In this paper we are discussing the question how a continuous quantum system can be simulated by mean field fluctuations of a finite number of qubits. On the kinematical side this leads to a convergence result which states that appropriately chosen fluctuation operators converge in a certain weak sense (i.e. we are comparing expectation values) to canonical position and momentum Q, P of one-degree of freedom, continuous quantum system. This result is substantially stronger than existing methods which rely either on central limit theorem arguments (and are therefore restricted to the Gaussian world) or are valid only if the states of the ensembles are close to the "fully polarized" state. Dynamically this relationship keeps perfectly intact (at least for small times) as long as the continuous system evolves according to a quadratic Hamiltonian. In other words we can approximate the corresponding (Heisenberg picture) time evolution of the canonical operators Q, P up to arbitrary accuracy by the appropriately chosen time evolution of fluctuation operators of the finite systems.In this paper we are discussing the question how a continuous quantum system can be simulated by mean field fluctuations of a finite number of qubits. On the kinematical side this leads to a convergence result which states that appropriately chosen fluctuation operators converge in a certain weak sense (i.e. we are comparing expectation values) to canonical position and momentum Q, P of one-degree of freedom, continuous quantum system. This result is substantially stronger than existing methods which rely either on central limit theorem arguments (and are therefore restricted to the Gaussian world) or are valid only if the states of the ensembles are close to the "fully polarized" state. Dynamically this relationship keeps perfectly intact (at least for small times) as long as the continuous system evolves according to a quadratic Hamiltonian. In other words we can approximate the corresponding (Heisenberg picture) time evolution of the canonical operators Q, P up to arbitrary accuracy by the appropriately chosen time evolution of fluctuation operators of the finite systems.

arXiv:1211.2173Nov 9, 20121 citation

Emulation of large-scale qubit registers with a phase-space approach

Christian de Correc, Denis Lacroix, Corentin Bertrand

A phase-space approach is used and benchmarked for the simulation of the continuous-time evolution of large registers of qubits. It is based on a statistical ensemble of independent mean-field trajectories, where mean field is introduced at the level of the qubits, substituting quantum fluctuations/correlations with classical ones. The approach only involves at worse a quadratic cost in the system size, allowing to simulate up to several thousands of qubits on a classical computer. It provides qualitatively accurate description of one-qubit observables evolutions, making it a useful reference in comparison to techniques limited to small qubit numbers. The predictive power is, however, less robust for multi-qubits observables. We benchmark the method on the $k$ -local transverse-field Ising model, considering a large variety of systems ranging from local to all-to-all interactions, and from weak to strong coupling regimes, with up to 2000 qubits. To showcase the versatility of the approach, simulations on 2D and 3D Ising models are also made.A phase-space approach is used and benchmarked for the simulation of the continuous-time evolution of large registers of qubits. It is based on a statistical ensemble of independent mean-field trajectories, where mean field is introduced at the level of the qubits, substituting quantum fluctuations/correlations with classical ones. The approach only involves at worse a quadratic cost in the system size, allowing to simulate up to several thousands of qubits on a classical computer. It provides qualitatively accurate description of one-qubit observables evolutions, making it a useful reference in comparison to techniques limited to small qubit numbers. The predictive power is, however, less robust for multi-qubits observables. We benchmark the method on the $k$-local transverse-field Ising model, considering a large variety of systems ranging from local to all-to-all interactions, and from weak to strong coupling regimes, with up to 2000 qubits. To showcase the versatility of the approach, simulations on 2D and 3D Ising models are also made.

quant-pharXiv:2602.10830APS Open ScienceFeb 12, 20260 citations

Ground state fluctuations of quantum mean field systems

A.K. Callebaut, M. Fannes

QP-PQ, quantum probability and white noise analysisJul 4, 19941 citation

Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics

Igor Rodnianski, Benjamin Schlein

arXiv:0711.3087Communications in Mathematical PhysicsJul 8, 2009231 citations

Quantum Dynamics with Mean Field Interactions: a New Approach

László Erdős, Benjamin Schlein

arXiv:0804.3774Journal of Statistical PhysicsMar 1, 200935 citations

Complex Langevin simulation of Bose-Einstein condensate under extreme conditions

智也早田

In this thesis, we perform ab-initio lattice simulations of the nonrelativistic and relativistic Bose-Einstein condensates under strong external fields such as the electric field, the magnetic field, and the rotation.All the lattice actions discussed in this thesis are complex, so that the conventional Monte Carlo method su↵ers from the notorious sign problem.The path integral measure is complex due to the complex action, so that the important sampling algorithm cannot be applied.To overcome this di culty, we adopt the complex Langevin method, which has been developed in the context of finite density quantum chromodynamics to attack the fermion sign problem.In particular, we perform ab-initio simulation of the quantum vortex nucleation by the magnetic field or the rotation.In dilute and low temperature system, quantum and thermal fluctuations are negligible.The system can be remarkably well described by the mean field approximation.However, when quantum or thermal fluctuation becomes large, it is highly nontrivial how the quantized vortices behave.Around the critical values of temperature, chemical potential, magnetic field, or angular velocity, the fluctuation grows and the mean-field description inevitably breaks down.To understand the quantum vortex nucleation in such situations, the ab-initio simulation is necessary as discussed in this thesis.The direct evidence of the quantum vortex nucleation is the quantization of circulation.We show that the circulation is quantized in the superconducting (superfluid) phase far from the critical chemical potential, but it is not just above the critical chemical potential.We observe that the quantized circulation is blurred by the quantum fluctuation as the chemical potential is getting close to its critical value.In our simulation, the quantum fluctuation of vortex number is observed, while the averaged circulation is clearly quantized.At first glance, these two facts seem to be incompatible.However, we show that the fluctuation of vortex number behaves as Gaussian and, as a result of cancellation, the average circulation becomes exact integer.iiiIn this thesis, we perform ab-initio lattice simulations of the nonrelativistic and relativistic Bose-Einstein condensates under strong external fields such as the electric field, the magnetic field, and the rotation.All the lattice actions discussed in this thesis are complex, so that the conventional Monte Carlo method su↵ers from the notorious sign problem.The path integral measure is complex due to the complex action, so that the important sampling algorithm cannot be applied.To overcome this di culty, we adopt the complex Langevin method, which has been developed in the context of finite density quantum chromodynamics to attack the fermion sign problem.In particular, we perform ab-initio simulation of the quantum vortex nucleation by the magnetic field or the rotation.In dilute and low temperature system, quantum and thermal fluctuations are negligible.The system can be remarkably well described by the mean field approximation.However, when quantum or thermal fluctuation becomes large, it is highly nontrivial how the quantized vortices behave.Around the critical values of temperature, chemical potential, magnetic field, or angular velocity, the fluctuation grows and the mean-field description inevitably breaks down.To understand the quantum vortex nucleation in such situations, the ab-initio simulation is necessary as discussed in this thesis.The direct evidence of the quantum vortex nucleation is the quantization of circulation.We show that the circulation is quantized in the superconducting (superfluid) phase far from the critical chemical potential, but it is not just above the critical chemical potential.We observe that the quantized circulation is blurred by the quantum fluctuation as the chemical potential is getting close to its critical value.In our simulation, the quantum fluctuation of vortex number is observed, while the averaged circulation is clearly quantized.At first glance, these two facts seem to be incompatible.However, we show that the fluctuation of vortex number behaves as Gaussian and, as a result of cancellation, the average circulation becomes exact integer.iii

Mar 24, 20150 citations

k

-local transverse-field Ising model, considering a large variety of systems ranging from local to all-to-all interactions, and from weak to strong coupling regimes, with up to 2000 qubits. To showcase the versatility of the approach, simulations on 2D and 3D Ising models are also made.