# The Variational Power of Quantum Circuit Tensor Networks

Reza Haghshenas<EMAIL_ADDRESS>Division of Chemistry and Chemical
Engineering, California Institute of Technology, Pasadena, California 91125,
USA  Johnnie Gray<EMAIL_ADDRESS>Division of Chemistry and Chemical
Engineering, California Institute of Technology, Pasadena, California 91125,
USA  Andrew C. Potter  Department of Physics, University of Texas at Austin,
Austin, TX 78712, USA  Department of Physics and Astronomy, and Quantum Matter
Institute, University of British Columbia, Vancouver, BC, Canada V6T 1Z1
Garnet Kin-Lic Chan<EMAIL_ADDRESS>Division of Chemistry and Chemical
Engineering, California Institute of Technology, Pasadena, California 91125,
USA

###### Abstract

We characterize the variational power of quantum circuit tensor networks in
the representation of physical many-body ground-states. Such tensor networks
are formed by replacing the dense block unitaries and isometries in standard
tensor networks by local quantum circuits. We explore both quantum circuit
matrix product states and the quantum circuit multi-scale entanglement
renormalization ansatz, and introduce an adaptive method to optimize the
resulting circuits to high fidelity with more than $10^{4}$ parameters. We
benchmark their expressiveness against standard tensor networks, as well as
other common circuit architectures, for the 1D/2D Heisenberg and 1D Fermi-
Hubbard models. We find quantum circuit tensor networks to be substantially
more expressive than other quantum circuits for these problems, and that they
can even be more compact than standard tensor networks. Extrapolating to
circuit depths which can no longer be emulated classically, this suggests a
region of advantage in quantum expressiveness in the representation of
physical ground-states.

##  I Introduction

Advances in digital quantum computing have led to renewed interest in quantum
circuit representations of many-body states. For this purpose, it is crucial
to understand the representational power and trainability of different circuit
architectures. Out of the many possible architectures, one promising choice is
circuits derived from tensor network states used in classical simulations of
quantum states with limited entanglement. Because of the close connections
between tensor networks and quantum circuits, and the significant numerical
experience using them in classical simulations, they provide a natural setting
to define the boundary between classical and quantum capabilities for quantum
simulation. The simplest measure of the classical complexity of a tensor
network is the tensor bond dimension. Consequently, one can construct a tensor
network that is hard to simulate classically by devising a quantum circuit
that maps to a tensor network with a very large bond dimension, in a small
number of circuit elements. For example, one can construct quantum circuits
that generate tensors with bond dimensions exponential in the circuit depth.
This is the idea behind “deep” or quantum circuit tensor networks which have
been of interest for different applications of quantum devices Kim and Swingle
(2017); Cong  _et al._ (2019); Huggins  _et al._ (2019); Foss-Feig  _et al._
(2020); Bauer  _et al._ (2020); Lubasch  _et al._ (2020); Barratt  _et al._
(2021). Further, when combined with mid-circuit measurements and qubit reuse,
some of these methods allow simulation of large-scale quantum systems with few
physical qubits Foss-Feig  _et al._ (2020, 2021); Chertkov  _et al._ (2021).

However, constructing a class of states that is hard to represent classically
is not sufficient to advance the simulation of physical systems. In the
context of physical quantum many-body states, we must address additional
questions (i) are sparsely parameterized quantum “circuit” tensor networks
capable of representing physical states more efficiently than the “dense”
tensor networks (i.e. where all elements of the tensors can be independently
varied) traditionally used in classical simulation? (ii) and for this task,
what are the optimal circuit architectures and optimization protocols? The
answers have potential implications not only for quantum simulations, but also
for classical simulations with tensor networks. For example an affirmative
answer to (i) would suggest that it may sometimes be better to classically
simulate with the quantum circuit tensor network directly, rather than via the
classical dense tensor network, so long as the circuit depth and tensor values
support efficient classical contraction and/or approximation.

Some work to answer questions (i) and (ii) has already appeared, such as in
Refs Uvarov  _et al._ (2020); Lin  _et al._ (2021); Slattery  _et al._ (2021);
Haghshenas (2021). In this work, our focus will be on establishing the
variational power of quantum tensor networks to capture quantum many-body
ground-states. This is an application where traditional dense tensor networks
do well, and is thus in some sense the hardest test for quantum circuit tensor
networks to pass. We will focus in particular on understanding the resources
(e.g. number of variational parameters) required, optimization strategies, and
influence of circuit architecture on the results. Because of the large number
of numerical experiments required, we will mainly focus on one-dimensional
quantum many-body states, although we present suggestive findings on two-
dimensional problems also. As we shall demonstrate, with careful optimization
strategies, quantum circuit tensor networks are very expressive, and in some
cases, even more expressive than classical dense tensor networks. This
suggests a regime where a quantum advantage in the sense of expressiveness may
be observed in physical ground-state simulations.

The structure of the paper is as follows. We first introduce quantum circuit
tensor networks, review the mapping between two common tensor networks, the
matrix product state and the multiscale entanglement renormalization
ansatzVidal (2008) (MERA), to block unitary circuits Barratt  _et al._ (2021);
Foss-Feig  _et al._ (2020), and introduce different architectures for the
local unitary representations. We also provide some intuition regarding the
expressiveness of different structures of quantum circuit ansätze. We then
examine optimization strategies for such circuits. We next carry out a
detailed comparison between the quantum circuit tensor networks, classical
dense tensor networks, and reference circuit classes studied in the
literature, to evaluate their variational efficiency and power, for both
energies and correlation functions. We finish with a discussion of our
findings in the context of computational quantum advantage.

Figure 1: (Color online) Schematic representation of a matrix product state
and its quantum circuit in (a) right canonical form for a system with $L=12$
(or equivalently $12$ qubits). The square tensors represent isometric tensors.
(b) A quantum circuit MPS with four bond qubits ($q=4$) and a local brickwall
circuit structure with depth $\tau=4$. The circle tensors are two-qubit
unitary gates, while the black tensors denote register qubits initialized in
the $|0\rangle$ state. Note that the arrows associated with the MPS and
quantum circuit MPS follow the tensor network convention rather than the
circuit convention, i.e. they are in the _opposite_ direction of circuit-
execution time.

##  II Tensor networks and quantum circuits

###  II.1 The Canonical Form of the Matrix Product State

A matrix product state (MPS)Affleck  _et al._ (1987); Verstraete  _et al._
(2006, 2008) is a tensor network consisting of a tensor for each site,
connected by bonds in a one-dimensional geometry. It is represented
diagrammatically by

(1)

where each square tensor  denotes a $D\times D\times d$-dimensional tensor.
The open bonds index the $d$-dimensional physical Hilbert space of the site.
(For example, if $d=2$, we can identify each site with a qubit). The connected
“virtual” bonds of dimension $D$ then control the number of parameters and
thus the variational power of the MPS (or more physically, the maximum
bipartite entanglement at each bipartition in the network).

The individual tensors in an MPS are not in unique correspondence with a given
quantum state due to gauge degrees of freedom: the state is invariant under
insertion of a gauge matrix and its inverse $G,G^{-1}$ between two tensors
(along a virtual bond). To fix the gauge degrees of freedom, a MPS can be
recast into a canonical formPerez-Garcia  _et al._ (2007). In canonical form,
the tensors satisfy additional isometric or normalization constraints, but for
a (normalized) MPS, this imposes no loss of representational power. A simple
algorithm to obtain the canonical form is to perform a sequence of $QR$ ($LQ$)
decompositions of the tensors; doing this from left to right (right to left)
brings an arbitrary MPS into left (right) canonical formVidal (2003). For
example, the right canonical form of the MPS in Eq. 1 can be represented by
the following diagram:

where every square tensor with arrows  is an isometric tensor. The isometric
condition is satisfied for contractions performed on the incoming bonds ,
where the left side of the equality is the identity matrix. The isometric
constraint fixes the gauge freedom up to permutations of bond indices. In a
similar way, the right canonical form is defined by tensors which satisfy an
isometric condition , while a “mixed canonical form” (central to the density
matrix renormalization group (DMRG) White (1992)) is obtained by combining
left and right canonical forms around a given central site, with the central
tensor satisfying the condition . When the tensors in a canonical MPS satisfy
isometric conditions, the MPS is an example of an “isometric” tensor network
Haghshenas  _et al._ (2019); Zaletel and Pollmann (2020). The number of
independent variational parameters in a canonical MPS of length $L$, with all
elements real, scales asymptotically as $\sim L\times D(3D-1)/2$. We will
refer to these standard MPS as “dense” MPS (dMPS), since the tensors assume
their most flexible parametrization, in contrast to the “sparse”
parametrization by a quantum circuit used later. Because of the close
relationship between the standard MPS formulation and the DMRG, we will
sometimes use the term DMRG.

###  II.2 Quantum circuit MPS

Given the canonical form of the MPS, the mapping to a block unitary quantum
circuit follows a simple procedure Schön  _et al._ (2007); Barratt  _et al._
(2021); Foss-Feig  _et al._ (2020). This is because isometric tensors can be
viewed as arising from the application of a unitary tensor to a partial set of
inputs. Concretely, the steps are as follows: $(i)$ The virtual bonds of
dimension $D$ (thick lines) are viewed as $q$ qubit bonds (aggregate dimension
$D=2^{q}$) (thin lines). Graphically, this is the relation . (Note that,
following standard MPS conventions, the arrows on these diagrams indicate
right-canonical form, and are opposite to the execution-time direction in the
associated quantum circuit). $(ii)$ The isometric tensors are viewed as
columns of a block unitary matrix with one fixed output qubit (denoted by a
black dot, here assuming $d=2$), i.e. . This mapping generates the MPS via a
block unitary circuit, where each block unitary is a matrix of dimension
$dD\times dD$.

In the above mapping, the variational space of normalized states generated by
the block unitary circuit and the standard “dense MPS” (dMPS) are exactly the
same. However, we can imagine replacing the block unitary by a local circuit
of two-qubit gates of given depth. One can then create a block unitary that
acts on a set of $q$ qubits, using as few as $O(q)$ two-qubit gates. This
allows us to generate an entangled state that lives in the variational space
of a $D=2^{q}$ dMPS, with as few as $O(q)$ variational parameters per site. We
refer to an MPS where the block unitary is encoded as a local circuit as a
“quantum circuit MPS” (qMPS).

There is much freedom to choose the structure of the local circuit. Here, we
explore several local circuit structures: (i) a brick-wall circuit, denoted
graphically as

with densely connected nearest-neighbour two-body unitary gates (circle
tensors ). Throughout, colors visually distinguish different circuit layers,
but each gate of a given color implements a distinct gate. For the brick-wall
circuit, we refer to a layer of even gates and a layer of odd gates as two
layers, thus the above circuit has a circuit depth of $\tau=6$. In a brick-
wall circuit, correlations spread slowly with increasing $\tau$, as the
effective correlation length $\xi$ grows linearly with circuit depth
$\xi\sim\tau$. (ii) A ladder circuit, for example,

where we label the circuit above as depth $\tau=4$. Correlations propagate
more efficiently in this structure: with $\tau=1$, the first and last qubits
are already correlated. (iii) A MERA structure

which represents a binary MERAEvenbly and Vidal (2009) with depth $\tau=5$.
Properties of MERA circuits in general are discussed more in the section
below. Note that here, however, the MERA structure is only being used for the
local circuit (i.e. a MERA quantum circuit, rather than a quantum circuit
MERA) and globally, the ansatz is still a qMPS. An example of the final
circuit structure of the qMPS using a local brick-wall circuit is shown in
Fig. 1. Corresponding figures for qMPS with local ladder and MERA circuits are
shown in Fig. 12.

Overall, the variational power of the qMPS ansatz is determined by three
factors: (i) number of qubits $q$ on which each block unitary circuit acts
(placing the qMPS in the variational space of a dense MPS with $D=2^{q}$),
(ii) the number of gates in the local circuit, (iii) the internal structure of
the local circuit. Note that the number of gates in each layer differs between
the local circuit structures, thus $\tau$ should not be directly compared
between the different structures. Instead, the number of gates (or
equivalently, variational parameters) asymptotically behaves as
$\sim\frac{1}{2}\tau L(q+1)$, $\sim\tau L(q+1)$ and $\sim 2\tau L(q+1)$ for
the brick-wall, ladder, and MERA structures, respectively. In the numerical
simulations, we will refer to these three kinds of qMPS circuits as qMPS-b,
qMPS-l, and qMPS-m, respectively.

Figure 2: (Color online) (a) A binary MERA wave function for $L=32$ qubits,
and (b) its quantum circuit, i.e. qMERA, where the local quantum circuit is
chosen to have a brick-wall structure with $q=2$ and $\tau=2$. The different
layers of the MERA are shown in different colors.

###  II.3 Quantum circuit MERA

Another commonly used tensor network is the MERA. This is a tensor network
state where the tensors are arranged to introduce bipartite entanglement at
multiple scales. In this ansatz, isometric tensors perform coarse-graining
while unitary disentanglers are applied to remove short-range entanglement at
the different length scales. We show a binary form of MERA in Fig. 2(a) with
unitary disentanglers and isometric coarse-graining tensors  distinguished by
different colors in each layer. We will refer to the standard form of MERA
where all tensors are assumed dense as dense MERA (dMERA).

Since MERAs are isometric tensor networks by construction, like in the MPS
canonical form, a quantum circuit MERA (i.e. a circuit whose global structure
is derived from the MERA) i.e. qMERA can be straightforwardly obtained by
decomposing both the block isometric and unitary tensors into local circuits
with a finite depth $\tau$ and given internal structure as already discussed.
A graphical illustration of a qMERA is shown in Fig. 2(b). An important
difference between a qMERA and a qMPS is the presence of a structured set of
long-range unitary gates. In 1D, this allows the qMERA to capture critical
entanglement beyond the area law with only a polynomial number of gates
Evenbly and Vidal (2011).

Similarly to the qMPS, the variational power of the qMERA ansatz is determined
by three factors, i.e. $q$, the number of gates, and the internal structure of
local circuits. In this work, we only consider qMERA with local brick-wall
circuits as depicted in Fig. 2(b)mer , which we will refer to as qMERA-b in
the numerical studies.

Figure 3: (Color online) Global quantum circuits with (a) brick-wall and (b)
ladder structures. Execution time flows from top to bottom.  Figure 4: (Color
online) The influence of the choice of optimization method. (a) The infidelity
$\bar{\mathcal{F}}$ versus iteration number for the global brick-wall ansatz
QC-b with $L=16,\tau=6$, optimized by CG, L-BFGS-B and local methods. The
solid and dashed lines represent two different initial starting states, chosen
from a uniform random distribution for the tensors. (b) The same plot for a
qMPS-b with $L=24,q=4,\tau=4$. The targeted wave function is the ground state
of Heisenberg model $\mathcal{H}_{\text{H}}$. Figure 5: (Color online)
Adaptive initialization of a QC-b circuit with (a) $\tau=6$ from an optimized
smaller circuit (b) with $\tau=4$. The gray gates denote identity operators
with some small random perturbations. Note that execution time flows from top
to bottom (opposite to the arrow direction).

###  II.4 Global quantum circuit ansatz

To place the performance of the quantum circuit tensor networks in context, we
also consider global brick-wall and ladder circuit ansätze, as depicted in
Fig. 3. These are referred to as QC-b and QC-l in the numerical studies below.

###  II.5 Properties of different quantum circuit ansatz

All the above ansatz are universal approximators in the sense that with
sufficient numbers of parameters (for example, controlled by $q$ and $\tau$ in
the quantum circuit tensor networks) they can represent any state. Certain
types of ground-state might be more compactly represented by one ansatz than
another, but it is difficult to make general statements without numerical
studies, as performed below. However, here we briefly provide some intuition
into the relationship between circuit structure and expressivity, and the
connection between the different ansatze.

We start with the relationship between the global quantum circuits and quantum
MPS. Both the global brick-wall and ladder circuits can be directly
transcribed into qMPS by grouping gates into block unitaries (Fig. 6). QC-b
and QC-l circuits with $\tau$ layers map to qMPS with $q=\tau-1$ and $q=\tau$
block unitaries, respectively. Each block unitary has the sparsest possible
entangled parametrization with $\tau$ gates arranged in a ladder structure,
with the ladder ascending in the opposite direction to the ascending direction
of the qMPS block unitaries. In fact, the only difference between the global
brick-wall and global ladder circuits when viewed from their corresponding
qMPS, is that the QC-b block unitaries overlap on only $q-1$ qubits rather
than the usual $q$ qubits. This choice of non-maximal overlap is what gives
rise to the specific brick-wall lightcone, where correlations cannot spread as
quickly as in a general qMPS or a global ladder circuit.

Mapping QC-b and QC-l to qMPS reveals that the circuit structures prioritize
reaching block unitaries of large size $q$ (large MPS bond dimension
$D=2^{q}$) at the expense of accurately representing each unitary, as each
block unitary is only minimally connected. If we assume, as seems likely, that
the part of the dense MPS variational space of bond dimension $D=2^{q}$
required to represent a large variety of quantum ground-states of physical
interest is not fully captured by these minimal local circuits, then QC-b and
QC-l do not efficiently cover the variational space. One can see the influence
of the block unitary circuit depth most dramatically in the expectation values
of operators acting on the leftmost site. Because of the circuit ordering,
such expectation values depend only on the parametrization of the first block
unitary. In the QC-b and QC-l ansätze one must increase the global circuit
depth in order to improve the leftmost local expectation values.

Figure 6: (Color online) Mapping from global quantum circuit ansatz to qMPS
for the (a) brick-wall and (b) ladder structures. Grouping the gates in the
global ansatz (top) defines block-unitaries in the qMPS (bottom). Execution
time flows from top to bottom.

In the more general form of the qMPS, $\tau$ and $q$ can be independently
varied. Thus qMPS is a superset of QC-b or QC-l and is more expressive,
although the balance between $\tau$ and $q$ will be problem specific. The
question is then whether the local circuit depth can be significantly reduced
from that required to exactly parametrize a block unitary over $q$ qubits,
which is exponential in $q$. There is room for optimism, as there are other
powerful variational states which map to sparse parametrizations of dense MPS.
For example, correlator product states Changlani  _et al._ (2009); Neuscamman
_et al._ (2011); Clark (2018); Al-Assam  _et al._ (2011), entangled plaquette
states Mezzacapo  _et al._ (2009); Thibaut  _et al._ (2019), and neural
network quantum states Glasser  _et al._ (2018) can be viewed as variational
states parametrized by non-unitary gates. These can map to dense MPS with
large $D$ (for example, capturing volume law scaling of entanglement with a
polynomial number of variational parameters). However, numerical studies have
shown that the number of variational parameters required in these ansatz for
physical ground state problems can be fewer than in a dense MPS Changlani  _et
al._ (2009).

Many of the above points also apply to the qMERA, in particular, the potential
for sparse circuit representation of the unitaries and isometries arising in
the dense MERA. In addition, the special geometric structure of the MERA means
that in 1D it spreads correlations while capturing logarithmic corrections to
the entanglement law. This is an important formal distinction from the other
quantum circuit structures considered here, although its importance for
capturing the energies and correlation functions of finite systems in
numerical studies must be established empirically.

Figure 7: (Color online) The role of the initial guess in the optimization.
(a) The relative error of the ground-state energy $\delta E$ (of Heisenberg
model $\mathcal{H}_{\text{H}}$) as a function of iteration number for a global
brickwall quantum circuit (QC-b) with depth $\tau$. When the $\tau=10$ ansatz
is initialized from the optimized $\tau=8$ parameters, we obtain a better
minimum than from a random initial guess. Notice that when using a random
initial guess the ansatz optimization can get stuck in a poor minimum, as seen
by the $\tau=8$ (random) and $\tau=10$ (random) results, which obtain the same
minimum. (b) A better initialization procedure using optimized circuit
parameters from smaller depths guarantees that the relative error decreases
monotonically when increasing circuit depth $\tau$. (c) Similar plot for a
qMPS ansatz with $q=4$. We similarly find the relative error drops
monotonically when increasing $\tau$ using initial guesses from a smaller
$\tau$ ansatz (good), while initializing from random guesses (rand) results in
optimizations which terminate at poor minima. Reference data from dense MPS
(DMRG) also shown.  Figure 8:  (Color online) Comparing the variational power
of quantum circuit tensor networks (qMPS, qMERA), dense tensor networks (dMPS
and dMERA), and global quantum circuits (QC). We show the relative energy
error $\delta E$ versus the number of variational parameters in the ansatz for
the Heisenberg (a,b,c) and Fermi-Hubbard (d, e, f) models with $L=32$. Indices
’b’ and ’l’ stand for brick-wall and ladder. (a, b) Comparison between qMPS
with local brick-wall and ladder circuits with DMRG and QC with brick-wall and
ladder structures. The largest circuit depth used for QC-b(l) and qMPS-b(l)
with $q=5$ is $\tau=14(9)$. (c) The performance of qMERA-b versus qMPS-b and
dMERA. The largest parameter sets used for the qMERA-b ansatz correspond to
$q=4,\tau=8$. (d, e, f) The same comparison for the Fermi-Hubbard model. The
largest circuit depths for qMPS-b with $q=8$ and qMERA-b with $q=3$ are
$\tau=32$ and $\tau=12$, respectively.

##  III Numerical optimization studies

###  III.1 Algorithms

The first question to answer in a numerical assessment of the variational
power of an ansatz is how to optimize it. In this section, we investigate how
to optimize the quantum circuit tensor network and global circuit ansatz
considered in this work, by optimizing the parameters of the two-body unitary
gates . We assume each two-body unitary is a general $SO(4)$ unitary (i.e.
real-valued unitary) with a 6 dimensional variational spacecou . We use two
optimization algorithms: (i) a local “DMRG”-like optimization, where we sweep
through the unitary gates , optimizing them one at a time while holding the
others fixed; (ii) a global gradient-based optimization, where all variational
parameters are updated at the same time. In the local optimization scheme, a
linearization of the problem, similar to the one used in MERA optimizations
Evenbly and Vidal (2009), is used to find locally optimal gates. In the global
gradient-based scheme, the global gradient (i.e. the first-order derivative
with respect to all variational parameters
$\partial(\cdots,\includegraphics[scale={0.005}]{ptwobodygate.pdf},\includegraphics[scale={0.005}]{twobodygate.pdf},\cdots)$)
is analytically calculated by automatic differentiation as supported in
quimbGray (2018). The energy is computed via exact contractionGray and Kourtis
(2021) of either a matrix product operator representation of the Hamiltonian
(MPS) or a sum of local terms (MERA). The unitary constraints are enforced by
differentiating through a function that maps the gate parameters to a unitary
matrix. The cost-function minimization is then performed using either the
Conjugate Gradient (CG) or Limited-Memory Broyden–Fletcher–Goldfarb–Shanno
(L-BFGS-B) algorithms Nocedal (1980); Nocedal and Wright (2006). The
algorithms are stopped once the relative change in energy is less than
$10^{-8}$.

###  III.2 Model Hamiltonians

We choose to study the 1D Heisenberg and Fermi-Hubbard models:

$\displaystyle\mathcal{H}_{\text{H}}$ $\displaystyle=$ $\displaystyle
J\sum_{i}\vec{S}_{i}\cdot\vec{S}_{i+1},$
$\displaystyle\mathcal{H}_{\text{FH}}$ $\displaystyle=$
$\displaystyle-t\sum_{i,\sigma}\left(c^{\dagger}_{i\sigma}c^{\vphantom{\dagger}}_{i+1\sigma}+h.c.\right)+U\sum_{i}c^{\dagger}_{i\uparrow}c^{\vphantom{\dagger}}_{i\uparrow}c^{\dagger}_{i\downarrow}c^{\vphantom{\dagger}}_{i\downarrow}-$
$\displaystyle-\mu\sum_{i\sigma}c^{\dagger}_{i\sigma}c^{\vphantom{\dagger}}_{i\sigma},$

where $\vec{S}$ are spin-$1/2$ operators and $c^{\dagger}_{\sigma},c_{\sigma}$
are spin-$1/2$ fermionic creation and annihilation operators respectively. For
the Heisenberg model we use $J=1$, and for the Hubbard model we use $t=1$,
$U=3$ and $\mu=U/10$. In both cases, the ground states are gapless in the
thermodynamic limit with algebraically decaying correlation functions,
although in practice we will simulate finite chains with open boundary
conditions.

###  III.3 Local Optimization versus Global Optimization

We first compare local “DMRG” style optimization versus global gradient-based
optimization for the infidelity cost function
${\bar{\mathcal{F}}}=1-|\langle\Psi|\psi\rangle|$, where $|\Psi\rangle$ is the
ground-state wavefunction of the model and $|\psi\rangle$ is the ansatz state.
$|\Psi\rangle$ is obtained by the standard DMRG algorithm using a dense MPS of
sufficiently large bond dimension ($D\sim 400$) so that any error in
$|\Psi\rangle$ is negligible. When $\bar{\mathcal{F}}=0$ then the circuit
ansatz is identical to the ground-state wave function.

The result of minimizing $\bar{\mathcal{F}}$ for the different ansatz and
optimization methods is shown in Fig. 4. We show data from the quantum circuit
MPS with a brick-wall local ansatz (qMPS-b) ($L=24$, $q=4$, $\tau=6$) and
global brick-wall quantum circuit (QC-b) ($L=16$, $\tau=6$) as representative
examples. We find that in all cases, the local DMRG style optimization
converges to the local minimum faster than the global gradient-based
optimization using either the CG or L-BFGS-B algorithms. In addition, we find
that in all cases, the L-BFGS-B algorithm converges more quickly than the CG
algorithm. However, we also see that both the speed of convergence as well as
the final converged result has some dependence on the initial guess. As
observed in Fig. 4, given a suitable initial guess, the global gradient-based
optimization eventually converges to a slightly lower minimum than that found
by the local DMRG optimization.

###  III.4 Initial Guess

The dependence of the optimization on the initial guess is well-known in
quantum circuit optimization, where poor initial guesses can sometimes give
rise to exponentially small gradients (the barren plateau problem McClean  _et
al._ (2018); Grant  _et al._ (2019); Volkoff and Coles (2021); Cerezo  _et
al._ (2021); Zhao and Gao (2021)). We can see a related problem in our
circuits. To illustrate this, we show results from optimizing the energy cost
function
$E=\min\langle\psi|\mathcal{H}|\psi\rangle,\,|\psi\rangle\in\text{qTN}$ and we
report the relative energy error $\delta E=E/E_{\text{exact}}-1$ versus
iteration number in Fig. 7 for several example circuits. We see in the top
panel (Fig. 7(a)) that when starting from a random initial guess for the
global QC-b ansatz, we converge to the same relative error for two different
circuit depths $\tau=8$ and $\tau=10$, despite the significantly larger number
of variational parameters for $\tau=10$ versus $\tau=8$.

To improve the initialization of larger circuits, we use optimized gates
obtained from a shallow circuit to initialize gates at larger circuit depth.
The heuristic for this adaptive initialization method is summarized as
follows: (i) optimal gates are obtained from a random initial guess for a low
depth circuit $\tau^{\prime}$, (ii) the initial guess for the ansatz with
depth $\tau>\tau^{\prime}$, is given by the optimized gates (from previous
calculations) for $\tau\geq\tau-\tau^{\prime}$ and the identity operator for
$\tau<\tau-\tau^{\prime}$, respectively (measuring depth from the register
qubits) (see Fig. 5 for an explicit example), (iii) small random perturbations
to all gates in the ansatz are applied to avoid getting stuck in a local
minimum, (iv) for a circuit ansatz with larger depth, we repeat steps (ii,
iii). Empirically, it is found that a gradual increase of the circuit depth by
$2-6$ layers works well, i.e. $\tau-\tau^{\prime}=2-6$. The identity
perturbation strength is also chosen to be of the same order as the local
unitary gradient norm.

In Fig. 7(a), we show that optimizing the global QC-b ansatz with $\tau=10$
starting from optimal gates from $\tau=8$, indeed results in a lower relative
error. In Fig. 7(b), we plot the results of optimizing the QC-b ansatz with
different $\tau$ (each initialized in the manner above, using lower depth
circuits as shown in Fig. 5(c)), which shows that we can now achieve a
systematic decrease in the relative error $\delta E$ as function of increasing
$\tau$. Notice that in all cases, it is necessary to first perform many
iterations to bring the ansatz out of the local minimum generated by the
smaller $\tau$ guess, before one observes a significant drop in the relative
error. Similar results are shown for the qMPS-b ansatz with fixed $q=4$ and
increasing $\tau$ for the local circuit (brick-wall ansatz) in Fig. 7(c); we
similarly see that we can achieve a systematic decrease in the relative error
when increasing $\tau$. Indeed, as we increase $\tau$ in the qMPS-b ansatz, we
obtain results that approach the dense MPS (DMRG) result with bond dimension
$D=2^{q}=16$. As this is the lower bound for the variational energy of any
qMPS with $q=4$, our optimization heuristic using adaptive initialization thus
fully realizes the variational power of the quantum circuit tensor network.

Figure 9:  (Color online) Relative error in the spin-spin correlation function
as a function of distance $r$. (a), (c) are for the Heisenberg model, while
(b) is for the Fermi-Hubbard model (both $L=32$). Both qMPS and qMERA use a
local brick-work circuit (qMPS-b, qMERA-b). In (a), the qMPS with $q=4$,
$\tau=16$ has fewer parameters than the dMPS with $D=16$, while in (b), the
qMPS with $q=5$, $\tau=30$ has fewer parameters than the dMPS with $D=32$. In
(c), the QC-l ansatz has a comparable numbers of parameters to the qMERA-b and
qMPS-b with $q=2$, $\tau=6$ and $q=4$, $\tau=5$, respectively. qMPS-b ($q=4$,
$\tau=12$) and qMERA-b ($q=3$, $\tau=10$) have comparable numbers of
variational parameters. QC-b correlation functions are generally worse than
those from the above ansatz.

##  IV Variational power of quantum circuit tensor networks, dense tensor
networks, and global quantum circuits

###  IV.1 Energies

Using the above optimization strategies, we can now systematically
characterize the variational power of the different ansätze discussed in this
work for ground-state representation. As the measure of expressiveness, we use
the relative energy error $\delta E$ as a function of the number of
variational parameters. (Related measures have been recently used to compare
different variational wavefunctions Stair and Evangelista (2020)). The various
ansatz are optimized using the global gradient scheme with the L-BFGS-B
algorithm, while the dense MPS results are obtained by the DMRG algorithm. The
largest parametrized circuit ansätze correspond to the following: (i) for
qMPS-b and qMERA-b, the number of bond qubits and local circuit depths are
$q=8,\tau=32$ and $q=4,\tau=8$ respectively, and (ii) for the global circuit
ansatz QC-b and QC-l, the largest circuit depths are $\tau=14$ and $\tau=9$
respectively. Despite the large number of circuit parameters, we find that the
optimization heuristics work to high accuracy, if sufficient iterations are
used. For example, the smallest relative energy errors we find using the qMPS
ansatz are $10^{-6}$ and $10^{-4}$ for the Heisenberg and Fermi-Hubbard models
with $L=32$, respectively, using $\sim 10^{6-7}$ iterations.

Ansatz |  Heisenberg, $(a,b)$ |  Hubbard, $(a,b)$
---|---|---
qMPS-b | $(20,4.0)$ | $(9,1.9)$
qMPS-l | $(14,3.1)$ | $(10,1.9)$
QC-b | $(4,1.4)$ | $(4.4,1.0)$
QC-l | $(8,2.2)$ | $(0.4,0.5)$
qMERA-b | $(15,3.1)$ | $(6.0,1.4)$
dMPS (DMRG) | $(15,2.9)$ | $(8.0,1.5)$
dMERA | $(3.5,1.2)$ | $(0.8,0.6)$
Table 1: Scaling coefficients $(a,b)$ in the form $\delta E\sim an^{-b}$ for
the various ansatz in the Heisenberg and Fermi-Hubbard models. The asymptotic
behaviour of the relative error $\delta E$ at large $n$ is controlled by $b$.

We benchmark the performance of the qMPS, qMERA, and global QC ansatz versus
the dense MPS (DMRG) and dense MERA for the 1D Heisenberg and Fermi-Hubbard
models with $L=32$ (2D results are discussed in a later section). The key
findings are as follows: (i) Comparing dense MPS with qMPS, we find that for
an equivalent number of variational parameters, qMPS achieves lower energies
than the dense MPS (see Figs. 8(a), (d)) in both the Heisenberg and Fermi-
Hubbard models. (ii) Similarly, comparing dense MERA with qMERA, we find that
for an equivalent number of variational parameters, qMERA achieves lower
energies than dense MERA. Taken together with the previous statement, this
implies that the _appropriate quantum circuit tensor networks are more compact
and expressive than their traditional dense counterparts_ for these problems.
In particular, the worst case possibility, that one requires an exponential
number of gates to accurately parametrize the local block unitary, does not
apply to these physical ground states. (iii) Such expressiveness is not shared
by the global brick-wall and ladder circuits (Figs. 8(b), (e)) which are
consistently less expressive than the qMPS. This is consistent with the
theoretical analysis in Section II.5, which identifies QC-b and QC-l as
constrained versions of the qMPS with minimal parametrizations of the block
unitaries. (iv) qMPS is somewhat more expressive than qMERA (Figs. 8(c), (f)).
This suggests that the formal ability to capture logarithmic corrections
(which exist in the thermodynamic limit of the 1D Heisenberg model) is either
unimportant for the energy or at the system size considered.

Empirically, we can also summarize the data by fitting the relative error to
the inverse polynomial $\delta E(n)\sim an^{-b}$, where $n$ is the number of
variational parameters. As shown in Fig. 8 in the log-log plot, this form fits
all the ansätze reasonably well, with some small systematic deviations, for
example in the case of DMRG at larger $n$. A linear fit to the log-log data
yields an estimate of $a$ and $b$, as shown in Table 1, where $b$ gives the
asymptotic scaling for large $n$. These results further support the findings
above: in the large $n$-limit, the qMPS ansatz is the most expressive ansatz
and outperforms the dense MPS, while qMERA outperforms the dense MERA, with
both also outperforming the global brick-wall and ladder ansatz. We see that
the brick-wall local circuit structure for qMPS yields a better overall
performance than local ladder/MERA structures (see also Appendix A). The qMERA
ansatz performs similarly to the dense MPS algorithm in both models with
$b_{\text{qMERA}}\approx b_{\text{dMPS}}$. The ratio
$b_{\text{qMPS}}/b_{\text{qMERA}}$ and $b_{\text{qMPS}}/b_{\text{dMPS}}$ is
$\sim 1.3$ in both models. Earlier studies of scale-invariant MERA and
infinite MPS have found a similar ratio, $b_{\text{MPS}}/b_{\text{MERA}}\sim
1.2$ Avella and Mancini (2013). Finally, depending on the model, either QC-b
or QC-l performs better: $b_{\text{QC-b}}<b_{\text{QC-l}}$
($b_{\text{QC-b}}>b_{\text{QC-l}}$) for the Heisenberg (Fermi-Hubbard) models,
respectively.

###  IV.2 Correlation functions

We next study how faithfully the different ansätze capture correlation
functions of the Heisenberg and Fermi-Hubbard models. We use the spin-spin
correlation function, defined as
$\langle\vec{S}_{0}\cdot\vec{S}_{r}\rangle-\langle\vec{S}_{0}^{2}\rangle$
(Heisenberg and Fermi-Hubbard) as a representative example.

In Fig. 9, we show errors in the correlation function $\delta C(r)$ (relative
to numerically exact data) for the various ansätze. In Fig. 9(a) (Heisenberg
model), we see that qMPS-b with $q=4,\tau=16$ produces the same algebraically
decaying correlation function as a dense MPS with $D=16$ over the full
distance range of $r<22$, despite having fewer variational parameters. In Fig.
9(b) (Hubbard model), we similarly find that qMPS-b with $q=5,\tau=30$
produces a similar quality correlation function to the dense MPS with $D=32$,
but again with fewer variational parameters. These results are consistent with
the greater expressiveness of the quantum circuit tensor network relative to
its dense counterpart.

In both models, we see that increasing either $\tau$ or $q$ leads to an
improvement of the correlation function. However, neither is a dominant factor
for convergence. For example, in the Heisenberg model, we find that using a
qMPS-b with $q=5,\tau=14$ provides a lower relative error compared to
$q=4,\tau=18$, despite having fewer variational parameters; but a qMPS-b with
$q=8,\tau=4$ performs similarly to $q=4,\tau=5$, despite having a large number
of variational parameters. Thus a balanced choice of $q,\tau$ is needed to
obtain the best result.

In Fig. 9(c), we show the qMERA-b, QC-l and QC-b correlation functions
alongside the qMPS-b correlation functions. One expects that qMPS-b will
accurately reproduce short-range correlations (up to the MPS correlation
length) while qMERA should perform better at long distances. Quantitatively,
we find that qMPS-b ($q=4,\tau=12$ and $\delta E=6\times 10^{-5}$) indeed
provides a lower relative error at short distances, while qMERA-b (with
$q=3,\tau=10$ and $\delta E=3\times 10^{-4}$) with a similar number of
variational parameters is more accurate at longer distances ($r>15$). In
addition, qMPS with $q=4,\tau=5$ outperforms QC-l with a similar number of
parameters, especially at short distances. At long range, qMERA-b with
$q=2,\tau=6$ is clearly better than QC-l, while at short distances it is
comparable. Overall, qMPS and qMERA thus appear to provide more faithful
representations of the correlation functions than the general quantum circuit
ansatz, again reinforcing the need to balance the spreading of entanglement
and the accuracy of the local block unitary representation. In addition, the
improved entanglement spreading structure of the qMERA is detected in the
correlation functions, even though it is not represented in the energy metric
of the previous section.

Figure 10: (Color online) Schematic representation of (a) a snake dMPS and (b) its quantum circuit (qMPS) used in the two-dimensional simulation of Heisenberg model with $L=6\times 6$ sites. The qMPS-b shown has four bond qubits $q=4$ with depth $\tau=4$. Figure 11: (Color online) Comparison between qMPS and dMPS for the two-dimensional Heisenberg model on a $6\times 6$ lattice. Ansatz |  2d Heisenberg, $(a,b)$
---|---
qMPS-b | $(4.7,1.1)$
dMPS (DMRG) | $(1.4,0.48)$
Table 2: Scaling coefficients $(a,b)$ for qMPS and dMPS in the two-dimensional
Heisenberg model on a $6\times 6$ lattice.

###  IV.3 Two dimensional systems

We have also carried out a limited set of studies on a two-dimensional
Heisenberg model using the qMPS ansatz arrange in a traditional snake through
the two-dimensional lattice as depicted in Fig. 11. In two-dimensional
systems, it is well known that a traditional dense MPS (arranged as a snake)
requires an exponentially large bond dimension in the system width to
represent area law ground states. Since states with exponentially large bond
dimension can be reached by qMPS with a polynomial circuit depth, one might
expect a larger expressivity difference between qMPS and dMPS in 2D. The
relative energy error $\delta E$ is plotted as a function of the number of
parameters in Fig. 11, and the corresponding fit of the relative error $\delta
E(n)\sim an^{-b}$ is shown in Table 2. From both of these, we indeed find that
the expressivity gap between qMPS and dMPS is greater in 2D, with
$b_{\text{qMPS}}/b_{\text{dMPS}}=1.4$ in 1D, but
$b_{\text{qMPS}}/b_{\text{dMPS}}=2.3$ in 2D. Of course, in two and more
dimensions, the possible choices of circuit architecture are richer, with many
types of isometric tensor networks to explore beyond the qMPS Haghshenas  _et
al._ (2019); Zaletel and Pollmann (2020); Slattery and Clark (2021);
MacCormack  _et al._ (2021).

##  V Discussion of quantum advantage

Our numerical results clearly show that for ground states of some commonly
considered physical models, there is a quantum advantage in the expressivity
of quantum circuit tensor networks versus the traditionally employed classical
dense tensor networks. The difference in scaling of the achieved accuracy as a
function of the number of parameters suggests that the advantage in
expressiveness will persist into a regime where the circuits can no longer be
contracted efficiently classically. Assuming standard tensor networks are the
best classical representation for these problems, this means that in the high
accuracy regime, quantum circuit tensor networks may have the (theoretical)
potential to achieve quantum advantage also in terms of computational cost.

To explicitly translate the advantage in representation to one of computation,
we must consider the cost to compute with the quantum circuit representation
(on a quantum device) versus the dense representation on a classical device.
We consider the case of MPS as an example. The cost to contract a classical
dense MPS is $O(D^{3})$, or $O(n^{3/2})$, where $n$ here denotes the number of
local parameters in the block unitary. (We ignore scaling with $L$ here and
below). For qMPS, the cost to run the circuit to prepare the state (assuming
gates are executed sequentially) is $O(n)$. We then imagine computing the
energy by sampling terms in the Hamiltonian; for a relative precision $\delta
E$, we require $O(1/(\delta E)^{2})$ measurements per term. Combining these
factors together with the empirical scaling of $\delta E$ with $n$, one finds
that the cost $T$ to compute the energy to an accuracy of $\delta E$ is
$T\sim\delta E^{0.52}$ (classical dMPS) and $T\sim\delta E^{2.25}$ (qMPS-b) in
the 1D Heisenberg model, and $T\sim\delta E^{3.1}$ (classical dMPS) and
$T\sim\delta E^{2.9}$ (qMPS-b) in the 2D Heisenberg model. A similar analysis
for MERA finds in the 1D Hubbard model, $T\sim\delta E^{3.75}$ (classical
dMERA) and $T\sim\delta E^{2.7}$ (qMERA). These small polynomial advantages
(where they appear) are perhaps reflective of the challenges of variational
quantum algorithms, and whether they are realizable, or persist with improved
classical techniques remains to be seen. However, it should be noted that the
asymptotic inefficiency of the quantum algorithm stems from the steep cost
associated with sampling expectation values. Techniques that trade coherence
for reduced sampling, for example reducing the number of repetitions to as few
as $O(\log(1/(\delta E)))$, with a measurement circuit depth proportional to
$O(1/(\delta E))$, therefore greatly affect this analysis of computational
advantage Wang  _et al._ (2019).

Of course to seize this potential advantage, one would also need to optimize
circuit parameters in this classically-intractable regime. In the case of
gradient optimization, classical algorithms obtain the gradient at the same
cost as the energy through backpropagation, but for quantum algorithms using
finite differences (for example using the parameter shift rule) Mitarai  _et
al._ (2018); Banchi and Crooks (2021), the energy evaluation must be repeated
$O(n)$ times. In this case, the above polynomial advantages will disappear
unless coherent expectation value techniques are used, which may be further
combined with coherent techniques for gradient evaluation Gilyén  _et al._
(2019). Also, we have assumed that the number of optimization iterations
needed to find the ground-state scales with $\delta E$ in a comparable way in
the quantum and classical computations. While the optimization heuristics
discussed in this work successfully find accurate ground-states with a
tractable number of optimization iterations, it remains to be seen whether
this scaling persists in very large circuits.

##  VI Conclusions

In this work we studied the variational power of quantum circuit tensor
networks, and in particular, quantum circuit matrix product states and the
quantum circuit multi-scale entanglement renormalization ansatz, for
representing the ground-states of quantum many-particle problems. As we
argued, this is a problem where standard tensor networks excel, and is thus a
high bar for quantum circuit tensor networks to meet. We found that quantum
circuit tensor networks outperform other common global quantum circuit ansatz
in variational power, requiring far fewer parametrized gates for a given
accuracy. In fact, they appear to be asymptotically even more expressive than
the standard tensor networks, in terms of the number of parameters to converge
to a comparable accuracy in the variational energy and correlation functions.
Our initial results in 2D suggest that this expressive advantage increases in
higher dimensions.

Although all simulations here were carried out classically, the difference in
expressiveness of the classical and quantum circuit tensor networks raises the
possibility of polynomial quantum advantage in the computation of certain
problems. The practical realization of such advantage critically depends both
on the performance of optimization heuristics (such as the one proposed here)
as well as the cost of estimating expectation values on quantum devices.
However, the variational power of the quantum circuit tensor networks provides
new motivation to improve the optimization strategies for this class of
circuits. It also provides impetus to study related types of ansatz in the
context of classical simulations, where they may provide the chance to improve
on long-standing tensor network paradigms.

###### Acknowledgements.

Primary funding for this work (RH, GKC) was provided by the US Department of
Energy, Office of Science, via DE-SC0019374. GKC acknowledges additional
support from the Simons Foundation via the Many-Electron Collaboration and via
the Simons Investigator program. ACP was supported by the US NSF Convergence
Accelerator Track C award 2040549. The numerical codes were implemented in the
quimb library Gray (2018) which is freely available qui ; qMP . Support for JG
and the development of the quimb library was provided by a gift from Amazon
Web Services, Inc.

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Figure 12: (Color online) Schematic representation of a qMPS circuit with (a)
ladder and (b) MERA internal local circuits, with local depth $\tau=2,4$ and
bond qubits $q=5,6$, respectively. Figure 13: (Color online) Comparison
between qMPS with MERA and brick-wall local circuits. The solid and dashed
lines represent two different initial starting states.

##  Appendix A qMPS with MERA local circuits

We benchmark the accuracy of the qMPS-m ansatz by studying the relative error
$\delta E$ for the Heisenberg model $\mathcal{H}_{\text{H}}$. In general, a
systematic study of the ansatz is difficult as there are many controlling
parameters: $q,q_{m},\tau$ where $q_{m}$ is the number of bond qubits for the
local MERA; and in addition, there is also the choice of the internal
structure of the local MERA. To simplify things, we fix the number of bond
qubits to $q=8$ and use a brick-wall structure for the MERA with $q_{m}=3$.
Empirically, we find that it is difficult to obtain converged results for
qMPS-m as it easily gets stuck in local minima. Thus, for the reported data,
two different initial states were chosen, one random and one obtained from a
smaller optimized circuit. In Fig. 13(a), we compare qMPS-m with qMPS-b for
different $\tau$. We find that the qMPS-b with $\tau=4$ easily outperforms
qMPS-m with $\tau=4$ (which has a larger number of variational parameters). We
see that increasing $\tau$ in qMPS-m from $4$ to $10$ only slightly improves
the relative error, i.e. from $9\times 10^{-3}$ to $6\times 10^{-3}$. Overall,
the complexity of this circuit structure may require additional improvements
in optimization strategy in order to realize its variational power.