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Entropy

Efficient Quantum Simulation of an Anti-P-Pseudo-Hermitian Two-Level System

Chao Zheng, Tian Jin, Daili Li, Jingwei Wen, Shijie Wei, Yansong Li

Published Jul 24, 2020• 18 citations
Publisher

Abstract

Besides Hermitian systems, quantum simulation has become a strong tool to investigate non-Hermitian systems, such as PT-symmetric, anti-PT-symmetric, and pseudo-Hermitian systems. In this work, we theoretically investigate quantum simulation of an anti-P-pseudo-Hermitian two-level system in different dimensional Hilbert spaces. In an arbitrary phase, we find that six dimensions are the minimum to construct the anti-P-pseudo-Hermitian two-level subsystem, and it has a higher success probability than using eight dimensions. We find that the dimensions can be reduced further to four or two when the system is in the anti-PT-symmetric or Hermitian phase, respectively. Both qubit-qudit hybrid and pure-qubit systems are able to realize the simulation, enabling experimental implementations in the near future.

DOI
10.3390/e22080812

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Besides Hermitian systems, quantum simulation has become a strong tool to investigate non-Hermitian systems, such as PT-symmetric, anti-PT-symmetric, and pseudo-Hermitian systems. In this work, we theoretically investigate quantum simulation of an anti-P-pseudo-Hermitian two-level system in different dimensional Hilbert spaces. In an arbitrary phase, we find that six dimensions are the minimum to construct the anti-P-pseudo-Hermitian two-level subsystem, and it has a higher success probability than using eight dimensions. We find that the dimensions can be reduced further to four or two when the system is in the anti-PT-symmetric or Hermitian phase, respectively. Both qubit-qudit hybrid and pure-qubit systems are able to realize the simulation, enabling experimental implementations in the near future.
\n\nIn this work, quantum simulation of the evolution of a two-level quantum system with a generalized anti-PT(APT)-symmetric Hamiltonian in arbitrary phase is investigated theoretically. The evolution is reinterpreted as a subsystem in a Hilbert space of higher dimensions, enabling it to construct any two-dimensional APT-symmetric Hamiltonian in a Hermitian quantum system. In some special phases, a two-qubit quantum computer is able to achieve the quantum simulation. However, a minimum six-dimensional Hilbert space is necessary for a general case, indicating that a qutrit is needed as an ancillary. We show how to realize the simulation process in a qubit-qutrit hybrid system using the duality quantum algorithm. We also give the flow chart and quantum circuit for a quantum computer of three qubits to realize the quantum simulation. Therefore, experimental realizations and applications can be expected in the future.\n
Non-Hermitian (NH) quantum theory has been attracting increased research interest due to its featured properties, novel phenomena, and links to open and dissipative systems. Typical NH systems include PT-symmetric systems, pseudo-Hermitian systems, and their anti-symmetric counterparts. In this work, we generalize the pseudo-Hermitian systems to their complex counterparts, which we call pseudo-Hermitian-φ-symmetric systems. This complex extension adds an extra degree of freedom to the original symmetry. On the one hand, it enlarges the non-Hermitian class relevant to pseudo-Hermiticity. On the other hand, the conventional pseudo-Hermitian systems can be understood better as a subgroup of this wider class. The well-defined inner product and pseudo-inner product are still valid. Since quantum simulation provides a strong method to investigate NH systems, we mainly investigate how to simulate this novel system in a Hermitian system using the linear combination of unitaries in the scheme of duality quantum computing. We illustrate in detail how to simulate a general P-pseudo-Hermitian-φ-symmetric two-level system. Duality quantum algorithms have been recently successfully applied to similar types of simulations, so we look forward to the implementation of available quantum devices.
\n\nThe evolution of a two-level quantum system with a general parity-time-reversal-invariant Hamiltonian is reinterpreted as a subsystem in a four-dimensional Hilbert space, which makes it possible to simulate any two-level parity-time-symmetric (PT-symmetric) quantum system using a conventional quantum mechanics system. We studied the ability to simulate and investigate novel quantum systems and phenomena using a nowadays-available Hermitian quantum computer or a duality quantum computer. The quantum circuits are given for a qubit and two qudit systems, and they would be useful for experimental implementation and applications in the future.\n
Abstract Parity-time-reversal (PT) symmetric quantum mechanics promotes the increasing research interest of non-Hermitian (NH) systems for the theoretical value, novel properties, and links to open and dissipative systems in various areas. Recently, anti-PT–symmetric systems and their featured properties have started to be investigated. In this work, we develop the PT and anti-PT symmetry to PT-arbitrary-phase symmetry (or PT- φ symmetry) for the first time, being analogous to bosons, fermions and anyons. It can also be seen as a complex extension of the PT symmetry, unifying the PT and anti-PT symmetries and having intermediate properties between them. Many of the established concepts and mathematics in the PT-symmetric system are still compatible. We mainly investigate quantum simulation of this novel NH system in two dimensions in detail and discuss for higher-dimensional cases in general using the linear combinations of unitaries in the scheme of duality quantum computing, enabling implementations and experimental investigations of novel properties on both small quantum devices and near-term quantum computers.
Different from the Hermitian case, non-Hermitian (NH) systems have novel properties and strongly relate to open and dissipative quantum systems. In this work, we investigate how to simulate τ-anti-pseudo-Hermitian systems in a Hermitian quantum device using linear combinations of unitaries and duality quantum algorithm. Specifying the τ to time-reversal (T) and parity-time-reversal (PT) operators, we construct the two NH two-level systems, design quantum circuits including three qubits, and decide the quantum gates explicitly in detail. We also calculate the success probabilities of the simulation. Experimental implementation can be expected in small quantum simulator.
Non-Hermitian Hamiltonians are a fascinating class of many-body models that describe the effective dynamics of quantum systems interacting with the environment through particle, energy, or information exchange. Although their theoretical framework is well established, the controlled engineering of such Hamiltonians in the context of quantum simulations remains challenging, even more so when the non-Hermitian part describes a kkk-body interaction. Qudit quantum simulators offer a compelling framework to implement such models. We theoretically investigate the dynamics of a one-dimensional chain of qudits undergoing hybrid unitary-projective evolution, where suitably designed measurements constrain the dynamics to a Zeno subspace. As we illustrate for the case of qutrits, within the Zeno subspace the dynamics is governed by an effective non-Hermitian Hamiltonian for an ensemble of pseudo-spins 1/21/21/2, which can inherit non-Hermitian two-body interactions with the same connectivity as the full qutrit chain. We derive an analytical relation linking the monitored qutrits' evolution to a desired target non-Hermitian Hamiltonian and validate the effective description through numerical simulations of a representative model. Our scheme provides a constructive route for the realization of a large class of interacting non-Hermitian many-body Hamiltonians in experimentally relevant multilevel quantum platforms, including trapped ions and superconducting circuits.