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arXiv:2207.01954 · Physical Review A

Incorporating encoding into quantum system design

Alastair Kay

Published Apr 9, 2024• 1 citation
arXivHTMLPDF

Abstract

When creating a quantum system whose natural dynamics provide useful computational operations, designers have two key tools at their disposal: the (constrained) choice of both the Hamiltonian and the the initial state of the system (an encoding). Typically, we fix the design and utilize encodings to tolerate experimental imperfections. In this paper we describe a vital insight that incorporates encoding into the design process, with radical consequences. This transforms the study of perfect state transfer from the unrealistic scenario of specifying the Hamiltonian of an entire system to the far more realistic situation of being given a Hamiltonian over which we had no choice in the design, and designing time control of just two parameters to still achieve perfect transfer. Published by the American Physical Society 2024

DOI
10.1103/physreva.109.042408

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When creating a quantum system whose natural dynamics provide useful computational operations, designers have two key tools at their disposal: the (constrained) choice of both the Hamiltonian and the the initial state of the system (an encoding). Typically, we fix the design and utilize encodings to tolerate experimental imperfections. In this paper we describe a vital insight that incorporates encoding into the design process, with radical consequences. This transforms the study of perfect state transfer from the unrealistic scenario of specifying the Hamiltonian of an entire system to the far more realistic situation of being given a Hamiltonian over which we had no choice in the design, and designing time control of just two parameters to still achieve perfect transfer. Published by the American Physical Society 2024
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Perfect transfer of a quantum state through a one-dimensional chain is now well understood, allowing one not only to decide whether a fixed Hamiltonian achieves perfect transfer but to design a suitable one. We are particularly interested in being able to design, or understand the limitations imposed upon, Hamiltonians subject to various naturally arising constraints such as a limited coupling topology with low connectivity (specified by a graph) and type of interaction. In this paper, we characterize the necessary and sufficient conditions for transfer through a network and describe some natural consequences such as the impossibility of routing between many different recipients for a large class of Hamiltonians and the limitations on transfer rate. We also consider some of the trade-offs that arise in uniformly coupled networks (both Heisenberg and XXXXXX models) between transfer distance and the size of the network as a consequence of the derived conditions.
We review the subject of perfect state transfer — how one designs the (fixed) interactions of a chain of spins so that a quantum state, initially inserted on one end of the chain, is perfectly transferred to the opposite end in a fixed time. The perfect state transfer systems are then used as a constructive tool to design Hamiltonian implementations of other primitive protocols such as entanglement generation and signal amplification in measurements, before showing that, in fact, universal quantum computation can be implemented in this way.
We introduce a simple yet versatile protocol to inverse engineer the time-dependent Hamiltonian in two- and three-level systems. In the protocol, by utilizing a universal SU(2) transformation, a given speedup goal can be obtained with large freedom to select the control parameters. As an illustration example, the protocol is applied to perform population transfer between nitrogen-vacancy (NV) centers in diamond. Numerical simulation shows that the speed of the present protocol is fast compared with that of the adiabatic process. Moreover, the protocol is also tolerant to decoherence and experimental parameter fluctuations. Therefore, the protocol may be useful for designing an experimental feasible Hamiltonian to engineer a quantum system.