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arXiv:2210.11371 · Quantum

Describing Trotterized Time Evolutions on Noisy Quantum Computers via Static Effective Lindbladians

Keith R. Fratus, Kirsten Bark, Nicolas Vogt, Juha Leppäkangas, Sebastian Zanker, Michael Marthaler, Jan-Michael Reiner

Published Sep 11, 2025• 3 citations
arXivHTMLPDF

Abstract

We consider the extent to which a Trotterized time evolution implemented on a quantum computer is altered by the presence of decoherence. Given a specific set of assumptions regarding the manner in which noise processes acting on such a device can be modeled at the circuit level, we show how the effects of noise can be reinterpreted as a shift to the dynamics of the original system being simulated. In particular, we find that this shift can be described through the use of static Lindblad noise terms, which act in addition to the original unitary dynamics. The form of these noise terms depends not only on the underlying noise processes occurring on the device, but also on the original unitary dynamics, as well as the manner in which these dynamics are simulated on the device, i.e., the choice of quantum algorithm. We call this effectively simulated open quantum system the noisy algorithm model. Our results are confirmed through numerical analysis.

DOI
10.22331/q-2025-09-11-1854

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We consider the extent to which a Trotterized time evolution implemented on a quantum computer is altered by the presence of decoherence. Given a specific set of assumptions regarding the manner in which noise processes acting on such a device can be modeled at the circuit level, we show how the effects of noise can be reinterpreted as a shift to the dynamics of the original system being simulated. In particular, we find that this shift can be described through the use of static Lindblad noise terms, which act in addition to the original unitary dynamics. The form of these noise terms depends not only on the underlying noise processes occurring on the device, but also on the original unitary dynamics, as well as the manner in which these dynamics are simulated on the device, i.e., the choice of quantum algorithm. We call this effectively simulated open quantum system the noisy algorithm model. Our results are confirmed through numerical analysis.
When modeling the effects of noise on quantum circuits, one often makes the assumption that these effects can be accounted for by individual decoherence events following an otherwise noise-free gate. In this work, we address the validity of this model. We find that under a fairly broad set of assumptions, this model of individual decoherence events provides a good approximation to the true noise processes occurring on a quantum device during the implementation of a quantum circuit. However, for gates which correspond to sufficiently large rotations of the qubit register, we find that the qualitative nature of these noise terms can vary significantly from the nature of the noise at the underlying hardware level. The bulk of our analysis is directed towards analyzing what we refer to as the separability ansatz, which is an ansatz concerning the manner in which individual quantum operations acting on a quantum system can be approximated. In addition to the primary motivation of this work, we identify several other areas of open research which may benefit from the results we derive here.
Noise in quantum devices is generally considered detrimental to computational accuracy. However, the recent proposal of noise-assisted simulation has demonstrated that noise can be an asset in digital quantum simulations of open systems on noisy intermediate-scale quantum (NISQ) devices. In this context, we introduce an optimized decoherence rate control scheme that can significantly reduce computational requirements by multiple orders of magnitude, in comparison to the original noise-assisted simulation. We further extend this approach to encompass Lindblad equations with time-dependent coefficients, using only quantum error characterization and mitigation techniques. This extension allows for the perturbative simulation of non-Markovian dynamics on NISQ devices, eliminating the need for ancilla qubits or midcircuit measurements. Our contributions are validated through numerical experiments on an emulated IBM Quantum device. Overall, our paper offers valuable optimizations that bring current quantum processors closer to effectively simulating realistic open systems.
Quantum computing is an emergent field promising the improvement of processing speed in key algorithms by reducing their exponential scaling to polynomial, thus enabling solutions to problems that exceed classical computational capabilities. Gate-based quantum computing is the most common approach but still faces high levels of noise and decoherence. Gates play the role of probability mixers codifying information settled in quantum systems. However, they are deviated from their programmed behaviour due to those decoherent effects as a hidden source modifies the desired probability flux. Their quantification of such unavoidable behaviours becomes crucial for quantum error correction or mitigation. This work presents an approach to decoherence in quantum circuits using the Lindblad master equation to model the impact of noise and thermalisation underlying the ideal programmed behaviour expected for processing gates. The Lindblad approach then provides a comprehensive tool to model both probability fluxes being present in the process, thus regarding the gate and the environment. It analyses the deviation of resulting noisy states from the ideal unitary evolution of some gates considered as universal, setting some operating regimes. Thermalisation considers a radiation bath where gates are immersed as a feasible model of decoherence. Numerical simulations track the information loss as a function of the decay rate magnitude. It also exhibits the minimal impact on decoherence coming from particular quantum states being processed, but a higher impact on the number of qubits being processed by the gate. The methodology provides a unified framework to characterise the processing probability transport in quantum gates, including noise or thermalisation effects.
In principle a quantum system could be used to simulate another quantum system. The purpose of such a simulation would be to obtain information about problems which are difficult to simulate on a classical computer due to the exponential increase of the Hilbert space with the size of the system, and which cannot be readily measured or controlled in an experiment. A quantum simulator is, however, an open quantum system that will interact with the surrounding environment, which in this case are other particles in the system, and will be implemented using imperfect controls, making it subject to noise. It has been suggested that noise does not need to be controlled to the same extent as it must be for general quantum information processing. However, the effects of noise in quantum simulations are not well understood and how best to treat them in most cases is not known. In the present work we study an existing quantum algorithm for the simulation of the one-dimensional Fano-Anderson model. This algorithm was proposed for a liquid-state NMR device. We examine models of noise in the evolution using different initial states in the original model. We also add interacting spins to simulate realistic situation where an environment of spins is present. We find that states which are entangled with their environment, and sometimes correlated but not necessarily entangled have an evolution which is described by maps which are not completely positive. We discuss the conditions for this to occur and also the implications.
The study of the impact of noise on quantum circuits is especially relevant to guide the progress of Noisy Intermediate-Scale Quantum (NISQ) computing. In this paper, we address the pulse-level simulation of noisy quantum circuits with the Quantum Toolbox in Python (QuTiP). We introduce new tools in qutip-qip , QuTiP's quantum information processing package. These tools simulate quantum circuits at the pulse level, leveraging QuTiP's quantum dynamics solvers and control optimization features. We show how quantum circuits can be compiled on simulated processors, with control pulses acting on a target Hamiltonian that describes the unitary evolution of the physical qubits. Various types of noise can be introduced based on the physical model, e.g., by simulating the Lindblad density-matrix dynamics or Monte Carlo quantum trajectories. In particular, the user can define environment-induced decoherence at the processor level and include noise simulation at the level of control pulses. We illustrate how the Deutsch-Jozsa algorithm is compiled and executed on a superconducting-qubit-based processor, on a spin-chain-based processor and using control optimization algorithms. We also show how to easily reproduce experimental results on cross-talk noise in an ion-based processor, and how a Ramsey experiment can be modeled with Lindblad dynamics. Finally, we illustrate how to integrate these features with other software frameworks.
Abstract Non-Hermitian quantum dynamics lie in an intermediate regime between unitary Hamiltonian dynamics and trace-preserving non-unitary open quantum system dynamics. Given differences in the noise tolerance of unitary and non-unitary dynamics, it is interesting to consider implementing non-Hermitian dynamics on a noisy quantum computer. In this paper, we do so for a non-Hermitian Ising Floquet model whose many-body dynamics gives rise to persistent temporal oscillations, a form of time crystallinity. In the simplest two qubit case that we consider, there is an infinitely longlived periodic steady state at certain fine-tuned points. Using a generalized Floquet analysis, we show that infinitely long-lived oscillations are generically lost for arbitrarily weak values of common types of noise and compute the corresponding damping rate; this result is confirmed numerically using IBM's Qiskit platform. We further show that the complex gaps responsible for oscillation damping are opened generically at first order in perturbation theory for any convex combination of depolarization, relaxation, and dephasing, establishing that no engineered mixture of these noise channels can stabilize the time-crystalline phase. Experiments on a real device (ibmq-lima) are consistent with rapid oscillation damping, though a significant mismatch between the manufacturer noise model and observed device behavior complicates quantitative comparison, illustrating the importance of rigorous analytical noise analysis.