SME solver guide

The stochastic master equation (SME) solver adds measurement-conditioned open-system evolution on top of the v0.5 Lindblad solver.

What it solves

For homodyne detection, QubitOS evolves the conditional density matrix

d rho_c = L(rho_c) dt + sqrt(eta) H[c](rho_c) dW

where:

L(.)     Lindblad drift term
eta      measurement efficiency in [0, 1]
H[c](.)  measurement innovation superoperator
dW       Wiener increment

When eta = 0, the solver dispatches to the existing Lindblad path exactly.

Python API

import numpy as np
from qubitos.lindblad import CollapseOperator
from qubitos.sme import SMEConfig, SMESolver

rho0 = np.array([[0.5, 0.5], [0.5, 0.5]], dtype=np.complex128)
hamiltonians = [np.zeros((2, 2), dtype=np.complex128) for _ in range(80)]
collapse_ops = [CollapseOperator.amplitude_damping(t1_us=50.0)]

config = SMEConfig(
    num_time_steps=80,
    duration_ns=160.0,
    measurement_efficiency=0.6,
    random_seed=123,
    store_measurement_record=True,
)

solver = SMESolver(config=config, collapse_ops=collapse_ops)
result = solver.solve_trajectory(rho0, hamiltonians)

Key result fields

final_density_matrix
    Final conditional state.

measurement_record
    Homodyne current samples for accepted substeps.

max_trace_deviation
    Largest unit-trace drift seen before renormalization.

positivity_violations
    Count of accepted steps whose minimum eigenvalue fell below the
    configured tolerance before any optional projection.

dt_history
    Accepted adaptive timesteps in seconds.

Ensembles

Use solve_ensemble(...) to average many conditional trajectories.

ensemble = solver.solve_ensemble(
    rho0,
    hamiltonians,
    num_trajectories=256,
)

The ensemble result reports:

mean_density_matrix
variance_real
variance_imag
mean_fidelity / std_fidelity
convergence_trace_distance

Validation

Every density-matrix surface emitted by qubitos.sme is validated at the module boundary. Trajectory outputs use a stochastic tolerance override; ensemble averages use the strict density-matrix tolerance.