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arXiv:0910.5498 · Physical Review Letters

Efficient Measurement of Quantum Dynamics via Compressive Sensing

Alireza Shabani, Robert L. Kosut, Masoud Mohseni, HERSCHEL A. RABITZ, Matthew A. Broome, Marcelo Pereira de Almeida, Alessandro Fedrizzi, Andrew G. White

Published Mar 7, 2011• 227 citations
arXivHTMLPDF

Abstract

The resources required to characterize the dynamics of engineered quantum systems--such as quantum computers and quantum sensors--grow exponentially with system size. Here we adapt techniques from compressive sensing to exponentially reduce the experimental configurations required for quantum process tomography. Our method is applicable to processes that are nearly sparse in a certain basis and can be implemented using only single-body preparations and measurements. We perform efficient, high-fidelity estimation of process matrices of a photonic two-qubit logic gate. The database is obtained under various decoherence strengths. Our technique is both accurate and noise robust, thus removing a key roadblock to the development and scaling of quantum technologies.

DOI
10.1103/physrevlett.106.100401

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The resources required to characterize the dynamics of engineered quantum systems--such as quantum computers and quantum sensors--grow exponentially with system size. Here we adapt techniques from compressive sensing to exponentially reduce the experimental configurations required for quantum process tomography. Our method is applicable to processes that are nearly sparse in a certain basis and can be implemented using only single-body preparations and measurements. We perform efficient, high-fidelity estimation of process matrices of a photonic two-qubit logic gate. The database is obtained under various decoherence strengths. Our technique is both accurate and noise robust, thus removing a key roadblock to the development and scaling of quantum technologies.
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The characterization of a decoherence process is among the central challenges in quantum physics. A major difficulty with current quantum process tomography methods is the enormous number of experiments needed to accomplish a tomography task. Here we present a highly efficient method for tomography of a quantum process that has a small number of significant elements. Our method is based on the compressed sensing techniques being used in information theory. In this new method, for a system with Hilbert space dimension n and a process matrix of dimension n × n with sparsity s, the required number of experimental configurations is O(s log n). This heralds a logarithmic advantage in contrast to other methods of quantum process tomography. More specifically, for q-qubits with n = 2, the scaling of resources is O(sq) — linear in the product of sparsity and number of qubits.
We develop an efficient and robust approach for quantum measurement of nearly sparse many-body quantum Hamiltonians based on the method of compressive sensing. This work demonstrates that with only O(sln(d))O(s\mathrm{ln}(d))O(sln(d)) experimental configurations, consisting of random local preparations and measurements, one can estimate the Hamiltonian of a ddd-dimensional system, provided that the Hamiltonian is nearly sss sparse in a known basis. The classical postprocessing is a convex optimization problem on the total Hilbert space which is generally not scalable. We numerically simulate the performance of this algorithm for three- and four-body interactions in spin-coupled quantum dots and atoms in optical lattices. Furthermore, we apply the algorithm to characterize Hamiltonian fine structure and unknown system-bath interactions.
We present methods that can provide an exponential savings in the resources required to perform dynamic parameter estimation using quantum systems. The key idea is to merge classical compressive sensing techniques with quantum control methods to significantly reduce the number of signal coefficients that are required for reconstruction of time-varying parameters with high fidelity. We show that incoherent measurement bases and, more generally, suitable random measurement matrices can be created by performing simple control sequences on the quantum system. Random measurement matrices satisfying the restricted isometry property can be used efficiently to reconstruct signals that are sparse in any basis. Because many physical processes are approximately sparse in some basis, these methods can benefit a variety of applications such as quantum sensing and magnetometry with nitrogen-vacancy centers.
Quantum Process Tomography (QPT) is one of the most important task of checking the functionality of quantum information processing devices. However, there is a significant difficulty that the required resources grow scaling exponentially with the number of qubits during the QPT. Recently, a compressed sensing QPT (CS QPT) is proposed that can reduce the required resource significantly. But the work did not consider the outlier noises caused by the disturbance of the system or channel. In this paper, we propose a joint sparse coding QPT aiming for the reconstruction of the two or more similar quantum channel corrupted by sparse outlier noises simultaneously. The simulations verify that the method is robust to outlier noises and the obtained reconstruction matrix is superior to the QPT using compressed sensing method.
A fundamental difficulty in demonstrating quantum state tomography is that the required resources grow exponentially with the system size. For pure states and nearly pure states, the task of tomography can be more efficient. We proposed two methods for state reconstruction, by (1) minimizing entropy and (2) maximizing likelihood. The algorithm of compressed sampling is employed to solve the optimization problem. Experiments are demonstrated considering 4-qubit photonic states. The results show that (1) much fewer measurements than the standard tomography are sufficient to obtain high fidelity, and (2) the method of maximizing likelihood is more accurate and noise robust than the original reconstruction method of compressed sampling. Furthermore, the physical meaning of the methods of minimizing entropy and maximizing likelihood is clear.