Self-consistent mean-field quantum approximate optimization

TL;DR

A self-consistent mean-field quantum optimization algorithm that approximates the ground state of classical Ising Hamiltonians and decomposes the problem into independent subproblems and treats the interactions between them in a mean-field manner enables the solution of problems that would otherwise exceed the qubit and gate counts of current quantum hardware.

Abstract

We introduce a self-consistent mean-field quantum optimization algorithm that approximates the ground state of classical Ising Hamiltonians. The algorithm decomposes the problem into independent subproblems and treats the interactions between them in a mean-field manner. These interactions are captured by a common environment, constructed self-consistently through a variational quantum circuit, and which modifies the subproblems to account for mutual influence while maintaining computational independence. Consequently, subproblems can be solved individually, avoiding the computational cost of the full problem. We explore the properties of the generated environment and assess the algorithm's performance through extensive numerical simulations on Sherrington-Kirkpatrick spin glasses. Furthermore, we apply it experimentally to a weighted maximum clique problem applied to molecular docking. This framework enables the solution of problems that would otherwise exceed the qubit and gate counts of current quantum hardware.

Figures (6)

• Figure 1: Schematic of the decomposition strategy. A problem is partitioned into $K=4$ subproblems of equal size. The interaction terms between subproblems are encoded in a mean-field environment, which is determined self-consistently via a variational quantum circuit.
• Figure 2: Analysis of self-consistency and solution landscapes. (a) Energy landscape for an $N=128$ SK problem instance decomposed into $K=4$ subproblems, plotted after environmental self-consistency is achieved for fixed QAOA parameters $\gamma$ and $\beta$. (b) Distribution of optimal QAOA parameters ($\gamma,\beta$) for hundreds of randomly generated SK instances with fixed $N$ and $K$ values. (c) Convergence to self-consistency for the environment, as defined in Eq. \ref{['eq:convergence']} for $N=128$ SK instances ($K=4$) using the $p=1$ QAOA, plotted against the number of iterations. (d) Convergence rate of the environment for typical SK problem instances at fixed, near-optimal parameters. (e) Distribution of energy obtained by solving the system of nonlinear equations (Eq. \ref{['eq:sys_nonlin_eq']}) for a single $N=128, K=4$ instance, using random initial guesses. The vertical line indicates the energy found by our iterative method starting from a null environment. (f) Probability that the iterative method (initialized with a null environment) converges to the lowest energy configuration found across random trials, averaged over SK instances with fixed parameters.
• Figure 3: Performance scaling on SK instances with $N$ degrees of freedom decomposed into $K$ subproblems. (a) Energy density obtained via the self-consistent mean-field QAOA with $p=1$ layer. The $K=1$ case corresponds to the standard QAOA results Farhi2022. $P^*$ denotes the exact ground state energy density for infinite-size SK instances (Parisi constant) PhysRevLett.50.1946. (b) Energy density for $K>2$, rescaled by a power of the average environment magnitude. (c) Relative contribution of intra-subproblem terms to the total energy. Data points represent averages over tens to hundreds of random instances; error bars indicate the standard error of the mean. Dotted lines are guides to the eye.
• Figure 4: Impact of circuit depth. Average energy density for SK instances ($N/K \approx 16$) as a function of QAOA layers $p$, using the self-consistent mean-field approach. The limit $N \rightarrow +\infty$ with $K=1$ corresponds to the standard QAOA Farhi2022. Data points represent averages over tens to hundreds of random instances; error bars indicate the standard error of the mean. Dotted lines are guides to the eye.
• Figure 5: Application to molecular docking, formulated as a weighted maximum clique problem. (a) Convergence of the environment and energy towards self-consistency (Eq. \ref{['eq:convergence']}) as a function of iterations, using $p=1$ QAOA with fixed variational parameters and $\Lambda=2$. (b) Relative energy versus penalty strength $\Lambda$, with random sampling and standard $p=1$ QAOA serving as baselines. Data points represent averages over $10^4$ samples; error bars indicate the standard error of the mean and dotted lines are guides to the eye. (c) Energy probability density for $10^4$ classically post-processed samples where the clique constraint is enforced. The color scheme matches panel (b), and the absolute ground state energy is indicated by $\langle\hat{C}\rangle_\textrm{gs}\simeq -25.86$.