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Synthesis lectures on engineering, science, and technology

Deterministic Simulation of Noise

Thomas Grurl, Jürgen Fuß, Robert Wille

Published Nov 29, 2024• 0 citations
Publisher
DOI
10.1007/978-3-031-71036-0_5

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We develop analytical and algorithmic techniques that enable efficient simulation of a broad class of noisy stabilizer circuits. We derive closed-form expressions of expectation values for tensor product of Paulis in circuits with non-deterministic Pauli measurements, yielding an efficient strong simulation method that avoids explicit density matrix construction and enables direct noise parameter sweeps. We introduce a circuit compression framework that reduces the per-sample cost of weak simulation in general noisy stabilizer circuits, including deterministic measurements, by separating parameter-independent preprocessing from sampling. Finally, we extend the analytical framework beyond its standard domain to include a small number of deterministic measurements, general rotations, and non-diagonal noise channels. Our results provide a unified framework for both strong and weak simulation of noisy stabilizer circuits and corresponds to an extension of the noisy stabilizer formalism introduced in \cite{PhysRevA.107.032424}. They offer applications ranging from calculation of the expectation values of entanglement witnesses, determination of reduced states, to energy evaluation.

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Abstract Quantum decoherence, conventionally modeled as a stochastic process, remains the primary obstacle to scalable quantum computation. In this work, a conceptual framework grounded in the hydrodynamic formulation of quantum mechanics (Madelung-Bohm formalism) is presented, which reinterprets decoherence as a process governed by deterministic flow dynamics rather than fundamental randomness. A comprehensive numerical simulation of a qubit coupled to an environmental oscillator at strong coupling ( c = 1.5) has been performed, from which an ensemble of Bohmian trajectories has been extracted and analyzed. The strong coupling regime ensures conclusive, unambiguous demonstration of complete decoherence within accessible simulation times, with system purity exhibiting clear, monotonic decay from a pure to a maximally mixed state without residual oscillations. The results provide direct visualization of decoherence as a continuous dynamical evolution while perfectly reproducing the statistical predictions of standard open quantum systems theory, including Born rule agreement and decoherence-induced pointer state formation. The key finding is that the environmental influence—the ‘decoherence force’ F dec = − ∇ Q derived from the quantum potential—possesses a deterministic, non-random structure with identifiable spectral signatures. This insight provides the theoretical foundation for a class of Noise-Informed Control Strategies . It is argued that by experimentally characterizing this deterministic signature via quantum noise spectroscopy, bespoke control protocols can be designed to suppress or manipulate decoherence. This work shifts the perspective from combating fundamentally random noise to controlling a structured, deterministic interaction, aligning with emerging paradigms in quantum error correction and the information-selection principles of quantum Darwinism.

Physica Scripta•Feb 4, 2026•1 citation
We develop analytical and algorithmic techniques that enable efficient simulation of a broad class of noisy stabilizer circuits. We derive closed-form expressions of expectation values for tensor product of Paulis in circuits with non-deterministic Pauli measurements, yielding an efficient strong simulation method that avoids explicit density matrix construction and enables direct noise parameter sweeps. We introduce a circuit compression framework that reduces the per-sample cost of weak simulation in general noisy stabilizer circuits, including deterministic measurements, by separating parameter-independent preprocessing from sampling. Finally, we extend the analytical framework beyond its standard domain to include a small number of deterministic measurements, general rotations, and non-diagonal noise channels. Our results provide a unified framework for both strong and weak simulation of noisy stabilizer circuits and corresponds to an extension of the noisy stabilizer formalism introduced in \cite{PhysRevA.107.032424}. They offer applications ranging from calculation of the expectation values of entanglement witnesses, determination of reduced states, to energy evaluation.
Simulations of quantum systems with Hamiltonian classical stochastic noise can be challenging when the noise exhibits temporal correlations over a multitude of time scales, such as for 1 / f noise in solid-state quantum information processors. Here we present an approach for simulating Hamiltonian classical stochastic noise that performs temporal coarse-graining by effectively integrating out the high-frequency components of the noise. We focus on the case where the stochastic noise can be expressed as a sum of Ornstein-Uhlenbeck processes. Temporal coarse-graining is then achieved by conditioning the stochastic process on a coarse realization of the noise, expressing the conditioned stochastic process in terms of a sum of smooth, deterministic functions and bridge processes with boundaries fixed at zero, and performing the ensemble average over the bridge processes. For Ornstein-Uhlenbeck processes, the deterministic components capture all dependence on the coarse realization, and the stochastic bridge processes are not only independent but taken from the same distribution with correlators that can be expressed analytically, allowing the associated noise propagators to be precomputed once for all simulations. This combination of noise trajectories on a coarse time grid and ensemble averaging over bridge processes has practical advantages, such as a simple concatenation rule, that we highlight with numerical examples.
Abstract Quantum decoherence, conventionally modeled as a stochastic process, remains the primary obstacle to scalable quantum computation. In this work, a conceptual framework grounded in the hydrodynamic formulation of quantum mechanics (Madelung-Bohm formalism) is presented, which reinterprets decoherence as a process governed by deterministic flow dynamics rather than fundamental randomness. A comprehensive numerical simulation of a qubit coupled to an environmental oscillator at strong coupling ( c = 1.5) has been performed, from which an ensemble of Bohmian trajectories has been extracted and analyzed. The strong coupling regime ensures conclusive, unambiguous demonstration of complete decoherence within accessible simulation times, with system purity exhibiting clear, monotonic decay from a pure to a maximally mixed state without residual oscillations. The results provide direct visualization of decoherence as a continuous dynamical evolution while perfectly reproducing the statistical predictions of standard open quantum systems theory, including Born rule agreement and decoherence-induced pointer state formation. The key finding is that the environmental influence—the ‘decoherence force’ F dec = − ∇ Q derived from the quantum potential—possesses a deterministic, non-random structure with identifiable spectral signatures. This insight provides the theoretical foundation for a class of Noise-Informed Control Strategies . It is argued that by experimentally characterizing this deterministic signature via quantum noise spectroscopy, bespoke control protocols can be designed to suppress or manipulate decoherence. This work shifts the perspective from combating fundamentally random noise to controlling a structured, deterministic interaction, aligning with emerging paradigms in quantum error correction and the information-selection principles of quantum Darwinism.