Qubit-efficient encoding schemes for binary optimisation problems

TL;DR

This paper introduces qubit-efficient variational quantum algorithms for binary optimization, enabling solutions with logarithmic qubit scaling and systematically increasing correlation complexity, demonstrated on problems with up to 64 variables.

Contribution

It presents a novel encoding scheme that captures increasing correlations among classical variables with fewer qubits, enhancing quantum optimization capabilities.

Findings

Successfully solved a 64-variable problem with 7 qubits.

Improved solution quality by incorporating two-body correlations.

Analyzed resource efficiency for near-term quantum hardware.

Abstract

We propose and analyze a set of variational quantum algorithms for solving quadratic unconstrained binary optimization problems where a problem consisting of $n_c$ classical variables can be implemented on $\mathcal O(\log n_c)$ number of qubits. The underlying encoding scheme allows for a systematic increase in correlations among the classical variables captured by a variational quantum state by progressively increasing the number of qubits involved. We first examine the simplest limit where all correlations are neglected, i.e. when the quantum state can only describe statistically independent classical variables. We apply this minimal encoding to find approximate solutions of a general problem instance comprised of 64 classical variables using 7 qubits. Next, we show how two-body correlations between the classical variables can be incorporated in the variational quantum state and how it can improve the quality of the approximate solutions. We give an example by solving a 42-variable Max-Cut problem using only 8 qubits where we exploit the specific topology of the problem. We analyze whether these cases can be optimized efficiently given the limited resources available in state-of-the-art quantum platforms. Lastly, we present the general framework for extending the expressibility of the probability distribution to any multi-body correlations.