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Abstract Variational quantum algorithms have the potential for significant impact on high-dimensional optimization, with applications in classical combinatorics, quantum chemistry, and condensed matter. Nevertheless, the optimization landscape of these algorithms is generally nonconvex, leading the algorithms to converge to local, rather than global, minima and the production of suboptimal solutions. In this work, we introduce a variational quantum algorithm that couples classical Markov chain Monte Carlo techniques with variational quantum algorithms, allowing the former to provably converge to global minima and thus assure solution quality. Due to the generality of our approach, it is suitable for a myriad of quantum minimization problems, including optimization and quantum state preparation. Specifically, we devise a Metropolis–Hastings method that is suitable for variational quantum devices and use it, in conjunction with quantum optimization, to construct quantum ensembles that converge to Gibbs states. These performance guarantees are derived from the ergodicity of our algorithm’s state space and enable us to place analytic bounds on its time-complexity. We demonstrate both the effectiveness of our technique and the validity of our analysis through quantum circuit simulations for MaxCut instances, solving these problems deterministically and with perfect accuracy, as well as large-scale quantum Ising and transverse field spin models of up to 50 qubits. Our technique stands to broadly enrich the field of variational quantum algorithms, improving and guaranteeing the performance of these promising, yet often heuristic, methods.
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Systematic improvement of neural network quantum states using Lanczos
The quantum many-body problem lies at the center of the most important open challenges in condensed matter, quantum chemistry, atomic, nuclear, and high-energy physics. While quantum Monte Carlo, when applicable, remains the most powerful numerical technique capable of treating dozens or hundreds of degrees of freedom with high accuracy, it is restricted to models that are not afflicted by the infamous sign problem. A powerful alternative that has emerged in recent years is the use of neural networks as variational estimators for quantum states. In this work, we propose a symmetry-projected variational solution in the form of linear combinations of simple restricted Boltzmann machines. This construction allows one to explore states outside of the original variational manifold and increase the representation power with moderate computational effort. Besides allowing one to restore spatial symmetries, an expansion in terms of Krylov states using a Lanczos recursion offers a solution that can further improve the quantum state accuracy. We illustrate these ideas with an application to the Heisenberg J1−J2 model on the square lattice, a paradigmatic problem under debate in condensed matter physics, and achieve state-of-the-art accuracy in the representation of the ground state.
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- Award ID(s):
- 2120501
- PAR ID:
- 10413834
- Editor(s):
- Koyejo, S.; Mohamed, S.; Agarwal, A.; Belgrave, D.; Cho, K.; Oh, A.
- Date Published:
- Journal Name:
- Advances in Neural Information Processing Systems
- Volume:
- 35
- Page Range / eLocation ID:
- 7490--7503
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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