Random phase product states for quantum Boltzmann machines at finite temperature

I will introduce a mathematical proof and numerical examples of the method of random phase product state (RPPS) [1] for calculating thermal average of a physical quantity A at inverse temperature β,〈Φ(w(β))| A |Φ(w(β))〉, with respect to thermal neural network wave functions (Boltzmann machines), |Φ(w(β))〉= exp[-βH/2] |Φ(w(0))〉=exp[-τH/2]ⁿ|Φ(w(0))〉, where w's are variational parameters, τ is a small imaginary time step. The initial state |Φ(w(0))〉is set to a neural network wave function of RPPS representing a state at infinitely high temperature and the imaginary time evolution exp[-τH/2] |Φ〉is approximated with the natural gradient [2]. This method is a natural extension of the RPPS method for matrix product states (MPS) [1] and the random state method for a full Hilbert space [3,4].

[1] T. Iitaka, Random Phase Product Sate for Canonical Ensemble, https://arxiv.org/abs/2006.14459 .

[2] J. Stokes et al., Quantum 4, 269 (2020) and references therein.

[3] T. Iitaka, and T. Ebisuzaki, Phys. Rev. Lett. 90, 047203 (2003).

[4] T. Iitaka, and T. Ebisuzaki, Phys. Rev. E 69, 057701 (2004).