Two-Stage Estimation and Variance Modeling for Latency-Constrained Variational Quantum Algorithms

TL;DR

A novel stochastic trust-region method derived from a derivative-free, adaptive sampling trust-region optimization method intended to efficiently solve the classical optimization problem in QAOA by explicitly taking into account the two mentioned characteristics.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) has enjoyed increasing attention in noisy intermediate-scale quantum computing due to its application to combinatorial optimization problems. Because combinatorial optimization problems are NP-hard, QAOA could serve as a potential demonstration of quantum advantage in the future. As a hybrid quantum-classical algorithm, the classical component of QAOA resembles a simulation optimization problem, in which the simulation outcomes are attainable only through the quantum computer. The simulation that derives from QAOA exhibits two unique features that can have a substantial impact on the optimization process: (i) the variance of the stochastic objective values typically decreases in proportion to the optimality gap, and (ii) querying samples from a quantum computer introduces an additional latency overhead. In this paper, we introduce a novel stochastic trust-region method, derived from a derivative-free adaptive sampling trust-region optimization (ASTRO-DF) method, intended to efficiently solve the classical optimization problem in QAOA, by explicitly taking into account the two mentioned characteristics. The key idea behind the proposed algorithm involves constructing two separate local models in each iteration: a model of the objective function, and a model of the variance of the objective function. Exploiting the variance model allows us to both restrict the number of communications with the quantum computer, and also helps navigate the nonconvex objective landscapes typical in the QAOA optimization problems. We numerically demonstrate the superiority of our proposed algorithm using the SimOpt library and Qiskit, when we consider a metric of computational burden that explicitly accounts for communication costs.

Figures (12)

• Figure 1: In the context of a combinatorial optimization problem with cost Hamiltonian $H_C$, QAOA iteratively updates a parameter vector $\bm{x}$ to minimize the objective function value in \ref{['eq:qaoa-problem']}. Once a sufficiently good solution, $\bm{x}_{opt}$, is achieved, QAOA proceeds to obtain a probability distribution by measuring the quantum state $\ket{\psi(\bm{x}_{opt})}$. In this distribution, the solution with the highest frequency corresponds to the optimal solution for the original combinatorial problem, $\bm{\theta}^*$.
• Figure 2: In this plot, we simulate in Qiskit Qiskit a depth-10 QAOA circuit for solving a maxcut problem on a toy graph on 6 nodes, for which the optimal solution to the maxcut problem is 6. We provide BOBYQApowell2009bobyqa with the deterministic expectation statevector value for this toy problem and record the improving sequence of incumbent solutions returned. We illustrate, on the same log scale, the optimality gap of the incumbents found, as well as the population variance associated with the statevector value. We observe that, as expected, population variance decays alongside the optimality gap, but not necessarily monotonically.
• Figure 3: A cartoon illustrating the effect of using the two local models. \ref{['fig:wovm']} illustrates the performance of history-informed ASTRO-DF without the inclusion of $M_k^v$. In this case, $\widetilde{\bm{X}}_{k+1}$ will be further drawn to the basin of a local minimum. \ref{['fig:wvm']} illustrates the performance of history-informed ASTRO-DF with the proposed use of two models, $M_k$ and $M_k^v$. In this case, the design point $\bm{X}_k^2$, the minimizer of $M_k^v$, will be selected as the next incumbent $\bm{X}_{k+1}$ We prefer this global optimality-seeking behavior.
• Figure 4: Contour plot of the expectation of \ref{['eq:himme-problem']}. Each labelled point corresponds to an experimented initial solution shown in Figures\ref{['fig:himme-diff-init']} and \ref{['fig:himme-diff-(3,3)']}. The global minimum is attained at $(3, 2)$.
• Figure 5: Performance of solvers on \ref{['eq:himme-problem']} provided with various initial points. Translucent bands represent a 95% confidence interval over 20 macro-replications and solid lines represent mean performance. provided with various initial points. The $y$-axis is on a logarithmic scale.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3