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arXiv:1207.0358 · Physical Review Letters

Scalable Reconstruction of Density Matrices

Tillmann Baumgratz, D. Gross, M. Cramer, Martin Bodo Plenio

Published Jul 11, 2013• 119 citations
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Abstract

Recent contributions in the field of quantum state tomography have shown that, despite the exponential growth of Hilbert space with the number of subsystems, tomography of one-dimensional quantum systems may still be performed efficiently by tailored reconstruction schemes. Here, we discuss a scalable method to reconstruct mixed states that are well approximated by matrix product operators. The reconstruction scheme only requires local information about the state, giving rise to a reconstruction technique that is scalable in the system size. It is based on a constructive proof that generic matrix product operators are fully determined by their local reductions. We discuss applications of this scheme for simulated data and experimental data obtained in an ion trap experiment.

DOI
10.1103/physrevlett.111.020401

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Recent contributions in the field of quantum state tomography have shown that, despite the exponential growth of Hilbert space with the number of subsystems, tomography of one-dimensional quantum systems may still be performed efficiently by tailored reconstruction schemes. Here, we discuss a scalable method to reconstruct mixed states that are well approximated by matrix product operators. The reconstruction scheme only requires local information about the state, giving rise to a reconstruction technique that is scalable in the system size. It is based on a constructive proof that generic matrix product operators are fully determined by their local reductions. We discuss applications of this scheme for simulated data and experimental data obtained in an ion trap experiment.
Based on recently introduced efficient quantum state tomography schemes, we propose a scalable method for the tomography of unitary processes and the reconstruction of one-dimensional local Hamiltonians. As opposed to the exponential scaling with the number of subsystems of standard quantum process tomography, the method relies only on measurements of linearly many local observables and either (a) the ability to prepare eigenstates of locally informationally complete operators or (b) access to an ancilla of the same size as the to-be-characterized system and the ability to prepare a maximally entangled state on the combined system. As such, the method requires at most linearly many states to be prepared and linearly many observables to be measured. The quality of the reconstruction can be quantified with the same experimental resources that are required to obtain the reconstruction in the first place. Our numerical simulations of several quantum circuits and local Hamiltonians suggest a polynomial scaling of the total number of measurements and post-processing resources.
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.
The principle of maximum likelihood reconstruction has proven to yield satisfactory results in the context of quantum state tomography for many-body systems of moderate system sizes. Until recently, however, quantum state tomography has been considered to be infeasible for systems consisting of a large number of subsystems due to the exponential growth of the Hilbert space dimension with the number of constituents. Several reconstruction schemes have been proposed since then to overcome the two main obstacles in quantum many-body tomography: experiment time and post-processing resources. Here we discuss one strategy to address these limitations for the maximum likelihood principle by adopting a particular state representation to merge a well established reconstruction algorithm maximizing the likelihood with techniques known from quantum many-body theory.
Quantum state tomography is a key technology for fully determining a quantum state. Unfortunately, standard quantum state tomography is intractable for general many-body quantum states, because the number of measurements and the post-processing time increase exponentially with the size of the system. However, for the matrix product states (MPSs), there exists an efficient method using linearly scaled local measurements and polynomially scaled post-processing times. In this study, we demonstrate the validity of the method in practice by reconstructing a four-photon MPS from its local two- or three-photon reduced-density matrices with the presence of statistical errors and systematical errors in experiment.
The reconstruction of quantum states from experimental measurements, often achieved using quantum state tomography (QST), is crucial for the verification and benchmarking of quantum devices. However, performing QST for a generic unstructured quantum state requires an enormous number of state copies that growsexponentiallywith the number of individual quanta in the system, even for the most optimal measurement settings. Fortunately, many physical quantum states, such as states generated by noisy, intermediate-scale quantum computers, are usually structured. In one dimension, such states are expected to be well approximated by matrix product operators (MPOs) with a matrix/bond dimension independent of the number of qubits, therefore enabling efficient state representation. Nevertheless, it is still unclear whether efficient QST can be performed for these states in general. In other words, there exist no rigorous bounds on the number of state copies required for reconstructing MPO states that scales polynomially with the number of qubits. In this paper, we attempt to bridge this gap and establish theoretical guarantees for the stable recovery of MPOs using tools from compressive sensing and the theory of empirical processes. We begin by studying two types of random measurement settings: Gaussian measurements and Haar random projective measurements. We show that the information contained in an MPO with a constant bond dimension can be preserved using a number of random measurements that depends onlylinearlyon the number of qubits, assuming no statistical error of the measurements. We then study MPO-based QST with Haar random projective measurements that can in principle be implemented on quantum computers. We prove that only apolynomialnumber of state copies in the number of qubits is required to guarantee bounded recovery error of an MPO state. Remarkably, such recovery can be achieved by measuring the state in each random basis only once, despite the large statistical error associated with the outcome of each measurement. Our work may be generalized to accommodate random local or t-design measurements that are more practical to implement on current quantum computers. It may also facilitate the discovery of efficient QST methods for other structured quantum states.
Quantum state tomography is a key technique for quantum information processing but is challenging due to the exponential growth of its complexity with the system size. In this work we propose an algorithm which iteratively finds the best non-negative matrix product state approximation based on a set of measurement outcomes whose size does not necessarily grow exponentially. Compared to the tomography method based on neural network states, our scheme utilizes a so-called tensor train representation that allows straightforward recovery of the unknown density matrix in the matrix product operator form. As applications, the effectiveness of our algorithm is numerically demonstrated to reconstruct the ground state of the XXZ spin chain under depolarizing noise.