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Australian Control Conference

A reduced complexity min-plus method for synchronisation of qubits

Zibo Miao, Peter M. Dower

Published Nov 1, 2016• 0 citations
Publisher

Abstract

Solutions to optimal control problems can be obtained via grid-based dynamic programming iterations. However, such an approach is prohibitively expensive for problems with higher state dimensions, due to the exponential increase in grid points required in approximating the state space. This issue is especially severe in quantum systems whose dimensions grow exponentially with the number of interacting subsystems therein. In this paper, we study state synchronisation of two coupled qubits as an optimal control problem by tuning the interaction Hamiltonian of the system, whose dynamic programming equation is solved via a min-plus curse-of-dimensionality-free method. An illustration in the context of numerical examples is also provided.

DOI
10.1109/aucc.2016.7868222

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Solutions to optimal control problems can be obtained via grid-based dynamic programming iterations. However, such an approach is prohibitively expensive for problems with higher state dimensions, due to the exponential increase in grid points required in approximating the state space. This issue is especially severe in quantum systems whose dimensions grow exponentially with the number of interacting subsystems therein. In this paper, we study state synchronisation of two coupled qubits as an optimal control problem by tuning the interaction Hamiltonian of the system, whose dynamic programming equation is solved via a min-plus curse-of-dimensionality-free method. An illustration in the context of numerical examples is also provided.
The process of obtaining solutions to optimal control problems via mesh based techniques suffers from the well known problem of the curse of dimensionality. This issue is especially severe in quantum systems whose dimensions grow exponentially with the number of interacting elements (qubits) that they contain. In this article we develop a modification of recently introduced curse-of-dimensionality-free max-plus techniques, to the control of quantum systems. This method yields a more manageable growth that is related to the cardinality of the control set. Its efficacy is demonstrated by obtaining an approximate solution to a previously intractable problem on a 2 qubit system.
Although quantum computers have the potential to efficiently solve certain problems considered difficult by known classical approaches, the design of a quantum circuit remains computationally difficult. It is known that the optimal gate-design problem is equivalent to the solution of an associated optimal-control problem; the solution to which is also computationally intensive. Hence, in this article, we introduce the application of a class of numerical methods (termed the max-plus curse of dimensionality-free techniques) that determine the optimal control, thereby synthesizing the desired unitary gate. The application of this technique to quantum systems has a growth in complexity that depends on the cardinality of the control-set approximation rather than the much larger growth with respect to spatial dimensions in approaches based on gridding of the space, which is used in previous research. This technique is demonstrated by obtaining an approximate solution for the gate synthesis on SU(4)---a problem that is computationally intractable by grid-based approaches.
This thesis focuses on the optimal control of a class of closed 1 quantum systems. It encompasses and addresses the following themes 1. The theoretical underpinnings of the dynamic programming method for quantum spin systems. 2. The efficient numerical implementation of algorithms to generate optimal control laws. The use of the dynamic programming principle from optimal control theory for the control of quantum systems is introduced, while taking into account the properties of the manifolds on which they evolve. This enables the unified application of dynamic programming to disparate problems which were previously approached using distinct tools. Moreover the approach herein is applicable, in principle, to an arbitrarily large quantum system - thereby overcoming the limitations present in some other commonly used methods in the literature, on the size of the quantum system being analyzed. As the optimal cost function is non-differentiable in several cases of interest, this thesis provides a rigorous interpretation of this cost function in terms of viscosity solutions of the associated Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) over the manifolds of interest. This framework is applied to explore optimal control strategies for drift and drift-less quantum systems that arise in various applications - thus demonstrating the efficacy of a unified approach to address problems on qualitatively different systems. This thesis also investigates, via two distinct avenues, the efficient numerical solution to optimal control problems of interest. Firstly, it brings together the Markov chain ap- proximation methods of Kushner-Dupuis and the triangulation of manifolds introduced by Munkres, in order to numerically solve the optimal control problem on quantum systems via a value iteration approach. Secondly, the exponential growth in computational resources required - termed the curse of dimensionality (COD) -inherent in any spatial discretization approach to solving the HJB PDE is drastically lessened via a new reduced complexity method motivated by the COD free max-plus technique recently developed by McEneaney et al. The dramatic speedup obtained by this approach enables an approximate solution to a previously intractable optimal control problem for a two qubit2 spin system evolving on a 15 dimensional Lie group.
Quantum state preparation is a critical component in many quantum algorithms, yet its complexity scales exponentially with the number of qubits and the desired state's complexity. This work presents a novel approach to quantum state preparation utilizing dynamic programming. We demonstrate that dynamic programming can efficiently determine optimal sequences of quantum gates to realize a target quantum state, significantly reducing the computational burden compared to exhaustive search methods. The core of our method lies in formulating the problem as a state-space optimization task, where each state represents a partial quantum state and the transition between states corresponds to applying a quantum gate. We detail the algorithm and provide a theoretical analysis of its performance, highlighting its potential to mitigate the exponential scaling issues inherent in traditional quantum state preparation techniques. This approach offers a practical and scalable solution for preparing complex quantum states, representing a significant advancement in the field.
Developing fast, robust, and accurate methods for optimal control of quantum systems comprising interacting particles is one of the most active areas of current science. Although a valuable repository of algorithms is available for numerical applications in quantum control, the high computational cost is somewhat overlooked. Here, we present a fast and accurate optimal control algorithm for systems of interacting qubits, QOALA (quantum optimal control by adaptive low-cost algorithm), which is predicted to offer O ( M 2 ) speedup for an M -qubit system, compared to the state-of-the-art exact methods, without compromising overall accuracy of the optimal solution. The method is general and compatible with diverse Hamiltonian structures. The proposed approach uses inexpensive low-accuracy approximations of propagators far from the optimum, adaptively switching to higher accuracy, higher-cost propagators when approaching the optimum. In addition, the utilization of analytical Lie algebraic derivatives that do not require computationally expensive matrix exponential brings even better performance.