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arXiv:1002.4632

Efficient Direct Tomography for Matrix Product States

Olivier Landon-Cardinal, Yi-Kai Liu, David Poulin

Published Feb 24, 2010• 1 citation
arXivHTMLPDF

Abstract

In this note, we describe a method for reconstructing matrix product states from a small number of efficiently-implementable measurements. Our method is exponentially faster than standard tomography, and it can also be used to certify that the unknown state is an MPS. The basic idea is to use local unitary operations to measure in the Schmidt basis, giving direct access to the MPS representation. This compares favorably with recently and independently proposed methods that recover the MPS tensors by performing a variational minimization, which is computationally intractable in certain cases. Our method also has the advantage of recovering any MPS, while other approaches were limited to special classes of states that exclude important examples such as GHZ and W states.

DOI
10.48550/arxiv.1002.4632

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In this note, we describe a method for reconstructing matrix product states from a small number of efficiently-implementable measurements. Our method is exponentially faster than standard tomography, and it can also be used to certify that the unknown state is an MPS. The basic idea is to use local unitary operations to measure in the Schmidt basis, giving direct access to the MPS representation. This compares favorably with recently and independently proposed methods that recover the MPS tensors by performing a variational minimization, which is computationally intractable in certain cases. Our method also has the advantage of recovering any MPS, while other approaches were limited to special classes of states that exclude important examples such as GHZ and W states.
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.
Quantum state tomography (QST) remains the gold standard for benchmarking and verification of near-term quantum devices. While QST for a generic quantum many-body state requires an exponentially large amount of resources, most physical quantum states are structured and can often be represented by a much smaller number of parameters, making efficient QST potentially possible. A prominent example is a matrix product state (MPS) or a matrix product density operator (MPDO), which is believed to represent most physical states generated by one-dimensional (1D) quantum devices. We study whether a general MPS/MPDO state can be recovered with bounded errors using only a number of state copies polynomial in the number of qubits, which is necessary for efficient QST. To make this question practically interesting, we assume only local measurements of qubits directly on the target state. By using a local symmetric informationally complete positive operator-valued measurement (SIC-POVM), we provide a positive answer to the above question for a variety of common many-body quantum states, including typical short-range entangled states, random MPS/MPDO states, and thermal states of one-dimensional Hamiltonians. In addition, we also provide an affirmative no answer for certain long-range entangled states such as a family of generalized GHZ states, but with the exception of target states that are known to have real-valued wavefunctions. Our answers are supported by a near-perfect agreement between an efficient calculation of the Cramer-Rao bound that rigorously bounds the sample complexity and numerical optimization results using a machine learning assisted maximal likelihood estimation (MLE) algorithm. This agreement also leads to an optimal QST protocol using local SIC-POVM that can be practically implemented on current quantum hardware and is highly efficient for most 1D physical states.
Quantum state tomography, the ability to deduce the density matrix of a quantum system from measured data, is of fundamental importance for the verification of present and future quantum devices. It has been realized in systems with few components but for larger systems it becomes rapidly infeasible because the number of quantum measurements and computational resources required to process them grow exponentially in the system size. Here we show that we can gain an exponential advantage over direct state tomography for quantum states typically realized in nature. Based on singular value thresholding and matrix product state methods we introduce a state reconstruction scheme that relies only on a linear number of measurements. The computational resources for the postprocessing required to reconstruct the state with high fidelity from these measurements is polynomial in the system size.
Quantum state tomography (QST) is a central technique to fully characterize an unknown quantum state. However, standard QST requires an exponentially growing number of quantum measurements against the system size, which limits its application to smaller systems. Here, we explore the sparsity of underlying quantum state and propose a QST scheme that combines the matrix product states’ representation of the quantum state with a supervised machine learning algorithm. Our method could reconstruct the unknown sparse quantum states with very high precision using only a portion of the measurement data in a randomly selected basis set. In particular, we demonstrate that the Wolfgang states could be faithfully reconstructed using around 25% of the whole basis, and that the randomly generated quantum states, which could be efficiently represented as matrix product states, could be faithfully reconstructed using a number of bases that scales sub-exponentially against the system size.
We present a novel method for performing quantum state tomography for many-particle systems, which are particularly suitable for estimating the states in lattice systems such as of ultra-cold atoms in optical lattices. We show that the need to measure a tomographically complete set of observables can be overcome by letting the state evolve under some suitably chosen random circuits followed by the measurement of a single observable. We generalize known results about the approximation of unitary two-designs, i.e. certain classes of random unitary matrices, by random quantum circuits and connect our findings to the theory of quantum compressed sensing. We show that for ultra-cold atoms in optical lattices established experimental techniques such as optical super-lattices, laser speckles and time-of-flight measurements are sufficient to perform fully certified, assumption-free tomography. This is possible without the need to address single sites in any step of the procedure. Combining our approach with tensor network methods—in particular, the theory of matrix product states—we identify situations where the effort of reconstruction is even constant in the number of lattice sites, allowing, in principle, to perform tomography on large-scale systems readily available in present experiments.
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.