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arXiv:1007.2395 · Quantum Information and Computation

Optimal estimation of quantum processes using incomplete information: variational quantum process tomography

Thiago O. Maciel, Reinaldo O. Vianna

Published May 1, 2012• 6 citations
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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 common quantum process tomography schemes. It is based on the variational quantum tomography method (VQT) proposed by Maciel \emph{et al.} in arXiv:1001.1793[quant-ph] \cite{VQT}.

DOI
10.26421/qic12.5-6-5

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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 common quantum process tomography schemes. It is based on the variational quantum tomography method (VQT) proposed by Maciel \emph{et al.} in arXiv:1001.1793[quant-ph] \cite{VQT}.
Full characterization of quantum states and processes is a fundamental requirement for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because of the exponential growing with the system size of the number of measurements and the amount of computational power required to process them. A new approach for quantum state tomography and quantum process tomography, requiring a limited number of measurements, has been recently introduced by Maciel et al. [Quantum Inf. Comput. 12, 0442 (2012); Int. J. Mod. Phys. C 22, 1361 (2011).], namely variational quantum-process tomography (VQT). Here we adopt the VQT approach for the experimental characterization of two-qubit quantum processes and compare the reconstructed maps with those obtained by standard tomographic methods. The results demonstrate the high performance of this approach and propose VQT as a powerful alternative to the standard quantum process tomography.
Whenever we do not have an informationally complete set of measurements, the estimate of a quantum state cannot be uniquely determined. In this case, among the density matrices compatible with the available data, the one commonly preferred is the one which is the most uncommitted to the missing information. This is the purpose of the maximum entropy estimation (MaxEnt) and the variational quantum tomography (VQT). Here, we propose a variant of VQT and show its relationship with MaxEnt methods in quantum tomographies with an incomplete set of measurements. We prove their equivalence in the case of eigenbasis measurements, and through numerical simulations we stress their similar behavior. Hence, in the modified VQT formulation we have an estimate of a quantum state as unbiased as in MaxEnt and with the benefit that VQT can be more efficiently solved by means of linear semidefinite programs.
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.
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.
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.