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Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems
Alexandra Nagy, Vincenzo Savona
TL;DR
Simulating many-body open quantum systems is difficult because density matrices grow exponentially with system size. This paper develops a neural-network variational Monte Carlo method whose stochastic reconfiguration integrates the quantum master equation and models a dissipative XYZ lattice.
Problem
Efficiently representing the density matrix of many-body open quantum systems remains challenging as the Hilbert-space dimension grows exponentially with system size.
Method
The method combines a neural-network density-matrix ansatz, variational Monte Carlo sampling, and stochastic reconfiguration in Liouville space.
Results
Under general assumptions, the stochastic-reconfiguration dynamics asymptotically converges to the non-equilibrium steady state.
Takeaways & Limitations
The approach provides a variational-subspace interpretation of real-time evolution generated by the Liouvillian.
Abstract
from arXiv · showhide
The possibility to simulate the properties of many-body open quantum systems with a large number of degrees of freedom is the premise to the solution of several outstanding problems in quantum science and quantum information. The challenge posed by this task lies in the complexity of the density matrix increasing exponentially with the system size. Here, we develop a variational method to efficiently simulate the non-equilibrium steady state of Markovian open quantum systems based on variational Monte Carlo and on a neural network representation of the density matrix. Thanks to the stochastic reconfiguration scheme, the application of the variational principle is translated into the actual integration of the quantum master equation. We test the effectiveness of the method by modeling the two-dimensional dissipative XYZ spin model on a lattice.
SUPPLEMENTAL MATERIAL · NEURAL NETWORK REPRESENTATION OF THE DENSITY MATRIX
The supplemental material represents mixed-state density matrices with a positive-semidefinite neural-network ansatz based on RBMs. It also specifies the parameterization and logarithmic derivatives required for stochastic reconfiguration.
- SUPPLEMENTAL MATERIAL: The density matrix is treated as a parameterized mapping ρχ that returns matrix elements between computational-basis states.The basis uses N two-level degrees of freedom, with σ_i ∈ {−1, 1}, covering spin-1/2 systems and qubits.
- NEURAL NETWORK REPRESENTATION OF THE DENSITY MATRIX: The mixed state is expressed as a statistical mixture of J wave functions with probabilities satisfying 0 ≤ p_j(χ) ≤ 1.The component states need not be mutually orthogonal, and J ≤ d with d = 2^N.
- NEURAL NETWORK REPRESENTATION OF THE DENSITY MATRIX: Each wave function is represented by a Restricted Boltzmann Machine with M = α × N hidden spin variables.The hidden variables take values q_m = ±1, while couplings and biases may be complex-valued.
- NEURAL NETWORK REPRESENTATION OF THE DENSITY MATRIX: A second hidden-spin set of size Lβ × N represents the mixture probabilities p_j(χ) in RBM form.These hidden nodes also enter the wave-function representation to index the components of the mixed state.
- NEURAL NETWORK REPRESENTATION OF THE DENSITY MATRIX: Because intra-layer connections are forbidden, the hidden variables can be traced out explicitly to obtain the neural-network density matrix.This yields the explicit RBM density-matrix form proposed in the supplemental material.
- NEURAL NETWORK REPRESENTATION OF THE DENSITY MATRIX: The variational parameters are χ = {a_i, b_m, X_mi, c_l, W_li}, with all parameters except c_l generally complex-valued.The ansatz distinguishes the real-valued c_l parameters from the remaining complex-valued variational parameters.
- NEURAL NETWORK REPRESENTATION OF THE DENSITY MATRIX: The RBM contains N_p = N[(α + β)(2N + 1) + α + 2] real-valued computational parameters when real and imaginary parts are counted separately.The supplemental material then introduces effective angles for the variational description.
- NEURAL NETWORK REPRESENTATION OF THE DENSITY MATRIX: Stochastic Reconfiguration is used to optimize the steady-state parameters by calculating logarithmic derivatives of ρχ with respect to all real and imaginary variational components.The derivatives are obtained from the explicit neural-network density-matrix expression.
STOCHASTIC RECONFIGURATION FOR OPEN QUANTUM SYSTEMS
The method extends stochastic reconfiguration to a neural-network density-matrix ansatz by matching variational parameter updates to Liouvillian real-time evolution. Regularization stabilizes the covariance-matrix solve when it is non-invertible.
- Variational representation: The density matrix is represented by a neural-network ansatz and vectorized so the Liouvillian becomes a d^2 × d^2 matrix.Here, d is the Hilbert-space dimension.
- Variational dynamics: Stochastic reconfiguration extends to open systems by equating the master-equation evolution with motion in the variational subspace spanned by {O_k|ρ_χ⟩}.The resulting dynamics asymptotically approaches the non-equilibrium steady state under general assumptions.
- Regularization: The covariance matrix defines the optimization metric: choosing the identity yields steepest descent, while non-invertibility requires explicit regularization.The covariance matrix may be any positive-definite matrix.
- Regularization: λ(n) decays as max(λ0b^n, λmin), with λ0 = 100, b = 0.998, and λmin = 10^-2 in the calculations.The regularization parameter depends on the iteration step n.
STOCHASTIC SAMPLING
Stochastic sampling represents vectorized density-matrix configurations with a Metropolis-Hastings Markov chain sampling |ρχ(σ, η)|², using Liouvillian-guided moves to prioritize important configurations. The move set includes hopping, excitation, dissipative, and jumper updates selected with uniform proposition probability, with excitation proposed at ten-percent probability.
- Sampling scheme: A Metropolis-Hastings Markov chain samples many-body basis configurations according to |ρχ(σ, η)|².The configurations span the vectorized density-matrix space and are generated as a chain of NMH states.
- Sampling scheme: Liouvillian-guided transitions prioritize high-importance configurations and reduce required thermalization and sampling steps.Moves are based on transitions generated by applying the Liouvillian superoperator L|x⟩.
- Move proposals: The move set includes column- and row-hopping updates in both σ and η, along with single-site excitation moves.Hopping flips neighboring sites, while excitation flips one site in the corresponding variable.
- Move proposals: Dissipator moves flip sites with σj = ηj using an asymmetrical transition ratio, while excitation moves are proposed with ten-percent probability.Dissipative moves are always proposed.
- Move proposals: Jumper moves flip two randomly chosen spins, and all move types are selected with uniform probability.The jumper update flips σl and ηm.
DETAIL ON NUMERICAL APPROACH
The VMC implementation was written in Python and parallelized across independent Metropolis–Hastings chains. CPU and GPU versions were developed, with GPU calculations accelerating expectation-value updates and SR linear-system solves.
- Implementation: The VMC code was written in Python, and Metropolis–Hastings sampling was parallelized across several independent chains using MVAPICH2.The chains were run in parallel.
- Implementation: Both CPU and GPU versions of the code were developed.
- GPU acceleration: GPU calculations significantly accelerated updates of Ok expectation values and iterative SR linear-system solutions using MINRES-QLP.MINRES-QLP was used for the iterative solution of the linear system in stochastic reconfiguration.