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Quantum Science and Technology

Quantum state tomography based on infidelity estimation

Yong Wang, Lijun Liu, Tong Dou, Li Li, Shuming Cheng

Published Nov 14, 2024• 2 citations
Publisher

Abstract

Abstract Quantum state tomography is a cornerstone of quantum information technologies to characterize and benchmark quantum systems from measurement statistics. In this work, we present an infidelity-based least-squares estimator, which incorporates the state purity information and provides orders of magnitude higher tomography accuracy than previous ones. It is further enhanced with the randomized toolbox of direct fidelity estimation, making it applicable to large-scale quantum systems. We validate the proposed estimators through extensive experiments conducted on the IBM Qiskit simulator. The results also demonstrate that our estimator admits an O ( 1 / N ) infidelity scaling with Pauli sample size N for (nearly) pure states. Further, it enables high-precision pure-state tomography for systems of up to 25-qubit states, given some state priors. Our method provides a novel perspective on the union of underlying tomography technique and state properties estimation.

DOI
10.1088/2058-9565/ad92a4

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Abstract Quantum state tomography is a cornerstone of quantum information technologies to characterize and benchmark quantum systems from measurement statistics. In this work, we present an infidelity-based least-squares estimator, which incorporates the state purity information and provides orders of magnitude higher tomography accuracy than previous ones. It is further enhanced with the randomized toolbox of direct fidelity estimation, making it applicable to large-scale quantum systems. We validate the proposed estimators through extensive experiments conducted on the IBM Qiskit simulator. The results also demonstrate that our estimator admits an O ( 1 / N ) infidelity scaling with Pauli sample size N for (nearly) pure states. Further, it enables high-precision pure-state tomography for systems of up to 25-qubit states, given some state priors. Our method provides a novel perspective on the union of underlying tomography technique and state properties estimation.
Adaptive tomography has been widely investigated with the aim of achieving faster state tomography processing of quantum systems. Infidelity of the nearly pure states in a quantum information process generally scales as O(1/N)O(1/\sqrt{N})O(1/N​), which requires a large number of statistical ensembles in comparison to the infidelity scaling of O(1/N)O(1/N)O(1/N) for mixed states. One previous report optimized the measurement basis in a photonic qubit system, whose state tomography uses projective measurements, to obtain an infidelity scaling of O(1/N)O(1/N)O(1/N). However, this dramatic improvement cannot be applied to indistinct measurement systems in which two quantum states cannot be distinguished with perfect measurement fidelity. We introduce in this work an optimal measurement basis to achieve fast adaptive quantum state tomography and a minimum magnitude of infidelity in an indistinct measurement system. We expect that the adaptive quantum state tomography protocol can lead to a reduction in the number of required measurements of approximately 33.74% via analytical expectation without changing the O(1/N)O(1/\sqrt{N})O(1/N​) scaling. Experimentally, we find a 15.06% measurement number reduction in a superconducting circuit system.
Adaptive tomography has been widely investigated to obtain faster state tomography process for quantum computation. Infidelity [1 − F(ρ,σ)] on nearly pure states in quantum information process scales generally as O(1/N)O\left( {1/\sqrt N } \right)O(1/N​), which requires a large number of statistical ensembles in comparison to the infidelity scaling on mixed states of O(1/N). Recently, optimization of a measurement basis in a photonic qubit system, whose state tomography uses projective measurements, reported improved infidelity scaled as O(1/N). However, this dramatic improvement cannot be applied to a weak-value based measurement system, which can be attributed to the fact that one cannot distinguish two quantum states with perfect measurement reliability. We introduce new optimal measurement basis to achieve a fast adaptive quantum state tomography and minimum magnitude of infidelity in this weak measurement system. This novel protocol allows us to accomplish approximately 80% error reduction without changing the scaling of O(1/N)O\left( {1/\sqrt N } \right)O(1/N​) in numerical simulations and realize approximately 90% error reduction in superconducting circuit QED experiments.
Adaptive tomography has been widely investigated to achieve faster state tomography processing of quantum systems. Infidelity of the nearly pure states in a quantum information process generally scales as O(1/sqrt(N) ), which requires a large number of statistical ensembles in comparison to the infidelity scaling of O(1/N) for mixed states. One previous report optimized the measurement basis in a photonic qubit system, whose state tomography uses projective measurements, to obtain an infidelity scaling of O(1/N). However, this dramatic improvement cannot be applied to weak-value-based measurement systems in which two quantum states cannot be distinguished with perfect measurement fidelity. We introduce in this work a new optimal measurement basis to achieve fast adaptive quantum state tomography and a minimum magnitude of infidelity in a weak measurement system. We expect that the adaptive quantum state tomography protocol can lead to a reduction in the number of required measurements of approximately 33.74% via simulation without changing the O(1/sqrt(N)) scaling. Experimentally, we find a 14.81% measurement number reduction in a superconducting circuit system.
We introduce a simple protocol for adaptive quantum state tomography, which reduces the worst-case infidelity [$1\ensuremath{-}F(\stackrel{^}{\ensuremath{\rho}},\ensuremath{\rho})$] between the estimate and the true state from O(1/N)O(1/\sqrt{N})O(1/N​) to O(1/N)O(1/N)O(1/N). It uses a single adaptation step and just one extra measurement setting. In a linear optical qubit experiment, we demonstrate a full order of magnitude reduction in infidelity (from 0.1% to 0.01%) for a modest number of samples ($N\ensuremath{\approx}3\ifmmode\times\else\texttimes\fi{}{10}^{4}$).
Multi-qubit state tomography is a key problem in quantum information technology, which has been studied extensively. In this work, we focus on multi-qubit state tomography based on neural network estimators and typical conventional estimation approaches. For multi-qubit pure states, fully connected neural networks and restricted Boltzmann machine networks are designed, respectively, to carry out state tomography of 2-qubit (low-dimensional) systems and 5-qubit (high-dimensional) systems. Comparisons are made with maximum likelihood estimation and least squares estimation, where performance indicators are selected as reconstruction accuracy, time cost and the number of parameters. Simulation results indicate that intelligent approaches have significant advantages over conventional approaches for state tomography of low-dimensional systems; for high-dimensional systems, however, the conventional approach is still dominant when the measurement is complete, while the restricted Boltzmann machine network can achieve higher reconstruction accuracy when the measurement is incomplete.
Quantum state tomography is a technique in quantum information science used to reconstruct the density matrix of an unknown quantum state, providing complete information about the quantum state. It is of significant importance in fields such as quantum computation, quantum communication, and quantum simulation. However, as the size of the quantum system increases, the number of measurement settings and sampling requirements for quantum state tomography grow exponentially with the number of qubits. This not only makes experimental design and implementation more complex, but also exacerbates the consumption of experimental resources. These limitations severely hinder the application of state tomography in large-scale quantum systems. To reduce measurement settings and improve sampling efficiency, this study proposes a state tomography method based on auxiliary systems. This method can be implemented through either entanglement between the quantum system to be measured and a quantum auxiliary system or through correlation between the quantum system and a probabilistic classical auxiliary system. Measurements on the entire joint system enable more efficient extraction of information about the quantum state to be measured. This method relies on standard quantum gate operations and requires only two measurement settings, with a total sampling complexity of O(d2)O(d^2)O(d2), significantly simplifying experimental operations and measurement processes. Additionally, this study provides two schemes for measuring purity based on the proposed circuit, one of which achieves measurement precision at the Heisenberg limit. This study validates the effectiveness of the proposed method through a detailed theoretical analysis, a series of numerical simulations, and experiments.