Instance independence of single layer quantum approximate optimization algorithm on mixed-spin models at infinite size

Abstract

The Quantum Approximate Optimization Algorithm (qaoa) is a variational quantum algorithm designed to give approximate solutions to unconstrained binary optimization problems [1]. While qaoa can be proven to give the optimal answer in the limit where the number of qaoa layers p goes to infinity, rigorous results on the performance of qaoa with finite p are difficult to obtain. In a recent paper, Farhi et al. [2] studied the application of the qaoa to the Sherrington-Kirkpatrick (SK) model, a spin-glass model with random all-to-all two-body couplings, in the limit of a large number of spins. Their paper demonstrated that for fixed p, the performance of the qaoa is independent of the specific instance of the SK model and can be predicted by explicit formulas. The paper also showed that the approximation ratio of the qaoa at p = 11 outperforms a large class of classical optimization algorithms (although not the best classical algorithm [3]). In the current paper, we generalize the result of Farhi et al. to mixed-spin SK models, which extends the two-body couplings of standard SK to random all-to-all q-body couplings. We demonstrate that for p = 1, the performance of the qaoa is again independent of the specific instance, and we provide an explicit formula for the expected performance. Our work provides a potential avenue to demonstrating the advantage of qaoa over classical algorithms, as the best known classical algorithms for mixed-spin SK models have an approximation ratio that is bounded away from 1 [4, 5].