)S(\lvert\text{\bf q}-\text{\bf k}\rvert)$:

$\displaystyle\left\langle n^{*}(\text{\bf k})n^{*}(\text{\bf q}-\text{\bf
k})\mathcal{Q}\Omega_{-v_{0}}\hat{u}_{\parallel}(\text{\bf q})\right\rangle$
(233) $\displaystyle=\left\langle n^{*}(\text{\bf k})n^{*}(\text{\bf
q}-\text{\bf k})\Omega_{-v_{0}}\hat{u}_{\parallel}(\text{\bf q})\right\rangle$
$\displaystyle-\left\langle n^{*}(\text{\bf k})n^{*}(\text{\bf q}-\text{\bf
k})\mathcal{P}\Omega_{-v_{0}}\hat{u}_{\parallel}(\text{\bf q})\right\rangle$
$\displaystyle=\left\langle\left[\Omega_{-v_{0}}^{\dagger}\left(n^{*}(\text{\bf
k})n^{*}(\text{\bf q}-\text{\bf k})\right)\right]\hat{u}_{\parallel}(\text{\bf
q})\right\rangle$ $\displaystyle-\left\langle n^{*}(\text{\bf
k})n^{*}(\text{\bf q}-\text{\bf k})n(\text{\bf
q})\right\rangle\frac{1}{NS(q)}\left\langle n^{*}(\text{\bf
q})\Omega_{-v_{0}}\hat{u}_{\parallel}(\text{\bf q})\right\rangle$
$\displaystyle=-iv_{0}\bigg{\langle}\sum_{i,j,k}(\hat{u}_{i,\alpha}\hat{q}_{\alpha})\left[\hat{u}_{j,\beta}k_{\beta}+\hat{u}_{k,\beta}(q_{\beta}-k_{\beta})\right]$
$\displaystyle\times\text{e}^{i\text{\bf k}\cdot\text{\bf
r}_{j}}\text{e}^{i(\text{\bf q}-\text{\bf k})\cdot\text{\bf
r}_{k}}\text{e}^{-i\text{\bf q}\cdot\text{\bf
r}_{i}}\bigg{\rangle}+i\frac{Nv_{0}q}{2}S(k)S(\lvert\text{\bf q}-\text{\bf
k}\rvert)$ $\displaystyle=i\frac{Nv_{0}}{2}\bigg{[}qS(k)S(\lvert\text{\bf
q}-\text{\bf k}\rvert)$
$\displaystyle-(\bm{\widehat{\textbf{q}}}\cdot\text{\bf k})S(\lvert\text{\bf
q}-\text{\bf k}\rvert)-\bm{\widehat{\textbf{q}}}\cdot(\text{\bf q}-\text{\bf
k})S(k)\bigg{]}$ $\displaystyle=\
i\frac{Nv_{0}\rho_{0}}{2}\left[(\bm{\widehat{\textbf{q}}}\cdot\text{\bf
k})c(k)+\bm{\widehat{\textbf{q}}}\cdot(\text{\bf q}-\text{\bf
k})c(\lvert\text{\bf q}-\text{\bf k}\rvert)\right],$

where the direct correlation function $\rho_{0}c(q)$ and the particle density
of the system $\rho_{0}$ were defined below Eq. (154) and above Eq. (12),
respectively.

We can now give the expression for the mode-coupling memory kernel. To this
end, we resort to the approximation

$\text{e}^{\mathcal{Q}\Omega_{-v_{0}}\mathcal{Q}t}\approx\text{e}^{\Omega_{-v_{0}}t},$
(234)

and use a Gaussian factorization for computing the two-body density
correlation. Moreover, due to the irreversibility of the dynamics, we have

$\left\langle n^{*}(\text{\bf k})n^{*}(\text{\bf q}-\text{\bf
k})\mathcal{Q}\Omega_{-v_{0}}\hat{u}_{\parallel}(\text{\bf
q})\right\rangle=\left\langle\hat{u}_{\parallel}(\text{\bf
q})\Omega_{-v_{0}}\mathcal{Q}n(\text{\bf k})n(\text{\bf q}-\text{\bf
k})\right\rangle.$ (235)

The memory kernel reads, in the mode-coupling approximation

$\begin{split}M(\text{\bf
q},t)&\approx\frac{v_{0}^{2}\rho_{0}}{4}\int\frac{d\text{\bf
k}}{(2\pi)^{2}}\\\ &\times\left[(\bm{\widehat{\textbf{q}}}\cdot\text{\bf
k})c(k)+\bm{\widehat{\textbf{q}}}\cdot(\text{\bf q}-\text{\bf
k})c(\lvert\text{\bf q}-\text{\bf k}\rvert)\right]^{2}\\\ &\times S(\text{\bf
k},t)S(\text{\bf q}-\text{\bf k},t).\end{split}$ (236)

A similar expression can be obtained for $M_{s}$, using as a starting point
the expansion

$\begin{split}\mathcal{Q}\Omega_{-v_{0}}\hat{u}_{\parallel,i}(\text{\bf
q})\Biggr{\rangle}&=\sum_{\text{\bf k}}\frac{\left\langle n^{*}(\text{\bf
k})n_{i}^{*}(\text{\bf q}-\text{\bf
k})\mathcal{Q}\Omega_{-v_{0}}\hat{u}_{\parallel,i}(\text{\bf
q})\right\rangle}{NS(k)}\\\ &\times n(\text{\bf k})n_{i}(\text{\bf
q}-\text{\bf k})\Biggr{\rangle}.\end{split}$ (237)

The final result reads

$M_{s}(\text{\bf q},t)\approx\frac{v_{0}^{2}\rho_{0}}{2}\int\frac{d\text{\bf
k}}{(2\pi)^{2}}\left[\bm{\widehat{\textbf{q}}}\cdot\text{\bf
k}c(k)\right]^{2}F(\lvert\text{\bf q}-\text{\bf k}\rvert,t)S(\text{\bf k},t).$
(238)

Note that the structure of the memory kernels $M$ and $M_{s}$ is the same as
the one obtained for equilibrium dynamics, their magnitude being now
determined by the magnitude of the self-propulsion speed $v_{0}$.

In the next sections we study the ergodicity breaking transition and the
behavior of the diffusion constant stemming from these approximate expressions
for the memory kernels.

####  VII.1.4 Long-time dynamics: acceleration and ergodicity breaking

To study the long-time dynamics we introduce the normalized dynamical
structure function $\phi(\text{\bf q},t)\equiv\frac{S(\text{\bf q},t)}{S(q)}$.
Its equation of motion reads

$\begin{split}\partial^{2}_{t}\phi(\text{\bf
q},t)&=-\frac{v_{0}^{2}q^{2}}{2S(q)}\phi(\text{\bf
q},t)-D_{r}\partial_{t}\phi(\text{\bf q},t)\\\ &-\int_{0}^{t}\differential\tau
M(\text{\bf q},t-\tau)\partial_{\tau}\phi(\text{\bf q},\tau).\end{split}$
(239)

We first study the ergodicity breaking transition. We define the long-time
limits of the normalized structure factor and the memory kernel

$\begin{split}\lim_{t\to+\infty}\phi(\text{\bf q},t)&=\phi_{\infty}(\text{\bf
q}),\\\ \lim_{t\to+\infty}M(\text{\bf q},t)&=M_{\infty}(\text{\bf
q}).\end{split}$ (240)

In the long-time limit, Eq. (239) becomes a self-consistent equation for the
plateau value of the correlation function:

$\begin{split}\frac{\phi_{\infty}(\text{\bf q})}{1-\phi_{\infty}(\text{\bf
q})}&=\frac{2S(q)\rho_{0}}{q^{2}}\int\frac{d\text{\bf k}}{(2\pi)^{2}}\\\
&\times\left[(\bm{\widehat{\textbf{q}}}\cdot\text{\bf
k})c(k)+\bm{\widehat{\textbf{q}}}\cdot(\text{\bf q}-\text{\bf
k})c(\lvert\text{\bf q}-\text{\bf k}\rvert)\right]^{2}\\\
&\times\phi_{\infty}(\text{\bf k})\phi_{\infty}(\text{\bf q}-\text{\bf
k}).\end{split}$ (241)

This equation is the same as the one obtained for overdamped equilibrium
dynamics [54]. It follows that ergodicity breaking is expected to occur at the
same critical density as in equilibrium.

To study the relaxation of the system in the ergodic regime, we introduce a
relaxation time

$\tau\equiv\int_{0}^{+\infty}\differential t\phi(\text{\bf q},t).$ (242)

Integration of Eq. (239) in $t$ from $0$ to $+\infty$ gives

$\tau=\frac{2S(q)D_{r}}{v_{0}^{2}q^{2}}+\frac{2S(q)\rho_{0}}{q^{2}}\int_{0}^{+\infty}\differential
t\int\frac{d\text{\bf
k}}{(2\pi)^{2}}\left[(\bm{\widehat{\textbf{q}}}\cdot\text{\bf
k})c(k)+\bm{\widehat{\textbf{q}}}\cdot(\text{\bf q}-\text{\bf
k})c(\lvert\text{\bf q}-\text{\bf k}\rvert)\right]^{2}\phi(\text{\bf
k},t)\phi(\text{\bf q}-\text{\bf k},t).$ (243)

This expression for $\tau$ is formally identical to the equilibrium case with
the bare diffusion coefficient $D_{0}$ now being replaced with
$\frac{v_{0}^{2}}{2D_{r}}=D_{0}$. While $D_{0}$ is slaved to the temperature
in equilibrium, $\frac{v_{0}^{2}}{2D_{r}}$ can be independently varied in the
lifted-ABP model, potentially leading to an accelerated dynamics.

####  VII.1.5 Diffusion constant

We obtain an equation of motion for the mean-squared displacement $\Delta(t)$
by substituting Eq. (217) in Eq. (230) and retaining only the leading term in
the limit $q\rightarrow 0$:

$\partial^{2}_{t}\Delta(t)=2v_{0}^{2}-D_{r}\partial_{t}\Delta(t)-\int_{0}^{t}d\tau
M_{s}(\text{\bf q}\to 0,t-\tau)\partial_{\tau}\Delta(\tau).$ (244)

We now assume a diffusive behavior at large times: $\Delta(t)\sim 4Dt$.
Substituting into Eq. (244) we obtain an expression for the diffusion constant
$D$:

$D=\frac{v_{0}^{2}}{2D_{r}}\frac{1}{1+\frac{v_{0}^{2}}{2D_{r}}m_{s}^{\infty}},$
(245)

where the memory kernel is given by

$m_{s}^{\infty}=\rho_{0}\int_{0}^{+\infty}\differential t\int\frac{d\text{\bf
k}}{(2\pi)^{2}}\left[\bm{\widehat{\textbf{q}}}\cdot\text{\bf
k}c(k)\right]^{2}F_{s}(k,t)S(k,t).$ (246)

The formal expression of $m_{s}^{\infty}$ is the same as in equilibrium, while
the one for $D$ is also formally similar, with the replacement $D_{0}$ by
$\frac{v^{2}_{0}}{2D_{r}}$. In the ergodic phase when $m_{s}^{\infty}$ is
finite, the diffusion constant $D$ can be enhanced with respect to equilibrium
by increasing $\frac{v^{2}_{0}}{2D_{r}}>D_{0}$, which plays the role of the
driving force. Differently from the transverse force case, the acceleration
survives the high-temperature limit, since $D=\frac{2v_{0}^{2}}{D_{r}}$ in
this limit. In the opposite limit of temperature approaching the dynamic
transition, $m_{s,\infty}$ diverges and thus $D$ exactly recovers its
equilibrium expression and the lifted-ABP dynamics is no longer felt. This
suggests that lifted dynamics acceleration plummets as the kinetic glass
transition is approached. This is in line with recent numerical results [73].

##  VIII Conclusion

The irreversible sampling of the Boltzmann distribution can generically be
faster than a conventional equilibrium dynamics. Our goal in this work was to
study theoretically the acceleration and the microscopic relaxation dynamics
obtained in the case of dense liquids approaching the glass transition when
irreversible dynamics are used [37, 38, 73]. Unlike our precedent effort [39]
where the problem was analysed exactly in the mean-field limit of liquids in
large dimensions, we developed here an approximate mode-coupling approach to
study the dynamics directly in finite dimension. These two approaches are thus
complementary [74].

To this end, we developed a novel mode-coupling approach specifically tailored
to address nonreciprocal transverse forces, which can be seen as one of the
simplest nonequilibrium drive achieving acceleration. Technically, this has
required a careful treatment of transverse and longitudinal modes. Physically,
we concluded that the location of the ergodicity breaking transition predicted
to occur in equilibrium without transverse forces is never affected by
transverse forces. However, the dynamics in the ergodic phase is
systematically accelerated by transverse forces, with a non-trivial
temperature dependence. These results are fully consistent with both the large
dimensional results [39] and computer simulations [38].

The emerging physical picture is that longitudinal diffusion gets arrested for
the same value of the control parameters as in equilibrium dynamics, but the
energy injected by the transverse forces is dissipated in an ever-going odd
motion, that we picture as a swirling motion within an environment that
becomes increasingly confining as the temperature is lowered. Approaching the
kinetic glass transition, the odd diffusion saturates to a constant value, the
odd mobility vanishes, and the odd viscosity diverges.

In a recent work [38] we further argued that transverse forces share the same
accelerating physical ingredient as their lifted counterparts in which the
nonequilibrium drive results from coupling the dynamics to extra degrees of
freedom. Our explicit mode-coupling approach clearly supports this intuition.
For the lifted processes explicitly considered in our final section, the
physical conclusions echo those drawn from considering transverse forces. We
believe that this analogy is a credible explanation of the observations of
[73] showing that event-chain Monte Carlo methods lose their edge over
conventional Metropolis Monte Carlo as the fluid enters more deeply into its
glassy regime.

Our work opens at least two immediate research directions. The first one has
to do with the collective behavior of dense assemblies of chiral active
particles [75, 76], for which a mode-coupling analysis will surely closely
follow the footsteps developed in our work. The second one is more subtle: it
is concerned with the numerical integration of the derived mode-coupling
equations. This will surely be a highly nontrivial task due to the loss of
isotropy. The absence of a fluctuation-dissipation theorem is additionally
responsible for increasing the number of coupled self-consistent integro-
differential equations to solve, which will add to the complexity of the
numerical solution.

###### Acknowledgements.

LB, FG and FvW acknowledge the financial support of the ANR THEMA AAPG2020
grant. GS acknowledges the support of NSF Grant No. CHE 2154241.

##  Appendix A Irreducible memory kernel

In this appendix we justify the choice for the irreducible memory kernel made
in Eq. (63). We first review how the irreducible memory kernel is introduced
in the equilibrium case, when $\gamma=0$. Following Kawasaki [51], the
starting point is the observation that the evolution operator $\Omega\equiv
D_{0}\sum_{i}\bm{\nabla}_{i}\left[\beta\text{\bf
F}_{i}+\bm{\nabla}_{i}\right]$ can be mapped to a Hermitian operator
$H\equiv\text{e}^{\frac{\beta\mathcal{H}}{2}}\Omega\text{e}^{\frac{-\beta\mathcal{H}}{2}}$.
It can be written in a manifestly Hermitian form as

$H=-\sum_{i}\text{\bf U}_{i}^{\dagger}\cdot\text{\bf U}_{i},$ (247)

with $\text{\bf
U}_{i}\equiv\sqrt{D_{0}}\left[-\bm{\nabla}_{i}+\frac{\beta}{2}\text{\bf
F}_{i}\right]$. Going back to the non-Hermitian representation we see that

$\Omega=-\sum_{i}\text{e}^{\frac{-\beta\mathcal{H}}{2}}\text{\bf
U}_{i}^{\dagger}\text{e}^{\frac{\beta\mathcal{H}}{2}}\cdot\text{e}^{\frac{-\beta\mathcal{H}}{2}}\text{\bf
U}_{i}\text{e}^{\frac{\beta\mathcal{H}}{2}}\equiv-\sum_{i}\text{\bf
O}^{\times}_{i}\cdot\text{\bf O}_{i},$ (248)

with

$\begin{split}\text{\bf O}_{i}^{\times}&=\sqrt{D_{0}}\bm{\nabla}_{i},\\\
\text{\bf O}_{i}&=\sqrt{D_{0}}\left[-\bm{\nabla}_{i}+\beta\text{\bf
F}_{i}\right].\end{split}$ (249)

At this point, one can insert any generic projection operator
$\mathcal{P}_{i}$ and its orthogonal counterpart
$\mathcal{Q}_{i}\equiv\mathcal{I}-\mathcal{P}_{i}$. For consistency with the
notation of the main text, we also enclose $\Omega$ between the orthogonal
projector $\mathcal{Q}=\mathcal{I}-\mathcal{P}$, defined from Eq. (53):

$\begin{split}\mathcal{Q}\Omega\mathcal{Q}&=-\mathcal{Q}\left(\sum_{i}\text{\bf
O}^{\times}_{i}\mathcal{P}_{i}\cdot\text{\bf O}_{i}-\sum_{i}\text{\bf
O}^{\times}_{i}\mathcal{Q}_{i}\cdot\text{\bf O}_{i}\right)\mathcal{Q}\\\
&\equiv-\delta\Omega+\Omega^{\text{irr}}.\end{split}$ (250)

What remains to do is to properly choose the projection operator
$\mathcal{P}$, in such a way that a renormalization of the memory kernel
occurs: $\widetilde{M}(q,z)=\frac{M(q,z)}{1+cM(q,z)}$, with $c$ some
$q$-dependent constant. Following Cichocki and Hess [53] we take

$\mathcal{P}_{i}\equiv\ldots\text{e}^{-i\text{\bf q}\cdot\text{\bf
r}_{i}}\rangle\langle\text{e}^{i\text{\bf q}\cdot\text{\bf r}_{i}}\ldots,$
(251)

leading to the following expression for $\delta\Omega$

$\delta\Omega=\frac{D_{0}\beta^{2}}{N}\ldots\mathcal{Q}\text{\bf f}(\text{\bf
q})\rangle\cdot\langle\text{\bf f}^{*}(\text{\bf q})\mathcal{Q}\ldots$ (252)

Note that this is different from the choice of Kawasaki, which is instead

$\mathcal{P}_{j}^{\text{Kawasaki}}=\ldots\text{\bf O}_{j}\text{e}^{-i\text{\bf
q}\cdot\text{\bf r}_{j}}\rangle\frac{1}{D_{0}q^{2}}\langle\text{e}^{i\text{\bf
q}\cdot\text{\bf r}_{j}}\text{\bf O}^{\times}_{j}\ldots$ (253)

which leads to, using the fact that particles are equivalent,

$\begin{split}\delta\Omega^{\text{Kawasaki}}&=\ldots\Omega n(\text{\bf
q})\rangle\frac{1}{\langle n^{*}(\text{\bf q})\Omega n(\text{\bf
q})\rangle}\langle n^{*}(\text{\bf q})\Omega\ldots\\\
&=\frac{D_{0}\beta^{2}}{N}\ldots\mathcal{Q}f_{\parallel}(\text{\bf
q})\rangle\langle f_{\parallel}^{*}(\text{\bf q})\mathcal{Q}\ldots\end{split}$
(254)

We see that in equilibrium, the operator $\delta\Omega^{\text{Kawasaki}}$
contains only contributions from the longitudinal currents.

We now turn to the general situation with $\gamma\neq 0$. In this case one can
see that an analogous decomposition for $\Omega_{\gamma}$ can be made, with an
extra term to take into account the presence of transverse forces:

$\Omega_{-\gamma}=-\text{\bf O}^{\times}\cdot\left(\mathbf{1}-\gamma\text{\bf
A}\right)\cdot\text{\bf O}.$ (255)

Using the projection operator $P_{i}$ defined in Eq. (251) we obtain

$\begin{split}\Omega^{\text{irr}}_{-\gamma}&=\mathcal{Q}\text{\bf
O}^{\times}\mathcal{Q}_{i}\cdot\left(\mathbf{1}-\gamma\text{\bf
A}\right)\cdot\text{\bf O}\\\
&=D_{0}\mathcal{Q}\sum_{j}\bm{\nabla}_{j}\mathcal{Q}_{j}\cdot\left(\mathbf{1}-\gamma\text{\bf
A}\right)\left[-\beta\text{\bf
F}_{j}+\bm{\nabla}_{j}\right]\mathcal{Q},\end{split}$ (256)

which is Eq. (63) shown in the main text.

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# Tailoring non-Abelian lattice gauge theory for quantum simulation

Indrakshi Raychowdhury<EMAIL_ADDRESS>Indian Association for the
Cultivation of Science, 2A and 2B Raja SC Mullick Road, Jadavpur, Kolkata
700032, India  Jesse R. Stryker<EMAIL_ADDRESS>Institute for Nuclear Theory,
University of Washington, Seattle, WA, 98195-1550, USA

###### Abstract

The prepotential formulation of non-Abelian lattice gauge theories is a
promising paradigm for digital quantum simulation due to its gauge invariant
basis, its integer-valued towers of states, and the simple action of the
Hamiltonian in this basis. In this letter, we introduce matter into a general
framework – valid in any dimension – that casts all Gauss law constraints into
Abelian form. We explicitly solve one problem of practical importance by
constructing Gauss law oracles for SU(2) theories, which can help filter out
unphysical errors in a simulation.

###### pacs:

11.15.Ha, 03.67.Ac, 03.65.Fd

††preprint: INT-PUB-18-062

## Introduction.

Quantum computation is expected to efficiently simulate quantum mechanical
many-body problems Benioff (1980); Feynman (1982); Lloyd (1996); Jordan  _et
al._ (2012), as they can handle the exponential growth of information in
entangled quantum systems that overwhelms classical computers. For lattice
gauge theories (LGTs), quantum computers offer hope for _ab initio_ studies of
non-zero density, topological properties, and real-time phenomena Byrnes and
Yamamoto (2006); Banerjee  _et al._ (2013); Marcos  _et al._ (2013); Mezzacapo
_et al._ (2015); Dalmonte and Montangero (2016); Muschik  _et al._ (2017),
which are exponentially hard to solve classically due to sign problems Troyer
and Wiese (2005); Aarts (2009); AuroraScience Collaboration  _et al._ (2012).
However, with the limitations faced by near-term quantum devices, especially
in the NISQ era Preskill (2018), it is crucial that LGTs be formulated
economically and tailored to the capabilities of quantum devices Tagliacozzo
_et al._ (2014); Zohar and Burrello (2015); Zache  _et al._ (2018); Klco  _et
al._ (2018).
Non-Abelian LGT is conventionally constructed in terms of a local Hamiltonian
and a Hilbert space of group representation states, supplemented by a non-
Abelian Gauss law for gauge invariance Creutz (1984); Smit (2002). States in
the Hilbert space are predominantly unphysical, and a noisy quantum computer
would get lost among them. In order to remove gauge redundancy, one must
impose Gauss’s law, which is nontrivial on a quantum computer as its color
components are not simultaneously diagonalizable. Apart from that, the
different representations to be mapped onto a register of qubits are on
different footings under (and mixed by) the action of the Hamiltonian;
crafting the action of the Hamiltonian in terms of quantum computer operations
– though straightforward in principle – seems rather unnatural to do. In
constrast, the prepotential formulation of LGT uses gauge invariant towers of
states, characterized by integer quantum numbers. The Hamiltonian acts as a
sum of ladder operators on those towers of states, which seems far more
natural to digitize using qubits.
Unlike conventional loop formulations of LGTs, where the manifestly gauge
invariant Hilbert space suffers from non-locality and a proliferation of loop
and string states, prepotentials Mathur (2005, 2006, 2007); Anishetty  _et
al._ (2010); Raychowdhury (2013) give an equivalent formulation of Hamiltonian
LGT in terms of local gauge invariant operators and states. An exact and
orthonormal basis is obtained via a set of local constraints, namely, Abelian
Gauss law constraints and Mandelstam constraints. However, Mandelstam
constraints are nonlinear, which would have been cumbersome to enforce on a
register of qubits. A new variant of the prepotential formulation Raychowdhury
(2018); Anishetty and Sreeraj (2018) circumvents that problem.
In this letter, we construct a theoretical framework based on prepotentials
that could define a whole new paradigm in digital quantum simulations of non-
Abelian gauge theories, making them a reality sooner. We introduce matter into
the already existing prepotential formalism in 2D SU(2) gauge theory and then
generalize to 3D. The local algebraic structure of this framework is equally
simple in any dimension. We particularly show how Gauss law’s and the
Hamiltonian dynamics bear a closer resemblance to their U(1) cousins,
presenting the novel opportunity for SU(2) simulations to benefit from
algorithmic developments made for U(1) theories. As a first exploration of the
advantages, we construct oracles for testing physicality of states that are
strikingly similar to those in Abelian gauge theories Stryker (2018). In 3D,
the SU(2) oracles are actually simpler than their U(1) counterparts.

## Hamiltonian formulation.

A lattice Hamiltonian for SU(2) gauge theory coupled to fundamental matter,
using units where the lattice spacing $a=1$, may be formulated as Kogut and
Susskind (1975)

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\underbrace{g^{2}\sum_{(x,i)}\hat{E}^{2}(x,i)}_{H_{E}}+\underbrace{\tfrac{1}{g^{2}}\sum_{\Box}\text{Tr}(2-\hat{U}_{\Box}-\hat{U}^{\dagger}_{\Box})}_{H_{B}}$
$\displaystyle+\underbrace{\sum_{(x,i)}\hat{\psi}^{\dagger}(x)\hat{U}(x,i)\hat{\psi}(x+e_{i})+m\sum_{x}(-)^{x}\hat{\psi}^{\dagger}(x)\hat{\psi}(x)}_{H_{M}}.$

Left and right color electric fields $\hat{E}_{L/R}^{\mathrm{a}}$
($\text{a}=1,2,3$) both form SU(2) Lie algebras. Along with the link operators
$\hat{U}$ ($2\times 2$ operator matrices) defined on links $(x,i)$, they
constitute canonical variables satisfying the algebra

$\displaystyle[\hat{E}_{L}^{\mathrm{a}}\,,\,\hat{U}]$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}\sigma^{\mathrm{a}}\hat{U}~{}~{},~{}~{}[\hat{E}_{R}^{\mathrm{a}}\,,\,\hat{U}]=\tfrac{1}{2}\hat{U}\sigma^{\mathrm{a}}\
.$ (2)

The Gauss law operators at any site $x$ are

$\hat{G}^{\mathrm{a}}(x)=\sum_{i=1}^{d}(\hat{E}^{\mathrm{a}}_{L,i}(x)+\hat{E}^{\mathrm{a}}_{R,i}(x))-\tfrac{1}{2}\hat{\psi}^{\dagger}_{\alpha}(x){\sigma^{\mathrm{a}}}_{\alpha\beta}\hat{\psi}_{\beta}(x),$
(3)

and physical states are those annihilated by every $\hat{G}^{\text{a}}(x)$. In
order to have $[\hat{H},\hat{G}^{\mathrm{a}}]=0$, the gauge invariant Casimirs
$\hat{E}^{2}\equiv\hat{E}_{L}^{\mathrm{a}}\hat{E}_{L}^{\mathrm{a}}=\hat{E}_{R}^{\mathrm{a}}\hat{E}_{R}^{\mathrm{a}}$
and plaquette operators $\hat{U}_{\Box}$ (products of link operators around
elementary loops) appear in $\hat{H}_{E}$ and $\hat{H}_{B}$. The matter
Hamiltonian $\hat{H}_{M}$ describes staggered fermions coupled to the gauge
field $\hat{U}_{\alpha\beta}$ in a gauge invariant combination, plus a fermion
mass term. Solutions to Gauss’s law describe gauge invariant, non-local loop
and string states, but these form an overcomplete basis. Mutually dependent
loops satisfy Mandelstam constraints Mandelstam (1979); Loll (1993); Watson
(1994), which are non-local and notoriously hard to solve.
In the prepotential formulation Mathur (2005, 2006, 2007); Anishetty  _et al._
(2010); Raychowdhury (2013), $\hat{U}$ and $\hat{E}$ are replaced by bilinears
of harmonic oscillator doublets $\hat{a}^{\dagger}_{\alpha}(L/R)$ (the
“prepotentials”) at each end $L\equiv x$ and $R\equiv x+e_{i}$ of every link
$(x,i)$ as

$\displaystyle\hat{E}_{L}^{\mathrm{a}}$ $\displaystyle\equiv$
$\displaystyle\tfrac{1}{2}\hat{a}^{\dagger}(L)\sigma^{\mathrm{a}}\hat{a}(L)\quad\,,\quad\,\,\,\hat{E}_{R}^{\mathrm{a}}\equiv\tfrac{1}{2}\hat{a}^{\dagger}(R)\sigma^{\mathrm{a}}\hat{a}(R),$
(4) $\displaystyle\hat{U}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{\hat{n}_{L}+1}}\left(\begin{array}[]{cc}\hat{a}^{\dagger}_{2}({L})&\hat{a}_{1}(L)\\\
-\hat{a}^{\dagger}_{1}(L)&\hat{a}_{2}(L)\end{array}\right)$ (10)
$\displaystyle\times\left(\begin{array}[]{cc}\hat{a}^{\dagger}_{1}(R)&\hat{a}^{\dagger}_{2}(R)\\\
\hat{a}_{2}(R)&-\hat{a}_{1}(R)\end{array}\right)\frac{1}{\sqrt{\hat{n}_{R}+1}}=U_{L}U_{R}\
.$

where $\hat{n}_{L}$ ($\hat{n}_{R}$) is the total occupation number for
prepotentials at the left (right) end of link $(x,i)$. With these definitions,
(2) follows, along with
$\hat{U}^{\dagger}\hat{U}=\hat{U}\hat{U}^{\dagger}=\hat{1}$. Moreover, using
(10) one can show that
$[\hat{U}_{\alpha\beta},\hat{U}_{\gamma\delta}]=[\hat{U}_{\alpha\beta},\hat{U}^{\dagger}_{\gamma\delta}]=0.$
Since the SU(2) representations situated at the ends of a link must have the
same Casimir $(\hat{E}_{L}^{2}=\hat{E}_{R}^{2})$, there is an Abelian Gauss
law constraint on each link:

$\displaystyle n_{L}(x,i)=n_{R}(x,i)\ .$ (11)

In terms of prepotentials, the link operator is broken into a left part
$U_{L}$ and a right part $U_{R}$, denoting the SU(2) group to be localized at
each end. In (Hamiltonian formulation.), the staggered fermions, which are
SU(2) doublets, also live on lattice sites. Thus, the prepotential operators
and the matter fields can all be uniquely associated with sites and they
transform in the same way. (The essential difference between matter and
prepotentials is their statistics.) This feature enables one to construct
local SU(2) invariants combining both prepotentials and matter, resulting in a
local loop (SU(2) invariant) basis. The local loop states, together with flux
flowing along the links following the Abelian Gauss law (11), yield non-local
Wilson loops.
In the next section we discuss the local loop and string states in the
simplest case of 1D with fundamental matter.

## 1D lattice with matter.

For SU(2) gauge theory defined on a one dimensional lattice,
$\hat{H}=\hat{H}_{E}+\hat{H}_{M}$ only. Each site is connected to two links
along directions, say, $1$ and $\bar{1}$ as in Fig. 1. In the prepotential
framework, $\hat{a}^{\dagger}_{\alpha}$ ($\hat{b}^{\dagger}_{\alpha}$) is
attached to the link along the direction $1$ ($\bar{1}$).

Figure 1: (a) 1D lattice with matter, denoted by circles at the sites. (b)
Gauge invariant states at a site. The state is characterized by a loop quantum
number (the continuous line) and string quantum numbers (the lines ending in
blobs) as given in (14).

A staggered fermion field
$\hat{\psi}^{\dagger}=(\hat{\psi}_{1}^{\dagger},\hat{\psi}_{2}^{\dagger})$
lives on the sites $x$. Using (10) in $\hat{H}_{M}$, the Hamiltonian involves
the following local SU(2) invariant operators and their adjoints:

$\displaystyle\hat{\mathcal{S}}^{++}_{1}$
$\displaystyle\equiv\hat{\psi}^{\dagger}_{\alpha}\hat{\tilde{a}}^{\dagger}_{\alpha},~{}~{}$
$\displaystyle~{}~{}\,\hat{\mathcal{S}}^{++}_{\bar{1}}$
$\displaystyle\equiv\hat{\psi}^{\dagger}_{\alpha}\hat{\tilde{b}}^{\dagger}_{\alpha},~{}~{}$
$\displaystyle~{}~{}\,\hat{\mathcal{L}}^{++}_{1\bar{1}}$
$\displaystyle\equiv\hat{a}^{\dagger}_{\alpha}\hat{\tilde{b}}^{\dagger}_{\alpha},$
(12) $\displaystyle\hat{\mathcal{S}}^{+-}_{1}$
$\displaystyle\equiv\hat{\psi}^{\dagger}_{\alpha}\hat{a}_{\alpha},~{}~{}$
$\displaystyle~{}~{}\hat{\mathcal{S}}^{+-}_{\bar{1}}$
$\displaystyle\equiv\hat{\psi}^{\dagger}_{\alpha}\hat{b}_{\alpha},~{}~{}$
$\displaystyle~{}~{}\hat{\mathcal{L}}^{+-}_{1\bar{1}}$
$\displaystyle\equiv\hat{a}^{\dagger}_{\alpha}\hat{b}_{\alpha},$ (13)

where $\tilde{a}_{\alpha}\equiv\epsilon_{\alpha\beta}a_{\beta}$ and
$\tilde{b}_{\alpha}\equiv\epsilon_{\alpha\beta}b_{\beta}$. The
$\hat{\mathcal{S}}$’s in (12) and (13) are string operators (matter coupled to
prepotentials) and the $\hat{\mathcal{L}}$’s are local loop operators (present
in the pure gauge theory).
A general gauge invariant state at site $x$ is defined by their action on the
local harmonic oscillator/fermionic vacua,

$\displaystyle|l_{1\bar{1}},s_{1},s_{\bar{1}}\rangle=(l_{1\bar{1}!})^{-1}(\hat{\mathcal{L}}_{1\bar{1}})^{l_{1\bar{1}}}(\hat{\mathcal{S}}_{1})^{s_{1}}(\hat{\mathcal{S}}_{\bar{1}})^{s_{\bar{1}}}|0\rangle_{x}\
,$ (14)

where $l_{1\bar{1}},s_{1},s_{\bar{1}}$ are non-negative integers. The local
vacuum is annihilated by operators in (13) and $\mathcal{L}^{\dagger}$. In
(14), $l_{1\bar{1}}$ is unbounded, as it is associated with a bosonic
operator. Being associated with fermionic variables, the string quantum
numbers $s_{1},s_{\bar{1}}$ are $0$ or $1$. The fermionic and bosonic
occupation numbers $n^{F},N^{B}_{1/\bar{1}}$ are related to the local loop-
string quantum numbers at a site by

$\displaystyle n^{F}=s_{1}+s_{\bar{1}}~{},~{}n^{B}_{1}$
$\displaystyle=s_{1}+l_{1\bar{1}}~{},~{}n^{B}_{\bar{1}}=s_{\bar{1}}+l_{1\bar{1}}\
,$ $\displaystyle\Rightarrow~{}~{}n^{B}_{1}-n^{B}_{\bar{1}}$
$\displaystyle=s_{1}-s_{\bar{1}}\ .$ (15)

(15) is the generalized Abelian Gauss law (11) for the direction on which the
matter field resides.
The actions of the loop-string operators in (12) and (13) on the states (14)
are given by

$\displaystyle\hat{\mathcal{S}}^{++}_{1}|l_{1\bar{1}},0,0\rangle$
$\displaystyle=$ $\displaystyle|l_{1\bar{1}},1,0\rangle\ ,$
$\displaystyle\hat{\mathcal{S}}^{++}_{\bar{1}}|l_{1\bar{1}},0,0\rangle$
$\displaystyle=$ $\displaystyle|l_{1\bar{1}},0,1\rangle\ ,$
$\displaystyle\hat{{\mathcal{S}}}^{+-}_{\bar{1}}|l_{1\bar{1}},0,0\rangle$
$\displaystyle=$ $\displaystyle|l_{1\bar{1}}-1,0,1\rangle\ ,$
$\displaystyle\hat{{\mathcal{S}}}^{-+}_{\bar{1}}|l_{1\bar{1}},0,1\rangle$
$\displaystyle=$ $\displaystyle(l_{1\bar{1}}+1)|l_{1\bar{1}}+1,0,0\rangle\ .$

The last two actions give rise to dynamical pair production and string
breaking phenomena, which can be simulated locally. Any other state is
annihilated by the matter field in the string operators.

## 2D lattice

Prepotential formulation of pure SU(2) gauge theory on a 2D lattice has been
studied in great detail in Mathur (2005, 2006, 2007); Anishetty  _et al._
(2010); Raychowdhury (2013). This particular formulation yields local loop
basis, however, that is again overcomplete and are associated with Mandelstam
constraints in a local form. Solving these constraints is nontrivial and
becomes increasingly difficult in higher dimensions.
In a recent development Raychowdhury (2018); Anishetty and Sreeraj (2018), a
virtual “point splitting” of lattice sites in 2D was found to be particularly
useful, as it bypasses the Mandelstam constraints and casts all constraints of
the theory into the Abelian form of (11). Below, we show that coupling to
matter and generalizing to higher dimensions introduces minimal extra
complication in this framework. This seems a vital step toward building
digital quantum simulators for 3D non-Abelian LGTs, which has never been
attempted so far.

## Virtual hexagonal lattice with matter.

A virtual splitting of a lattice site is shown in Fig. 2.

Figure 2: (a) A site $x$ with matter, denoted by a circle at that site. (b)
Virtual point splitting divides $x$ into the pair
$x^{\prime},x^{\prime\prime}$, and the matter lives on a new intermediate site
$x_{m}$. (c) Loop configurations at three virtual sites
$x^{\prime\prime},x_{m},x^{\prime}$.

In 2D, point splitting results in a hexagonal lattice. One can now formulate
prepotentials on this virtual hexagonal lattice as in (4) and (10). On this
hexagonal lattice, physical lattice directions are along $1$ and $2$, and only
the electric fields along these two directions contribute to $\hat{H}_{E}$.
However, in $\hat{H}_{B}$, the elementary loops are indeed hexagonal
plaquettes. The matter field, originally at sites $x$ of the square lattice,
now lives on sites $x_{m}$, as shown in Fig. 2. This is the middle of the
virtual link along $3$. We treat the sites $x^{\prime},x^{\prime\prime}$ on
the same footing as in pure gauge theory Raychowdhury (2018) (to be explained
below), whereas $x_{m}$, being connected to only two virtual links, is
effectively handled as a site on a 1D lattice as in Fig. 1.

## 2D pure gauge theory.

At each site $x$ (standing for either $x^{\prime}$ or $x^{\prime\prime}$) of
the hexagonal lattice, links emerge in three directions, labeled as $1,2,3$.
Each link is associated with prepotential creation and annihilation operators
$\\{\hat{a}^{\alpha}(x,i),\hat{a}^{\dagger}_{\alpha}(x,i)\\}$ for $i=1,2,3$
and $\alpha=1,2$. Using (10) in $\hat{H}_{B}$, one obtains the following gauge
invariant operators and their conjugates at each site,

$\displaystyle\hat{\mathcal{L}}^{++}_{ij}\equiv\hat{a}^{\dagger}_{\alpha}(i)\hat{\tilde{a}}^{\dagger}_{\alpha}(j)~{}~{},~{}~{}\hat{\mathcal{L}}^{+-}_{ij}$
$\displaystyle\equiv$
$\displaystyle\hat{a}^{\dagger}_{\alpha}(i)\hat{a}_{\alpha}(j)\,\,$ (16)

Above,
$\hat{\tilde{a}}^{\dagger\alpha}\equiv\epsilon^{\alpha\beta}\hat{a}^{\dagger}_{\beta}$,
$\hat{\tilde{a}}_{\alpha}\equiv\epsilon_{\alpha\beta}\hat{a}^{\beta}$, and
$i,j$ are direction indices with $i\neq j$. It is clear from the definitions
that when acting on $\ket{0}$ only $\hat{\mathcal{L}}^{++}_{ij}$ is non-zero
and will build up the local loop Hilbert space. For the 2D hexagonal lattice,
the local loop space is characterized by three independent linking numbers
$l_{ij}$ (cf. Fig 2) denoting the flux flowing along three $(ij)$ directions
($(12)$, $(23)$, and $(31)$). The action of these operators on a general
orthonormal loop state,

$|l_{12},l_{23},l_{31}\rangle\equiv\frac{(\hat{\mathcal{L}}^{++}_{12})^{l_{12}}(\hat{\mathcal{L}}^{++}_{23})^{l_{23}}(\hat{\mathcal{L}}^{++}_{31})^{l_{31}}}{(l_{12}+l_{23}+l_{31})!~{}l_{12}!~{}l_{23}!~{}l_{31}!}|0\rangle_{x}\
,$ (17)

is given by

$\displaystyle\hat{\mathcal{L}}^{++}_{ij}|l_{ij}\rangle$
$\displaystyle=\sqrt{(l_{ij}+1)(l_{12}+l_{23}+l_{31}+2)}|l_{ij}+1\rangle,$
(18) $\displaystyle\hat{\mathcal{L}}^{--}_{ij}|l_{ij}\rangle$
$\displaystyle=\sqrt{l_{ij}(l_{12}+l_{23}+l_{31}+1)}|l_{ij}-1\rangle,$ (19)
$\displaystyle\hat{\mathcal{L}}^{+-}_{ij}|l_{ij}\rangle$
$\displaystyle=-\sqrt{(l_{ik}+1)l_{jk}}|l_{jk}-1,l_{ik}+1\rangle.$ (20)

These simple local loop actions, glued together along the links following
(11), reproduce the non-local loops and strings of the original theory.
Moreover, these loop operators now act more like their U(1) counterparts: loop
operators in U(1) theories shift $E$ by unit increments along infinite tower
of states, although there the normalization factor is always trivial.

## 2D lattice with matter.

Putting matter at site $x$ corresponds to putting matter at the middle of the
virtual $3-\bar{3}$ link in the point-split picture. As indicated by Fig. 2,
the gauge invariant Hilbert space associated with the virtual sites is as
follows: (i) The two sites $x^{\prime},x^{\prime\prime}$ have only loop states
$|l_{12},l_{23},l_{31}\rangle_{x^{\prime}/x^{\prime\prime}}$, being treated
identically as in pure gauge theory. (ii) The third virtual site $x_{m}$ along
the $3-\bar{3}$ direction contains both local loop and string states
$|l_{3\bar{3}},s_{3},s_{\bar{3}}\rangle$, being structurally identical to a
site with matter in 1D. (iii) The Abelian Gauss laws along the three
directions of the hexagonal lattice are

$\displaystyle n_{1}(x)=n_{\bar{1}}(x+e_{1})$
$\displaystyle,~{}n_{2}(x)=n_{\bar{2}}(x+e_{2})\ ,$ (21) $\displaystyle
n_{3}(x)+s_{\bar{3}}$ $\displaystyle=n_{\bar{3}}(x+e_{3})+s_{3}\ ,$ (22)

where $n_{1}=l_{12}+l_{31}$, $n_{2}=l_{12}+l_{23}$, and $n_{3}=l_{23}+l_{31}$
count prepotential occupation numbers at each site.
The structure of the 2D theory with matter is therefore equivalent to pure
gauge theory in 2D, plus gauge fields coupled to matter in 1D.

## 3D lattice with matter.

The same scheme would continue as one generalizes to any arbitrary dimension.
For example, in 3D, each site splits into four three-point vertices and matter
is added at another intermediate two-point vertex, as shown in Fig. 3.

Figure 3: One site $x$ on a 3D square lattice is virtually split into four
sites
$x^{\prime},x^{\prime\prime},x^{\prime\prime\prime},x^{\prime\prime\prime\prime}$
connected by three intermediate links in virtual directions $4-\bar{4}$,
$5-\bar{5}$, and $6-\bar{6}$. Matter is added on direction $5-\bar{5}$ at site
$x_{m}$.

As shown for 2D, the local loop Hilbert space remains identical (three linking
numbers) to the pure gauge theory on all virtual sites in Fig. 3. Matter is
added at the middle of one virtual link, i.e., along $5-\bar{5}$ in Fig. 3,
such that the gauge invariant Hilbert space at the additional site has two
string numbers and one loop number, just like the $3-\bar{3}$ links in the 2D
or $1-\bar{1}$ links in 1D. The modified Abelian Gauss laws on the 3D lattice
are

$\displaystyle n_{i}(x)$ $\displaystyle=$ $\displaystyle n_{\bar{i}}(x+e_{i})\
,\quad\quad(i=1,2,3,4,6)$ (23) $\displaystyle n_{5}(x)+s_{\bar{5}}$
$\displaystyle=$ $\displaystyle n_{\bar{5}}(x+e_{5})+s_{5}\ .$ (24)

This scheme generalizes straightforwardly to any arbitrary dimension. All of
these can be checked straightforwardly on a quantum computer using oracles
described below.

Figure 4:  (a) A routine for measuring physicality, i.e., the Abelian Gauss
law, in our framework. The qubit registers correspond to a $3-\bar{3}$ link in
2D with matter. The net effect is that the “query” qubit $\ket{q}$ is flipped
if and only if the configuration physical. (b) A generic adder circuit $A$ for
$(y,z)\rightarrow(y,z+y)$. Adder circuits taking an incoming carry bit
$c_{0}=0,1$ can compute $z+y+c_{0}$ using exactly the same gates that compute
$z+y$. (c) $N$ CNOTs are used to compare two $N$-bit numbers $y$ and $z$.

## Physicality oracles for SU(2) gauge theory.

In Fig. 4(a) we construct an oracle for checking the Abelian Gauss law
constraints along a link. We define $F=1$ if the link satisfies Gauss’s law
and $F=0$ otherwise. The circuit specifically considers (22), i.e., a
$3-\bar{3}$ link in 2D with matter. Here, $\ket{l_{ij}}$ means an $N$-qubit
binary number in the computational basis. $s_{3}$ ($s_{\bar{3}}$) acts as an
incoming “carry qubit” Draper  _et al._ (2004); Cuccaro  _et al._ (2004) to
the addition subroutines $A$, which can have the effect of adding one unit to
the overall sum $l_{31}+l_{23}$ ($l_{\bar{3}\bar{1}}+l_{\bar{2}\bar{3}}$) [cf.
Fig. 4(b)]. The sums on each side of (22) are then compared bit-wise by a
string of $N$ CNOTs, as displayed in Fig. 4(c). A conditional phase -1 is
applied if and only if they are identical ($F=1$). An auxiliary query qubit
$\ket{q}$ is flipped if $F=1$, and the remaining gates restore the qubit
registers to their original configurations. Matterless links would have
$\ket{s_{3/\bar{3}}}\rightarrow\ket{0}$ and the appropriate $l_{ij}$’s as
inputs in order to check (21). For 3D, again the only difference is the input
$l_{ij}$’s and $s_{i}$’s, implying that the same simple routine can indeed
check all the constraints that define a physical loop configuration in SU(2)
lattice gauge theory, in any dimension. For $N$-qubit link registers, the
oracle can be constructed using no more than $17N+5$ CNOTs, $8N$ Toffolis, and
the $C^{N}(Z)$ gate Cuccaro  _et al._ (2004). This is likely to involve far
less T gates than any Trotter step unless $N=1$.
These routines are likely to be useful in digital simulations because non-
gauge invariant errors can easily arise from the Trotter approximation to
$e^{-it\hat{H}}$ or from quantum noise Stryker (2018). Previously these were
only constructed for Abelian theories. Because our framework has cast all
constraints into an Abelian form, we can benefit from the same techniques. An
analogous construction using the conventional group representation states is
much less straightforward because different components of the non-Abelian
Gauss law operator are not simultaneously diagonalizable. Constructing
physicality oracles for SU(3) is a goal for future work.

## Discussion.

We see several virtues of using the prepotential framework for simulating
lattice gauge theories. The immediate benefit is that non-Abelian gauge
redundancy is absent in the constructed Hilbert space, enabling one to attain
a higher cutoff on the physical Hilbert space than when working with all the
redundant gauge degrees of freedom. Abelian Gauss law constraints are still
required on the links, but these can be validated straightforwardly by
constructing Gauss oracles for SU(2), in any dimension. Another virtue is
working with infinite towers of states rather than multiplets of varying
dimensions, so it is far more obvious how to truncate to some dimensionality
that is a power of two and map them onto qubits. Operating on towers of states
more closely resembles U(1) gauge theory, so it is conceivable that other
algorithms developed for Abelian theories can also be ported over to SU(2).
We have noted that the point splitting technique increases the number of links
to be simulated and that plaquette operators must deal with more links; it is
our hope and expectation that this drawback is outweighed by the simpler
action of individual link operators in a plaquette. Our construction
nonetheless stands to more directly benefit from any progress made in
algorithms for implementing U(1) plaquette operators.
The framework presented in this letter can be generalized to SU(3) using
(Anishetty  _et al._ , 2010), as we have not explicitly relied on knowing any
Clebsch-Gordon coefficients specific to SU(2). Generalizing this work to SU(3)
will no doubt be a milestone towards quantum simulation of lattice QCD. Our
immediate goal is to characterize the simulation of time evolution using
prepotentials, which we expect to do by Trotterization. The work in this
direction is in progress.

## Acknowledgments.

We thank D. B. Kaplan and N. Klco for helpful discussions and feedback on the
manuscript. IR was supported in part by the Thomas L. and Margo G. Wyckoff
Endowed Faculty Fellowship. JRS was supported by DOE Grant No. DE-
FG02-00ER41132, and by the National Science Foundation Graduate Research
Fellowship under Grant No. 1256082.

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