Efficient Quantum State Sample Tomography with Basis-Dependent Neural Networks

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Efficient Quantum State Sample Tomography with Basis-Dependent Neural Networks

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

PRX Quantum 2, 020348 – Published 28 June 2021

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.

Received 18 August 2020
Accepted 1 June 2021

DOI:https://doi.org/10.1103/PRXQuantum.2.020348

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Machine learning Quantum benchmarking Quantum computation Quantum information processing Quantum metrology Quantum simulation Quantum tomography

Quantum Information, Science & Technology

Authors & Affiliations

Alistair W. R. Smith ^1,*, Johnnie Gray^1,2, and M. S. Kim¹

¹QOLS, Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom
²Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA

^*alistair.smith18@imperial.ac.uk

Popular Summary

We introduce a machine-learning-based approach to quantum state tomography—the identification of an unknown quantum state based on measurements. Exact quantum state tomography becomes exponentially more difficult to perform as systems get larger, particularly when these systems are noisy. It typically requires huge numbers of measurements to be performed followed by a large computational effort to identify the state, making it impractical even at the scale of currently available quantum computing devices. Each step of our protocol can be done efficiently, yielding an approximate but useful model of the state with far fewer measurements than would be needed for an exact description. We show the effectiveness of our scheme in real noisy devices by applying it to data taken on IBM Quantum devices.

Our model allows one to predict the corresponding distributions in arbitrary local bases by interpolating between observed measurement distributions. By design, it then allows these distributions to be efficiently sampled from, providing access to numerous useful quantities without having to explicitly calculate the distribution. To do this, our model combines two neural network architectures; the first learns to efficiently encode the observed measurement distributions in various local bases and the second learns how the first network’s parameters vary with the basis choice. Modeling a continuously varying probability distribution with a composition of two neural networks in this way demonstrates a natural link between quantum physics and machine learning.

This scheme has the potential to aid in numerous quantum algorithms, providing a computationally inexpensive way of estimating properties of the state in question. It also has potential applications in the benchmarking near-term noisy intermediate-size quantum devices as these systems are now reaching scales that are inaccessible with traditional tomographic approaches.

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Vol. 2, Iss. 2 — June - August 2021

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Figure 1
Illustration of direct QST versus SST. Here we show the differences between direct QST, in which the density matrix is inferred, and state sample tomography where we build a model to implicitly predict measurement outcome distributions and then sample from them efficiently. The end goal in both cases is the calculation of quantities such as observables $⟨ \hat{O} ⟩$ and state fidelities $F (ρ, ρ_{T})$ . Lines in red show stages that become intractable for large systems when using direct QST. (i) Measurements ${σ_{b}} ({σ_{r}})$ are taken of the quantum state in various bases ${b} ({r})$ —the choice of bases depends on the approach taken. Whether direct QST or SST is inherently more expensive in the required number of measurements is beyond the scope of this work but, in general, depends on the exact methods used and the state in question. (ii) Training (or reconstruction) stage: in SST a model is trained to map measurement settings $r$ to outcome distributions $p_{r}$ ; the BDRBM model that we present performs this stage efficiently provided each local measurement distribution can be tractably approximated by an RBM. Direct QST in this stage involves optimization over an exponentially large number of, potentially complex-valued, parameters to produce an exponentially large density matrix. (iii) Calculation stage: once the tomography has produced a model of either the state or its measurement distributions, this is used to calculate some desired quantities, which in the setting presented here are those that can be calculated efficiently from measurement results in local bases. SST allows these to be efficiently calculated by sampling from the predicted measurement distributions. Direct QST requires manipulation of large density matrices that are difficult to manipulate and sample from.
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Figure 2
Diagram of the BDRBM training process. (i) Measurements ${σ_{r}}$ are taken of the quantum system in local bases denoted by ${r}$ , the Bloch sphere coordinates of the each qubits’ measurement axis. The measurement bases are randomly chosen such that ${r}$ is uniformly distributed on the surface of the Bloch (upper hemi)sphere for each qubit. (ii) These measurement results are then used to train a RBM with the CD-1 algorithm; the RBM parameters are recorded as $λ_{r}^{obs}$ . (iii) A FFNN (here shown with one hidden layer) takes the basis setting as an input and outputs a set of predicted RBM parameters $λ_{r}^{pred}$ , thereby implicitly predicting the measurement distribution for this basis setting. (iv) The parameters in the FFNN are trained using gradient descent, minimizing the squared difference between $λ_{r}^{pred}$ and $λ_{r}^{obs}$ .
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Figure 3
Achieved classical fidelities for BDRBM tomography of six-site antiferromagnetic TFIM. These graphs show the average classical fidelities between the predicted measurement distributions of the BDRBM after training with (a) ideal simulated data, (b) data from the $i b m q_s i n g a p o r e$ device, and (c) data from ibmq_paris over a range of transverse fields (and $J_{z} = 1$ ). The training process is as described in Sec. 2a, with six hidden neurons in the RBM and a linear FFNN with directly connected input and output layers. Each graph shows four fidelities between the BDRBM’s predicted distributions and train/measurement, the observed measurements in the bases used for training; train/target, the exact simulated target distributions for the training bases; val/measurement, the withheld measurements in the bases used for validation; val/target, the exact simulated target distributions for the validation bases.
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Figure 4
Scaling of fidelities and overfitting with the number of training measurements. These graphs show how performance of our scheme changes as the BDRBM is given larger sets of training data. This is shown for a linear BDRBM trained with different numbers of simulated measurements (noiseless with 8192 shots per measurement basis) of the antiferromagnetic TFIM ground state with $J_{x} = J_{z} = 1$ (the quantum-critical point). These simulations are performed for ground states with between two to ten sites (indicated by the line color). The RBMs used have $n_{v} = n_{h}$ . (a) The scaling of the average predictive classical fidelity between the BDRBM’s predicted measurement distribution and the exact (calculated from the target state’s state-vector) distributions in randomly selected bases that are not used for training. (b) The scaling of the average reconstructive classical fidelity between the BDRBM’s predicted distribution and the exact distributions in the bases used for training. (c) The scaling of the difference between the average reconstructive and predictive classical fidelities—a measure of how much the network has overfitted to the training data.
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Figure 5
Filters found for linear FFNNs trained on antiferromagnetic TFIM data. These are visual representations of the component values of the FFNN filters found for some of the TFIM ground states in Sec. 2b. As a linear network is used, each component of a filter $M_{i j}$ gives the derivative of the predicted RBM parameter indexed $j$ (columns) with respect to a Bloch sphere coordinate $i$ (rows). Here the rows are grouped into coordinates $r_{k} = (x_{k}, y_{k}, z_{k})$ to indicate which basis coordinates are associated with each qubit. A range of transverse fields are shown: $J_{x} = 0$ being in the ordered, entangled phase; $J_{x} = 3$ in the disordered low-entanglement phase; and $J_{x} = 0.8$ , which is in the intermediate phase around the phase transition. The RBM parameters (columns in each image) are indexed as follows: 0–5 are the biases on each visible neuron (site), 6–11 are the biases on each hidden neuron, and 12–47 are the (flattened) weights between visible and hidden neurons.
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