Efficient and Large-Scale Semidefinite Programming with Quantum Neural Networks

Semidefinite programs are optimization methods with a wide array of applications, such as approximating difficult combinatorial problems. We introduce a variational quantum algorithm for semidefinite programs that uses only $n$ qubits, a constant number of circuit preparations, and $O(n^2)$ expectation values in order to solve semidefinite programs with up to $N=2^n$ variables and $M=2^n$ constraints. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxiliary qubit, a technique known as the Hadamard Test. The Hadamard Test enables us to optimize the objective function by estimating only a single expectation value of the ancilla qubit, rather than separately estimating exponentially many expectation values. Similarly, we illustrate that the semidefinite programming constraints can be effectively enforced by implementing a second Hadamard Test, as well as imposing $sim n^2/2$ Pauli string amplitude constraints. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm, which is a useful approximation for various NP-hard problems, such as MaxCut. Our method exceeds the performance of analogous classical methods on a diverse subset of well-studied MaxCut problems from the GSet library. Extremely large-scale graphs and implications for classical optimization are discussed. Experimental implementations are explored.