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arXiv:1701.03695 · Physical Review A

Fast reconstruction of high-qubit-number quantum states via low-rate measurements

Kai Li, Jiaojiao Zhang, Shuang Cong

Published Jul 27, 2017• 10 citations
arXivHTMLPDF

Abstract

Due to the exponential complexity of the resources required by quantum state tomography (QST), people are interested in approaches towards identifying quantum states which require less effort and time. In this paper, we provide a tailored and efficient method for reconstructing mixed quantum states up to 12 (or even more) qubits from an incomplete set of observables subject to noises. Our method is applicable to any pure or nearly pure state $\ensuremath{\rho}$ and can be extended to many states of interest in quantum information processing, such as a multiparticle entangled $W$ state, Greenberger-Horne-Zeilinger states, and cluster states that are matrix product operators of low dimensions. The method applies the quantum density matrix constraints to a quantum compressive sensing optimization problem and exploits a modified quantum alternating direction multiplier method (quantum-ADMM) to accelerate the convergence. Our algorithm takes $8,\phantom{\rule{0.16em}{0ex}}35$, and 226 seconds, respectively, to reconstruct superposition state density matrices of $10,11,\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}12$ qubits with acceptable fidelity using less than $1%$ of measurements of expectation. To our knowledge it is the fastest realization that people can achieve using a normal desktop. We further discuss applications of this method using experimental data of mixed states obtained in an ion trap experiment of up to 8 qubits.

DOI
10.1103/physreva.96.012334

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Due to the exponential complexity of the resources required by quantum state tomography (QST), people are interested in approaches towards identifying quantum states which require less effort and time. In this paper, we provide a tailored and efficient method for reconstructing mixed quantum states up to 12 (or even more) qubits from an incomplete set of observables subject to noises. Our method is applicable to any pure or nearly pure state \ensuremathρ\ensuremath{\rho}\ensuremathρ and can be extended to many states of interest in quantum information processing, such as a multiparticle entangled WWW state, Greenberger-Horne-Zeilinger states, and cluster states that are matrix product operators of low dimensions. The method applies the quantum density matrix constraints to a quantum compressive sensing optimization problem and exploits a modified quantum alternating direction multiplier method (quantum-ADMM) to accelerate the convergence. Our algorithm takes 8,358,\phantom{\rule{0.16em}{0ex}}358,35, and 226 seconds, respectively, to reconstruct superposition state density matrices of 10,11,and1210,11,\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}1210,11,and12 qubits with acceptable fidelity using less than 11%1 of measurements of expectation. To our knowledge it is the fastest realization that people can achieve using a normal desktop. We further discuss applications of this method using experimental data of mixed states obtained in an ion trap experiment of up to 8 qubits.
Reconstructing high-dimensional quantum state accurately with noise is a challenging problem, because of the probabilistic measurements of quantum states and the exponential growth of the computation in terms of the number of qubits. In this paper, an improved Alternating Direction Multiplier Method (ADMM) is proposed for the quantum state estimation based on the lower limit of the measurement rate inferred by Compressive Sensing (CS) theory. The proposed algorithm solves the density matrix correlation subproblems by introducing a proximal gradient step to avoid large-scale matrix inversion. Furthermore, it reduces the computational complexity by changing the order of operations. The algorithm we proposed can dramatically reduces the reconstruction time on the premise of achieving high reconstruction accuracy. When the number of qubits n = 14 and sampling rate η = 0.0819%, it takes about 3 hours to reconstruct the density matrix of a pure quantum state with the reconstruction error of 4.97e-2 and fidelity of 98.79%.
Based on the theoretical lower bounds of the number of measurements given by Compressive Sensing (CS), the quantum state estimation task can be formulated as a convex optimization problem subject to positive semi-definite and unit-trace Hermitian constraints. This paper proposes an improved Alternating Direction Multiplier Method (AD-MM) for high-dimensional quantum state reconstruction. The proposed method not only greatly reduces the computational complexity and speeds up the convergence, but also has a strong robustness to the noise. For the density matrix reconstruction of 11-qubit quantum system, the proposed algorithm achieves 4.54e-2 reconstruction error using 0.6% of total measurement data and 99.26 seconds.
The possible state space dimension increases exponentially with respect to the number of qubits. This feature makes the quantum state tomography expensive and impractical for identifying the state of merely several qubits. The recent developed approach, compressed sensing, gives us an alternative to estimate the quantum state with fewer measurements. It is proved that the estimation then can be converted to a convex optimization problem with quantum mechanics constraints. In this paper we present an alternating augmented Lagrangian method for quantum convex optimization problem aiming for recovering pure or near pure quantum states corrupted by sparse noise given observables and the expectation values of the measurements. The proposed algorithm is much faster, robust to outlier noises (even very large for some entries) and can solve the reconstruction problem distributively. The simulations verify the superiority of the proposed algorithm and compare it to the conventional least square and compressive quantum tomography using Dantzig method.
A pure or nearly pure quantum state can be described as a low-rank density matrix, which is a positive semidefinite and unit-trace Hermitian. We consider the problem of recovering such a low-rank density matrix contaminated by sparse components, from a small set of linear measurements. This quantum state estimation task can be formulated as a robust principal component analysis (RPCA) problem subject to positive semidefinite and unit-trace Hermitian constraints. We propose an efficient and fast inexact alternating direction method of multipliers (I-ADMM), in which the subproblems are solved inexactly and hence have closed-form solutions. We prove global convergence of the proposed I-ADMM, and the theoretical result provides a guideline for parameter setting. Numerical experiments show that the proposed I-ADMM can recover state density matrices of 5 qubits on a laptop in 0.69 s, with 6 × 10-4accuracy (99.38% fidelity) using 30% compressive sensing measurements, which outperforms existing algorithms.
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.
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.