Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE International Conference on Robotics and Biomimetics (ROBIO)

An improved quantum state estimation algorithm via compressive sensing

Shuang Cong, H. Zhang, K. Li

Published Dec 1, 2014• 1 citation
Publisher

Abstract

The dimension of the density matrix of a quantum system increases with the qubits of the quantum system, which makes the quantum state estimation time consuming and requires huge computation. In order to reduce the computational time in quantum state estimation, the problem of quantum state estimation based on compressive sensing is changed to the optimization problem with error constraint. In this paper, an improved Alternating Direction Method of Multipliers (ADMM) algorithm is proposed to design the optimization scheme of solving the pure state of quantum state estimation in the cases of with and without external noise. The experiments are implemented in the MATLAB. The comparison results between adaptive and fixed weight value indicate that the improved algorithm has better performances in both aspects of estimation accuracy and robustness to external disturbances. We also extend the quantum state estimation to the qubits of six and seven.

DOI
10.1109/robio.2014.7090687

Related papers

RECONSTRUCTING PURE 14-QUBIT QUANTUM STATES IN THREE HOURS USING COMPRESSIVE SENSING

Zhilin Hu, Kezhi Li, Shuang Cong +1 more

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%.

IFAC-PapersOnLine•Jan 1, 2019•4 citations

Fast algorithm of high-dimensional quantum state estimation via low measurement rates

Jiaojiao Zhang, Kezhi Li, Shuang Cong

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.

Asian Control Conference (ASCC)•Dec 1, 2017•2 citations

A Robust Compressive Quantum State Tomography Algorithm Using ADMM

Kezhi Li, Shuang Cong

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.

arXiv:1401.6533•Jan 25, 2014•0 citations

An Efficient and Fast Quantum State Estimator With Sparse Disturbance

Jiaojiao Zhang, Shuang Cong, Qing Hua Ling +1 more

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.

IEEE Transactions on Cybernetics•May 4, 2018•19 citations

A reconstruction algorithm for compressive quantum tomography using various measurement sets

Zheng Kai, Kezhi Li, Shuang Cong

Abstract Compressed sensing (CS) has been verified that it offers a significant performance improvement for large quantum systems comparing with the conventional quantum tomography approaches, because it reduces the number of measurements from O ( d 2 ) to O ( rd log( d )) in particular for quantum states that are fairly pure. Yet few algorithms have been proposed for quantum state tomography using CS specifically, let alone basis analysis for various measurement sets in quantum CS. To fill this gap, in this paper an efficient and robust state reconstruction algorithm based on compressive sensing is developed. By leveraging the fixed point equation approach to avoid the matrix inverse operation, we propose a fixed-point alternating direction method algorithm for compressive quantum state estimation that can handle both normal errors and large outliers in the optimization process. In addition, properties of five practical measurement bases (including the Pauli basis) are analyzed in terms of their coherences and reconstruction performances, which provides theoretical instructions for the selection of measurement settings in the quantum state estimation. The numerical experiments show that the proposed algorithm has much less calculating time, higher reconstruction accuracy and is more robust to outlier noises than many existing state reconstruction algorithms.

Scientific Reports•Dec 14, 2016•26 citations

Efficient and Fast Optimization Algorithms for Quantum State Filtering and Estimation

Kun Zhang, Shuang Cong, Jiao Ding +2 more

In this paper, based on Alternating Direction Multiplier Method (ADMM) and Compressed Sensing (CS), we develop three types of novel convex optimization algorithms for the quantum state estimation and filtering. Considering sparse state disturbance and measurement noise simultaneously, we propose a quantum state filtering algorithm. At the same time, the quantum state estimation algorithms for either sparse state disturbance or measurement noise are proposed, respectively. Contrast with other algorithms in literature, simulation experiments verify that all three algorithms have low computational complexity, fast convergence speed and high estimation accuracy at lower measurement rates.

International Conference on Intelligent Control and Information Processing•Dec 1, 2019•2 citations
The dimension of the density matrix of a quantum system increases with the qubits of the quantum system, which makes the quantum state estimation time consuming and requires huge computation. In order to reduce the computational time in quantum state estimation, the problem of quantum state estimation based on compressive sensing is changed to the optimization problem with error constraint. In this paper, an improved Alternating Direction Method of Multipliers (ADMM) algorithm is proposed to design the optimization scheme of solving the pure state of quantum state estimation in the cases of with and without external noise. The experiments are implemented in the MATLAB. The comparison results between adaptive and fixed weight value indicate that the improved algorithm has better performances in both aspects of estimation accuracy and robustness to external disturbances. We also extend the quantum state estimation to the qubits of six and seven.
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.
Abstract Compressed sensing (CS) has been verified that it offers a significant performance improvement for large quantum systems comparing with the conventional quantum tomography approaches, because it reduces the number of measurements from O ( d 2 ) to O ( rd log( d )) in particular for quantum states that are fairly pure. Yet few algorithms have been proposed for quantum state tomography using CS specifically, let alone basis analysis for various measurement sets in quantum CS. To fill this gap, in this paper an efficient and robust state reconstruction algorithm based on compressive sensing is developed. By leveraging the fixed point equation approach to avoid the matrix inverse operation, we propose a fixed-point alternating direction method algorithm for compressive quantum state estimation that can handle both normal errors and large outliers in the optimization process. In addition, properties of five practical measurement bases (including the Pauli basis) are analyzed in terms of their coherences and reconstruction performances, which provides theoretical instructions for the selection of measurement settings in the quantum state estimation. The numerical experiments show that the proposed algorithm has much less calculating time, higher reconstruction accuracy and is more robust to outlier noises than many existing state reconstruction algorithms.
In this paper, based on Alternating Direction Multiplier Method (ADMM) and Compressed Sensing (CS), we develop three types of novel convex optimization algorithms for the quantum state estimation and filtering. Considering sparse state disturbance and measurement noise simultaneously, we propose a quantum state filtering algorithm. At the same time, the quantum state estimation algorithms for either sparse state disturbance or measurement noise are proposed, respectively. Contrast with other algorithms in literature, simulation experiments verify that all three algorithms have low computational complexity, fast convergence speed and high estimation accuracy at lower measurement rates.