Exponentially accurate open quantum simulation via randomized dissipation with minimal ancilla

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 achieve 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 – $4+\lceil\log_2 M\rceil$ in the first case and $7$ 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 towards making open quantum system simulations more feasible on early fault-tolerant quantum computing devices.

💡 Research Summary

This paper tackles the long‑standing challenge of simulating open quantum systems governed by the Lindblad master equation with both high accuracy and low resource overhead. While recent fault‑tolerant algorithms achieve “exponential accuracy” – i.e., a circuit depth that scales only logarithmically with the desired error ε – they require a large number of ancilla qubits and complex controlled operations because all jump operators must be coherently encoded. The authors propose two randomized quantum algorithms that dramatically reduce ancilla requirements while preserving the logarithmic depth scaling.

The first algorithm (Theorem 2) works directly with the Lindblad generator expressed as a sum of Pauli strings: the Hamiltonian H = Σ_j α₀j P₀j and each jump operator L_k = Σ_j α_kj P_kj, where each L_k contains at most M Pauli terms. By expanding the dynamical map e^{t L} in a Taylor series of its transfer‑matrix representation, they obtain a linear combination of completely positive trace‑non‑increasing (CPTN) maps {f_{W_v}}. Crucially, they introduce a new decomposition that suppresses the total norm of the coefficients to a value close to one, avoiding the exponential blow‑up that would otherwise make random sampling infeasible. Each CPTN map can be implemented with a circuit acting on the n‑qubit system plus only 3 + ⌈log₂ M⌉ ancilla qubits, using mid‑circuit measurement and reset. Randomly sampling the terms according to their (rescaled) coefficients yields an unbiased estimator of the observable expectation Tr