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arXiv:0802.3892 · Physical Review A

Incomplete quantum process tomography and principle of maximal entropy

Mário Ziman

Published Sep 30, 2008• 22 citations
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Abstract

Our main goal is to extend and apply the principle of maximum entropy (MaxEnt) to incomplete quantum process estimation tasks. We will define a so-called process entropy function being the von Neumann entropy of the state associated with the quantum process via Choi-Jamiolkowski isomorphism. It will be shown that an arbitrary process estimation experiment can be reformulated in a unified framework and the MaxEnt principle can be consistently exploited. We will argue that the suggested choice for the process entropy satisfies a natural list of properties and it reduces to the state MaxEnt principle if applied to preparator devices.

DOI
10.1103/physreva.78.032118

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Our main goal is to extend and apply the principle of maximum entropy (MaxEnt) to incomplete quantum process estimation tasks. We will define a so-called process entropy function being the von Neumann entropy of the state associated with the quantum process via Choi-Jamiolkowski isomorphism. It will be shown that an arbitrary process estimation experiment can be reformulated in a unified framework and the MaxEnt principle can be consistently exploited. We will argue that the suggested choice for the process entropy satisfies a natural list of properties and it reduces to the state MaxEnt principle if applied to preparator devices.
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We propose an iterative algorithm for incomplete quantum process tomography with the help of quantum state estimation. The algorithm, which is based on the combined principles of maximum likelihood and maximum entropy, yields a unique estimator for an unknown quantum process when one has less than a complete set of linearly independent measurement data to specify the quantum process uniquely. We apply this iterative algorithm adaptively in various situations and so optimize the amount of resources required to estimate a quantum process with incomplete data.
The connection between the maximum entropy (MaxEnt) formalism and restricted Boltzmann machines (RBMs) is natural as both give rise to a Boltzmann-like distribution with constraints enforced by Lagrange multipliers, which correspond to RBM parameters. We integrate RBMs into quantum state tomography by using them as probabilistic models to approximate quantum states while satisfying MaxEnt constraints. Additionally, we employ polynomially efficient quantum sampling techniques to enhance RBM training, enabling scalable and high-fidelity quantum state reconstruction. This approach provides a computationally efficient framework for applying RBMs to MaxEnt-based quantum tomography. Furthermore, our method applies to the general and previously unaddressed case of reconstructing arbitrary mixed quantum states from incomplete and potentially noncommuting sets of expectations of observables while still ensuring maximal entropy.
Quantum tomography is a widely applicable tool for complete characterization of quantum states and processes. We develop a method for precision-guaranteed quantum process tomography. With the use of the Choi–Jamiołkowski isomorphism, we generalize the recently suggested extended norm minimization estimator for the case of quantum processes. Our estimator is based on the Hilbert–Schmidt distance for quantum processes. Specifically, we discuss the application of our method for characterizing quantum gates of a superconducting quantum processor in the framework of the IBM Q Experience.
Quantum state tomography provides a fundamental framework for reconstructing quantum states. When the measurement data are not informationally complete, the observed statistics admit multiple compatible density matrices, making the reconstruction problem inherently underdetermined and calling for the selection of a meaningful estimator. Two well-established approaches to address this ambiguity are Maximum Entropy (MaxEnt) and Variational Quantum Tomography (VQT). A variant of VQT, named VQT∞_\infty∞​, has been introduced to reproduce MaxEnt-like solutions. In this work, we generalize this approach by introducing a parametrized cost function that interpolates between the 1-norm and the infinity norm, thereby unifying VQT and VQT∞_\infty∞​ within a single framework. By tuning the associated hyperparameters, the proposed method enables controlled exploration of the set of compatible density matrices. We show that this interplay yields reconstructed states with higher fidelity to the MaxEnt solution than those obtained via VQT∞_\infty∞​ while preserving computational tractability.