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arXiv:1504.03926

Performance Measure for Optimal Quantum Control

Alexandre Coutinho Lisboa, José Roberto Castilho Piqueira

Published Apr 15, 2015• 1 citation
arXivHTMLPDF

Abstract

The problem concerning the minimum time for an initial state to evolve up to a target state plays an important role in the Classic Optimal Control theory. In the quantum context, as quantum states are so sensitive to environmental influences, the problem is more complex, but its formulation and solution are decisive to implement quantum information processing systems. As is well known, the decoherence phenomenon is unavoidable and the time-energy uncertainty relation must be used to study quantum dynamics. Here, the time-energy uncertainty relations are revisited, being fundamental to propose performance measures based on minimum time evolution. A minimum time performance measure is definedfor quantum control problems. Then, some practical examples are considered and the minimum time performance measure is applied providing results that are supposed to be useful for researchers to pursue strategies to optimize the control of the states of a quantum system.

DOI
10.48550/arxiv.1504.03926

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The problem concerning the minimum time for an initial state to evolve up to a target state plays an important role in the Classic Optimal Control theory. In the quantum context, as quantum states are so sensitive to environmental influences, the problem is more complex, but its formulation and solution are decisive to implement quantum information processing systems. As is well known, the decoherence phenomenon is unavoidable and the time-energy uncertainty relation must be used to study quantum dynamics. Here, the time-energy uncertainty relations are revisited, being fundamental to propose performance measures based on minimum time evolution. A minimum time performance measure is definedfor quantum control problems. Then, some practical examples are considered and the minimum time performance measure is applied providing results that are supposed to be useful for researchers to pursue strategies to optimize the control of the states of a quantum system.
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Abstract One of the most widely known building blocks of modern physics is Heisenberg’s indeterminacy principle. Among the different statements of this fundamental property of the full quantum mechanical nature of physical reality, the uncertainty relation for energy and time has a special place. Its interpretation and its consequences have inspired continued research efforts for almost a century. In its modern formulation, the uncertainty relation is understood as setting a fundamental bound on how fast any quantum system can evolve. In this topical review we describe important milestones, such as the Mandelstam–Tamm and the Margolus–Levitin bounds on the quantum speed limit , and summarise recent applications in a variety of current research fields—including quantum information theory, quantum computing, and quantum thermodynamics amongst several others. To bring order and to provide an access point into the many different notions and concepts, we have grouped the various approaches into the minimal time approach and the geometric approach , where the former relies on quantum control theory, and the latter arises from measuring the distinguishability of quantum states. Due to the volume of the literature, this topical review can only present a snapshot of the current state-of-the-art and can never be fully comprehensive. Therefore, we highlight but a few works hoping that our selection can serve as a representative starting point for the interested reader.
One of the most widely known building blocks of modern physics is Heisenberg's indeterminacy principle. Among the different statements of this fundamental property of the full quantum mechanical nature of physical reality, the uncertainty relation for energy and time has a special place. Its interpretation and its consequences have inspired continued research efforts for almost a century. In its modern formulation, the uncertainty relation is understood as setting a fundamental bound on how fast any quantum system can evolve. In this topical review we describe important milestones, such as the Mandelstam–Tamm and the Margolus–Levitin bounds on the quantum speed limit, and summarise recent applications in a variety of current research fields—including quantum information theory, quantum computing, and quantum thermodynamics amongst several others. To bring order and to provide an access point into the many different notions and concepts, we have grouped the various approaches into the minimal time approach and the geometric approach, where the former relies on quantum control theory, and the latter arises from measuring the distinguishability of quantum states. Due to the volume of the literature, this topical review can only present a snapshot of the current state-of-the-art and can never be fully comprehensive. Therefore, we highlight but a few works hoping that our selection can serve as a representative starting point for the interested reader.
Optimal control theory provides a powerful set of tools and concepts that can be applied to quantum control systems. This chapter presents the general formulation of optimal control problems and gives the basic necessary conditions of optimality, the Pontryagin maximum principle (PMP). It presents an outline of the derivation of the PMP in the case of a fixed final time and free final state. One of the major problems in the context is that quantum systems are very sensitive to the presence of the environment which often destroys the main features of quantum dynamics. The chapter examines how the problem of minimum time control for system specializes in the case where eL/eK is a symmetric space.
Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine which is the Hamiltonian. We show that, in general, this requires a time $\ensuremath{\Delta}t$ that obeys the uncertainty relation $\ensuremath{\Delta}t\ensuremath{\Delta}H\ensuremath{\gtrsim}1,$ where $\ensuremath{\Delta}H$ is a measure of how accurately the unknown Hamiltonian must be estimated. We apply this result to the problem of measuring the energy of an unknown quantum state. It has been previously shown that if the Hamiltonian is known, then the energy can, in principle, be measured with arbitrarily large precision in an arbitrarily short time. On the other hand, we show that if the Hamiltonian is not known then an energy measurement necessarily takes a minimum time $\ensuremath{\Delta}t$ which obeys the uncertainty relation $\ensuremath{\Delta}t\ensuremath{\Delta}E\ensuremath{\gtrsim}1,$ where $\ensuremath{\Delta}E$ is the precision of the energy measurement. Several examples are studied to address the question of whether it is possible to saturate these uncertainty relations. Their interpretation is discussed in detail.