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arXiv:2009.07601 · PRX Quantum

Efficient Quantum State Sample Tomography with Basis-Dependent Neural Networks

Alistair W. R. Smith, Johnnie Gray, M. S. Kim

Published Jun 28, 2021• 32 citations
arXivHTMLPDF

Abstract

We use a metalearning neural-network approach to analyze data from a measured quantum state.Once our neural network has been trained, it can be used to efficiently sample measurements of the state in measurement bases not contained in the training data.These samples can be used to calculate expectation values and other useful quantities.We refer to this process as "state sample tomography."We encode the state's measurement outcome distributions using an efficiently parameterized generative neural network.This allows each stage in the tomography process to be performed efficiently even for large systems.Our scheme is demonstrated on recent IBM Quantum devices, producing a model for a six-qubit state's measurement outcomes with a predictive accuracy (classical fidelity) greater than 95% for all test cases using only 100 random measurement settings as opposed to the 729 settings required for standard full tomography using local measurements.This reduction in the required number of measurements scales favorably, with training data in 200 measurement settings, yielding a predictive accuracy greater than 92% for a ten-qubit state where 59 049 settings are typically required for full local measurement-based quantum state tomography.A reduction in the number of measurements by a factor, in this case, of almost 600 could allow for estimations of expectation values and state fidelities in practicable times on current quantum devices.

DOI
10.1103/prxquantum.2.020348

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We use a metalearning neural-network approach to analyze data from a measured quantum state.Once our neural network has been trained, it can be used to efficiently sample measurements of the state in measurement bases not contained in the training data.These samples can be used to calculate expectation values and other useful quantities.We refer to this process as "state sample tomography."We encode the state's measurement outcome distributions using an efficiently parameterized generative neural network.This allows each stage in the tomography process to be performed efficiently even for large systems.Our scheme is demonstrated on recent IBM Quantum devices, producing a model for a six-qubit state's measurement outcomes with a predictive accuracy (classical fidelity) greater than 95% for all test cases using only 100 random measurement settings as opposed to the 729 settings required for standard full tomography using local measurements.This reduction in the required number of measurements scales favorably, with training data in 200 measurement settings, yielding a predictive accuracy greater than 92% for a ten-qubit state where 59 049 settings are typically required for full local measurement-based quantum state tomography.A reduction in the number of measurements by a factor, in this case, of almost 600 could allow for estimations of expectation values and state fidelities in practicable times on current quantum devices.
We use a meta-learning neural-network approach to predict measurement outcomes of a quantum state in arbitrary local bases and thus carry out an approximate quantum state tomography. Each stage of this procedure can be performed efficiently, allowing it to be used effectively on large systems. We demonstrate this approach on the most recent noisy intermediate-scale IBM Quantum devices, achieving an accurate generative model for a 6-qubit state's measurement outcomes with only 100 random measurement settings as opposed to the 729 settings required for full tomography. This reduction in the required number of measurements scales favourably, with around 200 measurement settings yielding good results for a 10 qubit state that would require 59,049 settings for full quantum state tomography. This reduction in the number of measurement settings coupled with the efficiency of the procedure could allow for estimations of expectation values and state fidelities in practicable times on current quantum devices. For suitable states, this could then help in increasing the speed of other optimization schemes when attempting to produce states on noisy quantum devices at a scale where traditional maximum likelihood based approaches are impractical.
An efficient state estimation model, neural network estimation (NNE), empowered by machine learning techniques, is presented for full quantum state tomography (FQST). A parameterized function based on neural network is applied to map the measurement outcomes to the estimated quantum states. Parameters are updated with supervised learning procedures. From the computational complexity perspective our algorithm is the most efficient one among existing state estimation algorithms for full quantum state tomography. We perform numerical tests to prove both the accuracy and scalability of our model.
Abstract Modern day quantum simulators can prepare a wide variety of quantum states but the accurate estimation of observables from tomographic measurement data often poses a challenge. We tackle this problem by developing a quantum state tomography scheme which relies on approximating the probability distribution over the outcomes of an informationally complete measurement in a variational manifold represented by a convolutional neural network. We show an excellent representability of prototypical ground- and steady states with this ansatz using a number of variational parameters that scales polynomially in system size. This compressed representation allows us to reconstruct states with high classical fidelities outperforming standard methods such as maximum likelihood estimation. Furthermore, it achieves a reduction of the estimation error of observables by up to an order of magnitude compared to their direct estimation from experimental data.
Abstract Quantum state tomography is a daunting challenge of experimental quantum computing, even in moderate system size. One way to boost the efficiency of state tomography is via local measurements on reduced density matrices, but the reconstruction of the full state thereafter is hard. Here, we present a machine-learning method to recover the ground states of kkk k -local Hamiltonians from just the local information, where a fully connected neural network is built to fulfill the task with up to seven qubits. In particular, we test the neural network model with a practical dataset, that in a 4-qubit nuclear magnetic resonance system our method yields global states via the 2-local information with high accuracy. Our work paves the way towards scalable state tomography in large quantum systems.
With increasing maturity of quantum computers, standard methods for characterizing global properties of their output quantum states via direct measurements and classical post-processing are becoming increasingly impractical due to large measurement costs. Although quantum neural networks could directly process quantum states to identify underlying characteristics with reduced measurement efforts, they often require deep quantum circuits that cannot be implemented on existing devices. To overcome these challenges, we introduce a hybrid quantum-classical neural network that consists of a shallow parameterized quantum circuit, measurements, and a classical neural network. The parameterized quantum circuit performs a nonlocal transformation of the measurement basis, which is jointly trained with the classical neural network to maximize the statistical distance between data obtained by measuring different quantum states. Using supervised learning, we demonstrate that the hybrid neural network distinguishes the topological phase of the surface code from a symmetry-enriched topological phase and random product states. Moreover, this hybrid neural network reduces both inference and training sample complexities of recognizing the topological phase by approximately one order of magnitude compared to a classical neural network trained on randomized Pauli measurements. As this hybrid neural network features a shallow quantum circuit that can be readily implemented on existing quantum computers, it enables the efficient characterization of complex quantum states.
Abstract It is believed that one of the first useful applications for a quantum computer will be the preparation of groundstates of molecular Hamiltonians. A crucial task involving state preparation and readout is obtaining physical observables of such states, which are typically estimated using projective measurements on the qubits. At present, measurement data is costly and time-consuming to obtain on any quantum computing architecture, which has significant consequences for the statistical errors of estimators. In this paper, we adapt common neural network models (restricted Boltzmann machines and recurrent neural networks) to learn complex groundstate wavefunctions for several prototypical molecular qubit Hamiltonians from typical measurement data. By relating the accuracy ɛ of the reconstructed groundstate energy to the number of measurements, we find that using a neural network model provides a robust improvement over using single-copy measurement outcomes alone to reconstruct observables. This enhancement yields an asymptotic scaling near ɛ −1 for the model-based approaches, as opposed to ɛ −2 in the case of classical shadow tomography.