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arXiv:2307.14932 · Open Systems & Information Dynamics

Wave Matrix Lindbladization I: Quantum Programs for Simulating Markovian Dynamics

Dhrumil Patel, Mark M. Wilde

Published Jun 1, 2023• 20 citations
arXivHTMLPDF

Abstract

Density Matrix Exponentiation is a technique for simulating Hamiltonian dynamics when the Hamiltonian to be simulated is available as a quantum state. In this paper, we present a natural analogue to this technique, for simulating Markovian dynamics governed by the well known Lindblad master equation. For this purpose, we first propose an input model in which a Lindblad operator [Formula: see text] is encoded into a quantum state [Formula: see text]. Then, given access to [Formula: see text] copies of the state [Formula: see text], the task is to simulate the corresponding Markovian dynamics for time [Formula: see text]. We propose a quantum algorithm for this task, called Wave Matrix Lindbladization, and we also investigate its sample complexity. We show that our algorithm uses [Formula: see text] samples of [Formula: see text] to achieve the target dynamics, with an approximation error of [Formula: see text].

DOI
10.1142/s1230161223500105

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Density Matrix Exponentiation is a technique for simulating Hamiltonian dynamics when the Hamiltonian to be simulated is available as a quantum state. In this paper, we present a natural analogue to this technique, for simulating Markovian dynamics governed by the well known Lindblad master equation. For this purpose, we first propose an input model in which a Lindblad operator [Formula: see text] is encoded into a quantum state [Formula: see text]. Then, given access to [Formula: see text] copies of the state [Formula: see text], the task is to simulate the corresponding Markovian dynamics for time [Formula: see text]. We propose a quantum algorithm for this task, called Wave Matrix Lindbladization, and we also investigate its sample complexity. We show that our algorithm uses [Formula: see text] samples of [Formula: see text] to achieve the target dynamics, with an approximation error of [Formula: see text].
In this paper, we investigate the problem of simulating open system dynamics governed by the well-known Lindblad master equation. In our prequel paper [Patel and Wilde, Open Sys. & Info. Dyn., 30:2350010 (2023)], we introduced an input model in which Lindblad operators are encoded into pure quantum states, called program states, and we also introduced a method, called wave matrix Lindbladization, for simulating Lindbladian evolution by means of interacting the system of interest with these program states. Therein, we focused on a simple case in which the Lindbladian consists of only one Lindblad operator and a Hamiltonian. Here, we extend the method to simulating general Lindbladians and other cases in which a Lindblad operator is expressed as a linear combination or a polynomial of the operators encoded into the program states. We propose quantum algorithms for all these cases and also investigate their sample complexity, i.e., the number of program states needed to simulate a given Lindbladian evolution approximately. Finally, we demonstrate that our quantum algorithms provide an efficient route for simulating Lindbladian evolution relative to full tomography of encoded operators, by proving that the sample complexity for tomography is dependent on the dimension of the system, whereas the sample complexity of wave matrix Lindbladization is dimension independent.
We present a quantum algorithm for simulating a family of Markovian master equations that can be realized through a probabilistic application of unitary channels and state preparation. Our approach employs a second-order product formula for the Lindblad master equation, achieved by decomposing the dynamics into dissipative and Hamiltonian components and replacing the dissipative segments with randomly compiled, easily implementable elements. The sampling approach eliminates the need for ancillary qubits to simulate the dissipation process and reduces the gate complexity in terms of the number of jump operators. We provide a rigorous performance analysis of the algorithm. We also extend the algorithm to time-dependent Lindblad equations, generalize the family of Markovian master equations it can be applied to, and explore applications beyond the Markovian noise model. A new error bound, in terms of the diamond norm, for second-order product formulas for time-dependent Liouvillians is provided that might be of independent interest.
The Markovian dynamics of open quantum many-body systems are typically governed by the Lindblad master equation, yet obtaining the reduced density matrix for bosonic and fermionic systems remains a formidable numerical challenge due to the exponential growth of the Hilbert space. Here, we introduce a computationally efficient framework for constructing the reduced density matrix by modeling the environment as a copy of the primary system with a monotonically decaying coupling that enforces unidirectional energy flow. Our method directly yields exact solutions that rigorously satisfy the Lindblad master equation for both bosonic and fermionic cases, bypassing the need for costly Liouvillian diagonalization. We validate our approach through applications to paradigmatic models, demonstrating accurate reproduction of dissipative dynamics across a broad parameter regime. This work provides a straightforward and powerful tool for simulating Markovian open-system evolution, with immediate applicability to quantum transport and control problems in many-body physics.
Abstract Density matrix exponentiation (DME) is a quantum algorithm that processes multiple copies of a program state σ to realize the Hamiltonian evolution e − i σ t . Wave matrix Lindbladization (WML) similarly processes multiple copies of a program state ψ L in order to realize a Lindbladian evolution. Both algorithms are prototypical sample-based quantum algorithms and can be used for various quantum information processing tasks, including quantum principal component analysis, Hamiltonian simulation, and Lindbladian simulation. In this work, we present detailed sample complexity analyses for DME and sample-based Hamiltonian simulation, as well as for WML and sample-based Lindbladian simulation. In particular, we prove that the sample complexity of DME is no larger than 4 t 2 / ε , where t is the desired evolution time and ɛ is the desired imprecision level, as quantified by the normalized diamond distance. We also establish a fundamental lower bound on the sample complexity of sample-based Hamiltonian simulation, which matches our DME sample complexity bound up to a constant multiplicative factor, thereby proving that DME is optimal for sample-based Hamiltonian simulation. Additionally, we prove that the sample complexity of WML is no larger than 3 t 2 d 2 / ε , where d is the dimension of the space on which the Lindblad operator acts nontrivially, and we prove a lower bound of 10 − 4 t 2 / ε on the sample complexity of sample-based Lindbladian simulation. These results prove that WML is optimal for sample-based Lindbladian simulation whenever the Lindblad operator acts nontrivially on a constant-sized system. Finally, we point out that the DME sample complexity analysis in Kimmel et al(2017 npj Quantum Inf. 3 13) and the WML sample complexity analysis in Patel and Wilde (2023 Open Syst. Inf. Dyn. 30 2350010) appear to be incomplete, highlighting the need for the results presented here.
Abstract The Lindblad equation generalizes the Schrödinger equation to quantum systems that undergo dissipative dynamics. The quantum simulation of Lindbladian dynamics is therefore non-unitary, preventing a naive application of state-of-the-art quantum algorithms. Here, we make use of an approximate correspondence between Lindbladian dynamics and evolution based on repeated interaction (RI) CPTP maps to write down a Hamiltonian formulation of the Lindblad dynamics and derive a rigorous error bound on the master equation. Specifically, we show that the number of interactions needed to simulate the Liouvillian e t L within error ε scales in most physical scenarios as ν ∈ O ( t 2 ‖ L ‖ 1 → 1 2 / ϵ ) . This is significant because the error in the Lindbladian approximation to the dynamics is not explicitly bounded in existing quantum algorithms for open system simulations. We then provide quantum algorithms to simulate RI maps using an iterative qubitization approach and Trotter–Suzuki formulas, and specifically show that for iterative qubitization the number of operations needed to simulate the dynamics (for a fixed value of ν) scales as O ( α 0 t + ν log ⁡ ( 1 / ϵ ) / log ⁡ log ⁡ ( 1 / ϵ ) ) in the limit where α 0 (the coefficient 1-norm for the system and bath Hamiltonians) asymptotically dominates over the corresponding factor for the interaction Hamiltonian, which is often the case in weak coupling. This scaling would appear to be optimal if the complexity of ν is not considered, which underscores the importance of considering the error in the Liouvillian that we reveal in this work.