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]
Motivated by the search for an experimentally realizable high density and strongly interacting one dimensional quantum liquid, we have performed quantum Monte Carlo simulations of bosonic helium-4 confined inside a nanopore with cylindrical symmetry. By implementing two numerical estimators of superfluidity corresponding to capillary flow and the rotating bucket experiment, we have simultaneously measured the finite size and temperature superfluid response of 4 He to the longitudinal and rotational motion of the walls of a nanopore. Within the two-fluid model, the portion of the normal liquid dragged along with the boundaries is dependent on the type of motion, and the resulting anisotropic superfluid density plateaus far below unity at T = 0.5 K. The origin of the saturation is uncovered by computing the spatial distribution of superfluidity, with only the core of the nanopore exhibiting any evidence of phase coherence. The superfluid core displays scaling behavior consistent with Luttinger liquid theory, thereby providing an experimental test for the emergence of a one dimensional quantum liquid.
As we enter a new era of quantum technology, it is increasingly important to develop methods to aid in the accurate preparation of quantum states for a variety of materials, matter, and devices. Computational techniques can be used to reconstruct a state from data, however the growing number of qubits demands ongoing algorithmic advances in order to keep pace with experiments. In this paper, we present an open-source software package called QuCumber that uses machine learning to reconstruct a quantum state consistent with a set of projective measurements. QuCumber uses a restricted Boltzmann machine to efficiently represent the quantum wavefunction for a large number of qubits. New measurements can be generated from the machine to obtain physical observables not easily accessible from the original data. Contents arXiv:1812.09329v2 [quant-ph] 16 May 2019 SciPost Physics Submission 4 Conclusion 14 A Glossary 15 References 171 IntroductionCurrent advances in fabricating quantum technologies, as well as in reliable control of synthetic quantum matter, are leading to a new era of quantum hardware where highly pure quantum states are routinely prepared in laboratories. With the growing number of controlled quantum degrees of freedom, such as superconducting qubits, trapped ions, and ultracold atoms [1-4], reliable and scalable classical algorithms are required for the analysis and verification of experimentally prepared quantum states. Efficient algorithms can aid in extracting physical observables otherwise inaccessible from experimental measurements, as well as in identifying sources of noise to provide direct feedback for improving experimental hardware. However, traditional approaches for reconstructing unknown quantum states from a set of measurements, such as quantum state tomography, often suffer the exponential overhead that is typical of quantum many-body systems.Recently, an alternative path to quantum state reconstruction was put forward, based on modern machine learning (ML) techniques [5][6][7][8][9][10]. The most common approach relies on a powerful generative model called a restricted Boltzmann machine (RBM) [11], a stochastic neural network with two layers of binary units. A visible layer v describes the physical degrees of freedom, while a hidden layer h is used to capture high-order correlations between the visible units. Given a set of neural network parameters λ, the RBM defines a probabilistic model described by the parametric distribution p λ (v). RBMs have been widely used in the ML community for the pre-training of deep neural networks [12], for compressing high-dimensional data into lower-dimensional representations [13], and more [14]. More recently, RBMs have been adopted by the physics community in the context of representing both classical and quantum many-body states [15,16]. They are currently being investigated for their representational power [17][18][19], their relationship with tensor networks and the renormalization group [20][21][22][23][24], and in other contexts in quantum many-bo...
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The entanglement entropy of a quantum critical system can provide new universal numbers that depend on the geometry of the entangling bipartition. We calculate a universal number called κ, which arises when a quantum critical system is embedded on a two-dimensional torus and bipartitioned into two cylinders. In the limit when one of the cylinders is a thin slice through the torus, κ parameterizes a divergence that occurs in the entanglement entropy sub-leading to the area law. Using large-scale Monte Carlo simulations of an Ising model in 2+1 dimensions, we access the second Rényi entropy, and determine that, at the Wilson-Fisher (WF) fixed point, κ2,WF = 0.0174(5). This result is significantly different from its value for the Gaussian fixed point, known to be κ2,Gaussian ≈ 0.0227998. arXiv:1904.08955v1 [cond-mat.str-el]
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