Source code for netket.experimental.driver.tdvp_schmitt

# Copyright 2020, 2021  The NetKet Authors - All rights reserved.
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# See the License for the specific language governing permissions and
# limitations under the License.

from typing import Callable, Union, Optional

from functools import partial

import jax
import jax.numpy as jnp

from netket import stats
from netket.operator import AbstractOperator
from netket.optimizer.qgt import QGTJacobianDense
from netket.optimizer.qgt.qgt_jacobian_dense import convert_tree_to_dense_format
from netket.vqs import VariationalState, VariationalMixedState, MCState
from netket.jax import tree_cast
from netket.utils import timing

from netket.experimental.dynamics import RKIntegratorConfig

from .tdvp_common import TDVPBaseDriver, odefun

[docs] class TDVPSchmitt(TDVPBaseDriver): r""" Variational time evolution based on the time-dependent variational principle which, when used with Monte Carlo sampling via :class:`netket.vqs.MCState`, is the time-dependent VMC (t-VMC) method. This driver, which only works with standard MCState variational states, uses the regularization procedure described in `M. Schmitt's PRL 125 <>`_ . With the force vector .. math:: F_k=\langle \mathcal O_{\theta_k}^* E_{loc}^{\theta}\rangle_c and the quantum Fisher matrix .. math:: S_{k,k'} = \langle \mathcal O_{\theta_k} (\mathcal O_{\theta_{k'}})^*\rangle_c and for real parameters :math:`\theta\in\mathbb R`, the TDVP equation reads .. math:: q\big[S_{k,k'}\big]\theta_{k'} = -q\big[xF_k\big] Here, either :math:`q=\text{Re}` or :math:`q=\text{Im}` and :math:`x=1` for ground state search or :math:`x=i` (the imaginary unit) for real time dynamics. For ground state search a regularization controlled by a parameter :math:`\rho` can be included by increasing the diagonal entries and solving .. math:: q\big[(1+\rho\delta_{k,k'})S_{k,k'}\big]\theta_{k'} = -q\big[F_k\big] The `TDVP` class solves the TDVP equation by computing a pseudo-inverse of :math:`S` via eigendecomposition yielding .. math:: S = V\Sigma V^\dagger with a diagonal matrix :math:`\Sigma_{kk}=\sigma_k` Assuming that :math:`\sigma_1` is the smallest eigenvalue, the pseudo-inverse is constructed from the regularized inverted eigenvalues .. math:: \tilde\sigma_k^{-1}=\frac{1}{\Big(1+\big(\frac{\epsilon_{SVD}}{\sigma_j/\sigma_1}\big)^6\Big)\Big(1+\big(\frac{\epsilon_{SNR}}{\text{SNR}(\rho_k)}\big)^6\Big)} with :math:`\text{SNR}(\rho_k)` the signal-to-noise ratio of :math:`\rho_k=V_{k,k'}^{\dagger}F_{k'}` (see `[arXiv:1912.08828] <>`_ for details). .. note:: This TDVP Driver uses the time-integrators from the `nkx.dynamics` module, which are automatically executed under a `jax.jit` context. When running computations on GPU, this can lead to infinite hangs or extremely long compilation times. In those cases, you might try setting the configuration variable `nk.config.netket_experimental_disable_ode_jit = True` to mitigate those issues. """
[docs] def __init__( self, operator: AbstractOperator, variational_state: VariationalState, integrator: RKIntegratorConfig, *, t0: float = 0.0, propagation_type: str = "real", holomorphic: Optional[bool] = None, diag_shift: float = 0.0, diag_scale: Optional[float] = None, error_norm: Union[str, Callable] = "qgt", rcond: float = 1e-14, rcond_smooth: float = 1e-8, snr_atol: float = 1, ): r""" Initializes the time evolution driver. Args: operator: The generator of the dynamics (Hamiltonian for pure states, Lindbladian for density operators). variational_state: The variational state. integrator: Configuration of the algorithm used for solving the ODE. t0: Initial time at the start of the time evolution. propagation_type: Determines the equation of motion: "real" for the real-time Schrödinger equation (SE), "imag" for the imaginary-time SE. error_norm: Norm function used to calculate the error with adaptive integrators. Can be either "euclidean" for the standard L2 vector norm :math:`w^\dagger w`, "maximum" for the maximum norm :math:`\max_i |w_i|` or "qgt", in which case the scalar product induced by the QGT :math:`S` is used to compute the norm :math:`\Vert w \Vert^2_S = w^\dagger S w` as suggested in PRL 125, 100503 (2020). Additionally, it possible to pass a custom function with signature :code:`norm(x: PyTree) -> float` which maps a PyTree of parameters :code:`x` to the corresponding norm. Note that norm is used in jax.jit-compiled code. holomorphic: a flag to indicate that the wavefunction is holomorphic. diag_shift: diagonal shift of the quantum geometric tensor (QGT) diag_scale: If not None rescales the diagonal shift of the QGT rcond : Cut-off ratio for small singular :math:`\sigma_k` values of the Quantum Geometric Tensor. For the purposes of rank determination, singular values are treated as zero if they are smaller than rcond times the largest singular value :code:`\sigma_{max}`. rcond_smooth : Smooth cut-off ratio for singular values of the Quantum Geometric Tensor. This regularization parameter used with a similar effect to `rcond` but with a softer curve. See :math:`\epsilon_{SVD}` in the formula above. snr_atol: Noise regularisation absolute tolerance, meaning that eigenvalues of the S matrix that have a signal to noise ratio above this quantity will be (soft) truncated. This is :math:`\epsilon_{SNR}` in the formulas above. """ self.propagation_type = propagation_type if isinstance(variational_state, VariationalMixedState): # assuming Lindblad Dynamics # TODO: support density-matrix imaginary time evolution if propagation_type == "real": self._loss_grad_factor = 1.0 else: raise ValueError( "only real-time Lindblad evolution is supported for " "mixed states" ) else: if propagation_type == "real": self._loss_grad_factor = -1.0j elif propagation_type == "imag": self._loss_grad_factor = -1.0 else: raise ValueError("propagation_type must be one of 'real', 'imag'") self.rcond = rcond self.rcond_smooth = rcond_smooth self.snr_atol = snr_atol self.diag_shift = diag_shift self.holomorphic = holomorphic self.diag_scale = diag_scale super().__init__( operator, variational_state, integrator, t0=t0, error_norm=error_norm )
# Copyright notice: # The function `_impl` below includes lines copied from the jVMC repository # found at and licensed according to # MIT License, Copyright (c) 2021 Markus Schmitt @timing.timed @partial(jax.jit, static_argnames=("n_samples")) def _impl(parameters, n_samples, E_loc, S, rhs_coeff, rcond, rcond_smooth, snr_atol): E = stats.statistics(E_loc) ΔE_loc = E_loc.reshape(-1, 1) - E.mean stack_jacobian = S.mode == "complex" O = S.O / jnp.sqrt(n_samples) # already divided by jnp.sqrt(n_s) if stack_jacobian: O = O.reshape(-1, 2, S.O.shape[-1]) O = O[:, 0, :] + 1j * O[:, 1, :] Sd = S.to_dense() ev, V = jnp.linalg.eigh(Sd) OEdata = O.conj() * ΔE_loc F = stats.sum(OEdata, axis=0) # Note: this implementation differs from Eq. 20 in Markus's paper, which I would # implement as `rho = mpi.mean(QEdata, axis=0)`. However, this is different from # changing the basis AFTER averaging over the samples, and leads to the wrong # normalisation of RHo. Q = jnp.tensordot(V.conj().T, O.T, axes=1).T QEdata = Q.conj() * ΔE_loc rho = V.conj().T @ F # Compute the SNR according to Eq. 21 snr = jnp.abs(rho) * jnp.sqrt(n_samples) / jnp.sqrt(stats.var(QEdata, axis=0)) # Discard eigenvalues below numerical precision ev_inv = jnp.where(jnp.abs(ev / ev[-1]) > rcond, 1.0 / ev, 0.0) # Set regularizer for singular value cutoff regularizer = 1.0 / (1.0 + (rcond_smooth / jnp.abs(ev / ev[-1])) ** 6) # Construct a soft cutoff based on the SNR regularizer2 = regularizer * (1.0 / (1.0 + (snr_atol / snr) ** 6)) # solve the linear system by hand eta_p = ev_inv * regularizer2 * rhs_coeff * rho # convert back to the parameter space update = V @ eta_p # remainder of the solution rmd = jnp.linalg.norm( - rhs_coeff * F) / jnp.linalg.norm(F) y, reassemble = convert_tree_to_dense_format(parameters, S.mode) update_tree = reassemble(update if jnp.iscomplexobj(y) else update.real) # If parameters are real, then take only real part of the gradient (if it's complex) dw = tree_cast(update_tree, parameters) return E, dw, rmd, snr @odefun.dispatch def odefun_schmitt(state: MCState, self: TDVPSchmitt, t, w, *, stage=0): # noqa: F811 # pylint: disable=protected-access state.parameters = w state.reset() op_t = self.generator(t) E_loc = state.local_estimators(op_t) self._S = QGTJacobianDense( state, diag_shift=self.diag_shift, diag_scale=self.diag_scale, holomorphic=self.holomorphic, ) self._loss_stats, self._dw, self._rmd, self._snr = _impl( state.parameters, state.n_samples, E_loc, self._S, self._loss_grad_factor, self.rcond, self.rcond_smooth, self.snr_atol, ) if stage == 0: # TODO: This does not work with FSAL. self._last_qgt = self._S return self._dw @partial(jax.jit, static_argnums=(3, 4)) def _map_parameters(forces, parameters, loss_grad_factor, propagation_type, state_T): forces = jax.tree_util.tree_map( lambda x, target: loss_grad_factor * x, forces, parameters, ) forces = tree_cast(forces, parameters) return forces