# Diagnosing local minima and accelerating convergence of variational quantum
eigensolvers with quantum subspace techniques

Mark R. Hirsbrunner<EMAIL_ADDRESS>Department of Physics, University
of Illinois at Urbana-Champaign, Urbana, IL 61801, USA  USRA Research
Institute for Advanced Computer Science, Mountain View, CA 94043, USA  J.
Wayne Mullinax  KBR, Inc., NASA Ames Research Center, Moffett Field, CA 94035,
USA  Intelligent Systems Division, NASA Ames Research Center, Moffett Field,
CA 94035, USA  Yizhi Shen  Applied Mathematics and Computational Research
Division, Lawrence Berkeley National Lab, Berkeley, CA 94720, USA  David B.
Williams-Young  Applied Mathematics and Computational Research Division,
Lawrence Berkeley National Lab, Berkeley, CA 94720, USA  Katherine Klymko
NERSC, Lawrence Berkeley National Lab, Berkeley, CA 94720, USA  Roel Van
Beeumen  Applied Mathematics and Computational Research Division, Lawrence
Berkeley National Lab, Berkeley, CA 94720, USA  Norm M. Tubman
<EMAIL_ADDRESS>Intelligent Systems Division, NASA Ames Research
Center, Moffett Field, CA 94035, USA

###### Abstract

Recent research has shown that wavefunction evolution in real- and imaginary-
time can generate quantum subspaces with significant utility for obtaining
accurate ground state energies. Inspired by these methods, we propose
combining quantum subspace techniques with the variational quantum eigensolver
(VQE). In our approach, the parameterized quantum circuit is divided into a
series of smaller subcircuits. The sequential application of these subcircuits
to an initial state generates a set of wavefunctions that we use as a quantum
subspace to obtain high-accuracy groundstate energies. We call this technique
the circuit subspace variational quantum eigensolver (CSVQE) algorithm. By
benchmarking CSVQE on a range of quantum chemistry problems, we show that it
can achieve significant error reduction compared to conventional VQE,
particularly for poorly optimized circuits, greatly improving convergence
rates. Furthermore, we demonstrate that when applied to circuits trapped at a
local minima, CSVQE can produce energies close to the global minimum of the
energy landscape, making it a potentially powerful tool for diagnosing local
minima.

##  I Introduction

A main goal of quantum algorithms for Hamiltonian simulation is generating
high fidelity groundstates for problems that are difficult to simulate
classically. Two important classes of promising algorithms for achieving
quantum advantage in Hamiltonian simulation are variational quantum algorithms
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 6, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] and quantum subspace
methods [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
72, 73, 74, 75]. Variational quantum algorithms are widely studied, especially
the variational quantum eigensolver (VQE), but quantum subspace methods are
only recently beginning to grow in popularity. Quantum subspace methods have
the potential to provide significant convergence improvements, but challenges
remain in the efficient generation of these subspaces on quantum hardware.

In this work, we consider the use of VQE circuits as a mean for generating
subspaces with accelerated convergence for groundstate energies of
Hamiltonians. We draw inspiration for our approach from recent work studying
subspaces generated using real- and imaginary-time evolution [76, 67, 77, 78].
We treat the VQE circuit itself as a time evolution operator in spirit, in
which the application of each gate in the circuit acts like a discrete time
evolution of the initial state. The initial state fed into the VQE circuit
traces out a path through the Hilbert space as it is “time-evolved” by the VQE
circuit, and it is from this path that we sample states to generate a
subspace. For an optimized VQE circuit, one might informally consider the
series of states generated by the circuit evolution to have similar
characteristics to imaginary time evolution, as both rapidly evolve an initial
state to the groundstate. By projecting the Hamiltonian for which the VQE
circuit is optimized into this circuit-generated subspace, we form a
generalized eigenvalue problem for the Hamiltonian. We call this method the
circuit subspace variational quantum eigensolver (CSVQE). In principle, the
subspace contains more information than the final state output by the VQE
circuit, and the goal of this work is to quantify how much advantage can be
extracted from this additional information.

##  II Circuit evolution subspace VQE

The inputs to the CSVQE algorithm are a VQE circuit and the Hamiltonian for
which the VQE circuit is optimized. The VQE circuit consists of an ordered
series of $N$ parameterized quantum gates acting on an initial wavefunction,

$\ket{\Psi_{N}}=U_{N}U_{N-1}\cdots U_{2}U_{1}\ket{\Psi_{0}}.$ (1)

We treat the parameters of these gates as constants throughout the CSVQE
algorithm, assuming them to be determined by some classical optimization
procedure. First we must generate the set of states explored by the circuit as
it evolves the initial wavefunction via the sequential application of the
quantum gates. Each of these states is obtained by the ordered application of
the first $M\leq N$ gates in the circuit to the initial wavefunction,

$\ket{\Psi_{M}}=U_{M}U_{M-1}\cdots U_{2}U_{1}\ket{\Psi_{0}}.$ (2)

The subspace of wavefunctions $\left\\{\ket{\Psi_{i}}\right\\}$, $i\in[0,N]$,
contains the initial wavefunction $\ket{\Psi_{0}}$, the final state produced
by the circuit, and many additional mid-circuit states.

Next we construct the generalized eigenvalue problem by calculating the
Hamiltonian matrix elements and the overlap matrix in this subspace,

$\displaystyle H_{ij}$
$\displaystyle=\matrixelement{\Psi_{i}}{\hat{H}}{\Psi_{j}}$ (3) $\displaystyle
S_{ij}$ $\displaystyle=\innerproduct{\Psi_{i}}{\Psi_{j}}.$ (4)

The resulting generalized eigenvalue problem,

$\sum_{j}H_{ij}x_{j}=E\sum_{j}S_{ij}x_{j},$ (5)

can be solved via classical algorithms for approximate eigenvalues of the
original Hamiltonian. We remark that the overlap matrix, $S$, can be rank
deficient, as is commonly encountered in other work on quantum subspace
methods [67, 79]. This occurs most often when the mid-circuit states from
nearby layers of the circuit have large overlaps with each other. In this
work, we address this problem by projecting out singular values of $S$ that
are less than some threshold, which we take to be $1\times 10^{-10}$. However,
this threshold has to be adjusted in general when accumulating data from noisy
hardware, as discussed in previous work [67]. The final wavefunction produced
by VQE is limited by the ansatz and initial state chosen and, in most cases,
is not capable of producing the exact groundstate. Solving the generalized
eigenvalue problem utilizes these subspace states to generate a new
groundstate wavefunction that has equal or larger overlap with the true
groundstate, and also a lower energy than any individual wavefunction in the
subspace.

In practice, the quantum circuit may contain an enormous number of gates,
making it impractical to calculate the matrix elements for the entire circuit-
generated subspace. Instead, some subset of $M<N+1$ states from the full
subspace must be chosen. However, choosing an optimal subset of states is a
challenging problem. To illustrate this for a circuit that has not been fully
optimized, in Fig. 1 we plot the groundstate energies obtained by CSVQE for
the molecule $\mathrm{C}_{2}$ using three heuristic approaches and a random
selection approach for choosing subsets of the full subspace. We describe
these state selection methods below, and describe the details of the circuit
used in the following section.

Figure 1: The error produced by the CSVQE algorithm when applied to the
circuit generated by the first optimization step for $\mathrm{C}_{2}$ as a
function of the number of intermediate states used. The dashed black line is
the error in the VQE energy, to which we compare the errors of the CSVQE
energies. The gray squares, blue pentagons, and teal triangles mark the error
produced by the “front-loaded”, “back-loaded”, and “even” state selection
methods. The green stars mark the lowest error obtained from 20,000 random
combinations of mid-circuit states, and $N=149$ for this circuit.

(a) $\mathrm{LiH}$

(b) $\mathrm{CH}_{4}$

(c) $\mathrm{H}_{2}\mathrm{O}$

(d) $\mathrm{NH}_{3}$

(e) $\mathrm{N}_{2}$

(f) $\mathrm{C}_{2}$

Figure 2: The error in the ground state energy produced by VQE and CSVQE at
each step in the classical optimization process for LiH, $\mathrm{CH}_{4}$,
$\mathrm{H}_{2}\mathrm{O}$, $\mathrm{NH}_{3}$, $\mathrm{N}_{2}$, and
$\mathrm{C}_{2}$ using the UCC ansatz. The energies produced by the CSVQE
algorithm using four, twelve, and all $(M=N+1)$ mid-circuit states are plotted
in green squares, blue pentagons, and gray triangles, respectively. The
reported errors represent the optimal values obtained by applying the CSVQE
algorithm with 1000 different random combinations of mid-circuit states.

The first method we consider is to choose states that are approximately evenly
distributed throughout the circuit, e.g. $\left\\{\ket{\Psi_{i}}\right\\}$
with $i$ in the set

$\left\\{0,\lfloor\frac{N}{M-1}\rfloor,\lfloor\frac{2N}{M-1}\rfloor,\dots,\lfloor\frac{(M-2)N}{M-1}\rfloor,N\right\\},$
(6)

which we refer to as the “even” method. We also consider the “back-loaded”
method that chooses states only from the end of the circuit,

$i\in\left\\{N-M+1,N-M+2,\dots,N-1,N\right\\},$ (7)

and the “front-loaded” method, in which states are taken from the beginning of
the circuit (also including the final state produced by the full circuit),

$i\in\left\\{0,1,\dots,M-3,M-2,N\right\\}.$ (8)

Finally, we consider a random search approach of sampling large numbers of
distinct random combinations of states from the subspace. In this way we can
try to access nearly optimal energies obtainable with CSVQE, although we note
this approach does not scale well and is impractical for use on real quantum
hardware. However, in each of these methods, we always include the final state
produced by the VQE circuit in the restricted subspace. In doing this we
ensure that the CSVQE energy is never higher than the VQE energy, at least
within statistical sampling error (shot noise) and biases from noisy hardware.

As shown by the results in Fig. 1, the “even” and “front-loaded” strategies
are comparably effective. Both reduce the error but do not quite achieve the
accuracy of the best result obtained from 20,000 distinct randomly sampled
state combinations. The “back-loaded” strategy obtains only minimal advantage
over the standard VQE result. The intuition behind these results is as
follows. We have ordered the gates in the circuit based on heuristic classical
approximations, such that the gates that make the largest changes to the
wavefunction are applied first in our construction of the circuit, which we
discuss in detail in the following section. States obtained early in the
circuit are therefore both the most distinct from the final state produced by
the circuit, and also more distinct from their neighboring states than those
later in the circuit. As such, they likely provide more utility when solving
for the groundstate than the states at the end of the circuit, which have very
large overlaps and contain nearly identical information. This is especially
true in the case that circuit optimization obtains an excited state with only
a small projection onto the groundstate, for which states near the beginning
of the circuit likely contain much more useful overlaps with the groundstate.

##  III Quantum chemistry benchmarking of CSVQE

In this section we study the effectiveness of the CSVQE algorithm in the
context of quantum chemistry, considering a range of molecules and employing
the unitary coupled cluster (UCC) ansatz. The UCC ansatz is the exponentiation
of the coupled cluster operator $\hat{T}$ acting on the Hartree–Fock reference
wave function $\ket{\Psi_{0}}$,

$\displaystyle\ket{\Psi_{\mathrm{UCC}}}=\exp(\hat{T}-\hat{T}^{\dagger})\ket{\Psi_{0}},$
(9)
$\displaystyle\hat{T}=\sum_{i}^{\mathrm{occ}}\sum_{a}^{\mathrm{vir}}\theta_{i}^{a}\hat{a}_{a}^{\dagger}\hat{a}_{i}+\sum_{ij}^{\mathrm{occ}}\sum_{ab}^{\mathrm{vir}}\theta_{ij}^{ab}\hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_{j}\hat{a}_{i}+\cdots\
.$ (10)

The second-quantized creation and annihilation operators $\hat{a}^{\dagger}$
and $\hat{a}$ act on the occupied (indexed by $i,j,\ldots$) and virtual
(indexed by $a,b,\ldots$) molecular orbitals in the reference wave function,
respectively.

We employ the factorized form of the UCC ansatz,

$\ket{\Psi_{\mathrm{UCC}}}=\prod_{ij\cdots}^{\mathrm{occ}}\prod_{ab\cdots}^{\mathrm{vir}}\hat{U}^{ab\cdots}_{ij\cdots}\ket{\Psi_{0}},$
(11)

for which the individual gates are defined as

$\begin{gathered}\hat{U}^{ab\cdots}_{ij\cdots}=\exp(\theta_{ij\cdots}^{ab\cdots}(\hat{a}_{ij\cdots}^{ab\cdots}\cdots-\hat{a}_{ab\cdots}^{ij\cdots}))\\\
\hat{a}_{ij\cdots}^{ab\cdots}=\hat{a}^{\dagger}_{a}\hat{a}^{\dagger}_{b}\dots\hat{a}^{\phantom{\dagger}}_{j}\hat{a}^{\phantom{\dagger}}_{i}.\end{gathered}$
(12)

For simplicity, we include only single- and double-excitation operators,
$\hat{U}_{i}^{a}$ and $\hat{U}_{ij}^{ab}$. We utilize a computationally
efficient implementation of the UCC ansatz that expresses the UCC factors
$\hat{U}^{ab\cdots}_{ij\cdots}$ in terms of sines and cosines of the
parameters $\theta$ [80]. We choose to order the individual UCC factors based
on the magnitude of the initial parameter values ($\absolutevalue{\theta}$),
which we generate from MP2 calculations using PySCF, such that the factor with
the largest coefficient acts first [81]. To reduce the computational resources
required, we employ a sparse wavefunction circuit simulator designed for
simulating the VQE algorithm [82, 83], which is based on techniques developed
for classical simulations [84, 85, 86, 87]. After each UCC factor is applied,
we check the number of determinants $N$ in the wave function. If $N$ is
greater than the desired number of determinants, $N_{\textrm{WF}}$, we sort
the amplitudes by magnitude and discard the determinants with the smallest
amplitudes such that only $N_{\textrm{WF}}$ determinants remain in the wave
function. We set $N_{\mathrm{WF}}$ to 50,000, which is sufficient to converge
the groundstate energy for each molecule tested here.

(a)

(b)

(c)

(d)

Figure 3: Top: The mean and minimum energies produced by applying CSVQE to the
(a) unoptimized and (b) fully optimized $\mathrm{C}_{2}$ circuits using 1000
random combinations of mid-circuit states, with gray shading indicating the
standard deviation. Bottom: Histograms of the energies obtained from 10,000
random combinations of 5 states for the (c) unoptimized and (d) fully
optimized $\mathrm{C}_{2}$ circuits. The red, black, and blue dashed vertical
lines denote the standard VQE error, the mean of the CSVQE error distribution,
and the minimum CSVQE error, respectively.

The results of our benchmarking are summarized in Fig. 2, in which we plot the
error of the groundstate energy produced by VQE circuits with and without the
CSVQE algorithm at each step in the optimization process for a range of
molecule. The details of our benchmark procedure are as follows:

  1. 1.

The set of molecules we consider is LiH, $\mathrm{CH}_{4}$,
$\mathrm{H}_{2}\mathrm{O}$, $\mathrm{NH}_{3}$, $\mathrm{N}_{2}$, and
$\mathrm{C}_{2}$.

  2. 2.

We use the STO-3G basis set for each molecule and employ experimental
geometries from the CCCBDB database [88].

  3. 3.

The circuits for each molecular are restricted to include at most fifty
doubles operators, choosing the ones with the largest initial parameter
values.

  4. 4.

The circuit parameters are optimized using the BFGS implementation from SciPy,
and we store the circuit obtained at each step of the optimization process
[89].

  5. 5.

We apply the CSVQE algorithm at each step in the optimization process to study
the effectiveness of the algorithm on circuits of varying quality. Rather than
deconstructing them into elementary gates, we treat each single and double
operator as a single circuit element and extract states after each operator.

  6. 6.

We calculate the error relative to the FCI energy and apply the CSVQE
algorithm using either four, twelve, or all mid-circuit states. Each data
point represents the lowest energy obtained by the CSVQE algorithm from 1000
distinct random combinations of $M$ states chosen from the subspace.

(a)  (b)  (c)

Figure 4: (a) The error in the energy relative to FCI produced by 30 randomly-
initialized UCC circuits for $\mathrm{H}_{2}$ using the 6-31G basis. The black
dots indicate the error in the fully optimized VQE energies, arranged in
ascending order. Four circuits obtained energies at or near the global minimum
of the circuit, and the remaining circuits ended the optimization process at
local minima or saddle points. Applying the CSVQE algorithm to these circuits
using 4 intermediate states, marked with green squares, slightly reduces the
energy. Applying CSVQE with 8 states, plotted with blue-green pentagons,
accomplishes significant error reduction, sometimes producing errors as low as
10-100 $\mu$Ha even for circuits that obtained local minima. Applying CSVQE
using 12 states achieves energies at parity with the global minimum for nearly
all circuits. The scale is logarithmic above the break and linear below. We
plot the convergence of the VQE and CSVQE energies as a function of the
optimization step for the worst (trial index 30) and best (trial index 1)
circuits in (b) and (c), respectively. In all CSVQE calculations we used
10,000 random combinations of mid-circuit states and report the lowest error
obtained.

The reduction in error accomplished by the CSVQE algorithm ranges widely
between both different molecules and different optimization steps of a given
circuit. For all molecules studied here, CSVQE accomplishes significant error
reduction by up to an order of magnitude early in the optimization process,
often achieving parity with the error produced by the fully optimized circuit.
However, the error reduction produced by the CSVQE algorithm diminishes as the
optimization proceeds, indicating that optimized circuits produce
wavefunctions with energies that cannot be improved via superposition with
mid-circuit states (at least within our numerical tests).

To gain insight into the distribution of the errors produced by these random
combinations of states, rather than only considering the minima obtained, we
plot in Figs. 3 (a) and (b) the mean, minimum, and standard deviation of the
CSVQE energies for the unoptimized and fully optimized $\mathrm{C}_{2}$
circuits as a function of $M$. We also plot in Figs. 3 (c) and (d) histograms
of the errors obtained for each circuit with $M=5$ using 10,000 random
combinations of states. The data indicate that the errors are relatively
flatly distributed around the average error, and the minimal errors occur as
rare events below the main cluster of errors. Fortunately, the minimal errors
are never significantly smaller than the bottom of the main distribution of
errors, indicating that large amounts of sampling are not necessary to obtain
good results. Furthermore, the width of the distribution narrows significantly
as the circuit is optimized.

##  IV Local Minima Analysis

One of the central challenges in implementing the VQE algorithm is handling
local minima of the energy landscape. Classical optimizers often get trapped
in these local minima, producing poor approximations of the groundstate energy
[90, 91]. Here we study the ability of the CSVQE algorithm to produce accurate
energies even when the VQE optimization terminates at a local minima.

We use the $\mathrm{H}_{2}$ molecule as a test case, employing the 6-31G basis
set and the experimental geometry [88], using the UCC ansatz described above
including all doubles operators with $N_{\text{WF}}=30,000$. We construct 30
circuits with random initial parameters and plot the error in the energy of
the optimized VQE wavefunctions in black dots in Fig. 4 (a). Four of these
circuits obtain energies at or near the true groundstate energy, but the
remaining circuits obtained local minima (or possibly saddle points),
producing energies on the order of tens or hundreds of milliHartree above the
true groundstate energy. We also apply the CSVQE algorithm to each of these
circuits using 10,000 random samples of $M=4,\,8,\mathrm{and}\,12$ mid-circuit
states and plot the resulting groundstate energy errors. The $M=4$ case
somewhat reduces the error produced by the circuits trapped at local minima,
but the $M=8$ and $M=12$ cases produce energies comparable to the global
minima. This result indicates that the CSVQE algorithm has significant
potential as a powerful tool for circumventing local minima in VQE circuit
optimization.

We also consider the performance of the CSVQE algorithm throughout the
optimization of both the best (trial index 1) and worst (trial index 30) of
these randomly initialized circuits. We plot the VQE and CSVQE energies of
circuit 1 and circuit 30 as a function of the optimization step in Fig. 4 (b)
and (c), respectively. For the energy before optimization, the initial error
of circuit 30 is roughly 1.5 Ha, an astronomical error compared to any other
reported in this work. Despite this enormous error, the CSVQE algorithm still
produces energies with errors on the order of 10 mHa using only four mid-
circuit states, or even on the scale of $\mu$Ha using twelve mid-circuit
states. Although this is a relatively small circuit for a relatively simple
system, it is remarkable that CSVQE can produce energies at or near the true
ground state energy even for randomized circuits. The result of applying CSVQE
to the best circuit is quite similar. For the unoptimized version of circuit
1, the error starts at the order of hundreds of mHa, but the CSVQE algorithm
produces errors on the scale of 10 $\mu$Ha to 10 mHa, depending on the number
of mid-circuit states used. The difference from circuit 30 is that the VQE and
CSVQE energies converge as the circuit is optimized to obtain the true
groundstate minimum.

##  V Conclusions

In this work we devised a modification of the VQE algorithm, dubbed circuit-
evolution VQE, that utilizes quantum spaces generated by sampling mid-circuit
states from VQE circuits. We applied the CSVQE algorithm to UCC circuits for a
range of molecules, studying the reduction in error that can be accomplished
with the technique at each step in the optimization process. We found that the
CSVQE algorithm accomplished significant reduction in error when applied to
partially optimized circuits, but the advantage gained is minimal for fully
optimized circuits. However, due to shot noise, hardware noise and difficult
optimization landscapes, optimization on quantum hardware is likely to be less
than ideal and the tools laid out here provide another route to improving
accuracy in such cases.

The CSVQE algorithm also shows significant promise for addressing the
challenge of local minima in the optimization of VQE circuits. We showed that
the CSVQE algorithm is able to produce energies comparable to the global
minimum when applied to circuits for which the optimization process obtains a
local minimum. Although this result was obtained for a relatively small
circuit, we believe it indicates that the CSVQE algorithm can be applied to
large circuits to diagnose cases in which the optimizer is stuck in a local
minima. These promising results merit additional study.

The utility of this algorithm is likely highly ansatz-dependent, and further
investigation of its utility beyond the UCC ansatz and quantum chemistry
problems is justified. One important shortcoming of the UCC ansatz utilized in
this work is that a large portion of the UCC parameters are quite small,
leading to a circuit evolution subspace containing many nearly-identical
wavefunctions. The CSVQE algorithm will likely perform significantly better on
more efficient ansatze in which each gate meaningfully changes the
wavefunction. This would also significantly lessen the challenge of designing
efficient state selection strategies for generating useful subspaces.

Acknowledgements.–We are grateful for support from NASA Ames Research Center.
We acknowledge funding from the NASA ARMD Transformational Tools and
Technology (TTT) Project. W.M. and N.T. acknowledge funding from the NASA ARMD
Transformational Tools and Technology (TTT) Project. This material is based
upon work supported by the U.S. Department of Energy, Office of Science,
National Quantum Information Science Research Centers, Superconducting Quantum
Materials and Systems Center (SQMS) under contract No. DE-AC02-07CH11359
(N.T.). M.R.H. participated in the Feynman Quantum Academy internship program.
Y.S., K.K., and R.V.B. acknowledge funding from the U.S. Department of Energy
(DOE) under Contract No. DE-AC0205CH11231, through the Office of Science,
Office of Advanced Scientific Computing Research (ASCR) Exploratory Research
for Extreme-Scale Science. This research used resources of the National Energy
Research Scientific Computing Center, a DOE Office of Science User Facility
supported by the Office of Science of the U.S. Department of Energy under
Contract No. DE-AC02-05CH11231 using NERSC award ASCR-ERCAP0024469.

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