Stochastic limits of Quantum repeated measurements

Abstract: We investigate quantum systems perturbed by noise in the form of repeated interactions between the system and the environment. As the number of interactions (aka time steps) tends to infinity, we show, following the works by Pellegrini, that this system converges to the solution of a Volterra stochastic differential equation. This development sets interesting future research paths at the intersection of quantum algorithms, stochastic differential equations, weak convergence and large deviations.

• The paper's main contribution is the rigorous derivation of stochastic differential equations, including the diffusive Belavkin equation, from quantum repeated measurements.
• It employs quantum stochastic calculus, weak convergence theory, and large deviations to bridge discrete quantum trajectories with continuous Markov and non-Markov models.
• The analysis reveals robustness to unitary imperfections and quantifies memory effects, offering practical insights for quantum error mitigation in noisy devices.

Stochastic Limits and Memory Effects in Quantum Repeated Measurements

Introduction

The paper "Stochastic limits of Quantum repeated measurements" (2512.11462) provides a comprehensive analysis of the asymptotic behavior of open quantum systems undergoing repeated indirect measurements. The principal mathematical focus is on the rigorous derivation of continuous stochastic models—including both Markovian and non-Markovian (memory) limits—from physically motivated discrete-time quantum trajectories. The analysis builds on quantum stochastic calculus, weak convergence theory, and large deviations, with direct relevance for quantum computation in NISQ devices, where realistic quantum circuits are influenced by adversarial noise and partial decoherence.

Discrete Quantum Trajectories and Convergence to Stochastic Equations

The fundamental setting involves a small quantum system described by a finite-dimensional Hilbert space, interacting sequentially with an infinite chain of identical environment units. Each interaction is mediated via a unitary operator, post-composed with a projective measurement on the environment, leading to a stochastic evolution of the system's state—realized as a Markov chain on the space of density operators.

Precise scaling limits are established by discretizing the time axis into intervals of size $\tau = \frac{1}{n}$, letting $n \to \infty$. By expanding the relevant block entries of the system-environment unitary and enforcing coherent Hamiltonian and dissipation generators, the authors show that the sequence of quantum trajectories converges—under Skorokhod topology and in expectation—to the solution of a stochastic differential equation (SDE) of Belavkin (stochastic Schrödinger) type. For generic non-diagonal measurement observables, one obtains the diffusive Belavkin equation:

$$d \rho_t = L(\rho_t) dt + [\rho_t C^\dagger + C\rho_t - \operatorname{Tr}(\rho_t(C+C^\dagger))\rho_t ] dW_t,$$

where $L$ is the associated Lindblad generator and $W_t$ a standard Brownian motion. Uniqueness in strong solutions is established.

This construction gives a rigorous foundation to the stochastic (Markovian) quantum trajectory model widely employed in quantum optics, quantum feedback, and quantum information theory. The discrete-to-continuum convergence is established in both expectation (to the Lindblad ODE) and in pathwise distribution (to the Belavkin SDE).

Perturbation Analysis and Robustness to Unitary Imperfections

The robustness of these stochastic limits with respect to perturbations in the unitary coupling is analyzed from two perspectives: alternation of different unitaries (modeling, e.g., fluctuating control fields or circuit compilation errors), and explicit quantum channel noise (as for erroneous gates in quantum hardware).

For alternating unitary dynamics (e.g., the interaction alternates deterministically between $U^+$ and $U^-$ at each step), the authors show that in the continuum limit the quantum trajectory solves a Belavkin-type SDE driven by combined Lindbladians and noise couplings:

$$d\rho_t = \tfrac{1}{2}[L^+(\rho_t) + L^-(\rho_t)]dt + \frac{1}{\sqrt{2}} \left(\Theta^+(\rho_t)dW_t^+ + \Theta^-(\rho_t)dW_t^- \right),$$

with $W_t^\pm$ independent Brownian motions.

For explicit noise modeled via quantum channels (convex combinations of unitaries and standard error models such as depolarizing, bit-flip, or phase-flip channels), the system is coupled with a noisy channel at each step. The authors precisely engineer the block decomposition of the system-environment unitary so that, in the scaling limit, the average state evolution converges (in $L^\infty$) to the semigroup generated by the quantum channel, and the fluctuations converge to a noisy SDE:

$$d\rho_t^\epsilon = \Omega^\epsilon(\rho_t^\epsilon)dt + \sqrt{\epsilon} \Xi(\rho_t^\epsilon) d\widetilde{B} dW_t,$$

where $\Omega^\epsilon$ implements the perturbed channel and $\Xi$ is constructed from Kraus operators. As $\epsilon \to 0$, strong control on the deviation between the perturbed and ideal processes is obtained; a large deviations principle quantifies the exponentially small likelihood of significant excursions in the quantum state from the noise-free trajectory.

Non-Markovian Limits: Quantum Memory and Volterra SDEs

A significant extension addressed by the paper is the rigorous derivation of quantum trajectory models where the future evolution depends not only on the current state, but also on the entire history—a scenario motivated by physical situations with non-negligible environmental memory (e.g., super-Ohmic environments, collision models with inter-environment interactions).

The discrete memory model relies on sequential swaps between neighboring environment units, introducing controlled memory with parameter $p=p(n)$. In the continuum limit, the reduced state evolution is governed by an integro-differential Volterra equation:

$$\phi_t = \Gamma \int_0^t e^{-\Gamma(t-s)} \mathcal{U}_{t-s}[\phi_s] ds + e^{-\Gamma t} \mathcal{U}_t[\rho_0]$$

(in the deterministic case) or, in the stochastic version driven by repeated measurement with noisy resetting,

$$\rho_t = \rho_0 + \int_0^t e^{-\Gamma(t-s)} L(\rho_s) ds + \int_0^t e^{-\Gamma(t-s)} \Theta_C(\rho_s) dW_s.$$

This SDE with memory kernel enriches the quantum trajectory framework by embedding non-Markovian effects, allowing for the modeling of colored noise, retardation, and environments with structured correlations. The derivations specify the functional analytic structure required for the coefficients and kernels to ensure well-posedness and convergence.

Practical and Theoretical Implications

The rigorous connection between physically realized discrete quantum processes and their stochastic (Markovian and non-Markovian) continuous models is germane for the simulation, analysis, and control of noisy quantum devices. The results elucidate how imperfections in gates, system-environment interactions, or even the protocol for measurement inject specific types of randomness and memory into the system, and precisely map these imperfections to terms in the limit SDE. The work establishes fluctuation bounds on noise-induced deviations, which can inform quantum error mitigation, the design of fault-tolerant protocols, and the benchmarking of stochastic simulation algorithms.

Of particular interest for future developments is the analytically tractable construction of non-Markovian quantum stochastic models from simple discrete protocols, suggesting novel routes for the simulation and (potentially) experimental realization of quantum systems with engineered memory. The large deviation results lay ground for rare-event sampling and error analysis beyond the standard weak noise regime.

Conclusion

The paper provides a mathematically robust and comprehensive analysis of the stochastic and memory-enriched limits of quantum repeated measurement models. The main contributions include the derivation of continuous-time Markovian and non-Markovian (Volterra) SDEs from microscopic repeated interaction quantum models, quantitative analysis of the effects of realistic imperfections in the quantum channels, and large deviation principles quantifying the probabilities of rare noise-induced deviations. These results furnish a rigorous toolkit for connecting quantum measurement protocols, stochastic open quantum dynamics, and the theory of quantum information under noise, with clear implications for the design and analysis of future quantum technologies.