Quantum-Trajectory-Inspired Lindbladian Simulation

TL;DR

Open quantum-system dynamics are described by the Lindblad master equation, and efficiently simulating $e^{\mathcal{L}t}$ is challenging when the number of jump operators $m$ is large. The paper introduces a quantum trajectory–inspired short-time approximation channel $\mathcal{E}$ and a structured linear combination of unitaries (LCU) framework to implement Lindbladian evolution with reduced $m$-dependence, via sampling and probabilistic purification. It presents two quantum algorithms: Algorithm 1 achieves $O\left(\frac{q n \tau^2}{\epsilon}\right)$ gate complexity with $m$-independence, while Algorithm 2 attains near-optimal $(t,\epsilon)$-dependence with an additional $\tilde{O}(m)$ factor, yielding $O\left(m q \tau \frac{(\log (mq \tau/\epsilon)+n)\log(\tau/\epsilon)}{\log\log(\tau/\epsilon)}\right)$ gates, where $\tau=t\|\mathcal{L}\|_{\mathrm{pauli}}$. A fractional-query variant further uses a generalized Hamming-weight cutoff to achieve $O\left(t\frac{\log^2(t/\epsilon)}{\log\log(t/\epsilon)}\right)$-type scaling with an $\tilde{O}(m)$ penalty. Together, these approaches enable efficient Lindbladian simulation for dissipative systems with many jump operators, with potential impact on open-system tasks and ground-state preparation via dissipation-driven cooling.

Abstract

Simulating the dynamics of open quantum systems is a crucial task in quantum computing, offering wide-ranging applications but remaining computationally challenging. In this paper, we propose two quantum algorithms for simulating the dynamics of open quantum systems governed by Lindbladians. We introduce a new approximation channel for short-time evolution, inspired by the quantum trajectory method, which underpins the efficiency of our algorithms. The first algorithm achieves a gate complexity independent of the number of jump operators, $m$, marking a significant improvement in efficiency. The second algorithm achieves near-optimal dependence on the evolution time $t$ and precision $ε$ and introduces only an additional $\tilde{O}(m)$ factor, which strictly improves upon state-of-the-art gate-based quantum algorithm that has an $\tilde O(m^2)$ factor. The improvement stems from the integration of the new approximation channel with a novel structured linear combination of unitaries method. In both our algorithms, the reduction of dependence on $m$ significantly enhances the efficiency of simulating practical dissipative processes characterized by a large number of jump operators.

Figures (22)

• Figure 1: In quantum trajectory method, the state of the system is viewed as a mixture of pure states, and a trajectory is a sequence of pure states $\ket{\psi_0},\ket{\psi_1},\ldots,\ket{\psi_t}$, the system states under a possible list of individual evolution operators. In this figure, an example Lindbladian with $H=H_1+H_2$ and jump operators $\{L_1, L_2\}$ is presented. The mixture of all possible trajectories provides a good approximation of the evolution. Inspired by the quantum trajectory method, we introduce a new approximation for the short-time evolution of LME that is composed of efficiently implementable individual channels. In this figure, the individual channels $\mathcal{F}_1,\mathcal{F}_2$ correspond to individual Hamiltonians $H_1,H_2$, and $\mathcal{E}_1,\mathcal{E}_2$ correspond to jump operators $L_1,L_2$.
• Figure 2: Gadget circuit for simulating $\mathcal{F}_l$ modified from BCCKS14.
• Figure 3: Gadget circuit for simulating $\mathcal{E}_j$, where $B_j \ket{0} = \sum_{k=1}^q \sqrt{T_{jk}} \ket{k}$.
• Figure 4: Example of binary-controlled unitary matrices where the set of unitary matrices is $U=\{V_0,V_1,V_2,V_3\}$.
• Figure 5: Hamiltonian simulation circuit without Hamming-weight cutoff. The circuit performs $(I-i\delta H)^r$ on the state register when measuring $\ket{0}$ in all control qubits. An $r=3$ example is presented.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Lemma 1: quantum trajectory method via quantum channel
• proof
• Definition 1
• Definition 2
• Theorem 3: structured LCU method
• proof
• Corollary 1
• proof