(347g) The Quantum Gaussian Process: Specialized Gaussian Process Modifications for Efficient Quantum-Classical Optimization

Quantum computing (QC) has been gaining traction due to its potential to address
computationally challenging problems spanning from chemical to financial applications [1,2].
QC has been experimentally shown to solve certain computational problems significantly
faster than what can be accomplished with classical computers [3]. More specifically to chemical
engineering, QC may improve our capacity for simulating quantum mechanical systems,
solving optimization problems, and addressing challenges for machine learning [1]. Although
QC is still nascent, it is being developed rapidly due to governmental and industrial interests.

QC success currently depends on hybrid quantum-classical optimization to build a quan-
tum circuit representing the problem of interest [4]. Being a developing technology, the number
of qubits, which are the QC analog to classical bits, are few in quantity and noisy. As a
result, building QCs with many stable qubits is an open engineering challenge. These hard-
ware limitations are why quantum-classical systems are necessary; quantum states can only
be maintained for a short amount of time, so optimizing the quantum circuit is off-loaded
onto the classical computer, while the QC handles the more complex simulations. The sim-
ulation evaluates the evolution of prepared qubits through the quantum circuit, which are
modified by quantum gates to produce an energy distribution representing the solution, and
then measures their quantum state by projecting it into classical bits. The classical com-
puter interprets the distribution by turning the distribution into many classical bits (0,1),
referred to as shots. Thus, the quantum-classical methods entail a feedback loop where a
classical optimization algorithm selects the parameters of the quantum circuit. Some ex-
amples include the variational quantum eigensolver (VQE) and the quantum approximate
optimization algorithm (QAOA).

VQE is a quantum-classical optimization algorithm for finding the ground state of a
molecular system [5]. The problem is constructed using a parameterized quantum circuit and
an ansatz. The ansatz provides an initial guess of the ground state; the Hartree-Fock method
is often used for molecular systems to find a good initial electronic structure. The parame-
terized circuit can be evaluated to provide an energy distribution, from which moments may
be used to find a better set of parameters. Physically, the ansatz parameters represent rota-
tion angles in the quantum gates, which modify the electronic structure of the qubits. Thus,
for the quantum circuit to provide accurate ground states, it must be sufficiently complex
(i.e., enough quantum gates to transform the initial qubit into an accurate distribution of the
ground state energy). This can be accomplished, for example, by adding layers to the circuit,
analogously to how layers in a deep neural network increase representational capabilities.

QAOA is another quantum-classical algorithm, which can be used to solve combinatorial
optimization problems [6]. The algorithm operates similarly to VQE, where the QAOA ansatz
is a Hamiltonian of the desired optimization problem. The Hamiltonian is an operator
explicitly designed for the problem of interest, whose eigenvalue, when applied to a system, is
its energy. These Hamiltonians then transform into ansaetze, encoded into the circuits. The
ansatz for QAOA is parameterized with the spin operators, similar to the VQE algorithm.

Hybrid quantum-classical optimization typically uses zeroth-order (or derivative-free) op-
timizers to find the optimal parameters for representing the physical system on a quantum
circuit using the variational principle [7]. These zeroth-order methods can only use function
observations to find a solution, whereas first-order (or second-order) methods can use gradi-
ents (or hessian) observations to find the solution. Although first-order strategies have also
been proposed to solve quantum-classical optimization, the computation of exact gradients
requires exponentially many shots to accurately compute the expectation at a particular
value. Alternatively, gradients may be approximated, e.g., with a parameter shift or finite
differences. While first-order methods are generally preferred over zeroth-order methods
since they require fewer function calls, the first-order methods for optimizing quantum cir-
cuits need a large number of shots at each iteration or multiple function calls, which in
turn results in a more expensive query at each iteration. As a result, it is unclear whether
approximate first or zeroth-order methods are posed to provide the most advantage in the
quantum-classical optimization paradigm, and both approaches require further investigation.

Bayesian Optimization (BO) is a family of sample-efficient zeroth-order optimizers and
has demonstrated success on various black-box problems (ones that do not have a closed-
form mathematical representation). BO’s sample efficiency results from using observations
from the quantum circuit to construct a statistical surrogate model known as a Gaussian
Process (GP). The GP’s ability to quantify the model uncertainty allows us to systematically
trade off between exploring the parameter space (e.g., learning how the parameters affect
the objective) and exploiting promising regions (e.g., trying to improve on the best-observed
parameters, often by sampling near the incumbent).

This work shows that traditional BO methods can outperform other zeroth-order and
some first-order optimizers on several benchmarking problems. Quantum optimization al-
gorithms, however, have several distinguishing differences from most classical optimization
problems. In particular, two key features can be exploited to improve BO. First, there is
a well-recognized 2Ï€ periodicity in the parameter space since the optimization parameters
are rotation angles on what is known as the Bloch sphere, represented pictorially. To the
best of our knowledge, there is no method for enforcing periodicity in a GP with a euclidean
kernel. To fully utilize the information collected from samples on a Bloch-sphere, we can
instead use a spherical manifold kernel in the GP, which endows the GP with knowledge of
the periodicity in the parameter space. The second feature is that executing the quantum
circuit is efficient enough to return a distribution, also addressing the inherently probabilistic
nature of the quantum mechanical description of these systems. Since traditional optimiza-
tion deals with values over distributions, QC literature often discusses metrics that may
represent the distribution as a single value (e.g., taking the expectation or conditional value
at risk). However, a principle assumption in fitting the GP to data is that the observations
are noisy, which requires estimating the noise variance. To this end, we recommend utilizing
the first two moments of the observed distribution, which provides an accurate value of the
measurement noise, instead of needing to learn it. Precisely, we may fit two GPs, one for
the expectation and a second conditioned on the observations of variance. We term the GP,
which exploits these two features as a quantum GP (QGP), and demonstrate its superior
performance over traditional BO and a myriad of other black-box optimizers commonly used
in the hybrid quantum-classical computation of the ground state of molecular hydrogen using
VQE and the solution to combinatorial optimization problems using QAOA.

References

(1) Bernal, D. E.; Ajagekar, A.; Harwood, S. M.; Stober, S. T.; Trenev, D.; You, F. Per-
spectives of quantum computing for chemical engineering. AIChE Journal 2022, 68,
e17651.
(2) Or ́us, R.; Mugel, S.; Lizaso, E. Quantum computing for finance: Overview and prospects.
Reviews in Physics 2019, 4, 100028.
(3) Arute, F. et al. Quantum supremacy using a programmable superconducting processor.
Nature 2019, 574, 505–510.
(4) Callison, A.; Chancellor, N. Hybrid quantum-classical algorithms in the noisy
intermediate-scale quantum era and beyond. Phys. Rev. A 2022, 106, 010101.
(5) Tilly, J.; Chen, H.; Cao, S.; Picozzi, D.; Setia, K.; Li, Y.; Grant, E.; Wossnig, L.; Rung-
ger, I.; Booth, G. H.; Tennyson, J. The Variational Quantum Eigensolver: A review of
methods and best practices. Physics Reports 2022, 986, 1–128, The Variational Quan-
tum Eigensolver: a review of methods and best practices.
(6) Farhi, E.; Goldstone, J.; Gutmann, S. A quantum approximate optimization algorithm.
arXiv preprint arXiv:1411.4028 2014,
(7) Pellow-Jarman, A.; Sinayskiy, I.; Pillay, A.; Petruccione, F. A comparison of various clas-
sical optimizers for a variational quantum linear solver. Quantum Information Processing
2021, 20, 202.