An Adaptive Optimizer for Measurement-Frugal Variational Algorithms

Jonas M. Kübler1,2, Andrew Arrasmith1, Lukasz Cincio1, and Patrick J. Coles1

1Theoretical Division, MS B213, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
2Max Planck Institute for Intelligent Systems, Max-Planck-Ring 4, 72076 Tübingen, Germany.

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Variational hybrid quantum-classical algorithms (VHQCAs) have the potential to be useful in the era of near-term quantum computing. However, recently there has been concern regarding the number of measurements needed for convergence of VHQCAs. Here, we address this concern by investigating the classical optimizer in VHQCAs. We introduce a novel optimizer called individual Coupled Adaptive Number of Shots (iCANS). This adaptive optimizer frugally selects the number of measurements (i.e., number of shots) both for a given iteration and for a given partial derivative in a stochastic gradient descent. We numerically simulate the performance of iCANS for the variational quantum eigensolver and for variational quantum compiling, with and without noise. In all cases, and especially in the noisy case, iCANS tends to out-perform state-of-the-art optimizers for VHQCAs. We therefore believe this adaptive optimizer will be useful for realistic VHQCA implementations, where the number of measurements is limited.

One of the major technological questions of our time is whether near-term quantum computers will have practical applications. A promising strategy to address this question is the paradigm of variational quantum algorithms, which combine classical optimization with evaluation of objective functions on a quantum device. These algorithms make the most of the limited resources of the comparatively small (hundreds of qubits), noisy quantum devices that will likely be available in the coming years. This potential makes variational algorithms arguably the best candidates for demonstrating the first advantage in computational speed or cost for quantum computers.

However, there is a legitimate concern over whether or not the number of times a quantum state must be prepared and measured (i.e. the number of “shots” taken on the quantum device) in order for the variational algorithm to converge will be prohibitive. This and related questions have sparked a recent interest in researching which classical optimizer should be used in variational algorithms.

In this work, we propose a shot-frugal optimization strategy for variational algorithms that dynamically adjusts the number of shots expended (and thus the precision) for each update step in a stochastic gradient descent procedure that we name iCANS (individual Coupled Adaptive Number of Shots). Allocating measurement resources separately for each component of the gradient estimates, iCANS takes advantage of very inexpensive update steps requiring few shots early in the optimization and smoothly increases the number of shots used in order to achieve a high precision optimization result. We present comparisons between iCANS and other optimizers that have been discussed for the context of variational algorithms and find that iCANS often performs better than these other methods.

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