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arXiv:0910.4609 · New Journal of Physics

Simplified quantum process tomography

Matthijs P. A. Branderhorst, J. Nunn, Ian A. Walmsley, Robert L. Kosut

Published Nov 13, 2009• 40 citations
arXivHTMLPDF

Abstract

We propose and evaluate experimentally an approach to quantum process tomography that completely removes the scaling problem plaguing the standard approach. The key to this simplification is the incorporation of prior knowledge of the class of physical interactions involved in generating the dynamics, which reduces the problem to one of parameter estimation. This allows part of the problem to be tackled using efficient convex methods, which, when coupled with a constraint on some parameters, allows globally optimal estimates for the Krauss operators to be determined from experimental data. Parameterizing the maps provides further advantages: it allows the incorporation of mixed states of the environment as well as some initial correlation between the system and environment, both of which are common physical situations following excitation of the system away from thermal equilibrium. Although the approach is not universal, in cases where it is valid it returns a complete set of positive maps for the dynamical evolution of a quantum system at all times.

DOI
10.1088/1367-2630/11/11/115010

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We propose and evaluate experimentally an approach to quantum process tomography that completely removes the scaling problem plaguing the standard approach. The key to this simplification is the incorporation of prior knowledge of the class of physical interactions involved in generating the dynamics, which reduces the problem to one of parameter estimation. This allows part of the problem to be tackled using efficient convex methods, which, when coupled with a constraint on some parameters, allows globally optimal estimates for the Krauss operators to be determined from experimental data. Parameterizing the maps provides further advantages: it allows the incorporation of mixed states of the environment as well as some initial correlation between the system and environment, both of which are common physical situations following excitation of the system away from thermal equilibrium. Although the approach is not universal, in cases where it is valid it returns a complete set of positive maps for the dynamical evolution of a quantum system at all times.
We address the problem of quantum process tomography with the preparators producing states correlated with the environmental degrees of freedom that play role in the system-environment interactions. We discuss the physical situations, in which the dynamics is described by nonlinear, or noncompletely positive transformations. In particular, we show that arbitrary mapping ϱin→ϱout\varrho_{\rm in}\to\varrho_{\rm out}ϱin​→ϱout​ can be realized by using appropriate set of preparators and applying the unitary operation SWAP. The experimental ``realization'' of perfect NOT operation is presented. We address the problem of the verification of the compatibility of the preparator devices with the estimating process. The evolution map describing the dynamics in arbitrary time interval is known not to be completely positive, but still linear. The tomography and general properties of these maps are discussed.
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.
A central aim of physics is to describe the dynamics of physical systems. Schrödinger's equation does this for isolated quantum systems. Describing the time evolution of a quantum system that interacts with its environment, in its most general form, has proved to be difficult because the dynamics is dependent on the state of the environment and the correlations with it. For discrete processes, such as quantum gates or chemical reactions, quantum process tomography provides the complete description of the dynamics, provided that the initial states of the system and the environment are independent of each other. However, many physical systems are correlated with the environment at the beginning of the experiment. Here, we give a prescription of quantum process tomography that yields the complete description of the dynamics of the system even when the initial correlations are present. Surprisingly, our method also gives quantitative expressions for the initial correlation.
The essential tool to characterize dynamics of an open quantum system is quantum process tomography (QPT). Although standard QPT methods are hard to be scalable, simplified QPT approach is available if we have the prior knowledge that the system Hamiltonian commutes with the system-environment interaction Hamiltonian. Using a nuclear magnetic resonance (NMR) quantum simulator, we experimentally simulate dephasing channels to demonstrate the simplified QPT as well as the standard QPT method as a comparison. The experimental results agree well with our predictions which confirm the validity and better efficiency of the simplified QPT.
A central aim of physics is to describe the dynamics of physical sys-tems. Schrödinger’s equation does this for isolated quantum systems. Describing the time evolution of a quantum system that interacts with its environment, in its most general form, has proved to be difficult because the dynamics is dependent on the state of the environment and the corre-lations with it. For discrete processes, such as quantum gates or chemical reactions, quantum process tomography provides the complete descrip-tion of the dynamics, provided that the initial states of the system and the environment are independent of each other. However, many physical systems are correlated with the environment at the beginning of the ex-periment. Here, we give a prescription of quantum process tomography that yields the complete description of the dynamics of the system even when the initial correlations are present. Surprisingly, our method also gives quantitative expressions for the initial correlation. There is a rich history to the studies of decoherence of quantum systems due to the interactions with the surrounding degrees of freedom. When the dynamics of the system (S) is Markovian it can be described by a master equation [1, 2, 3]. Nowadays many researchers are interested in systems that are non-Markovian, as there is mounting evidence that some natural systems of importance may be non-Markovian [4] and such features may allow to manipulate and control quantum systems in desired ways. There is also a great deal of interest in systems that are initially correlated with their environments (E) because non-Markovianity and initial system-environment (SE) correlations are intimately related [5, 6, 7]. 1 ar
Quantum Process Tomography (QPT) methods aim at identifying, i.e. estimating, a quantum process. QPT is a major quantum information processing tool, since it especially allows one to experimentally characterize the actual behavior of quantum gates, that may be used as the building blocks of quantum computers. We here consider unitary (possibly dense, i.e. without sparsity constraints) processes, which corresponds to isolated systems. Moreover, we develop a QPT method applicable to a significant number of qubits and hence to a high state space dimension, which allows one to tackle more complex problems. Using the unitarity of the process allows us to propose a method that first achieves part of QPT by performing an eigenanalysis of the estimated density matrix of a process output. We then complete the QPT by using another state differently. The relevance of our method is validated by means of simulations.