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arXiv:1904.00910 · Scientific Reports

A quantum algorithm for evolving open quantum dynamics on quantum computing devices

Zixuan Hu, Rongxin Xia, Sabre Kais

Published Feb 24, 2020• 163 citations
arXivHTMLPDF

Abstract

Designing quantum algorithms for simulating quantum systems has seen enormous progress, yet few studies have been done to develop quantum algorithms for open quantum dynamics despite its importance in modeling the system-environment interaction found in most realistic physical models. In this work we propose and demonstrate a general quantum algorithm to evolve open quantum dynamics on quantum computing devices. The Kraus operators governing the time evolution can be converted into unitary matrices with minimal dilation guaranteed by the Sz.-Nagy theorem. This allows the evolution of the initial state through unitary quantum gates, while using significantly less resource than required by the conventional Stinespring dilation. We demonstrate the algorithm on an amplitude damping channel using the IBM Qiskit quantum simulator and the IBM Q 5 Tenerife quantum device. The proposed algorithm does not require particular models of dynamics or decomposition of the quantum channel, and thus can be easily generalized to other open quantum dynamical models.

DOI
10.1038/s41598-020-60321-x

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Designing quantum algorithms for simulating quantum systems has seen enormous progress, yet few studies have been done to develop quantum algorithms for open quantum dynamics despite its importance in modeling the system-environment interaction found in most realistic physical models. In this work we propose and demonstrate a general quantum algorithm to evolve open quantum dynamics on quantum computing devices. The Kraus operators governing the time evolution can be converted into unitary matrices with minimal dilation guaranteed by the Sz.-Nagy theorem. This allows the evolution of the initial state through unitary quantum gates, while using significantly less resource than required by the conventional Stinespring dilation. We demonstrate the algorithm on an amplitude damping channel using the IBM Qiskit quantum simulator and the IBM Q 5 Tenerife quantum device. The proposed algorithm does not require particular models of dynamics or decomposition of the quantum channel, and thus can be easily generalized to other open quantum dynamical models.
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With the rapid progress in quantum hardware, there has been an increased interest in new quantum algorithms to describe complex many-body systems searching for the still-elusive goal of "useful quantum advantage."Surprisingly, quantum algorithms for the treatment of open quantum systems (OQSs) have remained underexplored, in part due to the inherent challenges of mapping non-unitary evolution into the framework of unitary gates.Evolving an open system unitarily necessitates dilation into a new effective system to incorporate critical environmental degrees of freedom.In this context, we present and validate a new quantum algorithm to treat non-Markovian dynamics in OQSs built on the ensemble of Lindblad's trajectories approach, invoking the Sz.-Nagy dilation theorem.Here we demonstrate our algorithm on the Jaynes-Cummings model in the strong-coupling and detuned regimes, relevant in quantum optics and driven quantum system studies.This algorithm, a key step towards generalized modeling of non-Markovian dynamics on a noisy-quantum device, captures a broad class of dynamics and opens up a new direction in OQS problems.
Research in open quantum system dynamics has received growing interest in recent years, and its ongoing investigation has uncovered many intriguing physics departing from closed systems. A particularly useful application of the open quantum system theory is to simulate quantum dynamical processes in condensed phase materials. As the quantum degree of freedom is constantly under the influence of its thermal environment, the accurate description of the dynamics often requires non-Markovian time evolution at finite temperature. Such calculation is usually quite challenging on classical computers as extensive memory storage is required. In this work, we focus on a quantum system linearly coupled to its harmonic bath and present a path integral based quantum algorithm that time-evolves the reduced density matrix under finite temperature non-Markovian dynamics. To treat the nonunitary time evolution, the Sz.-Nagy dilation scheme is used for the conversion to unitary dynamics that can be implementable on gate-based quantum computers. The modified Hadamard test is then used to retrieve all the information of the reduced density matrix. Complexity analysis shows that the memory requirement has exponential reduction on the quantum machine whereas the runtime complexity stays roughly the same as for the classical computer. It points to the possibility of using this algorithm to simulate multilevel and multisite non-Markovian quantum dynamics that are beyond the reach of classical computers. The algorithm makes no ad hoc assumptions and extends naturally beyond Markovian and weak coupling regimes. We validated the algorithm on the quantum simulator with the spin-boson model and demonstrated its excellent agreement with the classical computer benchmark. Published by the American Physical Society 2024
Electron transport in realistic physical and chemical systems often involves the nontrivial exchange of energy with a large environment, requiring the definition and treatment of open quantum systems. Because the time evolution of an open quantum system employs a nonunitary operator, the simulation of open quantum systems presents a challenge for universal quantum computers constructed from only unitary operators or gates. Here, we present a general algorithm for implementing the action of any nonunitary operator on an arbitrary state on a quantum device. We show that any quantum operator can be exactly decomposed as a linear combination of at most four unitary operators. We demonstrate this method on a two-level system in both zero and finite temperature amplitude damping channels. The results are in agreement with classical calculations, showing promise in simulating nonunitary operations on intermediate-term and future quantum devices.
Practical quantum systems are open systems due to interactions with their environment. Understanding the evolution of open systems dynamics is important for quantum noise processes , designing quantum error correcting codes, and performing simulations of open quantum systems. Here we proposed an efficient quantum algorithm for simulating the evolution of an open quantum system on a duality quantum computer. In contrast to unitary evolution in a usual quantum computer, the evolution operator in a duality quantum computer is a linear combination of unitary operators. In this duality algorithm, the time evolution of open quantum system is realized by using Kraus operators which is naturally realized in duality quantum computing. Compared to the Lloyd's quantum algorithm [Science.273, 1073(1996)] , the dependence on the dimension of the open quantum system in our algorithm is decreased. Moreover, our algorithm uses a truncated Taylor series of the evolution operators, exponentially improving the performance on the precision compared with existing quantum simulation algorithms with unitary evolution operations.
We present a quantum algorithm based on the generalized quantum master equation (GQME) approach to simulate open quantum system dynamics on noisy intermediate-scale quantum (NISQ) computers. This approach overcomes the limitations of the Lindblad equation, which assumes weak system-bath coupling and Markovity, by providing a rigorous derivation of the equations of motion for any subset of elements of the reduced density matrix. The memory kernel resulting from the effect of the remaining degrees of freedom is used as input to calculate the corresponding non-unitary propagator. We demonstrate how the Sz.-Nagy dilation theorem can be employed to transform the non-unitary propagator into a unitary one in a higher-dimensional Hilbert space, which can then be implemented on quantum circuits of NISQ computers. We validate our quantum algorithm as applied to the spin-boson benchmark model by analyzing the impact of the quantum circuit depth on the accuracy of the results when the subset is limited to the diagonal elements of the reduced density matrix. Our findings demonstrate that our approach yields reliable results on NISQ IBM computers.