Numerical and geometrical aspects of flow-based variational quantum Monte Carlo

Abstract

This article aims to summarize recent and ongoing efforts to simulate continuous-variable quantum systems using flow-based variational quantum Monte Carlo techniques, focusing for pedagogical purposes on the example of bosons in the field amplitude (quadrature) basis. Particular emphasis is placed on the variational real-and imaginary-time evolution problems, carefully reviewing the stochastic estimation of the time-dependent variational principles and their relationship with information geometry. Some practical instructions are provided to guide the implementation of a PyTorch code. The review is intended to be accessible to researchers interested in machine learning and quantum information science. Contents 1. Introduction 2. Notation 3. Preliminaries 4. Information geometry 4.1. Unnormalized wavefunctions and holomorphy 5. Time-dependent variational principles 5.1. Unnormalized wavefunctions and holomorphy 6. Stochastic estimation 6.1. VMC and t-VMC 6.2. Stochastic optimization and variance reduction 7. Additional comments on quantum flows 7.1. Group equivariant dynamics from flows 7.2. Universal approximation 8. Experiments 8.1. States and Hamiltonian 8.2. Experimental setup 8.3. Results 9. Discussion and future directions 10. Acknowledgements Appendix A. McLachlan's variational principles Appendix B. Justification for choice of baseline Appendix C. Quadratic Hamiltonian in one dimension Appendix D. Derivation of time-dependent variational principles Appendix E.

Quantum-inspired variational algorithms for partial differential equations: Application to financial derivative pricing

“…In summary, we introduced a generalization of McLachlan's variational principle applicable to generic timedependent PDEs as well as a quantum-inspired training algorithm based on neural-network quantum states which can be used to perform approximate time evolution in high dimensions, overcoming the curse-ofdimensionality. Although we focused on a mesh-based formulation in which the quantum state vector is represented by n qubits, it is clear that the mesh is not mandated by the formulation and it would be very interesting to pursue meshless variants based on continuous-variable neural-network quantum states including normalizing flows [24] and to address non-trivial boundary conditions. There exist a number of directions in which the results of this paper can be potentially improved.…”

“…Ref. [24] studied TDVPs through the lens of information geometry providing a unified perspective applicable to variational quantum algorithms (VQAs) and variational quantum Monte Carlo with normalized neural-network quantum states. In the following section, we further generalize TDVPs to include general time-dependent PDEs and establish additional connections with the VQA and numerical analysis literature.…”

“…In quantum physics applications, a suitable distance metric is the Fubini-Study metric, as previously argued both in VMC [5,24] and in VQA [25,2] literature. In this work we choose dist(•, •) to be the Euclidean norm.…”

“…The latter holds promise to accelerate a host of scientific computing problems including general-purpose solvers of partial differential equations (PDEs) using physics-informed neural-networks (PINNs) [20] and has already made substantial headway in solving the time-(in)dependent Schrödinger equation in high dimensions using variational quantum Monte Carlo (VMC) with neural-network quantum states [5]. It is noteworthy that there exist many shared parallels between the fields of VQAs, VMC [25,24] and PINNs. In particular, both VMC and VQAs hinge on the concept of adaptive stochastic estimation of the time-independent and time-dependent variational principles originally due to Rayleigh-Ritz and McLachlan [18], respectively.…”

NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems

“…Note, however, that there are also common architectures, such as autoregressive networks, that are not holomorphic. In the holomorphic case, the computational cost of differentiating the model, e.g., to compute the quantum geometric tensor (Section 3.5), can be reduced by exploiting the Cauchy-Riemann equations [46],…”