Inspired by the success of Boltzmann machines based on classical Boltzmann distribution, we propose a new machine-learning approach based on quantum Boltzmann distribution of a quantum Hamiltonian. Because of the noncommutative nature of quantum mechanics, the training process of the quantum Boltzmann machine (QBM) can become nontrivial. We circumvent the problem by introducing bounds on the quantum probabilities. This allows us to train the QBM efficiently by sampling. We show examples of QBM training with and without the bound, using exact diagonalization, and compare the results with classical Boltzmann training. We also discuss the possibility of using quantum annealing processors for QBM training and application.

In the face of mounting numerical evidence, Metlitski and Grover [arXiv:1112.5166] have given compelling analytical arguments that systems with spontaneous broken continuous symmetry contain a sub-leading contribution to the entanglement entropy that diverges logarithmically with system size. They predict that the coefficient of this log is a universal quantity that depends on the number of Goldstone modes. In this paper, we confirm the presence of this log term through quantum Monte Carlo calculations of the second Rényi entropy on the spin 1/2 XY model. Devising an algorithm to facilitate convergence of entropy data at extremely low temperatures, we demonstrate that the single Goldstone mode in the ground state can be identified through the coefficient of the log term. Furthermore, our simulation accuracy allows us to obtain an additional geometric constant additive to the Rényi entropy, that matches a predicted fully-universal form obtained from a free bosonic field theory with no adjustable parameters.Introduction -In condensed matter, the entanglement entropy of a bipartition contains an incredible amount of information about the correlations in a system. In spatial dimensions d ≥ 2, quantum spins or bosons display an entanglement entropy that, to leading order, scales as the boundary of the bipartition [1][2][3]. Subleading to this "area-law" are various constants and -particularly in gapless phases -functions that depend non-trivially on length and energy scales. Some of these subleading terms are known to act as informatic "order parameters" which can detect non-trivial correlations, such as the topological entanglement entropy in a gapped spin liquid phase [4][5][6][7]. At a quantum critical point, subleading terms contain novel quantities that identify the universality class, and potentially can provide constraints on renormalization group flows to other nearby fixed points [8][9][10][11][12][13][14].In systems with a continuous broken symmetry, evidence is mounting that the entanglement entropy between two subsystems with a smooth spatial bipartition contains a term, subleading to the area law, that diverges logarithmically with the subsystem size. First observed in spin wave [15] and finite-size lattice numerics [16], the apparently anomalous logarithm had no rigorous explanation until 2011, when Metlitski and Grover developed a comprehensive theory [17]. They argued that, for a finite-size subsystem with length scale L, the term is a manifestation of the two long-wavelength energy scales corresponding to the spin wave gap, and the "tower of states" arising from the restoration of symmetry in a finite volume [18][19][20][21]. Remarkably, their theory not only explains the subleading logarithm, but predicts that the * bkulchyt@uwaterloo.ca FIG. 1. Schematic energy level structure of the low energy tower of states for finite-size systems with spontaneous breaking of a continuous symmetry. The correction to the entanglement entropy may be approximated by the log of the number of quantum rotor states bel...

Generative modeling with machine learning has provided a new perspective on the data-driven task of reconstructing quantum states from a set of qubit measurements. As increasingly large experimental quantum devices are built in laboratories, the question of how these machine learning techniques scale with the number of qubits is becoming crucial. We empirically study the scaling of restricted Boltzmann machines (RBMs) applied to reconstruct ground-state wavefunctions of the one-dimensional transverse-field Ising model from projective measurement data. We define a learning criterion via a threshold on the relative error in the energy estimator of the machine. With this criterion, we observe that the number of RBM weight parameters required for accurate representation of the ground state in the worst case -near criticality -scales quadratically with the number of qubits. By pruning small parameters of the trained model, we find that the number of weights can be significantly reduced while still retaining an accurate reconstruction. This provides evidence that over-parametrization of the RBM is required to facilitate the learning process. arXiv:1908.07532v2 [quant-ph]