# Preparing Rernomalization Group Fixed Points on NISQ Hardware

Troy J. Sewell<EMAIL_ADDRESS>Joint Center for Quantum Information and
Computer Science, University of Maryland, College Park, MD 20742, USA  Stephen
P. Jordan  Quantum Systems, Microsoft, Redmond, WA 98052, USA

###### Abstract

Noisy intermediate-scale quantum (NISQ) hardware [1] is typically limited to
low-depth quantum circuits to limit the number of opportunities for
introduction of error by unreliable quantum gates. A less-explored alternative
approach is to repeatedly apply a quantum channel with a desired quantum state
as a stable fixed point. Increased circuit depth can in this case be
beneficial rather than harmful due to dissipative self-correction. The quantum
channels constructed from MERA circuits can be interpreted in terms of the
renormalization group(RG), and their fixed points are RG fixed points, i.e.
scale-invariant systems such as conformal field theories. Here, building upon
the theoretical proposal of Kim and Swingle [2], we numerically and
experimentally study the robust preparation of the ground state of the
critical Ising model using circuits adapted from the work of Evenbly and
White[3]. The experimental implementation exhibits self-correction through
renormalization seen in the convergence and stability of local observables,
and makes essential use of the ability to measure and reset individual qubits
afforded by the “quantum CCD” architecture of the Honeywell ion-trap[4]. We
also numerically test error mitigation by zero-noise extrapolation schemes
specially adapted for renormalization circuits, which are able to outperform
typical extrapolation schemes using lower gate overhead.

##  1 Introduction

The current period of experimentation on digital quantum devices that are
still quite limited in terms of size and error rates has been dubbed the Noisy
Intermediate-Scale Quantum (NISQ) era [1], and it is an ongoing question
within the field of quantum computation to find what techniques, if any, may
be leveraged to implement practical quantum algorithms on these early devices.
Prior to the development of fully fault-tolerant quantum computers, error
mitigation procedures will be needed to achieve robust quantum computation in
the presence of noise. In [2], Kim and Swingle proposed that the stable nature
of renormalization group fixed points could provide intrinsic robustness to a
class of quantum state-preparation algorithms based on a variant of the
Multiscale Entanglement Renormalization Ansatz (MERA) that they named Deep
MERA, or DMERA. Similar circuits were also considered for their noise-
resilience in [5]. MERA are a well-known variational tensor network ansatz for
describing ground states of scale-invariant quantum systems, such as at
critical points. The expectation values of local observables can be computed
in polynomial time classically using MERA. Unfortunately, the polynomial
scaling with bond dimension $\chi$ is of very high degree (e.g. $O(\chi^{8})$
for 1D and $O(\chi^{16})$ for 2D using the schemes proposed in [6]), making
MERA impractical for many classical computations. However, a MERA tensor
network may be interpreted directly as a quantum circuit for preparing the
state described. Thus MERA circuits can be used to calculate local observables
on a quantum computer with a polynomial but large scaling advantage relative
to classical computers. DMERA is a variant of MERA more directly tailored to
quantum state preparation applications by dealing with local quantum circuits
of variable depth rather than general unitary tensors of variable bond
dimension.

Given a quantum circuit for preparing a quantum state and a subset of the
qubits of the state to be prepared, one can define the causal cone of the
subset by discarding all qubits and gates at the earliest stage possible such
that the reduced density matrix on the chosen subset remains unchanged. A
noteworthy property of DMERA circuits is that geometrically local patches
selected from one-dimensional DMERA states have causal cones of constant
width, independently of the volume of the one dimensional state from which
they are taken. Viewed operationally, the causal cone can be regarded as a
quantum channel for preparing the reduced density matrix for the local patch
of the overarching state. This channel is composed by iterating a procedure in
which new qubits are introduced in the $|{0}\rangle$ state, unitary gates are
applied, and some qubits are discarded. From a renormalization group point of
view, this procedure can be regarded as zooming in on a smaller region of the
underlying scale-invariant system by fine-graining the lattice discretization,
and then discarding the qubits that move out of the field of view under this
zooming process [2]. The ability to measure local observables and correlation
functions by preparing the causal cone of a small subset of qubits from a
large state can dramatically reduce the resources needed and effects of noise
[7], both of which are crucial for developing realistic applications for NISQ
devices.

The quantum channel and renormalization group flow points of view are related
according to $\lambda=2^{-\Delta}$, where $\lambda$ is an eigenvalue of the
superoperator defining the local quantum channel being applied at each
iteration that scales the state by a factor of two, while $\Delta$ is the
scaling dimension of the associated operator in the underlying conformal field
theory[8]. If the scaling dimensions are all positive, as is generically the
case, then the quantum channel has a unique attractive fixed point toward
which any initial state will converge exponentially at a rate bounded by the
smallest scaling dimension. This provides the state-preparation procedure with
a degree of intrinsic self-correction. Dissipative quantum state preparation
and computation has a long history [9, 10, 11, 12, 13, 14, 15, 16, 17] and
within this context the DMERA perspective adds a systematic way of deriving
concrete quantum circuit implementations of the needed quantum channels for a
class of state preparation problems as well as an immediate way to predict
convergence rates based on known scaling dimensions of conformal field
theories.

Here, we experimentally and numerically investigate the application of DMERA
for the preparation of the ground state of the critical Ising model. Via the
Jordan-Wigner transformation, this model can be solved using techniques for
free-fermion models. Exploiting the structure provided by this exact solution,
Evenbly and White have derived wavelet-based analytic expressions for the
tensors by which a local free-fermion MERA can describe the ground state [3].
Due to the exactly solvable nature of this model, the circuit ansatz is an
example of a matchgate circuit which can be simulated efficiently classically
[18]. We thus do not propose this model for a demonstration of quantum
speedup. Rather, by choosing a model with analytically determined parameters
in the tensors, we can make a clean investigation of convergence and error-
robustness without confounding issues coming from optimization of variational
parameters. A modified version of the circuit with additional gates dual to
non-quadratic fermion operators would break the integrability of the circuit
dynamics. This would lead to structurally analogous circuits that are
challenging to simulate classically, and likewise able to prepare non-Gaussian
fermionic states which arise in interacting fermion models and nonintegrable
spin chains.

A noteworthy feature of this state preparation method is that it requires the
repeated introduction of fresh ancilla qubits in the $|{0}\rangle$ state and
subsequent discarding of qubits that are no longer needed. It is due to this
open-system nature of the procedure that entropy can be removed and self-
correction can be attained. This is a necessary feature for all dissipative
quantum algorithms, as well as for the “holographic” methods of [19, 20, 21],
but is not possible on most present-day quantum computing test beds. However,
measurement and resetting of individual qubits during the execution of a
quantum circuit is possible on the “quantum 2” architecture used by the
Honeywell HØ device on which our experiments were performed[4]. In principle,
the quantum channel derived from DMERA can be repeated indefinitely on a
quantum CCD device, such that the attractive nature of the fixed point
constitutes an active form of error correction which perpetually steers the
quantum state back toward its target. This is in stark contrast with most NISQ
algorithms which rely on limited circuit depth to limit the accumulation of
error.

##  2 Preliminaries

The DMERA circuits we consider consist of a depth $D$ layer with $D$
variational parameters which is iterated to drive the system into an
approximate ground state supported on a limited number of qubits. The circuit
appears to be local within one depth $D$ scale transformation. However, if we
fix the geometry of the qubits in the final state of the circuit, earlier
layers require interaction of pairs of geometrically distant qubits. This fact
favors implementation on near-term ion-trap computers where nonlocal gates can
be implemented natively without swap gates. Larger-depth implementations of
the quantum channel achieve smaller systematic error arising from the
limitations of the ansatz, but incur greater amount of error from imperfect
implementation because of the increased number of gates in the past causal
cone. We explore numerically the tension between approximation accuracy and
noise susceptibility to find the optimal channel depth for a given error rate.
This suggests that while the overall depth can be quite large with fixed
points stabilized by local dissipative dynamics, NISQ devices will still need
to focus on channels with relatively short depth to limit the effect of noise.
We find numerically that the approximation accuracy is exponential in $D$
while the noise susceptibility scales linearly with $D$, suggesting an optimal
depth such that $\epsilon\sim e^{-D}/D$ for gate error rate $\epsilon$. For
large enough $D$ this is just $D\sim\log(1/\epsilon)$, however, for the larger
error rates of NISQ devices we will be working in the regime of small depth
where subleading terms and constant factors are crucial for choosing the
optimal circuit for particular task at hand.

Each layer of the circuit implements one scale transformation by a spatially
homogeneous depth $D$ brickwork circuit of two-qubit gates. Within a single
scale the local gates are spatially homogeneous, built by one initial layer of
”isometry” gates $W(\theta_{1})$ and $D-1$ layers of gates $U(\theta_{i})$ for
depth $i$ within the layer, as seen in Fig. 1.

(a) $D=2$ layer with 3 qubit causal cone.

(b) $D=4$ layer with 2 qubit causal cone.

Figure 1: quantum channels on six qubits performing one scale of entanglement
renormalization, with a causal cone highlighted in purple for a local operator

The two-qubit gates $W$ and $U$ are matchgates and therefore preserve parity.
They are each parameterized in terms of a single angle, the rotation angle of
the even-parity subspace, which gives us $D$ variational parameters for the
angle at each depth within the scaling transformation. The odd-parity subspace
is fixed by permutation symmetry of the qubits involved in each gate. The
unitary gate $U$ is symmetric under the simultaneous permutation of incoming
and outgoing qubits, which fixes the action on the odd-parity subspace to be
the identity, up to a phase. The $W$ gates are in spirit isometries with only
one incoming qubit, since one of the two qubits is always introduced as a
$|{0}\rangle$ state, so they are symmetric under permutations of only the two
outgoing qubits. This has the effect of propagating an incoming $|{1}\rangle$
to the symmetric pair $|{01}\rangle+|{10}\rangle$, while an incoming
$|{0}\rangle$ state becomes an arbitrary even-parity state which is always
permutation-symmetric.

$W(\theta)=\begin{pmatrix}\cos(\theta-\frac{\pi}{4})&0&0&\sin(\theta-\frac{\pi}{4})\\\
0&\frac{1}{\sqrt{2}}&\frac{-1}{\sqrt{2}}&0\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\\
-\sin(\theta-\frac{\pi}{4})&0&0&\cos(\theta-\frac{\pi}{4})\end{pmatrix}\hskip
14.22636ptU(\theta)=\begin{pmatrix}\cos(\theta)&0&0&\sin(\theta)\\\ 0&1&0&0\\\
0&0&1&0\\\ -\sin(\theta)&0&0&\cos(\theta)\end{pmatrix}$ (1)

Since our state is scale-invariant we apply the same scale transformation
repeatedly, implementing identical steps of fine-graining to prepare the
renormalization UV fixed point defined by our local scaling operation. Since
the past causal cone of a local region is constant width, we can use this to
define a quantum channel on a constant number of qubits that implements the
local scaling operation on a fixed subsystem size. Since each scaling layer
introduces new qubits in order to fine-grain the state by zooming in on the
virtual lattice, the constant qubit channel also needs to discard qubits at
the edges which end up outside of the causal cone. This combination of
preparing new ancilla and discarding edge qubits is built into the definition
of the local scaling channel along with the unitary scale transformation,
making the channel non-unitary but physically implementable on a quantum
computer.

Since the unitary circuit implementing the channel is only invariant under
translation by an even number of sites, there are two different ways the
subregion of retained qubits can align with the circuit. The two choices of
subregion are akin to choosing to follow a subregion moving to the left or
right when descending down the state preparation circuit of a large spin
chain. Sampling over infinite sequences of these left and right channels would
correspond to an averaging of the location of our $n$ qubit subregion over the
the infiite sized spin chain. We implicitly perform this averaging of
subregion location by studying the channel which is the statistical mixture of
the left and right channels with equal probability, which ensures that the
fixed point energies measured are still variational for the infinite sized
system.

There is a single eigenvalue of the quantum channel with $\lambda=1$,
corresponding to the fixed point operator of the channel. All other operators
have eigenvalues with magnitude less than one, meaning those operators will
decay exponentially with successive iterations of the channel. The scaling
dimensions of these operators are given by $\Delta=-\log_{2}\lambda$. A local
$n$-qubit channel has $2^{2n}$ eigenvalues, the largest of which correspond to
the slowly decaying low energy modes described by the CFT with small scaling
dimension. We compare the scaling dimensions of the primary operators of the
CFT describing the critical Ising model with those derived from the local
quantum channels, seen in Table 1. Additional scaling dimensions are shown in
Fig. 2, where the smallest 32 scaling dimensions in the CFT are compared with
those derived from the largest 32 eigenvalues of the $D=2$ channel on three
qubits. The largest eigenvalues match well with the expected scaling
dimensions, suggesting that the low energy sector of the spin-chain is indeed
well described as a CFT, while the smallest eigenvalues that correspond to
rapidly decaying high energy modes that are not universal and do not
correspond to quantities of the CFT.

The positive scaling dimensions and unique fixed point imply that local
subsystems converge exponentially and only $\log(1/\epsilon)$ layers are
needed to prepare the target state with precision $\epsilon$. However, even
the exact fixed point is still just an approximation of the true ground state,
limited in precision by the depth $D$ of the ansatz used, and the actual fixed
point state prepared by a NISQ device will have perturbations from the ideal
fixed point introduced by noise.

| exact | $D=2$ | $D=4$
---|---|---|---
$E^{*}/L$ | -1.27323.. | -1.23948.. | -1.26757..
$\Delta_{I}$ | 0 | 0 | 0
$\Delta_{\sigma}$ | 0.125 | 0.136.. | 0.123..
$\Delta_{\epsilon}$ | 1 | 1 | 1
Table 1: Critical data extracted from quantum channels preparing
approximations of the critical Ising ground state, using wavelet-derived
angles of [3] compared with exact values. The exact ground state energy
density in the thermodynamic limit $E_{0}/L=-4/\pi$ is compared to the fixed
point state. The scaling dimensions of the three primary operators listed
above are derived from the three largest eigenvalues of the quantum channel.
Additional dimensions included in 2.  Figure 2: Scaling dimensions $\Delta$
extracted from the eigenvalues $\lambda$ of the quantum channel for $D=2$.
Exact values for the 32 smallest scaling dimensions of the Ising CFT compared
with those derived from the largest 32 eigenvalues of the noiseless channel.
Perturbations to these scaling dimensions in noisy versions of the channel are
also included.

We can understand why the DMERA circuit based on wavelets is able to prepare
the critical Ising state by considering the action of the circuit in terms of
fermion modes via a Jordan-Wigner transformation. The critical Ising model
spin Hamiltonian $H_{I}=-\sum_{i}X_{i}X_{i+1}+Z_{i}$ is transformed into the
majorana Hamiltonian $H_{M}=-i\sum_{i}\gamma_{i}\gamma_{i+1}$ 111For finite
size, the 1D Ising model with periodic boundary conditions is equivalent to a
free-fermion model with anti-periodic boundary conditions in the even parity
subspace and periodic boundary conditions in the odd parity subspace. We focus
on local subregions and observables of an infinite system where these
subtleties may be ignored., both of which have ground state energy density
$E_{0}/L=-4/\pi$. Being translation invariant, the eigenmodes of the massless
free-fermion Hamiltonian dual to the critical Ising model are plane-waves.
Matchgate circuits implement linear transformations on fermion modes, which
would require depth scaling linearly with the system size to implement an
exact transformation mapping local modes to the proper plane-wave modes via a
local quantum circuit. This can be cast as a QAOA circuit with depth $L/2$
with $O(L^{2})$ gates to prepare the exact critical Ising ground state for a
size $L$ system [22]. The wavelet transformation offers an alternate basis of
local wave-packets, which can be mapped to from local modes using a
$O(D\log(L))$ depth circuit with $O(DL)$ gates. Since the ground state is
invariant under transformations that do not mix the occupied and unoccupied
modes, it is conceivable that these wave-packet modes could be made from only
occupied modes and therefore prepare the exact ground state. In practice the
quasi-local modes found through variational optimization of a low-depth DMERA
circuit architecture only prepare an approximate ground state, albeit one
whose accuracy increases exponentially with the layer depth $D$. The analysis
of [3] considered the tight-binding model of 1D lattice of spinless fermions
at half-filling, which can be decoupled into two copies of the Hamiltonian
$H_{M}$ with local transformations. They designed the circuit and set of
angles to specify a wavelet transformation on the fermion modes that occupies
the Dirac sea of momentum modes with $|k|<\pi/2$, with vanishing support at
the Fermi points where $|k|=\pi/2$. Larger-depth variations of the circuit
then correspond to higher-order wavelets which are able to prepare a better
approximation by canceling higher derivatives of the mode occupation around
the Fermi points. These circuits can be directly adapted for the critical
Ising model as outlined in the Appendix of [3], which serves as the basis of
our analysis. Additional work on wavelet circuits has been done exploring
application to both free-fermion and free-boson systems [23, 24, 25].

The relation of different wavelet basis states by both translations and
scaling transformations makes them especially suited for critical models,
which have these operations as symmetries. Both the wavelet circuits and MERA
tensor networks reflect these symmetries in the spatial homogeneity of tensors
at a given depth in the circuit, and the homogeneity of the different circuit
layers in the hierarchical structure. This discrete scaling symmetry allows us
to interpret the past causal cone of a local region as the iteration of a
single quantum channel, and the translation invariance helps to reduce the
number of variational parameters needed to find good circuits.

Due to the inherently discrete nature of qubits, the continuum of a field
theory and its symmetry group must be discretized and broken to be represented
on digital quantum hardware. Additionally the staggered structure of the
individual quantum gates used to prepare the state and the fixed scaling by
two for each circuit layer break translation and scale invariance of the
unitary used to prepare a pure quantum state. These result in small deviations
from exact translation invariance and power-law scaling of the correlation
functions for the resulting ground state. These deviations can be improved by
averaging over different samples of the state transformed by these different
symmetries. This can be done implicitly by equally sampling over the past
causal cone from different locations of the global state, or symmetrizing the
measurement data by reflections or the global parity symmetry.

We find that the wavelet-based gates derived in [3] admit particularly
efficient decomposition in terms of Mølmer-Sørensen (MS) gates, the native
two-qubit gate set on ion trap devices, which we will sometimes denote
$XX(\theta)$. This conveniently allows the implementation of these circuits on
NISQ ion-trap computers with little gate overhead. MS gates can typically be
applied to any pair of qubits in an ion-trap and so the qubit architecture
within a trap is thought of as globally connected. Ion-trap computers engineer
this all-to-all connectivity in different ways, for instance by moving
individual ions around to bring pairs into interaction sites in the Honeywell
quantum CCD architecture [4].

$X_{i}X_{j}(\theta)=e^{-i\frac{\theta}{2}X_{i}X_{j}}$ (2)

The XX native interaction affords a simple implementation of our state
preparation circuit, with each gate $W$ and $U$ being decomposable in terms of
two MS gates and additional single qubit Z rotations
$Z(\phi)=\exp(i\frac{\phi}{2}Z)$. All of our gates are matchgates and thus
cannot form a universal gate set in a geometrically local circuit. The
addition of local $X$ or $Y$ rotations into the circuit would expand the gate
set to become universal. The non-local interaction of these gates is also
crucial for implementing a hierarchical algorithm like DMERA, where new qubits
need to be interleaved at each layer and would be costly to achieve with swap
gates. Importantly, the non-local gates are always performed over strings of
qubits in the all zero state, which means they can be performed with local
matchgates and fermionic swap gates, so they can still be considered local
matchgate circuits that are classically simulable. Local $Z$ rotations are
often implemented virtually via the internal clock of the hardware
Hamiltonian, making them practically noise free. We considered two possible
native gate decompositions of $W$ and $U$, denoted $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$, drawn out in Fig. 11.

$\displaystyle\begin{split}W_{\mathcal{C}_{1}}(\theta)&=XX(\pi/2)\cdot
Z_{1}(\theta)\cdot Z_{2}(\theta-\pi/2)\cdot XX(-\pi/2)\\\
U_{\mathcal{C}_{1}}(\theta)&=XX(\pi/2)\cdot Z_{1}(\theta)\cdot
Z_{2}(\theta)\cdot XX(-\pi/2)\\\
W_{\mathcal{C}_{2}}(\theta)&=Z_{1}(\pi/2)\cdot XX(-\theta)\cdot
Z_{1}(-\pi/2)\cdot Z_{2}(\pi/2)\cdot XX(\theta-\pi/2)\cdot Z_{2}(-\pi/2)\\\
U_{\mathcal{C}_{2}}(\theta)&=Z_{1}(\pi/2)\cdot XX(-\theta)\cdot
Z_{1}(-\pi/2)\cdot Z_{2}(\pi/2)\cdot XX(\theta)\cdot Z_{2}(-\pi/2)\end{split}$
(3)

To truly prepare an approximation of the critical Ising ground state, we would
need to apply the transitional layer of $XX_{\pi/2}$ gates to every pair of
nearest-neighbor qubits after the layers of DMERA circuit have been performed
[3]. Instead, we directly target the ground state of the Hamiltonian by
applying the transition layer to the usual Ising Hamiltonian $H_{I}$,
resulting in $H_{I^{*}}=\sum_{i}-X_{i}X_{i+1}+X_{i-1}Z_{i}X_{i+1}$.

##  3 Error Models

The largest single source of error is two-qubit gate infidelity, which we
model by imprecise rotation angle of MS gates given by a Gaussian distribution
with standard deviation $\sigma$. This noisy version of the MS gate, denoted
$\widetilde{XX}$, is equivalent to a quantum channel with Kraus operators
$K_{0}=\sqrt{\frac{1+e^{-\sigma^{2}/2}}{2}}XX(\theta)$ and
$K_{1}=\sqrt{\frac{1-e^{-\sigma^{2}/2}}{2}}XX(\theta-\pi)$, as shown in Eq. 4.
This is equivalent to a probability of $(1-e^{-\sigma^{2}/2})/2$ to flip both
bits after the ideal MS gate has been applied, which for small error is
$\approx\sigma^{2}/4$ to leading order.

This kind of noise is correlated between qubits, since the bit flips only
occur in pairs, and biased towards bit flip errors rather than other two-qubit
errors due to its source in gate imprecision.

$\displaystyle\begin{split}\widetilde{XX}_{\theta,\sigma}(\rho)&=\int^{\infty}_{-\infty}d\phi\frac{e^{-(\phi-\theta)^{2}/2\sigma^{2}}}{\sqrt{2\pi\sigma^{2}}}XX_{\phi}\rho
XX^{\dagger}_{\phi}\\\
&=\left(\frac{1+e^{-\sigma^{2}/2}}{2}\right)XX_{\theta}\rho
XX^{\dagger}_{\theta}+\left(\frac{1-e^{-\sigma^{2}/2}}{2}\right)XX_{\theta-\pi}\rho
XX^{\dagger}_{\theta-\pi}\\\
&=\left(\frac{1+e^{-\sigma^{2}/2}}{2}\right)XX_{\theta}(\rho)+\left(\frac{1-e^{-\sigma^{2}/2}}{2}\right)XXXX_{\theta}(\rho)XX\end{split}$
(4)

$\begin{split}\mathcal{D}_{p}(\rho)&=(1-p)\rho+\frac{p}{3}\left(X\rho X+Y\rho
Y+Z\rho Z\right)\\\
&=\left(1-\frac{4p}{3}\right)\rho+\frac{2p}{3}\mathbb{I}_{2}\end{split}$ (5)

Spontaneous logical errors from environmental noise can be modeled by
periodically applying a single qubit depolarizing channel $\mathcal{D}_{p}$
independently to each qubit, equivalent to applying a Pauli operator randomly
with small probability. This noise process is uncorrelated between the qubits
and invariant under single qubit rotations. The Kraus operators of single
qubit depolarizing noise, shown in Eq. 5, are $K_{0}=\sqrt{1-p}I$,
$K_{i}=\sqrt{p/3}\sigma_{i}$, for Pauli matrices $\sigma_{i}$. For the
depolarizing error model, a single qubit depolarizing channel with error
probability $p/2$ is applied to each qubit before and after the MS gate is
applied.

The depolarizing channel can be made anisotropic by biasing the probabilities
for each Pauli operator. This may arise from a particular device coupling to
the environment in a way that favors dephasing or bit-flip errors, while still
assuming no correlation of the noise process between qubits. Ion-trap devices
have relatively long coherence times and so environmental noise of this kind
is expected to be smaller than gate implementation error. With these local
error models we ignore state preparation and measurement errors, and make many
assumptions including that errors are homogeneous in space and time, lack
long-range correlations beyond the local gate associated to the error, and are
state-independent.

The leading term in the average gate infidelity $1-\overline{\mathcal{F}}$,
where $\overline{\mathcal{F}}$ is the average gate fidelity, scales as
$\sigma^{2}$ and $p$ respectively for the angle imprecision and depolarizing
models. These will be taken as the relevant noise parameters $\epsilon$ for
their respective error models. We also expect the fixed point infidelity and
local observable expectation values to scale linearly in these parameters to
leading order, which is confirmed by our numerics. The coefficient of this
scaling is determined by specific details of the noise models and circuit
implementation, including native gate sets and choices of gate decomposition.
The tensor network for the channel acting on the density matrix of three
qubits is given in Fig. 3 with the $\mathcal{C}_{1}$ gate decomposition in
terms of native $XX$ and $Z$ gates.

Figure 3: Tensor network diagram for the $\mathcal{C}_{1}$ circuit with $D=2$
on three qubit density matrices. Two copies of the unitary circuit are given
to act on each set of indices of the density matrix tensor, with initialized
zero states given by one-index tensors, and the discarded qubits given by
connecting the dangling legs from each circuit copy. Small red two-index
tensors are also given for each MS gate which stores the weights of the Kraus
operators which are summed over to implement a given local noise model for the
MS gate at that position in the circuit. The colors of all the single and two-
qubit gates are chosen to be identical for identical rotation angles. The
choice of open indices at the bottom of the diagram make this an instance of
the left channel. The left channel is averaged with the right channel,
constructed by choosing right three qubits as output, to form the variational
channel that implicitly averages over subregion location in the spin chain.

We see in Fig. 4 that errors from a prior layer tend to dilute evenly among
the qubits in subsequent layers. Here we apply one noisy scale transformation
to a global state on four bits, followed by multiple noiseless circuit layers
which each double the system size of the ground state on a circle. The global
state fidelity with the reference pure state prepared by the noiseless circuit
is constant as more layers are applied, and the subsystem entropy stays
roughly constant. Since prior errors mix and stabilize across the system as
the total size grows, the error on a constant size subsystem decreases
exponentially as more layers are applied. An individual gate error introduces
a perturbation to the fixed point state, which may be decomposed into the
channel eigenoperators. Each operator perturbing the fixed point state decays
exponentially according to its eigenvalue, which means the true convergence of
an expectation value to the asymptotic value is a sum of exponential decays.
In Fig. 5 we see that this can be fitted well by a single exponential with a
rate of convergence back to the fixed point value is given by a scaling
dimension of $\Delta\approx 1.89$. This convergence rate is consistent across
different noise strengths in the gate imprecision model.

Figure 4: Average fidelity $\mathcal{F}_{\ell}$ of size $\ell$ subsystems
between ideal DMERA circuit preparing $L$ spin ground state and states where
the initial layers have imprecise angles. As successive layers dilute early
circuit errors subsystem fidelities remain roughly constant as a fraction of
total system size, i.e. $\ell/L$. Figure 5: Response of the noiseless $D=2$
channel to noisy gates,the channel with $\sigma^{2}$ noise is applied once to
the noiseless fixed point state. The difference between the noisy energy
density $\tilde{E}$ after successive applications of the noiseless channel and
the fixed point energy density $E^{*}$ is plotted on a log scale, showing an
exponential convergence back to the fixed point energy density.

The stability of this channel against noise can be thought of in terms of an
equilibrium reached by the noisy channel by discarding error-laden qubits and
introducing new high fidelity qubits each layer. By introducing new qubits,
the average number of gates a given qubit has seen remains low. By discarding
qubits, a prior error may end up entirely on the discarded qubits, or in the
phase of the entanglement between the discarded qubits, both of which would
have no effect on observables or fidelities of the subregion we keep.

Given that an $n$-qubit channel has $O(nD)$ gates that each have a gate error
$\epsilon$, and the errors decay with average eigenvalue $\lambda$ under
action of the channel, we can expect the total error of an $n$-qubit fixed
point to be $\delta\sim
nD\epsilon+nD\epsilon\lambda+nD\epsilon\lambda^{2}+\ldots=nD\epsilon/(1-\lambda)$.

Figure 6: Percent error of fixed point energy densities for $D=2,3,4$ channels
using the angle imprecision error model in the $\mathcal{C}_{1}$(solid) and
$\mathcal{C}_{2}$ (dashed) gate decompositions. A modified model where angle
variance scales with the target rotation angle was also tried for
$\mathcal{C}_{2}$ (dotted), but would have no effect on $\mathcal{C}_{1}$ due
to fixed MS rotation angles.

We do find numerical support that the energy density does scale to leading
order in $\sigma^{2}$ and $p$ for our respective error models. In Fig. 6,
which shows the scaling with gate imprecision, we also see that the different
choices of gate decomposition can greatly affect the scaling of the noise,
with $\mathcal{C}_{1}$ scaling better than $\mathcal{C}_{2}$. Interestingly,
we also see that the $D=4$ channel unexpectedly scales better than the $D=3$
for the $\mathcal{C}_{1}$ gate decomposition, but not for $\mathcal{C}_{2}$.
These behaviors were not reflected in the local depolarizing noise model,
which had very similar scaling for different decompositions and increasing
error susceptibility with depth. This could be due to the uncorrelated and
local unitary invariance of the depolarizing noise, which may allow it to
propagate in a more regular manner within circuits and avoid the
irregularities seen in the gate imprecision model.

In [26] an alternate angle imprecision model is suggested for MS gates, where
the Gaussian variance is proportional to the angle of rotation. This angular
dependence of the variance would alter the error model of $\mathcal{C}_{2}$
gate decomposition, but not that of $\mathcal{C}_{1}$ decomposition used on
the Honeywell experiment where the MS gate angles are fixed. Shown dotted in
Fig. 6, this model has effectively lower noise since the MS gates have a small
rotation angle compared to the fixed angle of $\pi/2$ in the $\mathcal{C}_{1}$
decomposition.

Figure 7: Rate of error growth in the fixed point energy density,
$d\tilde{E}/d\epsilon$, versus channel depth $D$, using $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ circuit decompositions and the MS gate imprecision model
with $\epsilon=\sigma^{2}$. The error growth rate for each channel depth
averaged over many sets of angles with low energy fixed point, suggests a
linear growth in error susceptibility with channel depth.

Since the linear scaling of the energy density in the error parameter tends to
increase for larger-depth channels, we compare $d\tilde{E}/d\epsilon$ for
channels of different depth to find dependence on $D$ of the fixed point
error. To avoid potential peculiarities arising from particular sets of angles
having an atypical noise susceptibility, as seen with $\mathcal{C}_{1}$, we
averaged $d\tilde{E}/d\epsilon$ over many sets of angles for fixed depth $D$
in order to assess the typical behavior of channels of a given depth. The
different circuits sampled, however, were chosen so that they have a fixed
point energy density less than $-1.2$. This constraint is needed because we
are only interested in the behavior of circuits that provide reasonable ground
state approximations, and $d\tilde{E}/d\epsilon$ can be negative near
$\epsilon=0$ for channels with higher energy fixed points. After averaging the
noise susceptibility $d\tilde{E}/d\epsilon$ over different channels of the
same depth with low energy fixed point states, we find in Fig. 7 that the
circuit decomposition $\mathcal{C}_{2}$ does typically have worse scaling than
$\mathcal{C}_{1}$ under the gate imprecision model, and the fixed point energy
of both circuits seem to scale linearly with channel depth $D$.

##  4 Error Mitigation

Another kind of error modeling that is important for error mitigation is a
functional form to model the effect that error processes have on measurement
expectation values. The error mitigation technique of zero-noise extrapolation
works by assuming a functional form with certain free parameters, one
crucially being the noise-free expectation value, and solving for these
parameters by measuring the expectation value at different values of the noise
parameters [27]. In practice the noise parameters are not able to be freely
tuned, so what is often done is that certain noise sources are amplified by
inverting and repeating parts of the circuit $n$ times, $U(U^{\dagger}U)^{n}$,
so that the source of noise has been applied $2n+1$ times to a logically
equivalent circuit. The repeated unitary elements could be any unitary
subsection of the circuit, from individual gates to large sections of the
circuit. Since there are many methods of experimentally amplifying error
processes and functional forms to model the effects of errors on observables,
many zero-noise extrapolation schemes are possible with varying success.

In our numerical simulations of zero-noise extrapolation, gate repetitions are
implemented on individual MS gates. This could be performed everywhere in the
circuit, but would triple the total MS gate count for the lowest order
extrapolation. Since the DMERA channel acts to dissipate perturbations from
the fixed point state, errors from earlier layers of the circuit are
exponentially suppressed as seen in Fig. 5. To reduce the overhead of
inserting extra gates into the entire circuit getting the most of error
extrapolation we can focus on amplifying and subtracting noise from the final
layers of the circuit which have a greater impact on the measured expectation
value.

One way to model noise on an observable is as a polynomial perturbation. The
measured energy density $\tilde{E}$ is the noiseless asymptotic value $E^{*}$,
plus a polynomial perturbation in the unknown error parameters
$\varepsilon_{i}$ for some independent sources of error, which we assume
contribute independently without interaction to the measured energy
$\tilde{E}$. To fit a polynomial with $m$ parameters, a collection of at least
$m$ appropriate noise amplification measurements are needed in addition to the
unamplified measurement to solve for all the coefficients.

Chunking the error from each channel application into a single source, we
expect the error contribution $\varepsilon$ from each application to be
identical, but with error from earlier stages of the circuit reduced by the
scaling parameter $\lambda$. This suggests a polynomial model of the error
with three free parameters:

$\tilde{E}=E^{*}+\varepsilon+\varepsilon\lambda+\varepsilon\lambda^{2}+\ldots=E^{*}+\frac{\varepsilon}{1-\lambda}$
(6)

An alternate functional form for the error is a product of exponentials
applied to the target energy. Following the same logic, we get an analogous
functional form for the noisy energy as an exponential with three free
parameters.

$\tilde{E}=E^{*}e^{-\varepsilon}e^{-\varepsilon\lambda}e^{-\varepsilon\lambda^{2}}\cdots=E^{*}e^{\frac{-\varepsilon}{1-\lambda}}$
(7)

This multiplicative series model is equivalent to the additive series ansatz
when expanding the exponential to first order, since
$E^{*}(1-\frac{\varepsilon}{1-\lambda}+\dots)\approx
E^{*}-\frac{E^{*}\varepsilon}{1-\lambda}=E^{*}+\frac{\varepsilon^{\prime}}{1-\lambda}$,
and also has the correct behavior for large $\varepsilon$ where the energy
density should approach zero.

These model parameters can be fit by amplifying errors in the last and second
to last layers of the circuit independently. While this error mitigation
technique requires measuring three different versions of the circuit, the gate
count overhead for the amplified circuits is not large compared to amplifying
the entire circuit. Numerically the scheme also performs much better than full
linear extrapolation applied by amplifying individual gates in the entire
circuit (Fig. 8). The values of observables obtained through zero-noise
extrapolation is not guaranteed to lie between the noisy measurements and the
true noiseless value, so the extrapolated energies could lie below the
noiseless fixed point values and may not be a variational upper bound to the
ground state energy. In the examples considered in our numerics, however, no
extrapolation schemes resulted in energies below the noiseless fixed point
values.

Figure 8: Percent error of fixed point energy densities for $D=2$ and $D=3$
channels using the angle imprecision error model and the $\mathcal{C}_{1}$
gate decomposition. Unmitigated fixed point energies (solid) are considerably
improved through zero-noise extrapolation of MS gates using a geometric series
error ansatz with additive(dot-dash) or multiplicative(dash) error, both of
which outperform standard linear extrapolation of gates in the entire circuit
(dotted) that requires much larger gate overhead.

##  5 Experimental Data

Comparison to other experiments: Previous experiments using NISQ era ion-traps
have prepared critical ground states and thermofield double states for the
quantum transverse-field Ising model for small systems using the QAOA
variational ansatz [26]. It is known that the critical ground state on $L$
qubits can be prepared exactly with a depth $L/2$ QAOA circuit [22]. Since the
critical model has polynomially decaying correlations, it is expected that a
geometrically local QAOA ansatz would need depth scaling linearly with the
system size in order to prepare a good approximation to the ground state.
However, with non-local gates it could be possible to lower the required depth
to prepare a high fidelity ground state approximation. Using the DMERA ansatz
it is possible to prepare polynomially scaling correlations with only a
logarithmic-depth circuit by utilizing non-local gates. While the size of our
experiment is similarly limited, the DMERA state preparation is better thought
of as a finite subystem of the thermodynamic ground state, while the QOAO
protocol prepares the exact ground state of a small sized system.

We run our experiment on the Honeywell HØ ion-trap quantum computer. This
device has six qubits with global connectivity achieved by bussing the pair of
ions to an interaction site when applying the two-qubit gate. Each application
of the $n$-qubit channel requires twice as many to implement, so we are
limited to three qubit channels but we can apply the channel arbitrary number
of times with mid-circuit qubit reset.

The native MS gates of the Honeywell ion-trap are also given in terms of $ZZ$
rotations, rather than $XX$ rotations as used earlier in the paper. Rather
than sandwich each MS gate with explicit Hadamard gates to transform them to
the other convention, we will implicitly transform the circuits and states to
the convention native to the Honeywell trap by interchanging $X$ and $Z$
operators in the experimental implementation. This means our new Hamiltonian
will be $H_{I^{*}}=\sum_{i}-Z_{i}Z_{i+1}+Z_{i-1}X_{i}Z_{i+1}$ and new ancilla
will be prepared in the state $|{+}\rangle$ via a Hadamard transformation on
the reset qubits.

To see the effectiveness of the state preparation algorithm and stability
against noise, we measure the expectation value of local observables as a
function of the number of iterations of our three qubit channel. With limited
resources, we prioritized sampling as many different number of channel
iterations as possible to see convergence and stability, rather than repeating
individual measurements many times to get better measurement precision. The
longest circuits measured had twelve layers of the DMERA channel, each having
120 MS gates, 156 single qubit gates, and 33 qubits recycled mid-circuit, and
the average number of samples per measurement was 200.

For initial states we choose products of single qubit states
$|{\psi_{1}}\rangle$ and $|{\psi_{2}}\rangle$ since they can be prepared with
high fidelity. The product of the first state $|{\psi_{1}}\rangle$ is one of
the two mean-field ground states, i.e. a pure product state with the lowest
energy, which will actually be below the energy of our noisy fixed point. The
second state $|{\psi_{2}}\rangle$ is orthogonal to $|{\psi_{1}}\rangle$ and
serves as a higher energy initial state which will decrease in energy under
applications of the noisy channel. These initial states also have opposite
signs for the expectations of local $X$ and $Z$ observables which we see
converging towards their fixed point values experimentally.

$\displaystyle\begin{split}|{\psi_{1}}\rangle&=\sqrt{\frac{3+2\sqrt{2}}{6}}|{0}\rangle-\sqrt{\frac{3-2\sqrt{2}}{6}}|{1}\rangle\hskip
14.22636ptE_{1}/L=-\frac{32}{27}\\\
|{\psi_{2}}\rangle&=\sqrt{\frac{3-2\sqrt{2}}{6}}|{0}\rangle+\sqrt{\frac{3+2\sqrt{2}}{6}}|{1}\rangle\hskip
14.22636ptE_{2}/L=-\frac{16}{27}\end{split}$ (8)

Figure 9: Experimental convergence of energy density from initial product
states $|{\psi_{1}}\rangle$ and $|{\psi_{2}}\rangle$. Data from the Honeywell
HØ ion-trap computer plotted against simulation curves of noiseless(dashed)
and best fit noisy(solid) circuits.

(a)

(b)

(c)

(d)

Figure 10: Convergence of local expectation values, for single qubit X and Z
measurements (a),(b) and multi-spin Hamiltonian terms ZXZ and ZZ (c),(d).
Experimental datapoints from the Honeywell HØ ion-trap computer plotted
against noiseless and best fit noisy simulation curves, for initial product
states $|{\psi_{1}}\rangle$ and $|{\psi_{2}}\rangle$.

While there are some outlying data points we find general agreement with the
convergence of local observables to somewhat stable values after multiple
iterations of our quantum channel. We also see that the convergence rate can
vary greatly depending on the exact observable, with the $Z$ expectation
converging much slower than either $X$ or Hamiltonian terms $ZZ$ and $ZXZ$. We
include in the experimental data for the expectation values of energy density
(Fig. 9) and local observables (Fig.10) the simulated noiseless channel
(dashed) and the best fit value of the noise parameter $\sigma^{2}$ for the
angle imprecision model, which we found to be $\sigma^{2}=0.112$. This noise
model corresponds to a fixed point energy density of $-1.08$, 15.2% higher
than the true ground state energy density of $-4/\pi$.

##  6 Conclusions

We were able to numerically and experimentally demonstrate the convergence and
noise-resilience of local dissipative circuits that prepare subregions of
renormalization fixed point states, specifically using circuits derived from
the unitary wavelet transforms in [3] for the ground state of the critical
Ising model. Numerical noise models of gate imprecision and local depolarizing
noise support a linear scaling of the fixed point observables in the local
error parameters $\sigma^{2}$ and $p$, and the channel depth $D$. We also
showed evidence that different choices of gate decomposition may greatly
affect the noise susceptibility for some kinds of noise.

Error mitigation techniques such as zero-noise extrapolation are also seen to
dramatically improve the mitigated error rate of observables with relatively
little gate overhead when using extrapolation schemes designed using knowledge
about the error propagation for a dissipative channel. These effectively lower
noise levels achieved from classical post-processing can be leveraged to
obtain the approximation accuracy of larger-depth circuits even in regimes
where their raw output is noisier than shallower-depth circuits.

We are interested in how similar DMERA circuits which prepare states in a
noise-resilient way could be adapted for models away from a critical point,
for thermal states, or for non-integrable models. It has also yet to be seen
how the scaling with channel depth and benefits of our error mitigation scheme
hold up for real experimental data, but which may all be tested relatively
soon on NISQ devices.

There may also be interesting connections with our work and holographic models
of quantum gravity and error correction. This relation could arise from the
interpretation of MERA-like circuits as a description of the AdS/CFT
correspondence[28] and the recognition of error correcting properties in the
structure of the duality [29, 30, 31]. Alternatively, the robustness of our
circuit may be interpreted holographically in terms of the dilution of noise
in an expanding universe [2, 32].

##  7 Acknowledgements

We gratefully acknowledge support from the Department of Energy under award
number DE-SC0019139. We also thank Ning Bao, Aniruddha Bapat, and Ruben
Andrist for helpful discussions and comments.

## References

  * [1] J. Preskill.  Quantum Computing in the NISQ era and beyond.  Quantum, 2:79, Aug 2018.
  * [2] I. H. Kim and B. Swingle.  Robust entanglement renormalization on a noisy quantum computer.  2017\.  arXiv:1711.07500.
  * [3] G. Evenbly and S. R. White.  Entanglement Renormalization and Wavelets.  Physical Review Letters, 116:140403, Apr 2016.
  * [4] J. M. Pino, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, M. S. Allman, C. H. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer, and et al.  Demonstration of the trapped-ion quantum CCD computer architecture.  Nature, 592(7853):209–213, Apr 2021.
  * [5] J. Borregaard, M. Christandl, and D. S. Franca.  Noise-robust exploration of many-body quantum states on near-term quantum devices.  NPJ Quantum Information, 7:45, 2021.  arXiv:1909.04786.
  * [6] G. Evenbly and G. Vidal.  Algorithms for entanglement renormalization.  Physical Review B, 79:144108, 2009.  arXiv:0707.1454.
  * [7] Shehab O, H. Kim I, H. Nguyen N, Landsman K, H. Alderete C, Zhu D, C. Monroe, and M. Linke N.  Noise reduction using past causal cones in variational quantum algorithms.  2019\.  arXiv:1906.00476.
  * [8] V. Giovannetti, S. Montangero, and R. Fazio.  Quantum Multiscale Entanglement Renormalization Ansatz Channels.  Physical Review Letters, 101(18), Oct 2008.
  * [9] F.Verstraete, M. M. Wolf, and J. I. Cirac.  Quantum computation and quantum-state engineering driven by dissipation.  Nature Physics, 5:633–636, 2009.  arXiv:0803.1447.
  * [10] A. Beige.  Quantum computing using dissipation.  In Institute of Physics Conference Series 173, page 35, 2003.  arXiv:quant-ph/0306125.
  * [11] A. Beige, D. Braun, B. Tregenna, and P. L. Knight.  Quantum computing using dissipation to remain in a decoherence-free subspace.  Physical Review Letters, 85:1762, 2000.  arXiv:quant-ph/0004043.
  * [12] P. Zanardi, J. Marshall, and L. C. Venuti.  Dissipative universal Lindbladian simulation.  Physical Review A, 93:022312, 2016.  arXiv:1506.04311.
  * [13] M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert.  A dissipative quantum Church-Turing theorem.  Physical Review Letters, 107:120501, 2011.  arXiv:1105.3986.
  * [14] B. Kraus, H. P. Buchler, S. Diehl, A. Kantian, A. Micheli, and P. Zoller.  Preparation of entangled states by quantum Markov processes.  Physical Review A, 78:042307, 2008.  arXiv:0803.1463.
  * [15] J. Marshall, L. C. Venuti, and P. Zanardi.  Modular quantum information processing by dissipation.  Physical Review A, 94:052339, 2016.
  * [16] I. Sinayskiy and F. Pettruccione.  Dissipative quantum computing with open quantum walks.  In AIP Conference Proceedings 1633, page 186, 2014.
  * [17] I. Sinayskiy and F. Pettruccione.  Efficiency of open quantum walk implementation of dissipative quantum computing algorithms.  Quantum Information Processing, 11(5):1301–1309, 2012.  arXiv:1401.6658.
  * [18] R. Jozsa and A. Miyake.  Matchgates and classical simulation of quantum circuits.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 464(2100):3089–3106, Jul 2008.
  * [19] I. H. Kim.  Holographic quantum simulation.  2017\.  arXiv:1702.02093.
  * [20] M. Foss-Feig, D. Hayes, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, J. M. Pino, and A. C. Potter.  Holographic quantum algorithms for simulating correlated spin systems.  2020\.  arXiv:2005.03023.
  * [21] G. Anikeeva, I. H. Kim, and P. Hayden.  Recycling qubits in near-term quantum computers.  2020\.  arXiv:2012.01676.
  * [22] W. W. Ho and T. H. Hsieh.  Efficient variational simulation of non-trivial quantum states.  SciPost Physics, 6(3), Mar 2019.
  * [23] J. Haegeman, B. Swingle, M. Walter, J Cotler, G. Evenbly, and V. B. Scholz.  Rigorous Free-Fermion Entanglement Renormalization from Wavelet Theory.  Phys. Rev. X, 8:011003, Jan 2018.
  * [24] F. Witteveen, V. Scholz, B. Swingle, and M. Walter.  Quantum circuit approximations and entanglement renormalization for the Dirac field in 1+1 dimensions.  2019\.  arXiv:1905.08821.
  * [25] F. Witteveen and M. Walter.  Bosonic entanglement renormalization circuits from wavelet theory.  SciPost Physics, 10(6), Jun 2021.
  * [26] D. Zhu, S. Johri, N. M. Linke, K. A. Landsman, N. H. Nguyen, C. H. Alderete, A. Y. Matsuura, T. H. Hsieh, and C. Monroe.  Generation of Thermofield Double States and Critical Ground States with a Quantum Computer.  2020\.  arXiv:1906.02699.
  * [27] K. Temme, S. Bravyi, and J. Gambetta.  Error Mitigation for Short-Depth Quantum Circuits.  Physical Review Letters, 119:180509, Nov 2017.
  * [28] B. Swingle.  Entanglement Renormalization and Holography.  Physical Review D, 86(6), Sep 2012.
  * [29] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill.  Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence.  Journal of High Energy Physics, 2015(6), Jun 2015.
  * [30] A. Almheiri, X. Dong, and D. Harlow.  Bulk locality and quantum error correction in AdS/CFT.  Journal of High Energy Physics, 2015(4), Apr 2015.
  * [31] I. H. Kim and M. J. Kastoryano.  Entanglement renormalization, quantum error correction, and bulk causality.  Journal of High Energy Physics, 2017(4), Apr 2017.
  * [32] N. Bao, C. Cao, S. M. Carroll, and A. Chatwin-Davies.  De Sitter Space as a Tensor Network: Cosmic No-Hair, Complementarity, and Complexity.  Physical Review D, 96(12), Dec 2017.

Figure 11: Quantum circuit for causal cone of a three qubit observable using
three layers of the $D=2$ circuit ansatz starting from the all zeros state,
presented in two alternate gate decompositions, $\mathcal{C}_{1}$ (left) and
$\mathcal{C}_{2}$ (right), both constructed with two-qubit Mølmer-Sørensen
gates and single qubit Z rotations.