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Symmetry

Controlled State Transfer in Central Spin Models

Martiros Khurshudyan

Published Apr 17, 2024• 4 citations
Publisher

Abstract

In the recent literature, various aspects of the transfer of quantum states by spin chains have been thoroughly investigated. Part of the existing study is devoted to the problem of optimal control, with the goal of achieving a highly reliable information/state transfer for a given time T. In general, achieving this goal is not an easy task in the case of (open) quantum systems. Various approaches have been developed and applied, including Krotov’s method to study the problem. It is a gradient-based method used here to study the problem of state transfer control in central spin models. Our results show that with Krotov’s method, it is possible to find an optimal control form that allows for very-high-fidelity state transfer in the central spin models we have developed. Our results will be of interest for a better understanding of the non-trivial effects of the classical world on the quantum world, which have been discussed in the form of various new effects, including the Epstein effect, in the recent literature.

DOI
10.3390/sym16040489

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In the recent literature, various aspects of the transfer of quantum states by spin chains have been thoroughly investigated. Part of the existing study is devoted to the problem of optimal control, with the goal of achieving a highly reliable information/state transfer for a given time T. In general, achieving this goal is not an easy task in the case of (open) quantum systems. Various approaches have been developed and applied, including Krotov’s method to study the problem. It is a gradient-based method used here to study the problem of state transfer control in central spin models. Our results show that with Krotov’s method, it is possible to find an optimal control form that allows for very-high-fidelity state transfer in the central spin models we have developed. Our results will be of interest for a better understanding of the non-trivial effects of the classical world on the quantum world, which have been discussed in the form of various new effects, including the Epstein effect, in the recent literature.
Mathematical problems of optimal control of quantum systems attract high interest in connection with both fundamental questions and existing and prospective applications. An important problem is the development of methods for constructing controls for quantum systems. One of the widely used methods is the Krotov method. The article is the first part of the review devoted to mathematical aspects of this method for optimal control of quantum systems. Beyond quantum control this method was suggested in the articles [V.F. Krotov, I.N. Feldman, Eng. Cybern. 1983. Vol. 21, 123], [A.I. Konnov, V.F. Krotov, Automat. Remote Control, 1999], and then adopted to optimal quantum control in papers by V.A. Kazakov, V.F. Krotov, R. Kosloff, C.P. Koch, D.J. Tannor, and others. The review considers this method, including cases with constraints on control spectrum and on states of the quantum system, various ways for choosing the improvement function, in particular, linear or linear-quadratic form, etc. We also discuss GRAPE (GRadient Ascent Pulse Engineering) and CRAB (Chopped Random-Basis) methods, as well as combined methods such as the method suggested by I. Maday and G. Turinici for generalizing the 1st order Krotov method and the Zhu-Rabitz method. Nowadays, various versions of the Krotov method are used as a key tool in optimal quantum control, and the review describes applications of this method to mathematical control problems for molecular dynamics, Bose-Einstein condensate, nuclear-magnetic resonance, quantum gates generation, etc. The first part of the review is devoted to closed quantum systems.
This article (II) continues the research described in [Morzhin O.V. Gradient projection method and stochastic search for some optimal control models with spin chains. I (submitted)] (Article I), derives the needed finite-dimensional gradients corresponding to the infinite-dimensional gradients obtained in Article I, both for transfer and keeping problems at a certain NNN-dimensional spin chain, and correspondingly adapts a projection-type condition for optimality, gradient projection method (GPM). For the case N=3N=3N=3, the given in this article examples together with Example 3 in Article I show that: a) the adapted GPM and genetic algorithm (GA) successfully solved numerically the considered transfer and keeping problems; b) the two- and three-step GPM forms significantly surpass the one-step GPM. Moreover, GA and a special class of controls were successfully used in such the transfer problem that N=20N=20N=20 and the final time is not assigned.
Abstract The mathematics of optimal control of quantum systems is of great interest in connection with fundamental problems of physics as well as with existing and prospective applications to quantum technologies. One important problem is the development of methods for constructing controls for quantum systems. One of the commonly used methods is the Krotov method, which was initially proposed outside of quantum control theory in articles by Krotov and Feldman (1978, 1983). This method was used to develop a novel approach to finding optimal controls for quantum systems in [64] (Tannor, Kazakov, and Orlov, 1992), [65] (Somlói, Kazakov, and Tannor, 1993), and in many other works by various scientists. Our survey discusses mathematical aspects of this method for optimal control of closed quantum systems. It outlines various modifications with different forms of the improvement function (for example, linear or linear-quadratic), different constraints on the control spectrum and on the admissible states of the quantum system, different regularisers, and so on. The survey describes applications of the Krotov method to controlling molecular dynamics and Bose–Einstein condensates, and to quantum gate generation. This method is compared with the GRAPE (GRadient Ascent Pulse Engineering) method, the CRAB (Chopped Random-Basis) method, and the Zhu–Rabitz and Maday–Turinici methods. Bibliography: 158 titles.
Quantum optimal control (QOC) provides a systematic framework for achieving high-fidelity operations in quantum systems and plays a central role in tasks such as gate synthesis, state transfer, and pulse design. Existing QOC methods broadly fall into two categories: gradient-based and gradient-free algorithms. The associated optimization landscape is often high-dimensional, non-convex, and populated by numerous local minima, making efficient gradient-free search strategies essential. To address this, we introduce a gradient-free matrix product state/tensor train (MPS/TT) sampling heuristic for discrete quantum optimal control. In our approach, the MPS defines a score function over the space of discrete control parameters, which in turn induces a sampling distribution over candidate control sequences. This distribution is iteratively refined through selection of better performing sequences and local tensor updates to bias the search toward high-performing sequences. We evaluate the method on a range of benchmark problems, including single-qubit state transfer, Bell-pair preparation, qutrit gate implementation, and open-system population transfer. Across these tasks, the method exhibits stable convergence behavior and competitive empirical performance relative to established gradient-free baselines. These results suggest that tensor network sampling offers a viable heuristic framework for discrete quantum control.