# Parameter Redundancy in the Unitary Coupled-Cluster Ansatze for Hybrid
Variational Quantum Computing

Shashank G Mehendale<EMAIL_ADDRESS>Indian Institute of Science
Education and Research (IISER), Kolkata, West Bengal 741246, India  Bo Peng
<EMAIL_ADDRESS>Niranjan Govind<EMAIL_ADDRESS>Physical and
Computational Sciences Directorate, Pacific Northwest National Laboratory,
Richland, WA 99354, United States  Yuri Alexeev<EMAIL_ADDRESS>Computational
Science Division, Argonne National Laboratory, Lemont, IL 60439, United States

(August 27, 2024)

###### Abstract

One of the commonly used chemical-inspired approaches in variational quantum
computing is the unitary coupled-cluster (UCC) ansatze. Despite being a
systematic way of approaching the exact limit, the number of parameters in the
standard UCC ansatze exhibits unfavorable scaling with respect to the system
size, hindering its practical use on near-term quantum devices. Efforts have
been taken to propose some variants of UCC ansatze with better scaling. In
this paper we explore the parameter redundancy in the preparation of unitary
coupled-cluster singles and doubles (UCCSD) ansatze employing spin-adapted
formulation, small amplitude filtration, and entropy-based orbital selection
approaches. Numerical results of using our approach on some small molecules
have exhibited a significant cost reduction in the number of parameters to be
optimized and in the time to convergence compared with conventional UCCSD-VQE
simulations. We also discuss the potential application of some machine
learning techniques in further exploring the parameter redundancy, providing a
possible direction for future studies.

##  I Introduction

The variational quantum eigensolver (VQE) has arguably emerged as one of the
most promising algorithms for the ground state estimation of molecular
Hamiltonians in the noisy intermediate-scale quantum (NISQ) era. VQE starts
with a given Hamiltonian and a parameterized ansatz state. The expectation
value of the Hamiltonian with respect to the ansatz is taken to be the cost
function that is minimized by varying the parameters of the ansatz. The task
assigned to the quantum computer is to estimate the expectation value, while
the task assigned to the classical computer is to predict the new parameter
values based on the old expectation value. In this sense, they together form
the minimization routine and hence form a quantum-classical hybrid algorithm
appropriate for the NISQ era.

The first VQE experiment by Peruzzo and co-workers [1] utilized the unitary
coupled-cluster singles and doubles (UCCSD) ansatz derived from unitary
coupled-cluster theory [2, 3, 4, 5, 6, 7, 8, 9, 10]. The standard UCCSD ansatz
has an unfavorable scaling of the number of parameters with respect to the
system size [11], which translates to a large number of quantum gates and a
long circuit in the quantum simulations. The errors in these quantum gates and
circuits can quickly accumulate in the simulation, hindering the practical use
of the UCCSD ansatz for simulating large quantum systems on near-term quantum
devices.

To prepare ansatze with favorable scaling, researchers have proposed many
approaches (see Refs. 12, 13 for recent reviews). Usually, the ansatze are
classified into hardware-efficient type and chemical-inspired type with either
fixed or adaptive structures. Hardware-efficient ansatze are flexible and easy
to implement with the current quantum hardware [14, 15] but suffer from so-
called barren plateaus in the variational parameter landscape [16] and require
extra work to enforce the physical symmetries. The chemical-inspired ansatze,
on the other hand, are usually focused on some UCC variants with improved
scaling. Typical UCC variants with fixed circuit structures include the
unitary pair coupled cluster with generalized singles and doubles (k-UpCCGSD)
method [17] that provides “linear” scaling with the system size while
including an unknown prefactor $k$ that needs to be determined heuristically,
the orbital-optimized unitary coupled-cluster (OO-UCC) ansatz [18] introducing
Brueckner-type orbitals with slight performance improvement, and the double
unitary coupled-cluster (DUCC) ansatz [19, 20] naturally enabling the support
for more realistic active space approximations.

Between the chemical-inspired and hardware-efficient ansatze there also exists
the so-called qubit coupled-cluster (QCC) method introduced by Ryabinkin et
al. [21] resembling the coupled-cluster structure but working directly with
Pauli strings in the qubit space to reduce the number of two-qubit gates and
pursue the efficient use of quantum resources. In practice, however, since the
number of Pauli string candidates for the QCC ansatze have an exponential
scaling ($\mathcal{O}(4^{N}_{q})$, with $N_{q}$ the number of qubits
employed), QCC also requires a robust selection process based on an efficient
estimate of the contribution of the corresponding entanglers of these Pauli
strings to the correlation energy. Regarding chemical-inspired ansatze with
adaptive circuit structures, typical developments include the adaptive
derivative-assembled pseudo-Trotter ansatz variational quantum eigensolver
(ADAPT-VQE) [22], the qubit-excitation-based adaptative VQE (QEB-ADAPT-VQE)
[23], the iterative QCC approach [24], the unitary selective coupled-cluster
approach [25], sparse UCC ansatz [26], and their extensions. The key idea here
is to construct an ansatz that can recover most of the correlation energy with
the least number of operators and variational parameters, and the importance
of the operators is typically measured through the energy gradient with
respect to the corresponding variation parameters or simply their associate
correlation energy contribution.

In this paper we employ other approaches to explore the interconnections
between the parameters in the UCC ansatze that would possibly lead to a
performance improvement in the UCC ansatz preparation. In particular, we
employ spin adaption, small amplitude filtration, and entropy-based orbital
selection techniques in the UCCSD framework to explore parameter redundancy.
We also discuss the feasibility of applying machine learning techniques in
discovering parameter redundancy. The rest of this paper is organized as
follows. In Section II we give an introduction to the theoretical background
and numerical methodology. In Section III we present some numerical results
and the performance of our proposed UCC-VQE approaches for the quantum
simulations of some prototype molecular systems. In Section IV we discuss some
possible further improvements in this direction. In Section V we summarize our
conclusions and briefly present ideas for future work.

##  II Theory and method

###  II.1 Unitary coupled-cluster variational quantum eigensolver

In the UCC ansatze, the UCC operator is the exponential of the conventional
cluster operator minus its Hermitian conjugate. For molecular systems, the
cluster operator consists of all the excitation operators from the occupied
orbitals to the virtual orbitals, which, when acting on the single reference
state, produces a linear combination of all possible excited determinants. In
practice, the cluster operator is usually truncated to singles, doubles, or a
few more excitations. In this work we mainly use the UCCSD approach. The
reference state is usually taken to be the Hartree–Fock state of the molecular
system, $|\text{HF}\rangle$. Hence a parametrized UCC ansatz is given by

$\displaystyle\ket{\psi(\vec{\theta})}=e^{\mathbf{T}({\vec{\theta}})-\mathbf{T}^{\dagger}({\vec{\theta}})}\ket{\text{HF}},$
(1)

where for a UCCSD ansatz we have

$\displaystyle\mathbf{T}({\vec{\theta}})=\sum_{i,a}t_{i}^{a}({\vec{\theta}})a^{\dagger}_{a}a_{i}+\sum_{i<j,a<b}t_{ij}^{ab}({\vec{\theta}})a^{\dagger}_{a}a^{\dagger}_{b}a_{j}a_{i},$
(2)

with $i,j,\cdots$ being occupied spin orbital indices and $a,b,\cdots$ being
virtual spin orbital indices. Here $t_{i}^{a},t_{ij}^{ab}$ are the single and
double cluster amplitudes. The task of VQE is then to provide the minimum
value of the expectation value of the Hamiltonian $H$ with respect to the
parameterized state as

$\displaystyle E=\min_{\vec{\theta}}\expectationvalue{H}{\psi(\vec{\theta})}.$
(3)

###  II.2 Spin-adapted unitary coupled cluster

The first parameter simplification method we employ in this work is the well-
known technique of spin adaption [27, 28]. In a typical UCC-VQE minimization,
each parameter of the excitation operator is treated as an independent
variable. Thus, the resulting state from the minimization need not be an
eigenstate of the total spin, $S^{2}$, operator. In many cases, however, we
expect the ground state of the Hamiltonian to be an eigenstate of the $S^{2}$
operator. Hence, in order to make sure that we do not break the spin-symmetry,
some conditions on the excitation operators can be used. These ensure that the
cluster operator commutes with the $S^{2}$ operator and hence leads to an
eigenstate of the total spin. Consequently, the number of independent cluster
amplitudes is considerably reduced. In Eq. (2), if we relabel the subscripts
of cluster amplitudes to mean occupied/virtual spatial orbitals instead of
spin orbitals and specifically mention the flavor of the spin ($\alpha,\beta$)
besides these indices, then the spin adaption relations of a closed shell
electron configuration can be given as

$\displaystyle t_{i\alpha}^{a\alpha}$ $\displaystyle=t_{i\beta}^{a\beta}$ (4)
$\displaystyle t_{i\alpha j\alpha}^{a\alpha b\alpha}=t_{i\beta j\beta}^{a\beta
b\beta}$ $\displaystyle=t_{i\alpha j\beta}^{a\alpha b\beta}+t_{i\alpha
j\beta}^{a\beta b\alpha}$ (5) $\displaystyle t_{i\alpha j\beta}^{a\alpha
b\beta}=t_{i\beta j\alpha}^{a\beta b\alpha}\ \ $ $\displaystyle\&\ \
t_{i\alpha j\beta}^{a\beta b\alpha}=t_{i\beta j\alpha}^{a\alpha b\beta}.$ (6)

However, as has been noted in the literature [29], this procedure reduces only
the number of independent parameters supplied to the classical part of the
UCC-VQE routine and does not reduce the number of quantum gates and the
circuit depth. We also mention that even though we could implement these
restrictions to preserve spin symmetry, in a practical implementation we would
have to resort to Trotter decomposition, which might actually induce
additional errors and hence break the symmetry if the Trotter error is not
well controlled [29]. We will use the spin-adapted UCCSD or its notation SA-
UCCSD interchangeably in this paper.

###  II.3 Single orbital entropy

Another important concept used in this work is the single orbital entropy
[30]. The von Neumann entropy of a system that is represented by a density
matrix $\rho$ is given by

$\displaystyle S=-\rho\ln\rho=-\sum_{i}\lambda_{i}\ln\lambda_{i},$ (7)

where $\lambda_{i}$ is the $i$th eigenvalue of the density matrix. For a pure
state, the value of the entropy is zero, while for the maximally mixed state,
the value of the entropy is maximum. Calculation of entropy is useful for
evaluating the amount of entanglement in a system, and it is done by measuring
what is known as the entanglement entropy. Given a pure state of a system with
two subsystems $A$ and $B$, the entanglement entropy is defined as the von
Neumann entropy of the reduced density matrices of either of the subsystems.

Writing the electronic ground state as a density matrix, we can take any of
the molecular orbitals as a single subsystem and the remaining orbitals as
another subsystem and calculate the entanglement entropy of this molecular
orbital. This calculation gives the amount of entanglement shared between the
single molecular orbital and the rest of the molecular orbitals. This entropy
for each of the orbitals is called the single-orbital entropy. Mathematically,
let $\ket{\psi}$ be the ground state of the Hamiltonian in the Fock space

$\displaystyle\ket{\psi}=\sum_{n_{1},n_{2},\dots}N_{n_{1},n_{2},\dots}\ket{n_{1},n_{2},\dots},$
(8)

where $n_{i}$ corresponds to the occupation of the $i$th spatial orbital. That
is, $n_{i}$ can take one value from the set $\\{\\_\\_,\ \uparrow,\
\downarrow,\ \uparrow\downarrow\\}$ corresponding to a possible occupation
state: no electron, only $\alpha$ electron, only $\beta$ electron, and both
$\alpha$ and $\beta$ electrons, respectively. The corresponding density matrix
will be

$\displaystyle\rho$
$\displaystyle=\sum_{\begin{subarray}{c}n_{1},n_{2},\dots\\\
n^{\prime}_{1},n^{\prime}_{2},\dots\end{subarray}}N_{n_{1},n_{2},\dots}\outerproduct{n_{1},n_{2},\dots}{n^{\prime}_{1},n^{\prime}_{2},\dots}{N^{*}}_{n^{\prime}_{1},n^{\prime}_{2},\dots}.$
(9)

Then, for example, the reduced density matrix of the first spatial orbital can
be calculated as

$\displaystyle\rho_{1}$
$\displaystyle=\sum_{n^{\prime\prime}_{2},n^{\prime\prime}_{3},\dots}\sum_{\begin{subarray}{c}n_{1},n_{2},\dots\\\
n^{\prime}_{1},n^{\prime}_{2},\dots\end{subarray}}N_{n_{1},n_{2},\dots}\innerproduct{n^{\prime\prime}_{2},n^{\prime\prime}_{3},\dots}{n_{1},n_{2},\dots}$
$\displaystyle\hskip
50.00008pt\times\innerproduct{n^{\prime}_{1},n^{\prime}_{2},\dots}{n^{\prime\prime}_{2},n^{\prime\prime}_{3},\dots}{N^{*}}_{n^{\prime}_{1},n^{\prime}_{2},\dots}.$
(10)

Given that both $n_{1}$ and $n^{\prime}_{1}$ can take four different values,
$\rho_{1}$ will be a $4\times 4$ matrix. From this, we can calculate the
entanglement entropy or single-orbital entropy of the first spatial orbital as

$\displaystyle
S_{1}=-\rho_{1}\ln\rho_{1}=-\sum_{i=1}^{4}\lambda_{i}\ln\lambda_{i},$ (11)

where $\lambda_{i}$ is the $i$th eigenvalue of the reduced density matrix
$\rho_{1}$. The value of $S_{1}$ is a measure of entanglement between the
first spatial orbital and the rest of the orbitals. For example, it is zero
when the first orbital can be separated from all the others in Eq. (8).
Single-orbital entropy and/or the mutual information are usually employed as a
diagnostic measure of the correlation for the active space selection in the
complete active space calculations or system partitioning in the quantum
embedding calculations [31, 32, 33]. In this work we propose a way to utilize
the single-orbital entropy values to conduct orbital selection for correlation
energy calculation.

###  II.4 Numerical approach

Our proposed UCC-VQE numerical approach consists of three parts: spin
adaption, small amplitude filtration, and entropy-based orbital selection. The
aim is to minimize the computational cost of the conventional UCC-VQE
simulation through exploring and removing the redundancy in the parameter
space of the UCC ansatz. Spin adaption was briefly introduced above. The
details of the other two techniques used in our algorithm are explained below.

If a certain portion of the parameters stays negligibly small throughout the
minimization procedure, then the corresponding excitations can be
heuristically identified from the first $\kappa$ ($\kappa$ is a small integer
and $\kappa_{\min}=2$) iterations of the UCC-VQE approach and removed
afterward. We denote this approach as small amplitude filtration (SAF). We
combine this technique with spin adaption in the UCC-VQE approach to have a
more efficient UCC-VQE variant (denoted SA-SAF-UCC-VQE). The SA-SAF-UCC-VQE
procedure can be described as follows.

  1. 1.

Perform conventional UCC-VQE for $\kappa$ iterations.

  2. 2.

Apply two conditions on the parameters obtained from the $(\kappa-1)$th and
$\kappa$th iterations: (1) the absolute values of the parameters at the
$\kappa$th iteration, $|\vec{\theta}|$, need to be smaller than a small cutoff
$\epsilon_{1}$ and (2) the absolute value of the change in the parameter
values from the $(\kappa-1)$th to the $\kappa$th iteration,
$|\Delta\vec{\theta}|$, needs to be smaller than another small cutoff
$\epsilon_{2}$.

  3. 3.

Drop the parameters that satisfy conditions (1) and (2), and keep optimizing
the remaining parameters.

The resulting reduction in the number of parameters and execution time is
observed to be significant, as will be shown in Section III, which not only
improves the classical part of the VQE algorithm but also reduces the circuit
depth on the quantum computer.

Regarding the entropy-based orbital selection, we want to screen out the
orbitals that have a significantly lower entropy value, and correspondingly
negligible contributions to the total correlation, in comparison with other
orbitals. The efficiency of the UCC-VQE simulation can then be improved by
freezing these orbitals in the Hamiltonian to obtain an approximate yet
sufficiently accurate correlation energy for the whole system. Since the
orbital selection is based on the ordering of the orbital entropy values and
their relative discrepancy, the approximate reduced density matrices obtained
from some low-level correlation approaches can be sufficient for the entropy
calculations. In this sense, without explicitly mentioning it, the reduced
density matrices are computed at the Møller–Plesset (MP2) level in the present
study.

##  III Numerical Results

We applied the SA-SAF-UCCSD-VQE approach for computing the ground state
energies of five molecular systems— H4 (Linear), H4 (Ring), H6, LiH, and
H2O—in the STO-3G basis. We used the Qskit [34] module in Python to implement
our proposed approach. The aer_simulator_statevector backend was used to
perform the quantum computing emulation.

We will first explain the structures of the molecules considered. Next, we
will show the robustness of the SAF by comparing the number of free parameters
in the excitation operator and the time to convergence in the UCCSD-VQE ground
state simulations of the five molecules with and without employing spin
adaption and/or small amplitude filtration. Then, we will show the accuracy of
the algorithm by comparing the converged energy for each of the three
algorithms presented here. After that, we will discuss the improvement offered
by the entropy calculations.

For each of the molecules discussed below, unless mentioned otherwise, the
SciPy [35] minimizer L-BFGS-B has been used for the classical minimization
routine that constitutes the classical part of the UCC-VQE algorithm. The
initial parameter values for all the amplitudes of the ansatz have been chosen
to be zero. The Hartree–Fock state has been used as the reference for the
ansatz. The Jordan–Wigner [36] transformation is employed to convert Fermionic
operators to Pauli operators.

###  III.1 Molecular Structures

Figure 1: H4 on a ring.

  * •

The molecule H4 consists of 4 electrons and 8 spin orbitals in the STO-3G
basis and can be considered in different spatial structures. In this work we
focus mainly on two structures, the linear and the ring. In the linear
structure, all the hydrogen atoms are present on the same axis, and we have
taken the bond length between every adjacent hydrogen atom to be the same. For
the ring structure of the H4 molecule, we consider the four hydrogen atoms
forming a rectangle with the vertices to be on a ring, as depicted in Fig. 1.

  * •

We have considered only the linear structure of the H6 molecule for this work.
Here all the hydrogen atoms are present on the same axis, and we have taken
the distance between adjacent hydrogen atoms to be the same. We vary the H-H
bond length from 0.5 Å to 3.2 Å. H6 consists of 6 electrons and 12 spin
orbitals in the STO-3G basis.

  * •

The geometry of the LiH molecule is simple, comprising just two atoms. The
bond length has been varied from 0.8 Å to 4 Å. LiH has 4 electrons and 8 spin
orbitals in the STO-3G basis.

  * •

H2O is a planar molecule with the H-O-H bond making an angle of
$104.5^{\circ}$. We have taken O-H bond distances from 0.65 Å to 1.70 Å. The
molecule consists of 10 electrons in 14 spin orbitals in the STO-3G basis.

###  III.2 Performance of SA-SAF-UCCSD-VQE

Figure 2: Performance of the UCCSD-VQE (blue), SA-UCCSD-VQE (orange), and SA-
SAF-UCCSD-VQE (green) approaches for obtaining the ground states of five
molecules in terms of (a) the number of the parameters and (b) the time to
convergence. In the SA-SAF-UCCSD-VQE approach, we set $\kappa=2$,
$\epsilon_{1}=10^{-4}$, and $\epsilon_{2}=10^{-5}$. In all the UCCSD-VQE
calculations, the convergence is reached when the energy variant is less than
$10^{-6}$ a.u.

To understand the improvement offered by spin adaption and small amplitude
filtration, we first look at the number of free parameters reduced by
employing these schemes in the UCCSD-VQE approach and the time to convergence
in the ground state simulations of the five molecules. We collected the number
of free parameters required in three approaches UCCSD-VQE, SA-UCCSD-VQE, and
SA-SAF-UCCSD-VQE. The comparison is given in Fig. 2a. As can be seen, compared
with the conventional UCCSD-VQE, spin adaption in SA-UCCSD-VQE can reduce the
number of parameters in the optimization by roughly half; and combining both
spin adaption and small amplitude filtration as in SA-SAF-UCCSD-VQE further
reduces the number of free parameters to be only roughly one-quarter of that
in the conventional UCCSD-VQE. Since the number of parameters that need to be
optimized is greatly reduced, the VQE execution time is also significantly
reduced. As shown in Fig. 2b, as the system size grows and bond length
elongates (the latter usually indicates the electron correlation gets
stronger), the time saved by combining spin adaption and small amplitude
filtration is more significant than that by employing only spin adaption in
the UCCSD-VQE approach. A typical example can be observed in the VQE quantum
simulations of H2O at O$-$H=1.4 $\AA$, where the execution time of SA-UCCSD-
VQE is almost the same as that of UCCSD-VQE, while SA-SAF-UCCSD-VQE reduces
the execution time by almost an order of magnitude. We mention that in the SA-
SAF-UCCSD-VQE simulations, we set $\kappa=\kappa_{\min}=2$. Figure 3 shows the
average values of the amplitudes (identified as the small amplitudes after
$\kappa=2$ UCCSD-VQE iterations) at the end of the UCCSD-VQE simulations of
the five molecules at different bond lengths. As can be seen, the average
values of the small amplitudes at the end of the UCCSD-VQE simulations remain
mostly less than $10^{-5}$ throughout the bond length range, indicating that
their changes and impact on the minimization are trivial.

Figure 3: Average values of the small amplitudes at the end of UCCSD-VQE
simulations of the five molecules at different bond lengths. Each molecule is
scanned over a range of bond lengths that have been discretized to 12 bond
length indices.

We further compare the potential energy surfaces (PES’s) computed by different
UCCSD-VQE noise-free simulations discussed here, as well as by the exact
diagonalization. The results are shown in Fig. 4. As can be seen, all the
PES’s obtained from the UCCSD-VQE noise-free simulations agree with the ideal
PES’s very well, and the differences are well below the chemical accuracy
($\sim 1.6\times 10^{-3}$ Hartree). Looking more closely at the extent of
overlap, we see that the second column of Fig. 4 shows the error between the
UCCSD-VQE energies and the SA-UCCSD-VQE energies, and the third column shows
the error between the SA-UCCSD-VQE energies and SA-SAF-UCCSD-VQE energies. We
can clearly see that all the errors are extremely small (with the highest
being on the order of $10^{-5}$ for H2O), proving that removing the small
amplitudes makes a negligible difference in the converged energies as compared
with the approach that retains these amplitudes, and the small amplitude
filtering is reliable for calculating the ground states of the studied
molecules. We mention that the small amplitude filtering approach is
completely independent of spin adaption and hence a general scheme that can be
applied to any UCC-VQE approach at any stage.

Figure 4: Potential energy surfaces of the five molecules computed by three
UCCSD-VQE approaches. Left: the PES computed by conventional UCCSD-VQE and
exact diagonalization. Middle: the difference between the SA-UCCSD-VQE
energies and the UCCSD-VQE energies for all five molecules at different bond
lengths. Right: the difference between the SA-SAF-UCCSD-VQE energies and the
SA-UCCSD-VQE energies for all five molecules at different bond lengths.

Note that in these simulations we did not take symmetry into account. If
symmetry is taken into account, the classical minimization procedure can be
further improved. Take LiH computed at HF/STO-3G as an example. The
excitations including its SA-SAF-UCCSD ansatz are

$\displaystyle\left\\{\tau_{0}^{2},\tau_{0}^{5},\tau_{1}^{2},\tau_{1}^{5},\tau_{0,1}^{2,5},\tau_{0,7}^{2,11},\tau_{0,7}^{2,8},\tau_{0,7}^{3,9},\tau_{0,7}^{4,10},\tau_{0,7}^{5,11},\tau_{0,6}^{2,11},\right.$
$\displaystyle\left.\leavevmode\nobreak\ \leavevmode\nobreak\
\tau_{1,7}^{2,11},\tau_{0,6}^{2,8},\tau_{0,6}^{3,9},\tau_{0,6}^{4,10},\tau_{0,6}^{5,11},\tau_{1,7}^{2,8},\tau_{1,7}^{3,9},\tau_{1,7}^{4,10},\tau_{1,7}^{5,11}\right\\},$
(12)

where the orbital indices #0$-$#5 correspond to $\alpha$ spin orbitals with
#0$and$#1 occupied and others unoccupied and the indices #6$-$#11 correspond
to $\beta$ spin orbitals with #6 and #7 occupied and others unoccupied. Since
the unoccupied $\pi$ orbitals #3, #4, #9, and #10 are degenerate, the two sets
$\left\\{\tau_{0,7}^{3,9},\tau_{0,6}^{3,9},\tau_{1,7}^{3,9}\right\\}$ and
$\left\\{\tau_{0,7}^{4,10},\tau_{0,6}^{4,10},\tau_{1,7}^{4,10}\right\\}$
become interchangeable, and only one set needs to be optimized in the
minimization procedure. Note that this treatment does not help reduce the load
on the quantum computer since all the operators in (12) still need to be
implemented on the quantum side to form the UCCSD ansatz.

###  III.3 Improvement through orbital selection

Figure 5: (a) Single orbital entropies of the H2O molecule computed at the
MP2/STO-3G level. (b) Error in energy induced by freezing the different
orbitals.

Selecting a portion of the orbitals in the UCC-VQE calculations while still
maintaining the accuracy at a desired level not only reduces the number of
free parameters but also reduces the number of qubits required to encode the
problem on a quantum computer. In this section we use the H2O molecule as an
example to demonstrate the potential improvement in the computational
efficiency brought by the entropy-based orbital selection. The computed
single-orbital entropies of the H2O molecule over the studied O$-$H bond
length are shown in Fig. 5a, from which several observations can be made.

  * •

The entropies of orbitals #0 and #1 do not exhibit significant change over the
entire bond length range. In particular, the entropies of orbital #0 are
sufficiently small ($\sim 10^{-5}$) in comparison with the entropies of other
orbitals ($\geq 0.01$).

  * •

Regardless of the bond length, the entropies of orbitals #5 and #6 are always
the largest in comparison with the entropies of other orbitals, and they keep
increasing with the increasing bond length. Therefore, the entropy discrepancy
(for example, between orbitals #5 (or #6) and #0) becomes larger with the
increasing bond length.

  * •

The entropy ordering of orbitals #2, #3, and #4 does not stay the same over
the studied bond length range. Although the ordering exhibits crossovers over
the entire bond length range, the smallest one is very close to the entropy of
orbital #0 (the discrepancy is $<0.01$).

From these observations, since orbitals #0 and #2/#3/#4 become less important
in contributing to the total correlation, they can be frozen from the original
Hamiltonian. Specifically, (i) if the bond length $<1.4\leavevmode\nobreak\
\AA$, orbitals #0 and/or #4 can be frozen, (ii) if the bond length is between
$1.4\leavevmode\nobreak\ \AA$ and $1.5\leavevmode\nobreak\ \AA$, orbitals #0
and/or #3 can be frozen, and (iii) if the bond length
$>1.5\leavevmode\nobreak\ \AA$, orbitals #0 and/or #2 can be frozen. Figure 5b
compares the ground state energy of the Hamiltonian with and without freezing
these orbitals through exact diagonalization. As can be seen, the error
brought by freezing these orbitals is well below the chemical accuracy ($\sim
1.6\times 10^{-3}$ a.u.), and the error gets smaller with the increasing bond
length. In terms of improving the UCC-VQE efficiency, for the studied H2O
molecule, by removing one or two occupied orbitals, we reduce the number of
qubits required from 14 to 12 or 10, respectively, and at the same time reduce
the number of parameters in the UCCSD ansatz from 140 to 92 or 54,
respectively.

##  IV Discussion

While the results described above demonstrate the efficiency of the approach
proposed in this work, we are considering further improvement. For example, in
the case of SA-SAF-UCCSD-VQE, the cutoffs, $\epsilon_{1}$, and $\epsilon_{2}$,
and the heuristic number of iterations, $\kappa$, can be tweaked to balance
the efficiency and accuracy.

We can further speed up our approach by analyzing the relationship between the
excitation parameters. For example, in Eq. (5), to implement spin adaption for
the closed-shell electronic configuration, we have three choices: we can take
any two of the three excitation coefficients to be independent and write the
third one as a function of the remaining two. In such a situation, if we apply
small amplitude filtration after a spin adaption, the third parameter could be
filtered out while the remaining two parameters, despite being close to each
other, are minimized as independent parameters. Thus, we can perform small
amplitude filtration before applying spin adaption. If the filtered parameters
turn out to be one of the three quantities in Eq. (5), then only one of the
remaining two could be explicitly included in the minimizer.

A more involved theoretical analysis could also be done to further reveal the
connection between the parameters. Possible ways including employing the
mutual information [37, 38, 39] or evaluating the entropy of the wavefunction
using the determinant amplitudes [26]. For example, besides calculating the
single-orbital entropies, we could also calculate two-orbital entropies from a
16$\times$16 two-body reduced density matrix, where the trace would be taken
over by all except two orbitals. Then, the mutual information could be
calculated through

$\displaystyle I_{i,j}=\dfrac{1}{2}(S_{i}+S_{j}-S_{ij}).$

The value of the mutual information could be used in predicting possible
dependencies among the parameters. A similar idea has been employed in the
ADAPT-VQE algorithm [40] and could also be used for exploring the
interconnection between the excitation operators involving more than two
spatial orbitals.

Furthermore, one might anticipate the existence of other hidden relations
among the remaining parameters. A way to explore such hidden relations has
been reported recently (see Refs. 41, 42), where it has been observed through
phase analysis that the coupled-cluster amplitudes could be divided into
principal and auxiliary categories and that a synergistic relationship between
them exists during the iteration dynamics. To find such a classification as
well as the synergistic relationship, one could rely on some machine learning
(ML) techniques. Here, as an exploratory attempt, we have prototyped an ML-
assisted UCC-VQE approach, whose procedure can be summarized as follows.

  1. 1.

Perform the UCC-VQE calculation for $n$ iterations.

  2. 2.

Build a regression model based on the existing data obtained from step 1:

    1. (a)

Label the parameters in the $n^{\rm th}$ UCC-VQE iteration into principal and
auxiliary amplitudes according to a predefined absolute value cutoff
$\epsilon$.

    2. (b)

Using the labels in step 1 to collect the principal and auxiliary parameters
from $n$ iterations to form the subsets $\mathbf{X}$ and $\mathbf{Y}$,
respectively.

    3. (c)

Treat subset $\mathbf{X}$ as a set of independent variables and subset
$\mathbf{Y}$ as a set of dependent variables, and solve the regression problem
$\mathcal{F}:\mathbf{X}\rightarrow\mathbf{Y}$.

  3. 3.

Once the regression function is developed, proceed to new iterations, where
only the new principal amplitudes are evaluated. The new auxiliary amplitudes
are generated by applying the regression function to the new principal
amplitudes.

  4. 4.

Once the minimizer in step 3 reaches convergence, the resulting amplitudes
will be taken out and supplied back to the UCC operator of step 1 to perform
another $n$ UCC-VQE iterations.

  5. 5.

Repeat steps 2 to 4 until convergence is reached. The convergence is marked by
one of two scenarios: either the UCC-VQE simulation in step 1 terminates
within $n$ iterations, or step 4 reaches convergence in a single iteration.

Figure 6: Efficiency of the ML-assisted SA-SAF-UCCSD-VQE simulations for the
LiH molecule at different bond lengths in terms of (a) the energy difference
and (b) the time-to-convergence in comparison with the non-ML SA-SAF-UCCSD-VQE
simulations. In the ML-assisted simulations at each bond length, $\sim$30%–40%
of the total amplitudes from $n=4$ SA-SAF-UCCSD-VQE iterations are taken to be
the principal amplitudes to construct a nonlinear regression model employing
Eq. (17) as the kernel with $\gamma=1,c_{0}=0$, and $d=3$.

If one is considering a linear regression model, then the regression model
will be given by

$\displaystyle\mathbf{Y}=\mathcal{F}(\mathbf{X})=\mathbf{X}\cdot\beta,$ (13)

and the “slope” $\beta$ can be obtained by minimizing the cost function

$\displaystyle\mathcal{C}=\|\mathbf{Y}-\mathcal{F}(\mathbf{X})\|_{F}^{2}+\lambda\mathcal{R}(\beta)$
(14)

through

$\displaystyle\frac{\partial\mathcal{C}}{\partial\beta}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\beta=\mathbf{X}^{T}(\mathbf{X}\mathbf{X}^{T}+\lambda\mathbf{I})^{-1}\mathbf{Y}.$
(15)

Here the subscript $F$ denotes the Frobenius norm, and $\mathcal{R}(\beta)$ is
a slope-dependent ridge term, which is added to enhance the robustness of the
regression function on the test data. In the practical implementation, since
the combination of Eqs. (13) and (15) includes only the so-called kernel term

$\displaystyle\mathcal{K}=\mathbf{X}\mathbf{X}^{T},$ (16)

there is no need to explicitly compute $\beta$. To build a more general non-
linear regression model, one can replace the kernel term by, for example, its
polynomial form

$\displaystyle\mathcal{K}(x_{i},x_{j})$ $\displaystyle=(\gamma
x_{i}x_{j}^{T}+c_{0})^{d},$ (17)

where $x_{i}$ is the $i^{\rm th}$ row vector of subset $X$, $d$ is the order
of the polynomial, and $\gamma$ and $c_{0}$ are free parameters.

We have performed the ML-assisted SA-SAF-UCCSD-VQE simulation for the LiH
molecule. The energy differences and the time to convergence compared with the
non-ML SA-SAF-UCCSD-VQE simulations are given in Fig. 6. As can be seen, the
energy differences between the ML and non-ML versions of the SA-SAF-UCCSD-VQE
simulations are mostly on the order of $10^{-4}$ at most of the bond lengths.
The time to convergence between the two versions at smaller bond lengths is
comparable, but the ML-assisted simulation is clearly less time-consuming than
the non-ML version at larger bond lengths (i.e., stronger correlation). We
note that here we have explored a polynomial dependence of the auxiliary
amplitudes on the principal ones in a heuristic manner. Other types of kernel
functions could also be tested. Generally speaking, a kernel function that is
more physically motivated would predict the relationship among the parameters
more accurately.

##  V Conclusion

In this work we have shown that the conventional UCC-VQE calculations can be
improved significantly even after considering the spin adaption technique. The
corresponding noise-free UCC-VQE simulations of some small molecules at
various bond lengths show that a significant portion of cluster amplitudes are
sufficiently small throughout the simulations, and these amplitudes can be
identified heuristically from the first few UCC-VQE iterations. Thus,
eliminating these amplitudes improves both quantum and classical calculations.
Next, we proposed a way to freeze molecular orbitals based on their single-
orbital entropy values calculated from a low-level theory such as MP2. Using
the H2O molecule as an example, we demonstrated that freezing the orbitals
that have low enough entropy values compared with the rest can further
simplify the ansatz with well-controlled error in energy. Furthermore, we
explored the feasibility of employing some ML techniques for identifying other
possible hidden relationships between the amplitudes. This work opens up some
possible future directions. For example, one direction is how to build a more
appropriate, physically motivated kernel that, when combined with the right
pair of principal and auxiliary amplitude subsets, would give a better result.
Although here we have used a single kernel, the entire amplitude set could be
divided into multiple principal and auxiliary amplitude subsets with a unique
kernel for each pair of subsets, which might be able to further improve the
UCC-VQE simulations.

## Acknowledgment

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, and National Quantum Information Science Research Centers.
Y.A. acknowledges support from the U.S. Department of Energy, Office of
Science, under contract DE-AC02-06CH11357 at Argonne National Laboratory.

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