[Paper Review] Many-body quantum state tomography with neural networks

Giacomo Torlai, Guglielmo Mazzola+4 more|arXiv (Cornell University)|Mar 15, 2017

Quantum Information and CryptographyComputer Science31 references203 citations

TL;DR

The paper demonstrates neural-network quantum state tomography using restricted Boltzmann machines to reconstruct amplitudes and phases of many-body quantum states from experimental measurements, enabling estimation of entanglement and other observables.

ABSTRACT

The experimental realization of increasingly complex synthetic quantum systems calls for the development of general theoretical methods, to validate and fully exploit quantum resources. Quantum-state tomography (QST) aims at reconstructing the full quantum state from simple measurements, and therefore provides a key tool to obtain reliable analytics. Brute-force approaches to QST, however, demand resources growing exponentially with the number of constituents, making it unfeasible except for small systems. Here we show that machine learning techniques can be efficiently used for QST of highly-entangled states, in both one and two dimensions. Remarkably, the resulting approach allows one to reconstruct traditionally challenging many-body quantities - such as the entanglement entropy - from simple, experimentally accessible measurements. This approach can benefit existing and future generations of devices ranging from quantum computers to ultra-cold atom quantum simulators.

Motivation & Objective

• Motivate the need for scalable quantum-state tomography in highly entangled, many-body systems.
• Propose a neural-network representation (RBM) to efficiently encode quantum states.
• Demonstrate reconstruction from limited experimental data across 1D and 2D systems and through dynamics.
• Show recovery of observables and entanglement measures from reconstructed states.

Proposed method

• Use a restricted Boltzmann machine (RBM) with a visible layer for physical qubits and hidden layers to represent amplitudes and phases of the wave function.
• Parametrize the RBM wave-function as | ψ_{λ,μ}(x) = sqrt(p_{λ}(x)/Z_{λ}) e^{i φ_{μ}(x)/2}, where p_{λ} and φ_{μ} encode amplitude and phase information.
• Train the RBM on datasets of density measurements in multiple bases to minimize the total divergence Ξ(κ) = sum_b KL(P_b || |ψ_{κ}(σ^{[b]})|^2).
• Optimize amplitudes (λ) via KL divergence in the reference basis and then phases (μ) using auxiliary bases.
• Employ gradient-based optimization (stochastic gradient descent for amplitudes; natural gradient for phases) with Fisher information regularization.
• Demonstrate reconstruction on W states, phase-augmented W states, and many-body Hamiltonians (TFIM and XXZ), including ground states and quench dynamics.
• Estimate entanglement entropy using a ratio-trick sampling on the RBM-generated wave-function.

Experimental results

Research questions

• RQ1 Can RBM-based neural networks efficiently perform tomography for highly entangled many-body quantum states from limited measurement data?
• RQ2 To what extent can RBM-QST reproduce both amplitudes and phases across different bases, including dynamics and phase information?
• RQ3 How accurately can RBM-QST reproduce observables and non-local quantities such as entanglement entropy compared to conventional methods?
• RQ4 Is the approach scalable to 1D and 2D systems and to states arising from unitary evolution under realistic spin models?

Key findings

• RBM-based QST can reconstruct target states from density measurements in multiple bases, including real and complex wave-function coefficients.
• The method achieves high overlap with target W states and phase-augmented W states using comparatively few samples.
• RBM-QST accurately reproduces diagonal and off-diagonal observables for ground states of TFIM and XXZ models in 1D and 2D, and for quench dynamics.
• The approach yields good agreement for two-point correlations and non-local spin-spin correlations with exact or QMC benchmarks.
• Entanglement entropy (second Renyi) can be estimated from RBM-reconstructed states with overall good agreement to exact results.