Improved sample complexity bound for sample-based Lindbladian simulation

Abstract: We establish improved sample-complexity bounds for sample-based Lindbladian simulation based on the Wave Matrix Lindbladization (WML) algorithm. For a jump operator $L$ with dimension $d$, we derive an explicit non-asymptotic sample complexity bound $n_d^*(t,\varepsilon) \le \left( \frac{2d+3}{8} \right) |L|\infty^2 \left( \frac{t^2}{\varepsilon} \right)$, holding for simulation time $t$ and error $\varepsilon$. This refines the dimension dependence of the best previously known bound, $O(d^2 t^2/\varepsilon)$, from [Go et al., Quantum Sci. Tech. 10, 045058 (2025)]. Remarkably, we show that this dimensional overhead can be entirely avoided when $| L|\infty^2 = O(1/d)$, a condition satisfied with high probability for random Lindblad operators, yielding a typical-case sample complexity of $O(t^2/\varepsilon)$. On the other hand, in the worst case, we show that WML necessarily requires $Ω(dt^2/\varepsilon)$ samples by constructing an explicit example with a rank-one Lindblad operator. Our results reveal a sharp dichotomy between typical and adversarial sample complexities in Lindbladian simulation, thereby strengthening the theoretical foundations of sample-based quantum algorithms.

• The paper establishes an explicit upper bound for sample-based Lindbladian simulation using the WML protocol, reducing the prior quadratic dependence on dimension to linear.
• It leverages algebraic techniques and concentration inequalities to show that, in typical cases, the sample complexity can become essentially dimension-independent.
• The improved bounds highlight a strict separation from state tomography needs, offering practical advantages for NISQ implementations and privacy-preserving quantum simulations.

Improved Sample Complexity Bounds for Sample-Based Lindbladian Simulation

Introduction

The simulation of open quantum systems governed by Lindbladian dynamics is a critical task for quantum information science, spanning quantum chemistry, dissipative state engineering, and error mitigation. Unlike Hamiltonian (unitary) evolution, Lindbladian simulation must accommodate non-unitary, Markovian noise, whose general structure is captured by the GKSL equation. While gate-based digital quantum simulation and various Trotterization and decomposition methods have achieved notable successes, sample-based techniques—where one only has access to copies of an unknown program state encoding the Lindbladian—are of distinctive interest due to their utility in privacy-preserving and NISQ-relevant scenarios.

The Wave Matrix Lindbladization (WML) protocol, generalizing the density matrix exponentiation paradigm, enables sample-based Lindbladian simulation by repeated application of a fixed channel utilizing program state copies. Prior to this work, the tightest upper bound on simulation sample complexity was $O(d^2 t^2/\varepsilon)$ for a Hilbert space of dimension $d$ [go2025SamplebasedHamiltonianLindbladian]. However, this scaling leaves an undesirable gap with the known lower bound $\Omega(t^2/\varepsilon)$, both theoretically and for practical scaling as $d$ grows. This paper addresses this gap by refining the analysis of the WML protocol and characterizing the dependence on both $d$ and Lindblad operator properties.

Main Results

The paper establishes a new explicit upper bound on the sample complexity for sample-based Lindbladian simulation of a single dissipative channel with jump operator $L$, valid for arbitrary system size: $$n_d^*(t, \varepsilon) \leq \frac{2d+3}{8} \|L\|_\infty^2 \frac{t^2}{\varepsilon}$$ This result immediately improves the previous $O(d^2)$ dependence in two crucial senses:

1. Dimension dependence is reduced to linear, i.e., $O(d)$, dictated by the operator norm $\|L\|_\infty^2$.
2. For large random Lindblad operators (drawn from e.g., Frobenius-normalized Ginibre ensembles), the typical-case scaling can become dimension-independent, with $d$0, resulting in $d$1 sample complexity with high probability.

Conversely, the analysis demonstrates the fundamental necessity of $d$2 scaling in adversarial cases: for a rank-one jump operator $d$3 the sample complexity lower bound for WML matches the upper bound up to constants.

Technical Approach

The authors use explicit calculations of the WML channel, leveraging the algebraic structure of the WML jump operator and higher-order expansions, to obtain non-asymptotic analytic error bounds. The analysis for random Lindblad operators relies on concentration inequalities for the normalized operator norm in the complex Ginibre ensemble, demonstrating that for large $d$4, the worst-case $d$5 scaling typically vanishes.

Furthermore, the gap between the complexity of simulation and that of state tomography—$d$6 for a $d$7-dimensional program state—becomes unavoidably strict, reinforcing the privacy advantages of sample-based simulation.

Theoretical and Practical Implications

Complexity Separation

By improving the upper bound to $d$8, the work sharpens the separation between the sample complexity of simulation and that of quantum state/program tomography. This gives a principled justification for the use of sample-based methods even in high-dimensional settings, as it guarantees that practical Lindbladian simulation can be exponentially more sample-efficient than reconstructing the program state.

Typical vs. Worst-Case Dichotomy

The sharp dichotomy established—$d$9 for random/typical Lindbladians versus $\Omega(t^2/\varepsilon)$0 for adversarial/rank-one cases—has significant ramifications for both quantum algorithm design and operational interpretations of Lindbladian complexity. For realistic (i.e., not worst-case) physical noise and dynamics, simulation is much more tractable than previously realized.

Relevance to NISQ and Privacy-Preserving Simulation

The results underscore that WML-based simulation is advantageous for NISQ devices: it enables accurate quantum channel implementation with fewer, easier-to-prepare resource states, and robustly preserves privacy since full tomography is avoidable. Additionally, the strict sample complexity separation supports cryptographic and data-hiding applications in quantum information, following the logic of quantum copy-protection [aaronson2009QuantumCopyProtectionQuantum].

Extensions and Future Directions

The explicit, non-asymptotic nature of the bounds paves the way toward extension to Lindbladians with multiple incoherent terms and direct incorporation of unitary evolution (i.e., Hamiltonian + Lindbladian simulation). There is a conjectured irreducibility in the $\Omega(t^2/\varepsilon)$1 gap between algorithm-specific and general (algorithm-independent) lower bounds, closely tied to operator norm distinguishability trade-offs inherent in open quantum system dynamics.

In addition, the methods are readily extendable to more general subgaussian ensembles, as demonstrated in the appendices, suggesting that these results are robust to a broad class of physically relevant noise models.

Conclusion

The paper provides a rigorous, non-asymptotic improvement in the sample complexity bound for sample-based simulation of Lindbladian dynamics using the WML approach (2605.30301). By establishing both tight upper bounds in worst-case and typical-case settings, it elucidates the complexity landscape of open-system simulation and establishes crucial separations from state tomography—a result with direct implications for quantum algorithmics, complexity theory, and quantum cryptography. These insights are expected to inform the development of more efficient simulators and resource theories for open quantum dynamics, as well as privacy-preserving protocols for programmable quantum devices.