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De Gruyter eBooks

Large deviations from classical paths and the classical limit of quantum stochastic flows

Ph. Blanchard, Ph. Combe, M. Siruge, M. Sirugue

Published Dec 31, 1987• 0 citations
Publisher
DOI
10.1515/9783112319000-016

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Dynamical fluctuations of classical and quantum square dimer models

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The work in this thesis is split into two main parts; the dynamical study of the fully packed classical dimer model, and the development of advanced large deviation techniques applied to the classical dimer model, highlighting the connection to the quantum dimer model. The classical dimer model on the square lattice is a paradigmatic example of a system subject to strong local constraints. We study its behavior under local stochastic dynamics, by means of Monte Carlo simulations and theoretical arguments. We observe clear signatures of correlated dynamics in both global and local observables and over a broad range of time scales, indicating a breakdown of the simple continuum description that approximates well the statics. We show that this collective dynamics can be understood in terms of one-dimensional “strings” of high mobility. We introduce a coarse-grained description of the strings, which leads to exact results in the limit of low string density and provides a detailed qualitative understanding of the dynamics in all flux sectors. We then go on, still using the fully packed dimers on the square lattice as a paradigmatic system, to study the connection between the phase behaviour of the ground state of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point, a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the quantum Hamiltonian maps to a non-stochastic "tilted" master operator of the classical stochastic problem. This implies a direct relation between the phase behaviour of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. Using transition path sampling (TPS) -- supplemented by trajectory umbrella sampling -- we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behaviour of the classical and quantum systems. Finally, we take these ideas further by developing a feedback-based modified dynamics used for umbrella sampling alongside TPS. This feedback approach was first used in the cloning (or population dynamics) method. We compare different modified dynamics within TPS and show how much more efficient the feedback-modified dynamics can be than the original dynamics, for the purpose of rare event sampling. We also discuss how our approach may generalise to other problems with slow complex dynamics.

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Stochastic Pauli-path simulator for large-scale quantum optimization

Kaining Zhang, Xinbiao Wang, Kunsheng Li +4 more

Pauli-based simulators offer a promising route to large-scale classical simulation of quantum circuits in the low-magic regime. Yet their applicability remains largely limited to forward simulation, making them inadequate for optimization-driven quantum tasks such as variational state preparation and parameter initialization. Existing approaches either lack native support for gradient-based optimization or suffer from severe gradient bias. Here we propose the stochastic Pauli-path simulator (SPPS), a computational framework for large-scale quantum optimization that enables unbiased stochastic gradient estimation via Pauli-path sampling across optimization iterations. Our theoretical analysis shows that the proposed simulator yields unbiased gradient estimates and admits provable convergence guarantees. We systematically evaluate our proposal, including quantum eigensolver benchmarks with up to 100 qubits and quantum neural network benchmarks with up to 40 qubits. Across these tasks, SPPS faithfully tracks optimization dynamics, converges within minutes, and broadens the role of Pauli-based simulation from forward estimation to large-scale quantum optimization.

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Quantum stochastic flows

R. L. Hudson

De Gruyter eBooks•Dec 31, 1990•0 citations
The work in this thesis is split into two main parts; the dynamical study of the fully packed classical dimer model, and the development of advanced large deviation techniques applied to the classical dimer model, highlighting the connection to the quantum dimer model. The classical dimer model on the square lattice is a paradigmatic example of a system subject to strong local constraints. We study its behavior under local stochastic dynamics, by means of Monte Carlo simulations and theoretical arguments. We observe clear signatures of correlated dynamics in both global and local observables and over a broad range of time scales, indicating a breakdown of the simple continuum description that approximates well the statics. We show that this collective dynamics can be understood in terms of one-dimensional “strings” of high mobility. We introduce a coarse-grained description of the strings, which leads to exact results in the limit of low string density and provides a detailed qualitative understanding of the dynamics in all flux sectors. We then go on, still using the fully packed dimers on the square lattice as a paradigmatic system, to study the connection between the phase behaviour of the ground state of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point, a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the quantum Hamiltonian maps to a non-stochastic "tilted" master operator of the classical stochastic problem. This implies a direct relation between the phase behaviour of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. Using transition path sampling (TPS) -- supplemented by trajectory umbrella sampling -- we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behaviour of the classical and quantum systems. Finally, we take these ideas further by developing a feedback-based modified dynamics used for umbrella sampling alongside TPS. This feedback approach was first used in the cloning (or population dynamics) method. We compare different modified dynamics within TPS and show how much more efficient the feedback-modified dynamics can be than the original dynamics, for the purpose of rare event sampling. We also discuss how our approach may generalise to other problems with slow complex dynamics.
Pauli-based simulators offer a promising route to large-scale classical simulation of quantum circuits in the low-magic regime. Yet their applicability remains largely limited to forward simulation, making them inadequate for optimization-driven quantum tasks such as variational state preparation and parameter initialization. Existing approaches either lack native support for gradient-based optimization or suffer from severe gradient bias. Here we propose the stochastic Pauli-path simulator (SPPS), a computational framework for large-scale quantum optimization that enables unbiased stochastic gradient estimation via Pauli-path sampling across optimization iterations. Our theoretical analysis shows that the proposed simulator yields unbiased gradient estimates and admits provable convergence guarantees. We systematically evaluate our proposal, including quantum eigensolver benchmarks with up to 100 qubits and quantum neural network benchmarks with up to 40 qubits. Across these tasks, SPPS faithfully tracks optimization dynamics, converges within minutes, and broadens the role of Pauli-based simulation from forward estimation to large-scale quantum optimization.