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arXiv:1212.6452

A control protocol of finite dimensional quantum systems using square pules

Jianju Tang, H. C. Fu

Published Dec 28, 2012• 0 citations
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Abstract

The control protocols of two types of finite dimensional quantum systems are proposed. The feasibility of each protocol is possible and an arbitrary target state can be achieved from initial state by a constant field. The control parameters which are time periods of interaction between systems and control fields in each cycles are connected with the probability amplitudes of target states via trigonometric functions and can be determined analytically.

DOI
10.48550/arxiv.1212.6452

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The control protocols of two types of finite dimensional quantum systems are proposed. The feasibility of each protocol is possible and an arbitrary target state can be achieved from initial state by a constant field. The control parameters which are time periods of interaction between systems and control fields in each cycles are connected with the probability amplitudes of target states via trigonometric functions and can be determined analytically.
Control protocol to drive finite dimensional quantum systems to an arbitrary target state using square pulses is proposed explicitly. It is a multi-cycle control process and in each cycle we apply square pulses to cause single or a few transitions between energy levels. Systems with equal energy gaps except the first one, four dimensional system with equal first and third energy gaps and different second energy gap, and systems with all equal energy gaps of dimension three, are investigated in detail. The control parameters, the interaction time between systems and control fields and free evolution times between cycles, are connected with the probability amplitudes of target states via trigonometric functions and are determined analytically.
Complete controllability of finite-dimensional quantum systems with energy level degeneracy is investigated using two different approaches. One approach is to apply a weak constant field to eliminate the degeneracy and then control it using techniques developed for non-degenerate quantum systems. Conditions for the elimination of degeneracy are found and the issues of influence of relaxation time of a constant external field on the target state are addressed through the fidelity. Another approach is to control the degenerate system by a single control field directly. It is found that the system with twofold degenerate excited states and non-degenerate ground state is completely controllable except for the two-level system. Conditions of complete controllability are found for both systems with different energy gaps and with equal energy gaps.
A central problem in quantum computing is the finding of an unknown target state that encodes the solution of a certain computational task. To accomplish this goal, the evolution from a given initial state is performed with an associated total Hamiltonian, which is a time-dependent combination of two time-independent Hamiltonians: the problem-Hamiltonian, whose ground state is the unknown target, and a driving-Hamiltonian, whose ground state is the initial state. Here, we analyze this computational problem in the light of optimal control theory, considering each Hamiltonian modulated by an independent control function. For bounded controls, there is a minimum total evolution time beyond which the target state can be exactly prepared. We show that below this minimum time a possible optimal solution consists of both controls constantly tuned at their upper bound, provided the gradients of the associated control Hamiltonian with respect to the controls (the switching functions) are positive. We refer to this type of solution as the double-bang solution. We show that the double-bang solution is optimal for a teleportation protocol up to the minimum time. Additionally, we combine the double-bang solution and the adiabatic gate teleportation protocol to implement universal quantum computing. This approach to quantum computing is very appealing because of its simplicity and experimental feasibility. To corroborate our analytical results, we propose the use of a numerical quantum optimal control technique adapted to limit the amplitude of the controls, which converges to the double-bang solution when the final evolution time is shorter than the minimum time. We compare the fidelity of the teleported state obtained for the numerically optimized two-control functions with the usual one-control function scheme and with the quantum approximate optimization algorithm (QAOA). We find that the two-control approach has a better performance than the other approaches. Moreover, we investigate the energetic cost and the robustness against systematic errors in the teleportation protocol, considering different schemes.
Quantum phenomena of interest in connection with applications to computation and communication often involve generating specific transfers between eigenstates, and their linear superpositions. For some quantum systems, such as spin systems, the quantum evolution equation (the Schrödinger equation) is finite-dimensional and old results on controllability of systems defined on on Lie groups and quotient spaces provide most of what is needed insofar as controllability of non-dissipative systems is concerned. However, in an infinite-dimensional setting, controlling the evolution of quantum systems often presents difficulties, both conceptual and technical. In this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues. In particular, we analyze controllability for infinite-dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre-defined subspaces. This result, which we call the Finite Controllability Theorem, provides a set of sufficient conditions for controllability in an infinite-dimensional setting. We consider specific physical systems that are of interest for quantum computing, and provide insights into the types of quantum operations (gates) that may be developed.