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arXiv:0812.4323

Quantum Process Tomography via L1-norm Minimization

Robert L. Kosut

Published Dec 23, 2008• 13 citations
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Abstract

For an initially well designed but imperfect quantum information system, the process matrix is almost sparse in an appropriate basis. Existing theory and associated computational methods (L1-norm minimization) for reconstructing sparse signals establish conditions under which the sparse signal can be perfectly reconstructed from a very limited number of measurements (resources). Although a direct extension to quantum process tomography of the L1-norm minimization theory has not yet emerged, the numerical examples presented here, which apply L1-norm minimization to quantum process tomography, show a significant reduction in resources to achieve a desired estimation accuracy over existing methods.

DOI
10.48550/arxiv.0812.4323

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For an initially well designed but imperfect quantum information system, the process matrix is almost sparse in an appropriate basis. Existing theory and associated computational methods (L1-norm minimization) for reconstructing sparse signals establish conditions under which the sparse signal can be perfectly reconstructed from a very limited number of measurements (resources). Although a direct extension to quantum process tomography of the L1-norm minimization theory has not yet emerged, the numerical examples presented here, which apply L1-norm minimization to quantum process tomography, show a significant reduction in resources to achieve a desired estimation accuracy over existing methods.
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.
Quantum process tomography is the task of reconstructing unknown quantum channels from measured data. In this work, we introduce compressed sensing-based methods that facilitate the reconstruction of quantum channels of low Kraus rank. Our main contribution is the analysis of a natural measurement model for this task: We assume that data is obtained by sending pure states into the channel and measuring expectation values on the output. Neither ancillary systems nor coherent operations across multiple channel uses are required. Most previous results on compressed process reconstruction reduce the problem to quantum state tomography on the channel's Choi matrix. While this ansatz yields recovery guarantees from an essentially minimal number of measurements, physical implementations of such schemes would typically involve ancillary systems. A priori, it is unclear whether a measurement model tailored directly to quantum process tomography might require more measurements. We establish that this is not the case.Technically, we prove recovery guarantees for three different reconstruction algorithms. The reconstructions are based on a trace, diamond, and ℓ2 -norm minimization, respectively. Our recovery guarantees are uniform in the sense that with one random choice of measurement settings all quantum channels can be recovered equally well. Moreover, stability against arbitrary measurement noise and robustness against violations of the low-rank assumption is guaranteed. Numerical studies demonstrate the feasibility of the approach.
Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol has been carried out with success to characterize the evolution of a quantum system that is well described by a trace-preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantum state, which naturally arises in the presence of imperfect devices and system-environment interactions, in the context of quantum information science or quantum dynamics control. In that case, we show with the aid of a priori information on the losses structure of the quantum channel that the SEQPT reconstruction can be adapted to reconstruct the non-trace-preserving map. We explicitly describe how to implement the reconstruction in an arbitrary Hilbert space of finite dimension ddd. The method is experimentally verified on a superconducting quantum processor provided by IBM Quantum services, by estimating several non-trace-preserving quantum processes in dimensions up to d=6d=6d=6. Our results show that it is possible to efficiently reconstruct non-trace-preserving processes, with high precision, and with significantly higher fidelity than when the process is assumed to be trace preserving.
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 study quantum process tomography given the prior information that the map is a unitary or close to a unitary process. We show that a unitary map on a ddd-level system is completely characterized by a minimal set of d2+d{d}^{2}+dd2+d elements associated with a collection of POVMs, in contrast to the ${d}^{4}\ensuremath{-}{d}^{2}$ elements required for a general completely positive trace-preserving map. To achieve this lower bound, one must probe the map with particular sets of ddd pure states. We further compare the performance of different estimators inspired by compressed sensing, to reconstruct a near-unitary process from such data. We find that when we have accurate prior information, an appropriate compressed sensing method reduces the required data needed for high-fidelity estimation, and different estimators applied to the same data are sensitive to different types of noise. Convex optimization inspired by the techniques of compressed sensing can therefore be used both as indicators of error models and to validate the use of the prior assumptions.
Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol has been carried out with success to characterize the evolution of a quantum system that is well described by a trace preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantum state, which naturally arises in the presence of imperfect devices and system-environment interactions, in the context of quantum information science or quantum dynamics control. In that case, we show that with the aid of {\it a priori} information on the losses structure of the quantum channel, the SEQPT reconstruction can be adapted to reconstruct the non-trace-preserving map. We explicitly describe how to implement the reconstruction in an arbitrary Hilbert space of finite dimension ddd. The method is experimentally verified on a superconducting quantum processor of the IBM Quantum services, by estimating several non trace-preserving quantum processes in dimensions up to d=6d=6d=6. Our results show that it is possible to efficiently reconstruct non trace-preserving processes, with high precision, and with significantly higher fidelity than when the process is assumed to be trace-preserving.