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arXiv:1705.01524 · Physical Review A

Optimal design of measurement settings for quantum-state-tomography experiments

Jun Li, Shilin Huang, Zhihuang Luo, Keren Li, Dawei Lu, Bei Zeng

Published Sep 6, 2017• 25 citations
arXivHTMLPDF

Abstract

Quantum state tomography is an indispensable but costly part of many quantum experiments. Typically, it requires measurements to be carried out in a number of different settings on a fixed experimental setup. The collected data are often informationally overcomplete, with the amount of information redundancy depending on the particular set of measurement settings chosen. This raises a question about how one should optimally take data so that the number of measurement settings necessary can be reduced. Here, we cast this problem in terms of integer programming. For a given experimental setup, standard integer-programming algorithms allow us to find the minimum set of readout operations that can realize a target tomographic task. We apply the method to certain basic and practical state-tomographic problems in nuclear-magnetic-resonance experimental systems. The results show that considerably fewer readout operations can be found using our technique than by using the previous greedy search strategy. Therefore, our method could be helpful for simplifying measurement schemes to minimize the experimental effort.

DOI
10.1103/physreva.96.032307

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Quantum state tomography is an indispensable but costly part of many quantum experiments. Typically, it requires measurements to be carried out in a number of different settings on a fixed experimental setup. The collected data are often informationally overcomplete, with the amount of information redundancy depending on the particular set of measurement settings chosen. This raises a question about how one should optimally take data so that the number of measurement settings necessary can be reduced. Here, we cast this problem in terms of integer programming. For a given experimental setup, standard integer-programming algorithms allow us to find the minimum set of readout operations that can realize a target tomographic task. We apply the method to certain basic and practical state-tomographic problems in nuclear-magnetic-resonance experimental systems. The results show that considerably fewer readout operations can be found using our technique than by using the previous greedy search strategy. Therefore, our method could be helpful for simplifying measurement schemes to minimize the experimental effort.
Given an experimental setup and a fixed number of measurements, how should one take data to optimally reconstruct the state of a quantum system? The problem of optimal experiment design (OED) for quantum state tomography was first broached by Kosut et al. [R. Kosut, I. Walmsley, and H. Rabitz, e-print arXiv:quant-ph/0411093 (2004)]. Here we provide efficient numerical algorithms for finding the optimal design, and analytic results for the case of `minimal tomography'. We also introduce the average OED, which is independent of the state to be reconstructed, and the optimal design for tomography (ODT), which minimizes tomographic bias. Monte Carlo simulations confirm the utility of our results for qubits. Finally, we adapt our approach to deal with constrained techniques such as maximum-likelihood estimation. We find that these are less amenable to optimization than cruder reconstruction methods, such as linear inversion.
Quantum device characterization via state tomography plays an important role in both validating quantum hardware and processing quantum information, unfortunately with the exponential number of the measurements. As one of the main stream quantum platforms, superconducting quantum computing (SQC) system at least requires 3n3^n3n measurement settings consisted of single-qubit readout operators in reconstructing a nnn-qubit state. In this work, I add the 2-qubit evolutions as the readout operators, and propose an optimal tomographic scheme with the cost-reduced measurements using the integer programming. In detail, I present the minimum number of required measurements to fully reconstruct a state for the Nearest-Neighbor, 2-Dimensional, and All-to-All connectivities on SQC qubits. It is shown that this method can reduce the number of measurements by over 60% compared with the previous state tomography on SQC systems. It is expected that the experimentalist from SQC field can directly utilize the ready-made results for reconstructing quantum states involved in their research. Besides, this method can be applied to reduce the complexity of traditional state tomography in some quantum platforms including but not limited to SQC systems.
Quantum device characterization via state tomography plays an important role in both validating quantum hardware and processing quantum information, but it needs the exponential number of the measurements. For the systems with XX+++YY-type couplings and ZZZ readouts, such as superconducting quantum computing (SQC) systems, traditional quantum state tomography (QST) using single-qubit readout operations at least requires 3n{3}^{n}3n measurement settings in reconstructing an nnn-qubit state. In this paper, I propose an improved QST by adding two-qubit evolutions as the readout operations and obtain an optimal tomographic scheme using the integer programming optimization. I, respectively, apply the alternative scheme on SQC systems with the nearest-neighbor, two-dimensional, and all-to-all connectivities on qubits. It shows that this method can reduce the number of measurements by over 60% compared with the traditional QST. In addition, comparison with the traditional scheme in the experimental feasibility and robustness against errors was made by numerical simulation. It is found that the alternative scheme has good implementability and can achieve comparable or even better accuracy than the traditional scheme. It is expected that the experimentalist from related fields can directly utilize the ready-made results for reconstructing quantum states involved in their research.
Quantum tomography approaches typically consider a set of observables that we wish to measure, designing a measurement scheme that measures each of the observables and then repeats the measurements as many times as necessary. We show that instead of considering only the simple set of observables, one should consider a multiset of the observables taking into account the required repetitions, to minimize the number of measurements. This leads to a graph theoretic multicoloring problem. We show that the multiset method offers at most quadratic improvement but it is achievable. Furthermore, despite the NP-hard optimal coloring problem, the multiset approach with greedy coloring algorithms already offers asymptotically quadratic improvement in test cases.
Quantum state tomography (QST) is a crucial technique for characterizing the state of quantum systems. However, traditional QST requires a large number of measurements, leading to significant experimental overhead in terms of time, resources, and data acquisition. This work proposes a novel approach to QST by leveraging compressed sensing (CS) techniques. CS allows for accurate reconstruction of the quantum state from a significantly reduced set of measurements, thereby mitigating the experimental burden. We employ sparsity regularization, a key component of CS, to minimize the number of measurements needed. The theoretical framework and the underlying principles are presented, focusing on the mathematical formulation of the problem and the optimization strategies involved. This method offers a pathway to more efficient and cost-effective QST, particularly for complex quantum states or when limited experimental resources are available. We demonstrate the potential of this approach through a detailed mathematical analysis.