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arXiv:1404.7830 · Physical Review A

Single-scan quantum process tomography

Abhishek Shukla, T. S. Mahesh

Published Nov 4, 2014• 23 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.90.052301

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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 \ensuremathε\ensuremath{\varepsilon}\ensuremathε 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 \ensuremathχ\ensuremath{\chi}\ensuremathχ matrix characterizing the process \ensuremathε\ensuremath{\varepsilon}\ensuremathε. 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 standard method of quantum state tomography (QST) relies on the measurement of a set of noncommuting observables, realized in a series of independent experiments. Ancilla-assisted QST (AAQST) proposed by Nieuwenhuizen and co-workers [Phys. Rev. Lett. 92, 120402 (2004)] greatly reduces the number of independent measurements by exploiting an ancilla register in a known initial state. In suitable conditions AAQST allows mapping out density matrix of an input register in a single experiment. Here we describe methods for explicit construction of AAQST experiments in multiqubit registers. We also report nuclear magnetic resonance studies on AAQST of (i) a two-qubit input register using a one-qubit ancilla in an isotropic liquid-state system and (ii) a three-qubit input register using a two-qubit ancilla register in a partially oriented system. The experimental results confirm the effectiveness of AAQST in such multiqubit registers.
The standard method of Quantum State Tomography (QST) relies on the measurement of a set of noncommuting observables, realized in a series of independent experiments. Ancilla Assisted QST (AAQST) proposed by Nieuwenhuizen and co-workers (Phys. Rev. Lett., 92, 120402 (2004)) greatly reduces the number of independent measurements by exploiting an ancilla register in a known initial state. In suitable conditions AAQST allows mapping out density matrix of an input register in a single experiment. Here we describe methods for explicit construction of AAQST experiments in multi-qubit registers. We also report nuclear magnetic resonance studies on AAQST of (i) a two- qubit input register using a one-qubit ancilla in an isotropic liquid-state system and (ii) a three-qubit input register using a two-qubit ancilla register in a partially oriented system. The experimental results confirm the effectiveness of AAQST in such many-qubit registers.
Quantum process tomography (QPT) is a fundamental task to characterize the dynamics of quantum systems. In contrast to standard QPT, ancilla-assisted process tomography (AAPT) framework introduces an extra ancilla system such that a single input state is needed. In this paper, we extend the two-stage solution, a method originally designed for standard QPT, to perform AAPT. Our algorithm hasO(MdA2dB2)O(Md_{A}^{2}d_{B}^{2})O(MdA2​dB2​)computational complexity whereMMMis the type number of the measurement operators,dAd_{A}dA​is the dimension of the quantum system of interest, anddBd_{B}dB​is the dimension of the ancilla system. Then we establish an error upper bound and further discuss the optimal design on the input state in AAPT. A numerical example on a phase damping process demonstrates the effectiveness of the optimal design and illustrates the theoretical error analysis.
The field of quantum information relies on the crucial issue of characterizing quantum states from measurements. This is performed through a process called quantum state tomography (QST). However, QST requires a large number of measurements, each derived from a different physical observable corresponding to a different experimental setup. Changing the setup results in unwanted changes to the data, prolongs the measurement, and impairs assumptions made about noise. Here, we propose to overcome these drawbacks by performing QST with a single observable. A single observable can often be realized by a single setup, thereby considerably reducing the experimental effort. However, the information contained in a single observable is insufficient for full QST. To overcome the lack of sufficient measurements in a single observable, we increase the system dimension by adding an ancilla that couples to the information in the system and exploit the fact that the sought state is often close to a pure state. We demonstrate our approach on multiphoton states by recovering structured quantum states from a single observable in a single experimental setup.
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.
A major quantum information processing tool is Quantum Process Tomography (QPT), which consists of methods that aim at identifying, i.e. estimating, a quantum process. QPT may especially be applied to quantum gates, that are widely used as the building blocks of quantum computers, in order to experimentally characterize their actual behavior. In this paper, we address unitary processes (corresponding to isolated systems), that may be dense, i.e. that do not have sparsity constraints. The proposed approaches to QPT require one to first perform Quantum State Tomography (QST). Afterwards, the core of our QPT methods applies to a significant number of qubits and thus to a high state space dimension and to complex problems. Our methods first achieve part of QPT by performing an eigenanalysis of the density matrix of a process output estimated with QST, by taking advantage of the unitarity of the process. Our first resulting class of complete algorithms uses only one such eigendecomposition and is thus "single-stage". We then extend our approach to algorithms that are "multiple-stage", i.e. that perform several eigendecompositions. This allows one to handle high-dimensional state spaces with a reduced sensitivity to the estimation errors that result from the use of an arbitrary QST algorithm as a preprocessing step of our QPT methods. Our multi-stage methods first include two-stage versions and then dichotomic versions, with a number of stages that increases with the considered state space dimension. Simulations were successively performed with different magnitudes for QST errors / noise. They provide evidence of the relevance of all proposed methods. Their single-stage and two-stage versions are efficient up to 13 qubits when executed on a very simple platform: an 8-year old PC with only 16 GB of RAM. An even higher accuracy (Normalized Mean Square Error down to 10−310^{-3}10−3 ) is obtained with multi-stage methods.