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Quantum Information and Computation

Optimal estimation of quantum processes using incomplete information

O MacielThiago, O ViannaReinaldo

Published May 1, 2012• 0 citations
Publisher

Abstract

We develop a quantum process tomography method, which variationally reconstruct the map of a process, using noisy and incomplete information about the dynamics. The new method encompasses the most ...
DOI
10.5555/2230996.2231001

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Abstract In quantum-state tomography on sources with quantum degrees of freedom of large Hilbert spaces, inference of quantum states of light for instance, a complete characterization of the quantum states for these sources is often not feasible owing to limited resources. As such, the concepts of informationally incomplete state estimation becomes important. These concepts are ideal for applications to quantum channel/ process tomography, which typically requires a much larger number of measurement settings for a full characterization of a quantum channel. Some key aspects of both quantumstate and quantum-process tomography are arranged together in the form of a tutorial review article that is catered to students and researchers who are new to the field of quantum tomography, with focus on maximum-likelihood related techniques as instructive examples to illustrate these ideas.
It is assumed partial knowledge about the state of a quantum system. This knowledge is represented by the results of some measurements, spanning a certain subspace. We refer to this subspace as the known subspace, and to its complement as the unknown one. Then we introduce a variational method to estimate the expectation value of every observable. This estimate is optimal and reliable, in the sense that it is always smaller, in absolute value, than the true expectation value (the one that would be obtained if directly measured, of if the quantum state were known exactly). As a by-product of this approach, we obtain a method of reconstruction of the quantum state, using incomplete information. Our results make clear that it is typically not necessary to perform an informationally complete set of measurements, which grows exponentially with the Hilbert space size, in order to obtain certified optimal lower bounds to properties of a quantum system. The algorithm that implements our method is a linear convex problem, which has exact solution in terms of well known semidefinite programs.
Quantum process tomography is a useful tool for characterizing quantum processes. This task is essential for the development of different areas, such as quantum information processing. In this work, we present a protocol for selective continuous-variable quantum process tomography. Our proposal allows one to selectively estimate any element of an unknown continuous-variable quantum process in the position representation, without requiring the complete reconstruction of the process. By resorting to controlled squeezing and translation operations, and adaptatively discretizing the process, a direct measure of an estimate of any process element can be obtained. Furthermore, we show, supported by numerical simulations, how the protocol can be used to partially reconstruct on a region a continuous-variable quantum process.
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.
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.