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arXiv:2311.10009 · Physical Review Research

Efficient quantum algorithm to simulate open systems through a single environmental qubit

Giovanni Di Bartolomeo, Michele Vischi, Tommaso Feri, Angelo Bassi, Sandro Donadi

Published Dec 26, 2024• 15 citations
arXivHTMLPDF

Abstract

We present an efficient algorithm for simulating open quantum systems dynamics described by the Lindblad master equation on quantum computers, addressing key challenges in the field. In contrast to existing approaches, our method achieves two significant advancements. First, we employ a repetition of unitary gates on a set of n system qubits and, remarkably, only a single ancillary bath qubit representing the environment. It follows that, for the typical case of m locality of the Lindblad operators, we reach an exponential improvement of the number of ancilla in terms of m and up to a polynomial improvement in ancilla overhead for large n with respect to other approaches. Although stochasticity is introduced, requiring multiple circuit realizations, the sampling overhead is independent of the system size. Second, we show that, under fixed accuracy conditions, our algorithm enables a reduction in the number of Trotter steps compared to other approaches, substantially decreasing circuit depth. These advancements hold particular significance for near-term quantum computers, where minimizing both width and depth is critical due to inherent noise in their dynamics. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.043321

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We present an efficient algorithm for simulating open quantum systems dynamics described by the Lindblad master equation on quantum computers, addressing key challenges in the field. In contrast to existing approaches, our method achieves two significant advancements. First, we employ a repetition of unitary gates on a set of n system qubits and, remarkably, only a single ancillary bath qubit representing the environment. It follows that, for the typical case of m locality of the Lindblad operators, we reach an exponential improvement of the number of ancilla in terms of m and up to a polynomial improvement in ancilla overhead for large n with respect to other approaches. Although stochasticity is introduced, requiring multiple circuit realizations, the sampling overhead is independent of the system size. Second, we show that, under fixed accuracy conditions, our algorithm enables a reduction in the number of Trotter steps compared to other approaches, substantially decreasing circuit depth. These advancements hold particular significance for near-term quantum computers, where minimizing both width and depth is critical due to inherent noise in their dynamics. Published by the American Physical Society 2024
Simulating open quantum systems is an essential technique for understanding complex physical phenomena and advancing quantum technologies. Some quantum algorithms simulate Lindblad dynamics exponentially accurately, i.e., they achieve logarithmically short circuit depth in terms of accuracy, but they need to coherently encode all possible jump operators with a large ancilla consumption. Minimizing the gate and ancilla counts while achieving such a logarithmic scaling in accuracy remains an important challenge. In this work, we present two randomized quantum algorithms for simulating general Lindblad dynamics with multiple jump operators aimed at an observable estimation that achieves a circuit depth with not only logarithmic scaling in accuracy but also either partial or complete independence from the parameters specifying the Lindbladian. This is based on a novel random circuit compilation method that leverages dissipative processes with only a single jump operator, leading to the proposed methods using minimal ancilla qubits— 4 + ⌈ log 2 M ⌉ in the first case and seven in the other, where each single jump operator has at most M Pauli strings. In addition, we numerically demonstrate the practical advantage over existing approaches by providing a detailed analysis of the required gate and ancilla counts. This work represents a significant step toward making open quantum system simulations more feasible on early fault-tolerant quantum computing devices.
Simulating physical systems on quantum devices is one of the most promising applications of quantum technology. Current quantum approaches to simulating open quantum systems are still practically challenging on NISQ-era devices, because they typically require ancilla qubits and extensive controlled sequences. In this work, we propose a hybrid quantum-classical approach for simulating a class of open system dynamics called random-unitary channels. These channels naturally decompose into a series of convex unitary evolutions, which can then be efficiently sampled and run as independent circuits. The method does not require deep ancilla frameworks and thus can be implemented with lower noise costs. We implement simulations of open quantum systems up to dozens of qubits and with large channel ranks. Published by the American Physical Society 2024
Quantum computers promise efficient simulation of Lindbladian dynamics in open quantum systems, with broad applications in quantum chemistry, quantum error correction, and quantum state preparation. However, simulating open systems is more challenging than simulating closed systems, inherently requiring nonunitary operations. Existing approaches face unique challenges: methods with higher theoretical efficiency demand numerous ancillae and multiqubit operations that are experimentally challenging, while more experimentally feasible methods incur deep quantum circuits to control simulation errors. Here, we introduce a powerful framework, the linear combinations of superoperators, to systematically compensate for simulation errors and overcome these challenges. Utilizing simple gates and experimentally accessible Trotter decompositions, our approach achieves an exponential reduction in circuit depth with respect to the simulation precision, if only two ancillas are allowed to be used. We further extend the approach to time-dependent Lindbladians, achieving for the first time logarithmic depth in precision. Numerical simulations demonstrate significant performance advantages, establishing our method as a practical and scalable solution for simulating open quantum systems on near-term quantum hardware.
We present a novel method to simulate the Lindblad equation, drawing on the relationship between Lindblad dynamics, stochastic differential equations, and Hamiltonian simulations. We derive a sequence of unitary dynamics in an enlarged Hilbert space that can approximate the Lindblad dynamics up to an arbitrarily high order. This unitary representation can then be simulated using a quantum circuit that involves only Hamiltonian simulation and tracing out the ancilla qubits. There is no need for additional postselection in measurement outcomes, ensuring a success probability of one at each stage. Our method can be directly generalized to the time-dependent setting. We provide numerical examples that simulate both time-independent and time-dependent Lindbladian dynamics with accuracy up to the third order. Published by the American Physical Society 2024
Noise and decoherence are ubiquitous in the dynamics of quantum systems coupled to an external environment. In the regime where environmental correlations decay rapidly, the evolution of a subsytem is well described by a Lindblad quantum master equation. In this work, we introduce a quantum algorithm for simulating unital Lindbladian dynamics by sampling unitary quantum channels without extra ancillas. Using ancillary qubits we show that this algorithm allows approximating general Lindbladians as well. For interacting dephasing Lindbladians coupling two subsystems, we develop a decoupling scheme that reduces the circuit complexity of the simulation. This is achieved by sampling from a time-correlated probability distribution - determined by the evolution of one subsystem, which specifies the stochastic circuit implemented on the complementary subsystem. We demonstrate our approach by studying a model of bosons coupled to fermions via dephasing, which naturally arises from anharmonic effects in an electron-phonon system coupled to a bath. Our method enables tracing out the bosonic degrees of freedom, reducing part of the dynamics to sampling an IQP circuit. The sampled bitstrings then define a corresponding fermionic problem, which in the non-interacting case can be solved efficiently classically. We comment on the computational complexity of this class of dissipative problems, using the known fact that sampling from IQP circuits is believed to be difficult classically.
We consider the natural generalization of the Schrödinger equation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an nnn-qubit system for time ttt within precision εεε. If the Lindbladian consists of poly(n)\mathrm{poly}(n)poly(n) operators that can each be expressed as a linear combination of poly(n)\mathrm{poly}(n)poly(n) tensor products of Pauli operators then the gate cost of our algorithm is O(t polylog(t/ε)poly(n))O(t\, \mathrm{polylog}(t/ε)\mathrm{poly}(n))O(tpolylog(t/ε)poly(n)). We also obtain similar bounds for the cases where the Lindbladian consists of local operators, and where the Lindbladian consists of sparse operators. This is remarkable in light of evidence that we provide indicating that the above efficiency is impossible to attain by first expressing Lindblad evolution as Schrödinger evolution on a larger system and tracing out the ancillary system: the cost of such a \textit{reduction} incurs an efficiency overhead of O(t2/ε)O(t^2/ε)O(t2/ε) even before the Hamiltonian evolution simulation begins. Instead, the approach of our algorithm is to use a novel variation of the "linear combinations of unitaries" construction that pertains to channels.