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Flexible quantum state tomography

Daniel Uzcátegui Contreras, Dardo Goyeneche

Published Dec 12, 2019• 0 citations
OpenAlex

Abstract

We present an efficient algorithm that solves the quantum state tomography problem from an arbitrary number of projective measurements in any finite dimension $d$. The algorithm is flexible enough to allow us to impose any desired rank $r$ to the state to be reconstructed, ranging from pure ($r=1$) to full rank $(r=d)$ quantum states. The method exhibits successful and fast convergence under the presence of realistic errors in both state preparation and measurement stages, and also when considering overcomplete sets of observables. We demonstrate that the method outperforms semidefinite programming quantum state tomography for some sets of physically relevant quantum measurements in every finite dimension.

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We present an efficient algorithm that solves the quantum state tomography problem from an arbitrary number of projective measurements in any finite dimension ddd. The algorithm is flexible enough to allow us to impose any desired rank rrr to the state to be reconstructed, ranging from pure (r=1r=1r=1) to full rank (r=d)(r=d)(r=d) quantum states. The method exhibits successful and fast convergence under the presence of realistic errors in both state preparation and measurement stages, and also when considering overcomplete sets of observables. We demonstrate that the method outperforms semidefinite programming quantum state tomography for some sets of physically relevant quantum measurements in every finite dimension.
We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rdlog²d) measurement settings, compared to standard methods that require d² settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable against noise, and can be applied to states that are only approximately low rank. The acquired data can be used to certify that the state is indeed close to pure, so no a priori assumptions are needed.
We consider the task of performing quantum state tomography on a ddd-state spin qudit, using only measurements of spin projection onto different quantization axes. By an exact mapping onto the classical problem of signal recovery on the sphere, we prove that full reconstruction of arbitrary qudit states requires a minimal number of measurement axes, rdminr_d^{\mathrm{min}}rdmin​, that is bounded by 2d−1≤rdmin≤d22d-1\le r_d^{\mathrm{min}}\le d^22d−1≤rdmin​≤d2. We conjecture that rdmin=2d−1r_d^{\mathrm{min}}=2d-1rdmin​=2d−1, which we verify numerically for all d≤200d\le200d≤200. We then provide algorithms with O(rd3)O(rd^3)O(rd3) serial runtime, parallelizable down to O(rd2)O(rd^2)O(rd2), for (i) computing a priori upper bounds on the expected error with which spin projection measurements along rrr given axes can reconstruct an unknown qudit state, and (ii) estimating a posteriori the statistical error in a reconstructed state. Our algorithms motivate a simple randomized tomography protocol, for which we find that using more measurement axes can yield substantial benefits that plateau after r≈3dr\approx3dr≈3d.
Abstract We present an iterative method to solve the multipartite quantum state estimation problem. We demonstrate convergence for any informationally complete set of generalized quantum measurements in every finite dimension. Our method exhibits fast convergence in high dimensions and strong robustness under the presence of realistic errors both in state preparation and measurement stages. In particular, for mutually unbiased bases and tensor product of generalized Pauli observables it converges in a single iteration.
With nowadays steadily growing quantum processors, it is required to develop new quantum tomography tools that are tailored for high-dimensional systems. In this work, we describe such a computational tool, based on recent ideas from non-convex optimization. The algorithm excels in the compressed-sensing-like setting, where only a few data points are measured from a low-rank or highly-pure quantum state of a high-dimensional system. We show that the algorithm can practically be used in quantum tomography problems that are beyond the reach of convex solvers, and, moreover, is faster than other state-of-the-art non-convex approaches. Crucially, we prove that, despite being a non-convex program, under mild conditions, the algorithm is guaranteed to converge to the global minimum of the problem; thus, it constitutes a provable quantum state tomography protocol.
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.
It has been recently shown that a state generated by a one-dimensional noisy quantum computer is well approximated by a matrix product operator with a finite bond dimension independent of the number of qubits. We show that full quantum state tomography can be performed for such a state with a minimal number of measurement settings using a method known as tensor train cross approximation. The method works for reconstructing full rank density matrices and only requires measuring local operators, which are routinely performed in state-of-art experimental quantum platforms. Our method requires exponentially fewer state copies than the best known tomography method for unstructured states and local measurements. The fidelity of our reconstructed state can be further improved via supervised machine learning, without demanding more experimental data. Scalable tomography is achieved if the full state can be reconstructed from local reductions.