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arXiv:1911.06122 · Physical Review A

Objective compressive quantum process tomography

Yong Siah Teo, G. I. Struchalin, E. V. Kovlakov, D. Ahn, Hyunseok Jeong, Stanislav S. Straupe, S. P. Kulik, Gerd Leuchs, Luis Lorenzo Sánchez-Soto

Published Feb 26, 2020• 29 citations
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Abstract

We present a compressive quantum process tomography scheme that fully characterizes any rank-deficient completely positive process with no spurious a priori information. It uses randomly chosen input states and adaptive output von Neumann measurements. Both entangled and tensor-product configurations are flexibly employable in our scheme, the latter of which are especially compatible with many-body quantum computing. Two main features of this scheme are the certification protocol that verifies whether the accumulated data uniquely characterize the quantum process and a compressive reconstruction method for the output states. We emulate multipartite scenarios with high-order transverse modes and optical fibers to demonstrate that, in terms of measurement resources, our assumption-free compressive strategy can reconstruct quantum processes almost equally efficiently using all types of input states and basis measurements.

DOI
10.1103/physreva.101.022334

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We present a compressive quantum process tomography scheme that fully characterizes any rank-deficient completely positive process with no spurious a priori information. It uses randomly chosen input states and adaptive output von Neumann measurements. Both entangled and tensor-product configurations are flexibly employable in our scheme, the latter of which are especially compatible with many-body quantum computing. Two main features of this scheme are the certification protocol that verifies whether the accumulated data uniquely characterize the quantum process and a compressive reconstruction method for the output states. We emulate multipartite scenarios with high-order transverse modes and optical fibers to demonstrate that, in terms of measurement resources, our assumption-free compressive strategy can reconstruct quantum processes almost equally efficiently using all types of input states and basis measurements.
Quantum state tomography is both a crucial component in the field of quantum information and computation and a formidable task that requires an incogitable number of measurement configurations as the system dimension grows. We propose and experimentally carry out an intuitive adaptive compressive tomography scheme, inspired by the traditional compressed-sensing protocol in signal recovery, that tremendously reduces the number of configurations needed to uniquely reconstruct any given quantum state without any additional a priori assumption whatsoever (such as rank information, purity, etc.) about the state, apart from its dimension.
Quantum process tomography is the task of reconstructing unknown quantum channels from measured data. In this work, we introduce compressed sensing-based methods that facilitate the reconstruction of quantum channels of low Kraus rank. Our main contribution is the analysis of a natural measurement model for this task: We assume that data is obtained by sending pure states into the channel and measuring expectation values on the output. Neither ancillary systems nor coherent operations across multiple channel uses are required. Most previous results on compressed process reconstruction reduce the problem to quantum state tomography on the channel's Choi matrix. While this ansatz yields recovery guarantees from an essentially minimal number of measurements, physical implementations of such schemes would typically involve ancillary systems. A priori, it is unclear whether a measurement model tailored directly to quantum process tomography might require more measurements. We establish that this is not the case.Technically, we prove recovery guarantees for three different reconstruction algorithms. The reconstructions are based on a trace, diamond, and ℓ2 -norm minimization, respectively. Our recovery guarantees are uniform in the sense that with one random choice of measurement settings all quantum channels can be recovered equally well. Moreover, stability against arbitrary measurement noise and robustness against violations of the low-rank assumption is guaranteed. Numerical studies demonstrate the feasibility of the approach.
In the light of the progress in quantum technologies, the task of verifying the correct functioning of processes and obtaining accurate tomographic information about quantum states becomes increasingly important. Compressed sensing, a machinery derived from the theory of signal processing, has emerged as a feasible tool to perform robust and significantly more resource-economical quantum state tomography for intermediate-sized quantum systems. In this work, we provide a comprehensive analysis of compressed sensing tomography in the regime in which tomographically complete data is available with reliable statistics from experimental observations of a multi-mode photonic architecture. Due to the fact that the data is known with high statistical significance, we are in a position to systematically explore the quality of reconstruction depending on the number of employed measurement settings, randomly selected from the complete set of data, and on different model assumptions. We present and test a complete prescription to perform efficient compressed sensing and are able to reliably use notions of model selection and cross validation to account for experimental imperfections and finite counting statistics. Thus, we establish compressed sensing as an effective tool for quantum state tomography, specifically suited for photonic systems.
This review serves as a concise introductory survey of modern compressive tomography developed since 2019. These are schemes meant for characterizing arbitrary low-rank quantum objects, be it an unknown state, a process or detector, using minimal measuring resources (hence compressive) without any a priori assumptions (rank, sparsity, eigenbasis, etc.) about the quantum object. This paper contains a reasonable amount of technical details for the quantum-information community to start applying the methods discussed here. To facilitate the understanding of formulation logic and physics of compressive tomography, the theoretical concepts and important numerical results (both new and cross-referenced) shall be presented in a pedagogical manner.
High-dimensional optical modes usually require a large number of measurements to be characterized— typically of the order O ( d 2 ) where d is the dimension of the underlying Hilbert space. Recently developed compressive tomography techniques can reconstruct low-rank signals with minimum measurements and no a priori assumptions [1] , [2] , [3] , [4] . We discuss results of randomized compressive tomography (RCT) implemented with a quantum pulse gate (QPG), a device that performs flexible optical measurements in the time-frequency domain [5] , [6] . In our experiments, we perform different random von Neumann basis measurements with a QPG that exploits dispersion-engineered sum-frequency generation in lithium niobate waveguides to implement projections onto arbitrary user-specified temporal mode superpositions. Based solely on the measurement data and nothing else, one can deterministically compute a positive number s cvx that monotonically estimates the size of the convex set of signal states consistent with the measurement data. For a sufficient number of informationally complete (IC) QPG projections, the convex set converges to a single point ( s cvx = 0), where the signal is uniquely reconstructed.