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arXiv:1011.6443

Economical standard quantum process tomography

Xiaohua Wu, Ke Xu

Published Nov 30, 2010• 0 citations
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Abstract

Recently, Bendersky \emph{et al.} developed a method to complete the task of characterizing an arbitrary $χ$ matrix element in a scalable way, Phys. Rev. Lett. Vol. \textbf{100}, 190403(2008), where an auxiliary system was needed. In present work, we shall show that the same task can also be completed within the scheme of standard quantum process tomography (SQPT) where there is no requirement for ancilla. Our method depends on two observations: With the elaborately chosen operators basis, the SQPT may have an economical form where a single run of experiment, in which we measure the expectation value of a chosen operator in the outport of the quantum channel with a known input, is sufficient to characterize a selected $χ$ matrix element; With the progress recently achieved in quantum entanglement detection, we also find that the number of the experimental settings to realize the experiment for the selected $χ$ matrix element does not exceed 2N for the N-qubits system. For practice, our scheme can be applied for the cases where the controlled two-body interaction is neither available nor desirable.

DOI
10.48550/arxiv.1011.6443

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Recently, Bendersky \emph{et al.} developed a method to complete the task of characterizing an arbitrary χχχ matrix element in a scalable way, Phys. Rev. Lett. Vol. \textbf{100}, 190403(2008), where an auxiliary system was needed. In present work, we shall show that the same task can also be completed within the scheme of standard quantum process tomography (SQPT) where there is no requirement for ancilla. Our method depends on two observations: With the elaborately chosen operators basis, the SQPT may have an economical form where a single run of experiment, in which we measure the expectation value of a chosen operator in the outport of the quantum channel with a known input, is sufficient to characterize a selected χχχ matrix element; With the progress recently achieved in quantum entanglement detection, we also find that the number of the experimental settings to realize the experiment for the selected χχχ matrix element does not exceed 2N for the N-qubits system. For practice, our scheme can be applied for the cases where the controlled two-body interaction is neither available nor desirable.
Characterization of quantum dynamics is a fundamental problem in quantum physics and quantum-information science. Several methods are known which achieve this goal, namely standard quantum-process tomography (SQPT), ancilla-assisted process tomography, and the recently proposed scheme of direct characterization of quantum dynamics (DCQD). Here, we review these schemes and analyze them with respect to some of the physical resources they require. Although a reliable figure-of-merit for process characterization is not yet available, our analysis can provide a benchmark which is necessary for choosing the scheme that is the most appropriate in a given situation, with given resources. As a result, we conclude that for quantum systems where two-body interactions are not naturally available, SQPT is the most efficient scheme. However, for quantum systems with controllable two-body interactions, the DCQD scheme is more efficient than other known quantum-process tomography schemes in terms of the total number of required elementary quantum operations.
In this paper we describe in detail and generalize a method for quantum process tomography that was presented by Bendersky et al. [Phys. Rev. Lett. 100, 190403 (2008)]. The method enables the efficient estimation of any element of the $\ensuremath{\chi}$ matrix of a quantum process. Such elements are estimated as averages over experimental outcomes with a precision that is fixed by the number of repetitions of the experiment. Resources required implementing it scale polynomially with the number of qubits of the system. The estimation of all diagonal elements of the $\ensuremath{\chi}$ matrix can be efficiently done without any ancillary qubits. In turn, the estimation of all the off-diagonal elements requires an extra clean qubit. The key ideas of the method, which is based on efficient estimation by random sampling over a set of states forming a 2-design, are described in detail. Efficient methods for preparing and detecting such states are explicitly shown.
The standard procedure for quantum process tomography (QPT) requires a series of experiments. Each experiment involves initialization of the system to a particular basis state, applying the quantum process $\ensuremath{\varepsilon}$ on the system, and finally characterizing the output state by quantum state tomography (QST). The output states collected for a complete set of basis states enable us to calculate the $\ensuremath{\chi}$ matrix characterizing the process $\ensuremath{\varepsilon}$. The standard procedure for QST itself requires independent experiments, each involving measurement of a set of commuting observables. Thus QPT procedure demands a number of independent measurements and, moreover, this number increases rapidly with the size of the system. However, in ensemble systems, the total number of independent measurements can be greatly reduced with the availability of ancilla qubits. Ancilla-assisted process tomography (AAPT) has earlier been shown to require a single QST of system-ancilla space. Ancilla-assisted quantum state tomography (AAQST) has also been shown to perform QST in a single-scan measurement of an ensemble system. Here we combine AAPT with AAQST to realize a single-scan QPT (SSPT), a procedure to characterize a general quantum process in a single ensemble measurement. We demonstrate experimental SSPT by characterizing several single-qubit processes using a three-qubit NMR quantum register. Furthermore, using the SSPT procedure, we experimentally characterize the twirling process and compare the results with theory.
The characterization of the evolution of a quantum system is one of the main tasks to accomplish to achieve quantum information processing. The standard quantum process tomography (SQPT) has the unique property that it can be applied without introducing any additional quantum resources. In present work, we shall focus on the following two topics about the SQPT. At first, in the SQPT protocol for a ddd-dimensional system, one should encounter a problem in solving of a set of d4d^4d4 linear equations in order to get the matrix containing the complete information about the unknown quantum channel. Until now, the general form of the solution is unknown. And a long existed conviction is that the solutions are not unique. Here, we shall develop a self-consistent scheme, in which bounded linear operators are presented by vectors, to construct the set of linear equations. With the famous Cramer's rule for the set of linear equations, we are able to give the general form of the solution and prove that it is unique. In second, the central idea of the SQPT is to prepare a set of linearly independent inputs and measuring the outputs via the quantum state tomography (QST). Letting the inputs and the measurements be prepared by two sets of the rank-one positive-operator-valued measures [POVMs], where each POVM is supposed to be linearly independent and informationally complete (IC), we observe that SQPT now is equivalent to deciding a unknown state with a set of product IC-POVM in the d2d^2d2-dimensional Hilbert space. Following the general linear state tomography theory, we show that the product symmetric IC-POVM should minimize the mean-square Hilbert-Schmidt distance between the estimator and the true states. So, an optimal SQPT can be realized by preparing both the inputs and the measurements as the symmetric IC-POVM.
With the general theorem of SQPT, we shall develop a scheme to detemine an arbitrary matrix element of chichichi, which is expanded with the Choi operators, in a scalbe way.
The characterization of quantum processes is a key tool in quantum information processing tasks for several reasons: on one hand, it allows one to acknowledge errors in the implementations of quantum algorithms; on the other, it allows one to characterize unknown processes occurring in nature. Bendersky, Pastawski, and Paz [A. Bendersky, F. Pastawski, and J. P. Paz, Phys. Rev. Lett. 100, 190403 (2008); Phys. Rev. A 80, 032116 (2009)] introduced a method to selectively and efficiently measure any given coefficient from the matrix description of a quantum channel. However, this method heavily relies on the construction of maximal sets of mutually unbiased bases (MUBs), which are known to exist only when the dimension of the Hilbert space is the power of a prime number. In this article, we lift the requirement on the dimension by presenting two variations of the method that work on arbitrary finite dimensions: one uses tensor products of maximal sets of MUBs, and the other uses a dimensional cutoff of a higher prime power dimension.