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Physical Review A

Numerical strategies for quantum tomography: Alternatives to full optimization

Max S. Kaznady, Daniel F. V. James

Published Feb 9, 2009• 61 citations
Publisher

Abstract

We examine a variety of strategies for numerical quantum-state estimation from data of the sort commonly measured in experiments involving quantum-state tomography. We find that, in some important circumstances, an elaborate and time-consuming numerical optimization to obtain the optimum density matrix corresponding to a given data set is not necessary and that cruder, faster numerical techniques may well be sufficient; in other words, ``the best'' is the enemy of ``good enough.''

DOI
10.1103/physreva.79.022109

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We examine a variety of strategies for numerical quantum-state estimation from data of the sort commonly measured in experiments involving quantum-state tomography. We find that, in some important circumstances, an elaborate and time-consuming numerical optimization to obtain the optimum density matrix corresponding to a given data set is not necessary and that cruder, faster numerical techniques may well be sufficient; in other words, the best'' is the enemy of good enough.''
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Quantum tomography is a critically important tool to evaluate quantum hardware, making it essential to develop optimized measurement strategies that are both accurate and efficient. We compare a variety of strategies using nearly pure test states. Those that are informationally complete for all states are found to be accurate and reliable even in the presence of errors in the measurements themselves, while those designed to be complete only for pure states are far more efficient but highly sensitive to such errors. Our results highlight the unavoidable trade-offs inherent in quantum tomography.
When working with quantum states, analysis of the final quantum state generated through probabilistic measurements is essential. This analysis is typically conducted by constructing the density matrix from either partial or full tomography measurements of the quantum state. While full tomography measurement offers the most accurate reconstruction of the density matrix, limited measurements pose challenges for reconstruction algorithms, often resulting in nonphysical density matrices with negative eigenvalues. This is often remedied using maximum likelihood estimators, which have a high computing time or by other estimation methods that decrease the reconstructed fidelity. In this paper, we show that when restricting the measurement sample size, improvement over existing algorithms can be achieved. Our findings underline the multiplicity of solutions in the reconstruction problem, depending upon the generated state and measurement model utilized, thus motivating further research towards identifying optimal algorithms tailored to specific experimental contexts.
We propose, to the best of our knowledge, the first online algorithm for maximum-likelihood quantum state tomography. Suppose the quantum state to be estimated corresponds to a D D D-by-D D D density matrix. The per-iteration computational complexity of the algorithm is O(D3) O ( D ^ 3 ) O(D3), independent of the data size. The expected numerical error of the algorithm is O((1/T)Dlog⁡D)O(\sqrt{ ( 1 / T ) D \log D })O((1/T)DlogD​), where TTT denotes the number of iterations. The algorithm can be viewed as a quantum extension of Soft-Bayes, a recent algorithm for online portfolio selection (Orseau et al. Soft-Bayes: Prod for mixtures of experts with log-loss. \textit{Int. Conf. Algorithmic Learning Theory}. 2017).
Quantum tomography requires repeated measurements of many copies of the physical system, all prepared by a source in the unknown state. In the limit of very many copies measured, the often-used maximum-likelihood (ML) method for converting the gathered data into an estimate of the state works very well. For smaller data sets, however, it often suffers from problems of rank deficiency in the estimated state. For many systems of relevance for quantum information processing, the preparation of a very large number of copies of the same quantum state is still a technological challenge, which motivates us to look for estimation strategies that perform well even when there is not much data. After reviewing the concept of minimax state estimation, we use minimax ideas to construct a simple estimator for quantum states. We demonstrate that, for the case of tomography of a single qubit, our estimator significantly outperforms the ML estimator for small number of copies of the state measured. Our estimator is always full-rank, and furthermore, has a natural dependence on the number of copies measured, which is missing in the ML estimator.