Continuous optimization by quantum adaptive distribution search

TL;DR

This work tackles global optimization of continuous functions on quantum devices by combining Grover Adaptive Search (GAS) with covariance-matrix adaptation—evolution strategy (CMA-ES) to form Quantum Adaptive Distribution Search (QuADS). QuADS replaces the uniform initial state in GAS with an adaptive multivariate normal distribution, prepared via $\mathcal{G}_{\mu,\Sigma}$, and refines both the distribution and a search threshold using CMA-ES-inspired updates and amplitude amplification, achieving a favorable $O(1/\sqrt{p})$ oracle-sampling cost. Across simulations up to $D=3$ quantumly (and higher dimensions via classical estimation), QuADS consistently outperforms GAS and CMA-ES in expected oracle calls to reach the global optimum, with notable gains in high-dimensional settings and certain multimodal landscapes. This approach marks a significant step toward practical quantum-accelerated continuous optimization by exploiting function structure through adaptive probabilistic priors and quantum search.

Abstract

In this paper, we introduce the quantum adaptive distribution search (QuADS), a quantum continuous optimization algorithm that integrates Grover adaptive search (GAS) with the covariance matrix adaptation - evolution strategy (CMA-ES), a classical technique for continuous optimization. QuADS utilizes the quantum-based search capabilities of GAS and enhances them with the principles of CMA-ES for more efficient optimization. It employs a multivariate normal distribution for the initial state of the quantum search and repeatedly updates it throughout the optimization process. Our numerical experiments show that QuADS outperforms both GAS and CMA-ES. This is achieved through adaptive refinement of the initial state distribution rather than consistently using a uniform state, resulting in fewer oracle calls. This study presents an important step toward exploiting the potential of quantum computing for continuous optimization.

Figures (12)

• Figure 1: Structure of the quantum circuit for the search part of QuADS. $\mathcal{G}_{\mu, \Sigma}$ gate represents the circuit for preparing a normal distribution. The dashed block is repeated $r$ times, where $r$ represents the rotation number for amplitude amplification. In QuADS, we iteratively perform sampling from this circuit using random $r$ and subsequently update the parameters $\mu$ and $\Sigma$ by the samples.
• Figure 2: Typical optimization process of QuADS. The solid line represents the $1\sigma$ region of the distribution, and the points represent the samples from amplitude amplification at each iteration. The star mark represents the global optimum point.
• Figure 3: Expected oracle call counts for finding global optimum for each method (Eq. \ref{['eq:expected_total_calls']}) when $D = 3$. We utilized the bootstrap re-sampling method to generate the error bars in these figures. This method involves randomly selecting data from the pool of 100 simulations and recalculating Eq. \ref{['eq:expected_total_calls']} for each of these selected data sets. We present the estimated 5th and 95th percentiles of the expected oracle evaluation count as confidence intervals. Confidence intervals outside the figure indicate that upper bounds cannot be estimated from bootstrap.
• Figure 4: Optimization processes for the three-dimensional rastrigin function (\ref{['fig:quantum_rastrigin_3']}) and schwefel function (\ref{['fig:quantum_schwefel_3']}). The upper figure presents the smallest function value obtained up to a specific number of oracle calls in each trial. A circle marker at the end of each trial represents the trial found a global solution, and a cross marker indicates local convergence. The lower figure shows the proportion of trials that found a global solution (solid line) and that reached the terminal condition (dashed line) up to the number of oracle calls. The number of oracle calls required for local convergence can be larger than those for finding the global optimum. This arises from the algorithms' termination criteria: they terminate immediately upon locating the global optimum, but may continue until convergence of $\sigma$ if the global optimum remains unfound.
• Figure 5: The consistency verification between classical estimation and quantum simulation regarding the expected number of oracle calls across the functions in Fig. \ref{['fig:high-dim-exp']}. The dotted line represents $o_{\rm total} = \tilde{o}_{\rm lower}$. The classical estimation results ($\tilde{o}_{\rm lower}$) serve as a lower bound to the quantum simulation results ($o_{\rm total}$), which is consistent with theoretical predictions. The results of the quantum simulations are approximately twice that of the classical estimations, as indicated by the solid line $o_{\rm total}=2.3\tilde{o}_{\rm lower}$ for both QuADS and GAS.