Hefei National Laboratory, University of Science and Technology of China,
Hefei 230088, China Hefei National Laboratory, University of Science and
Technology of China, Hefei 230088, China

# Quantum circuit matrix product state ansatz for large-scale simulations of
molecules

Yi Fan  Hefei National Research Center for Physical Sciences at the
Microscale, University of Science and Technology of China, Hefei, Anhui
230026, China  Jie Liu  Hefei National Laboratory, University of Science and
Technology of China, Hefei 230088, China  Zhenyu Li<EMAIL_ADDRESS>Hefei
National Research Center for Physical Sciences at the Microscale, University
of Science and Technology of China, Hefei, Anhui 230026, China  Jinlong Yang
Hefei National Research Center for Physical Sciences at the Microscale,
University of Science and Technology of China, Hefei, Anhui 230026, China

###### Abstract

As in the density matrix renormalization group (DMRG) method, approximating
many-body wave function of electrons using a matrix product state (MPS) is a
promising way to solve electronic structure problems. The expressibility of an
MPS is determined by the size of the matrices or in other words the bond
dimension, which unfortunately should be very large in many cases. In this
study, we propose to calculate the ground state energies of molecular systems
by variationally optimizing quantum circuit MPS (QCMPS) with a relatively
small number of qubits. It is demonstrated that with carefully chosen circuit
structure and orbital localization scheme, QCMPS can reach a similar accuracy
as that achieved in DMRG with an exponentially large bond dimension. QCMPS
simulation of a linear molecule with 50 orbitals can reach the chemical
accuracy using only 6 qubits at a moderate circuit depth. These results
suggest that QCMPS is a promising wave function ansatz in the variational
quantum eigensolver algorithm for molecular systems.

##  1 Introduction

Electronic structure theory is essential in studying properties of chemical
systems. While classical methods such as configuration interaction (CI),
Møller-Plesset perturbation theory and coupled cluster (CC) theory provide
systematical ways to approach accurate ground states, the computational cost
grows rapidly and quickly goes beyond the capability of current computers.
Advances in quantum technology provide a promising pathway to solve the
electronic structure problem, which is expected to be one of the fields to
demonstrate practical quantum advantage in the near future1, 2, 3, 4, 5, 6. By
encoding the electronic wave function into the quantum state of qubits, the
many-body Schrödinger equation can be solved using quantum algorithms such as
quantum phase estimation (QPE) and variational quantum eigensolver (VQE).7, 2,
5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 Subject to
limited qubit resources and noisy realizations of near-terms quantum devices8,
25, the quantum-classical hybrid VQE method is more preferable for noisy
intermediate scale quantum (NISQ) devices. In recent years, variational
quantum circuit ansatzes such as the unitary coupled-cluster (UCC)26, 27, 28
and hardware-efficient ansatz (HEA)17 have been successfully applied in
quantum chemistry applications for small molecules on most leading quantum
hardware platforms12, 13, 14, 15, 16, 17, 18. Circuit optimization techniques
including operator reduction29, 30, 31, 32, 33 and adaptive algorithms34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46 have also been proposed to further
reduce circuit depth and lower optimization overhead.

In most VQE ansatzes, an orbital-to-qubit mapping is implemented, where
orbital occupations are mapped to $|0\rangle$ and $|1\rangle$ states of
qubits. The number of qubits required to encode the eigen states of an
electronic Hamiltonian is then determined by the number of basis functions,
which can easily exceed the capacity of NISQ devices. The recently proposed
quantum circuit tensor network (TN) ansatz provides a possible way to solve
this problem47, 48. By encoding a classical TN state into a structured
parametric quantum circuit with mid-circuit measurements, the quantum circuit
TN ansatz is capable of generating tensors with an exponentially large bond
dimension using a small number of qubits. Liu et al. demonstrated that matrix
product state (MPS) and projected entangled pair state (PEPS) represented by
quantum circuits are capable of simulating ground states of the Heisenberg
model with a high fidelity using much fewer qubits than the system size47.
Haghshenas et al. further investigated the variational expressibility of
quantum circuit MPS (QCMPS) against classical MPS and suggested that QCMPS can
be more parameter-efficient than its classical counterpart48. Therefore, it is
interesting to see whether it is possible to use the qubit-efficient QCMPS to
simulate large molecules which are difficult to be handled by conventional VQE
ansatzes due to the high qubit requirement and also by classical MPS methods,
such as the density matrix renormalization group (DMRG) algorithm, 49, 50 due
to the high bond-dimension requirement.

Implementing QCMPS based VQE algorithms in molecular systems will face two
challenges. A classical MPS is composed of dense matrices which are fully
flexible, while in QCMPS each matrix is represented by a structured circuit
block and the parameter space is thus constrained by the specific circuit
structure. Such a sparse parameterization certainly has influences on its
variational expressibility. Previous studies used nearest-neighbour entangled
blocks, such as the SU(2) symmetry circuit structures47 and the brick-wall
structure48, which may be inefficient to capture electron correlations
accurately. Therefore, the first challenge is to choose a suitable circuit
structure to build QCMPS for molecular systems. On the other hand, although
the number of qubits in QCMPS does not explicitly depend on the size of the
system or more specifically the number of orbitals, simulating a larger system
may still require a rapidly growing bond dimension thus more qubits to achieve
certain accuracy due to the violation of the area law51 of electronic
Hamiltonian. Therefore, the second challenge is to suppress the increase in
bond dimension. Similar to orbital localization in DMRG, a proper way to
change the basis for the electronic Hamiltonian is expected to be necessary in
QCMPS.

In this work, we developed a QCMPS ansatz for variational optimization of
ground states of molecular systems. The accuracy of the QCMPS is investigated
with different local circuit structures and molecular orbital basis. A
linearly entangled structure is compared with the fully entangled blocks. The
influence of orbital localization on the performance of QCMPS is analyzed. As
a demonstration, QCMPS is successfully applied in simulating ground states of
hydrogen chains with up to 50 orbitals, reaching chemical accuracy using a
very small number of qubits. Results presented here open a new avenue for VQE
simulation for large molecular systems.

##  2 Methods

###  2.1 Variational Quantum Eigensolver

(a)

(b)

Figure 1: (a) A quantum circuit representing
$\exp{[\theta(a^{\dagger}_{3}a_{0}-a^{\dagger}_{0}a_{3})]}|\Psi_{HF}\rangle$
under the Jordan-Wigner transformation. HY is defined as
$\texttt{HY}=\sqrt{2}/2\times(\texttt{Z}+\texttt{Y})$. (b) Hadamard test
circuit to calculate the expectation value of
$\hat{P}_{0}=\hat{\sigma}^{0}_{x}\hat{\sigma}^{1}_{y}\hat{\sigma}^{2}_{z}\hat{\sigma}^{3}_{x}$,
where $q_{\text{anc}}$ stands for the bottom ancillary qubit.

VQE computes expectation values of a given Hamiltonian on a quantum computer
and performs variational optimization of wave function parameters using a
classical algorithm. For a qubit Hamiltonian which is expressed as a linear
combination of Pauli strings:

$\hat{H}=\sum_{i}{c_{i}\hat{P}_{i}}$ (1)

where $\hat{P}_{i}$ is product of Pauli operators
$\\{\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z},\hat{\sigma}_{I}\\}$,
the expectation value of each Pauli string can be obtained through quantum
measurements, and the expectation value of $\hat{H}$ is thus expressed as a
summation:

$\displaystyle E(\theta)$
$\displaystyle=\langle\Psi(\theta)|\hat{H}|\Psi(\theta)\rangle=\langle\Psi(\theta)|\sum_{i}{c_{i}\hat{P}_{i}}|\Psi(\theta)\rangle$
(2)
$\displaystyle=\sum_{i}{c_{i}\langle\Psi(\theta)|\hat{P}_{i}|\Psi(\theta)\rangle}.$

An important part of VQE is the encoding of target wave function
$|\Psi(\theta)\rangle$ using a parametric quantum circuit. Conventional VQE
ansatzes usually use an orbital-to-qubit mapping, which is a straightforward
strategy to represent the eigen states of $\hat{H}$ using the quantum states
of qubits. For example, under the Jordan-Wigner mapping52, the orbital
occupation is represented directly by the qubit states $|0\rangle$ and
$|1\rangle$. As shown in Figure 1a, a two-electron Hartree Fock (HF) wave
function $|0011\rangle$ can be simply obtained using a four-qubit quantum
circuit by applying two X gates on the first two qubits, and the unitary
excitation operator
$\exp{[\theta(a^{\dagger}_{3}a_{0}-a^{\dagger}_{0}a_{3})]}$ corresponds to a
W-shape circuit which is parameterized by the Rz gates. Using such a protocol,
it is convenient to construct the parametric quantum circuit which encodes the
wave function from orbital-based physically motivated ansatzes such as unitary
coupled cluster (UCC). The expectation value of an operator $\hat{P}$ can also
be easily obtained by standard measurement techniques such as direct
measurement on qubits corresponding to orbital indices, or alternatively the
widely used Hadamard test as illustrated in Figure 1b.

With the orbital-to-qubit mapping, calculating wave function based properties
such as energy or reduced density matrices is straightforward. However, the
number of qubits required to simulate the wave function itself has a linear
dependence on the number of basis functions, which prohibits simulations of
larger systems or the use of large basis sets on NISQ devices. For example, a
simulation of H2 using cc-pVTZ basis set requires 56 qubits which are already
more than the qubits used in any quantum computing experiment for chemical
systems53. A moderate calculation for a typical periodic system using gaussian
basis set requires approximately $10^{3}$ qubits54, 55, 56, and this number
can be further increased by 10-100 times if the plane-wave basis set is
used57, leading to a more significant shortage in qubit resources.

###  2.2 Matrix Product State

The requirement for qubit resources is generally recognized as space
complexity of a quantum algorithm. On a classical computer, space complexity
is also an important issue. In traditional quantum chemistry, the electronic
wave function is first approximated as a single Slater determinant which is
easy to calculate. Then, electron correlation is recovered by including more
determinants. The number of all possible determinants increases exponentially
with the number of orbitals. However, as a consequence of locality, weights of
these determinants are likely to be parameterized by only a small amount of
information proportional to system size58. By encoding locality using a
connected graph of tensors, tensor network states can be used to reduce
computational overhead.59, 60, 61, 62, 63, 64, 65 MPS is one typical class of
tensor network states66 which have been widely used to solve quantum chemistry
problems.

The electronic wave function can be represented by a CI expansion:

$\left|\Psi\right\rangle=\sum_{i_{1}i_{2}\ldots i_{N}}{c_{i_{1}i_{2}\ldots
i_{N}}|i_{1}i_{2}\ldots i_{N}\rangle},$ (3)

where $N$ is the number of spin orbitals, and $|i_{1}i_{2}\ldots i_{N}\rangle$
is the computation basis marked by binary number string $i_{1}i_{2}\ldots
i_{N}$. The coefficients $\\{c_{i_{1}i_{2}\ldots i_{N}}\\}$ form an
$N$-dimensional tensor which contains $2^{N}$ amplitudes. The MPS ansatz
factorizes this rank-$N$ coefficient tensor into lower rank tensors
$\\{^{k}T\\}$ which can be written as:

$c_{i_{1}i_{2}\ldots i_{N}}=\sum_{u_{0}\ldots
u_{N}}{{{}^{1}T^{i_{1}}_{u_{0}u_{1}}}{{}^{2}T^{i_{2}}_{u_{1}u_{2}}}\ldots{{}^{N}T^{i_{N}}_{u_{N-1}u_{N}}}}$
(4)

where $\\{^{k}T^{i_{k}}_{u_{k-1}u_{k}}\\}$ represents elements of the rank-$3$
tensor ${{}^{k}T}$ at the $k$-th site, with $i_{k}$ called the physical index
and $u_{k}$ the auxiliary index. The maximum size of the auxiliary indices is
defined as the bond dimension of the MPS, which is denoted as $D=\max_{0\leq
k\leq N}{\\{u_{k}\\}}$. In classical calculations, a canonical form of the MPS
is often helpful to make the algorithm numerically more stable. A left-
canonical MPS satisfies

$\sum_{i_{k}}{\sum_{u_{k-1}}{({{}^{k}T^{i_{k}}_{u_{k-1}u^{{}^{\prime}}_{k}}})^{*}\times{{}^{k}T^{i_{k}}_{u_{k-1}u_{k}}}}}=\delta_{u^{{}^{\prime}}_{k}u_{k}}$
(5)

for every $0\leq k\leq N$. Similarly, the right-canonical form is ensured if

$\sum_{i_{k}}{\sum_{u_{k}}{{{}^{k}T^{i_{k}}_{u^{{}^{\prime}}_{k-1}u_{k}}}\times({{}^{k}T^{i_{k}}_{u_{k-1}u_{k}}})^{*}}}=\delta_{u^{{}^{\prime}}_{k-1}u_{k-1}}.$
(6)

Using the MPS formalism, the orbitals are encoded into tensors with size
$\mathcal{O}(D^{2})$, and the $N$-orbital electronic wave function is
parameterized using a total of $\mathcal{O}(D^{2}\times N)$ parameters which
lowered the space complexity from exponential to polynomial. The bond
dimension $D$ determines the expressibility of an MPS, and it significantly
influences the performance of MPS-based algorithms such as DMRG. If $D$ is
allowed to grow exponentially, the MPS can represent any wave function using
Equation 4. A wave function is said to be efficiently represented by an MPS if
$D$ grows sub-exponentially with the system size. Constant bond dimension can
be used if the system satisfies the one-dimensional area law, which is however
usually violated by electronic Hamiltonians. Using the MPS formalism combined
with proper orbital optimization methods, the DMRG algorithm is capable of
surpassing the gold-standard CCSD(T) for typical chemical systems67, 68, 69
and pushes the scale of high-level ab-initio calculations up to 200 orbitals.
However, the high bond-dimension requirement still makes a DMRG simulation
prohibitive for large systems.

(a)

(b)

(c)

Figure 2: (a) A schematic QCMPS circuit. The top qubit corresponds to the
physical indices $\\{i_{k}\\}$. A circuit block encoding a rank-3 MPS site
tensor is marked with a dashed rectangle. (b) A Hartree-Fock state represented
by a QCMPS circuit. (c) The equivalent global circuit of (a), where {Ui} and
{U${}_{i}^{{}^{\prime}}$} differs only by a few SWAP gates.

###  2.3 Quantum Circuit MPS Ansatz

The main idea of QCMPS is using quantum circuit blocks to represent the rank-3
tensors in MPS. Then, the number of blocks is determined by the number of
orbitals or in other words the size of the system. The size of the block and
thus the number of qubits is determined by the bond dimension instead of the
number of basis functions. As shown in Figure 2a, in a QCMPS with $N_{q}$
qubits, a circuit block corresponds to a specific orbital, which is similar to
a rank-3 tensor in classical MPS with a bond dimension $2^{N_{q}-1}$. Within
such a framework, the wave function represented by an MPS is encoded into a
quantum circuit. Typically, the Hartree-Fock wave function can be represented
with a $D=1$ QCMPS using only a single qubit (Figure 2b).

The orbital indices $\\{i_{k}\\}$ in the rank-3 tensors are not involved in
the horizontal contraction of MPS. Accordingly, the $q_{0}$ qubit which
corresponds to the orbital index should be reset after each circuit block.
Such a qubit reset or sometimes called qubit reuse technique is realized by a
mid-circuit measurement followed by conditioned X gate, as marked by a
measurement operation plus a $|0\rangle$ in Figure 2. It is also possible to
avoid mid-circuit measurement by using more qubits to construct an equivalent
global version of QCMPS. As shown in Figure 2c, in global QCMPS circuit, the
entanglement blocks $\\{U_{i}^{{}^{\prime}}\\}$ are consecutively applied on
$\\{q_{0},q_{1},q_{2}\\}$, $\\{q_{1},q_{2},q_{3}\\}$,
$\\{q_{2},q_{3},q_{4}\\}$, $\\{q_{3},q_{4},q_{5}\\}$ respectively, and the
output states are on the first 4 qubits $\\{q_{0},q_{1},q_{2},q_{3}\\}$ at the
end of the circuit. The blocks in QCMPS and global QCMPS differs only by a few
SWAP gates.

Calculating expectation values of physical quantities on the top of QCMPS is
slightly different from conventional orbital-to-qubit VQE ansatzes. Figure 3a
gives an example of measuring the expectation value of a 4-qubit operator
using Hadamard test. The controlled gates are always applied on $q_{0}$ which
is to be reset after each circuit block, and the measurement results from the
ancillary qubit $q_{anc}$ are used to calculate expectation values.

(a)

(b)

Figure 3:  (a) Circuit to calculate the expectation value of a Pauli string
with a QCMPS using Hadamard test. An ancillary qubit $q_{anc}$ is introduced.
(b) QCMPS is a right-canonicalized MPS.

Since the evolution of quantum states can be regarded as tensor contractions,
QCMPS corresponds to a right-canonical MPS as defined in Equation 6, as
illustrated in Figure 3b. Due to the sparse parameterization, the variational
expressibility of QCMPS is not only determined by the number of qubits (bond
dimension) but also constrained by specific implementation of circuit blocks.
For example, a circuit block with all qubit pairs entangled is expected to
perform better than a nearest-neighbour-entangled block, while a general
$N_{q}$-qubit unitary containing $\mathcal{O}(4^{N_{q}})$ parameters is
expected to be over redundant. Therefore, choosing a suitable circuit
structure is important in constructing QCMPS for chemical applications.

(a)

(b)

Figure 4: QCMPS circuit constructed with (a) LU or (b) AU layers.

Generally, a circuit block contains some single-qubit gates and $N_{l}$
entangling layers (Figure 4). In this work, two general types of entangling
layers are studied, including linearly entangled layers which entangles
neighbouring qubits consecutively (termed as LU) and fully entangled layers
which entangles all qubit pairs (termed as AU). An extension of AU terms as G2
is also used by replacing the 3-parameter controlled-U gate with a generic
$4\times 4$ unitary matrix, which can be decomposed using approximately 19
elementary one- or two-qubit quantum gates.70 The notation
$T\text{-}N_{q}$($N_{l}$) is used to represent an $N_{q}$-qubit QCMPS with
$N_{l}$ layers of type $T$ structure in each block, where
$T\in\\{\texttt{LU},\texttt{AU},\texttt{G2}\\}$.

###  2.4 Orbital Localization and Orbital Interaction

In classical MPS-based algorithms such as DMRG, localized orbitals are often
used to lower the requirement on bond dimension71. It is thus expected that a
proper orbital localization method can also be helpful for QCMPS. Starting
from a set of canonical molecular orbitals (CMO) which are expressed as linear
combinations of atomic orbitals:

$\phi_{s}(\bm{r})=\sum_{\mu}^{N}{\chi_{\mu}(\bm{r})C_{\mu s}},$ (7)

a set of localized orbitals can be expressed as linear combinations of the
CMOs:

$\tilde{\phi_{p}}(\bm{r})=\sum_{s}^{N}{\phi_{s}(\bm{r})U_{sp}},$ (8)

where the coefficients $\\{U_{sp}\\}$ form a unitary matrix. Several orbital
localization approaches have been proposed, such as the natural atomic
orbitals (NAO)72, Foster-Boys73, Pipek-Mezey74 and Edmiston-Ruedenberg 75
methods. Orbital transformation can also be regarded as applying an orbital
rotation operator $\hat{R}$ on the wave function:

$\displaystyle\hat{R}=\exp{(-\hat{\kappa})},$ (9)

where $\hat{\kappa}$ can be written as:

$\displaystyle\hat{\kappa}=\sum_{pq}{\kappa_{pq}\hat{a}^{\dagger}_{p}\hat{a}_{q}}.$
(10)

The anti-hermitian matrix $\kappa$ is the generator of $U$ in Equation 8,
which satisfies $U=\exp{(-\kappa)}$. This orbital rotation operator $\hat{R}$
is similar to a unitary coupled-cluster wave function with only single
excitations and can be implemented as a quantum circuit ansatz.76

In different orbital basis, there are generally distinct interactions between
orbitals. Using quantum information theory, the orbital interaction can be
characterized based on von Neumann entropies:77, 78

$I_{pq}=\frac{1}{2}(s^{2}_{pq}-s^{1}_{p}-s^{1}_{q})(1-\delta_{pq}).$ (11)

where $\\{p,q|p\neq q\\}$ are indices of orbitals, and the one- or two-orbital
entropies $s^{1}_{p}$ or $s^{2}_{pq}$ can be calculated from eigenvalues of
one- or two-orbital reduced density matrices $\\{\omega_{\alpha_{p}}\\}$ and
$\\{\omega_{\alpha_{pq}}\\}$ :

$\displaystyle
s^{1}_{p}=-\sum_{\alpha_{p}=1}^{4}{\omega_{\alpha_{p}}\ln{\omega_{\alpha_{p}}}},$
(12) $\displaystyle
s^{2}_{pq}=-\sum_{\alpha_{pq}=1}^{16}{\omega_{\alpha_{pq}}\ln{\omega_{\alpha_{pq}}}}.$

A large value of $I_{pq}$ indicates that interactions between orbital $p$ and
$q$ are significant. If $I_{pq}$ form a diagonal dominant matrix, the wave
function is locally entangled and can thus be simulated using an MPS with a
small bond dimension. In contrast, if there are large off-diagonal $I_{pq}$
elements, usually a larger or even exponentially growing bond dimension is
required. It should be noted that geometrically more localized orbitals does
not necessarily result in a more MPS-friendly wave function. As will be
discussed in Section 3, orbital localization in certain cases tends to give an
orbital interaction matrix with even more scattered pattern compared to the
case of Hartree-Fock orbitals.

##  3 Numerical Results

We first study the performance of different circuit structures. Then, the
influence of orbital localization is studied by calculating the orbital
interaction matrices. Finally, we use linear hydrogen chains to study the
expressibility of QCMPS by comparing with DMRG results. All calculations are
performed using the Q2Chemistry package79 interfaced with several open-source
software, including PySCF80 for one- and two-electron integrals and SciPy81
for the Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimizer. The STO-3G basis
set is used for H4, H6 and H8, and STO-6G is used for the hydrogen chains.
Full configuration interaction (FCI) results are obtained by direct
diagonalization of the qubit Hamiltonian. DMRG calculations are performed
using an in-house developed code based on PyTorch.82 The optimization of QCMPS
is carried out with constrains on particle number and total spin.

(a)

(b)

(c)

(d)

Figure 5: Geometries of (a) linear H4, (b) rectangular H6, (c) cubic H8 molecules and (d) one-dimensional hydrogen chain. H-H distance $d$ in H4, H6 and H8 molecules is $2.0$ Å. $\Delta E$ |  LU-4(1) |  LU-4(3) |  LU-4(7) |  AU-4(1) |  AU-4(3) |  G2-4(1) |  G2-4(3)
---|---|---|---|---|---|---|---
H4 | 201.893 | 1.302 | 1.302 | 1.301 | 1.301 | 1.301 | 1.301
H6 | 295.652 | 158.263 | 126.361 | 127.359 | 123.361 | 123.346 | 115.449
H8 | 363.896 | 269.661 | 232.374 | 244.085 | 226.038 | 228.248 | 211.035
$N_{p}$ |  LU-4(1) |  LU-4(3) |  LU-4(7) |  AU-4(1) |  AU-4(3) |  G2-4(1) |  G2-4(3)
H4 | 264 | 600 | 1272 | 480 | 1248 | 1664 | 4992
H6 | 396 | 900 | 1908 | 720 | 1872 | 2496 | 7488
H8 | 528 | 1200 | 2544 | 960 | 2496 | 3328 | 9984
Table 1: Error with respect to FCI in kcal/mol ($\Delta E$) and number of
variational parameters in QCMPS circuits ($N_{p}$) for H4, H6 and H8
molecules.

Effects of circuit structures are studied using linear H4, rectangular H6 and
cubic H8 molecules (Figure 5a-c). Since the purpose of this test is to study
the performance of different types of circuit blocks, we use four qubits
$N_{q}=4$ for all molecules, which does not converge the results to chemical
accuracy. As shown in Table 1, with a fixed number of qubits, the accuracy can
be improved by increasing the number of layers $N_{l}$ before it goes to
saturation. For the linear H4 molecule, three LU layers are capable of
approaching the accuracy of a single AU layer, which already makes LU
structure less favorable regarding the number of parameters. In the case of
non-linear molecules H6 and H8, seven LU layers are required for an accuracy
comparable to a single-layer AU structure. In these cases, to achieve similar
accuracy, LU structures leads to a two- to three-fold larger number of
parameters to be optimized compared to AU structures. These results indicate
that AU blocks are more efficient in capturing electron correlations in
complex systems than LU. Therefore, we mainly use the fully-entangled AU
structure in our QCMPS circuits. If G2 blocks are used, the accuracy can be
further improved on top of the AU circuit with the same $N_{l}$ but more
parameters.

(a)

(b)

Figure 6: (a) Potential energy curve of H4 molecule with R(H-H) ranging from
0.6 to 2.5 Å and (b) the corresponding errors with respect to FCI energies.
Asterisk marks indicate that NAOs are used, otherwise CMOs are used. Shaded
area represents the chemical accuracy ($\Delta E\leq 1.0$ kcal/mol).

Another flexibility in the QCMPS model is the basis set to construct the wave
function. Typically, Hartree-Fock orbitals are commonly used to construct
correlated wave function. However, such delocalized CMOs can be suboptimal for
MPS-like ansatzes. It is therefore interesting to further study the influence
of orbital localization. Four-qubit QCMPS results for the H4 molecule obtained
using CMOs and NAOs are shown in Figure 6. DMRG results with an equivalent
bond dimension $D=8$ are also given as a reference. For the strongly
correlated region where bond length R(H-H) is large, using NAOs for QCMPS
significantly outperforms canonical HF orbitals and is able to reduce the
errors by orders of magnitude. However, NAO is not universally preferable
across the potential energy surface, as shown for the results with
R(H-H)$<1.50$ Ångstrom.

(a)

(b)

(c)

(d)

Figure 7: Orbitals and the corresponding orbital interaction matrices for
QCMPS wave functions of the H4 molecule. (a) CMOs and (b) NAOs at R(H-H)=2.5
Å. (c) CMOs and (d) NAOs at R(H-H)=0.6 Å.

To understand such a degradation at small bond lengths, we calculate orbital
interaction matrices for ground state wave functions of the H4 molecule at
R(H-H)$=$0.6 and 2.5 Å. As shown in Figure 7, NAOs successfully concentrate
entanglement onto neighbouring orbitals at R(H-H)=2.5Å, which leads to a
QCMPS-friendly wave function. However, at R(H-H)=0.6Å, the interactions
between NAOs are still scattered and the values of $I_{pq}$ are even larger
than CMOs even if the NAOs are geometrically more localized. Although NAOs
greatly improve the accuracy of QCMPS at regions where static correlations are
significant, they fail to give consistently accurate results across the
potential energy curve. An inappropriate localization method can lead to large
orbital entanglement and even scatters strongly interacting orbitals far
apart, which makes the wave function beyond the expressibility of QCMPS with a
small number of qubits. An optimal choice of orbital localization methods for
such an MPS-inspired ansatz is usually system-specific and requires a
benchmark for complex systems83. Previous studies show that the general
structure of $I_{pq}$ are barely affected by the bond dimension78. Therefore,
in QCMPS simulations, it is possible to optimize orbital rotation parameters
($\\{\kappa_{pq}\\}$ in Equation 10) using a small $N_{q}$ before an accurate
calculation.

Number of orbitals | 24 | 32 | 40 | 50
---|---|---|---|---
$N_{q}=2$ | 1 | 1 | 1 | 1
$N_{q}=3$ | 2 | 2 | 2 | 2
$N_{q}=4$ | 3 | 3 | 3 | 3
$N_{q}=5$ | 4 | 5 | 5 | 4
$N_{q}=6$ | 6 | 6 | 6 | 6
Table 2: Number of layers $N_{l}$ used in the QCMPS simulations of hydrogen
chains. The $N_{l}$ is selected such that the energy difference of
G2-$N_{q}(N_{l})$ and G2-$N_{q}(N_{l}-1)$ is smaller than $2.0\times 10^{-5}$
Hartree. Figure 8: Error of QCMPS and DMRG simulations of H chains with
respect to well converged DMRG energies obtained at a bond dimension of
$D=128$. The numbers before ’QCMPS’ and ’DMRG’ give the number of orbitals in
hydrogen chains. Shaded area represents the chemical accuracy ($\Delta
E\leq$1.0 kcal/mol).

To demonstrate the power of QCMPS, we first calculate the potential energy
curve of H2 with CMOs. The chemical accuracy can be readily obtained with an
AU-2(1) circuit (Figure S1) as expected. Then, the ground state energies of
more complicated one-dimensional hydrogen chains with alternate H-H distances
(Figure 5d) are calculated. Here, the number of orbitals $N_{o}$ equals to the
number of hydrogen atoms. We consider four chains with $N_{o}=$ 24, 32, 40 and
50, respectively. Due to the excellent performance in the strongly correlated
region, NAOs are used in these systems. As listed in Table 2, adequate number
of G2 entanglement layers are used in the QCMPS circuit blocks. Therefore, the
number of qubits $N_{q}$ becomes the only hyperparameter of the QCMPS model,
which corresponds to a bond dimension $D=2^{N_{q}-1}$ in DMRG simulations.

As shown in Figure 8, errors in QCMPS simulations systematically decrease with
the number of qubits in a similar way as the errors decrease with the
corresponding bond dimensions in the DMRG calculations. The result that QCMPS
reaches a comparable accuracy to DMRG with bond dimension close to
$2^{N_{q}-1}$ is not trivial, since the rank-3 tensors are sparse with
constrains from circuit structure in QCMPS, while they contains a full
flexibility in DMRG. In the hydrogen chain case, such a difference between
QCMPS and DMRG starts to have an effect only when the number of qubits and the
corresponding bond dimension is already large and the chemical accuracy has
been reached by both methods.

In this example, we demonstrate that a large molecule can be simulated with
QCMPS using only a very small number of qubits. For a quantum circuit, another
important parameter with practical significance is the circuit depth. As shown
in Table 2, the required number of entanglement layers in this example is
roughly equal to $N_{q}$, which leads to a quantum circuit of
$\mathcal{O}(N_{q}^{3}\times N)$ gates where $N$ is the number of orbitals. In
this way, QCMPS is not only qubit efficient but also gate efficient compared
with commonly used ansatzes such as UCCSD. For example, in the 50-orbital
case, the total number of elementary one- and two-qubit gates used in G2-6(6)
is approximately $3.4\times 10^{5}$, which is orders of magnitude smaller than
that of the first-order Trotterized UCCSD ($5.9\times 10^{8}$) which has a
$\mathcal{O}(N^{4})$ scaling in circuit depth.

##  4 Conclusion

In this study, we have presented a qubit-efficient QCMPS ansatz for simulating
chemical systems. QCMPS encodes right-canonical MPS tensors into quantum
circuit blocks. The number of qubits controls the bond dimension of the
tensors and the number of circuit blocks is determined by the number of
orbitals in the system. A fully-entangled circuit structure is preferred in
QCMPS for molecular systems. QCMPS is efficient in qubit resource for large-
scale simulations. As a demonstration, QCMPS can use a maximum number of 6
qubits to simulate linear hydrogen chains with up to 50 orbitals, reaching
similar accuracies compared to those achieved by DMRG with an exponentially
large bond dimension. The QCMPS ansatz presented in this study thus represents
an attractive way to perform quantum simulations for large molecular systems
with limited qubit resources.

This work is supported by the Innovation Program for Quantum Science and
Technology (2021ZD0303306), the National Natural Science Foundation of China
(21825302), the Fundamental Research Funds for the Central Universities
(WK2060000018), and the USTC Supercomputing Center.

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