Universal quantum simulation of single-qubit nonunitary operators using duality quantum algorithm

Abstract

Quantum information processing enhances human’s power to simulate nature in quantum level and solve complex problem efficiently. During the process, a series of operators is performed to evolve the system or undertake a computing task. In recent year, research interest in non-Hermitian quantum systems, dissipative-quantum systems and new quantum algorithms has greatly increased, which nonunitary operators take an important role in. In this work, we utilize the linear combination of unitaries technique for nonunitary dynamics on a single qubit to give explicit decompositions of the necessary unitaries, and simulate arbitrary time-dependent single-qubit nonunitary operator F(t) using duality quantum algorithm. We find that the successful probability is not only decided by F(t) and the initial state, but also is inversely proportional to the dimensions of the used ancillary Hilbert subspace. In a general case, the simulation can be achieved in both eight- and six-dimensional Hilbert spaces. In phase matching conditions, F(t) can be simulated by only two qubits. We illustrate our method by simulating typical non-Hermitian systems and single-qubit measurements. Our method can be extended to high-dimensional case, such as Abrams–Lloyd’s two-qubit gate. By discussing the practicability, we expect applications and experimental implementations in the near future.

Simulating open quantum dynamics on an NMR quantum processor using the Sz.-Nagy dilation algorithm

“…In some cases, the noise from open dynamics can contribute significantly to errors in the computational output, leading to lower experimental fidelity and a reduction in the quality of the quantum device [10]. A duality quantum algorithm for simulating Hamiltonian evolution of an open quantum system was proposed where the time evolution is realized using Kraus operators [11,12]. A quantum algorithm was proposed to simulate a general finite-dimensional Lindblad master equations without needing to engineer system-environment interactions [13].…”

Quantum State Preparation and Non-Unitary Evolution with Diagonal Operators

“…Current quantum devices are typically unitary-gate-based, so non-unitary operators must be cast as unitary in order to be practically implementable. There are a variety of algorithms which have been developed to bypass this obstacle, including explicit mathematical dilations [8][9][10][11][12][13][14], quantum imaginary time evolution [15], duality [16,17], the variational principal [18], collision models [19], analog simulation [20], and others [21][22][23][24][25][26][27][28][29][30][31][32]. The majority of these algorithms rely on some form of dilation, either mapping the operator to a larger Hilbert space, or adding ancilla qubits.…”

A comparative study of universal quantum computing models: Toward a physical unification

“…For instance, in QCM unitary gates are often of product form and temporally ordered. Instead of product, a duality quantum computer uses superposition of gate operations, inspired by the wave‐particle duality principle, 21,22 which has led to quantum algorithms based on linear combination of unitary gates 68‐70 (also see Section 6.2). From the channel‐state duality, 71 gates can also be viewed as states hence acted upon by the so‐called superchannels, 72‐74 which has led to a quantum computing model without causal order 23 .…”