Source-linked AI summary
Neural-Network Approach to Dissipative Quantum Many-Body Dynamics
Michael J. Hartmann, Giuseppe Carleo
TL;DR
Computing expectation values in the variational treatment requires careful sampling because certain variance estimates can diverge. The paper uses Monte Carlo sampling and stochastic reconfiguration, establishing that the resulting solution minimizes the update measure under a positive-semidefinite matrix condition.
Problem
Sampling choices for estimating expectation values require scrutiny because variance estimates can diverge when off-diagonal density-matrix elements approach zero.
Method
The approach computes expectation values stochastically via Monte Carlo sampling and constructs stochastic-reconfiguration equations using a local Liouvillian.
Results
The derivation shows that the stochastic-reconfiguration solution minimizes the update measure because the matrix S is positive semidefinite.
Takeaways & Limitations
The supported derivation provides a minimum-based update rule for the stochastic-reconfiguration treatment of the dynamics.
Takeaways & Limitations
The sampling-based estimator can become problematic because its variance may diverge as off-diagonal density-matrix elements approach zero.
Abstract
from arXiv · showhide
In experimentally realistic situations, quantum systems are never perfectly isolated and the coupling to their environment needs to be taken into account. Often, the effect of the environment can be well approximated by a Markovian master equation. However, solving this master equation for quantum many-body systems, becomes exceedingly hard due to the high dimension of the Hilbert space. Here we present an approach to the effective simulation of the dynamics of open quantum many-body systems based on machine learning techniques. We represent the mixed many-body quantum states with neural networks in the form of restricted Boltzmann machines and derive a variational Monte-Carlo algorithm for their time evolution and stationary states. We document the accuracy of the approach with numerical examples for a dissipative spin lattice system.
SUPPLEMENTAL MATERIAL · Proof of Minimum
The supplemental material outlines a proof that the solution to Eq. (5) minimizes δ. The argument uses the real coefficients ˙α_k and shows that the matrix S is positive semidefinite.
- Proof of Minimum: The proof begins by expressing S and f⃗ in the stated form.
- Proof of Minimum: The coefficients ˙α_k are real in the proof.
- Proof of Minimum: The argument considers any vector v_k with real elements.
- Proof of Minimum: The derivation includes the term ∂⃗ρ ∂α_k v_k.
- Proof of Minimum: The matrix S is therefore positive semidefinite.
- Proof of Minimum: The proof concludes that the solution to Eq. (5) is indeed the minimum of δ.
- Proof of Minimum: The vector ν⃗ is defined through the expression ν⃗ = P k.
Fubini-Study norm
The Fubini–Study norm provides an alternative variational formulation to the 2-norm, yielding the same parameter-evolution ODE structure with modified S and f definitions.
- Fubini-Study norm: The approximation can alternatively be derived by minimizing the Fubini–Study norm rather than the 2-norm.The passage introduces this as an alternative variational approach.
- Fubini-Study norm: This formulation leads to the same system of ODEs as Eq. (7), but with different expressions for S and f.The parameter definitions include k ˙αkOk⃗ρ and µ = L⃗ρ.
Logarithmic derivatives
The logarithmic derivatives involve sums of exponentially many terms, so their expectation values are computed stochastically using Monte Carlo sampling.
- Logarithmic derivatives: The logarithmic derivatives involve sums of exponentially many terms.
- Logarithmic derivatives: All expectation values are computed stochastically via Monte Carlo sampling.
Sampling for expectation values
Expectation values require sampling from the distribution formed by the density matrix’s diagonal elements, rather than from p(⃗l,⃗m). Sampling p(⃗l,⃗m) can be problematic because its variance may diverge when off-diagonal elements vanish.
- Sampling for expectation values: Expectation values of observables require sampling from the distribution formed by the density matrix’s diagonal elements.The passage states that sampling over p(⃗l,⃗m) would not work for computing these expectation values.
- Sampling for expectation values: Sampling over p(⃗l,⃗m) would require finding its normalization, which may be flawed by statistical behavior.
- Sampling for expectation values: The variance could diverge for |ρ⃗l,⃗m| →0, creating problems under the Schwarz inequality |ρ⃗l,⃗m|2 ≤ ρ⃗l,⃗lρ⃗m,⃗m.
Local estimator of the Liouvillian
The local Liouvillian is introduced as the quantity needed to compute the stochastic-reconfiguration matrices S and f. Its Hamiltonian-commutator contribution can be evaluated efficiently for k-local Hamiltonians, with one-dimensional expressions given for the model considered.
- Local estimator: The stochastic-reconfiguration matrices S and f require evaluation of the local Liouvillian.The construction begins with the contribution −i[H, ρ].
- Local estimator: For k-local Hamiltonians, the local Liouvillian can be computed with the same complexity as the local energy in standard ground-state variational calculations.This establishes an efficient estimator for the Hamiltonian contribution.
- Local estimator: For the Hamiltonian in Eq. (14), the authors provide a one-dimensional form using spin variables n_j = ±1.The subsequent notation uses Kronecker deltas and represents the density-matrix indices with n = 2N.
Moves of the Metropolis Sampling
The Metropolis sampler for p(l⃗,r⃗) used four move types, while sampling q(l⃗,l⃗) used three. Local flips were equally likely within each scheme, whereas uniform-resampling moves occurred 100 times less often.
- Moves of the Metropolis Sampling: Sampling p(l⃗,r⃗) used four moves: single-index flips, same-site left/right flips, neighboring-index flips, and uniform redraws.The neighboring-index move flips either adjacent left indices or adjacent right indices.
- Moves of the Metropolis Sampling: For p(l⃗,r⃗), moves 1–3 had the same probability, while move 4 was chosen 100 times less often.Move 4 draws a new configuration from a uniform distribution.
- Moves of the Metropolis Sampling: Sampling q(l⃗,l⃗) used three moves: single-index flips, neighboring-index flips, and uniform redraws.The local moves flip one index or two neighboring indices.
- Moves of the Metropolis Sampling: For q(l⃗,l⃗), the uniform-redraw move was chosen to occur 100 times less often than the other two.The other two moves were the single-index and neighboring-index flips.