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QuTiP 5.3 Time Evolution

Research and API verification date: 2026-07-23. All signatures and examples target qutip==5.3.0.

Solver selection

Solver	State/model	Main uncertainty
sesolve	Schrödinger equation, pure state, no collapse channels	ODE and model approximation
mesolve	Lindblad master equation, mixed state, or Liouvillian	channel validity and ODE error
mcsolve	quantum-jump trajectories	ODE error plus sampling error
brmesolve	microscopic weak-coupling bath spectra	Born-Markov/secular approximation and positivity
ssesolve	stochastic Schrödinger equation	diffusive record sampling
smesolve	stochastic master equation with monitored/unmonitored channels	diffusive record sampling
fsesolve/fmmesolve	periodic Floquet dynamics	period, basis, harmonic and bath approximations

Do not use mcsolve merely to make a deterministic calculation parallel. Do not use brmesolve as a generic replacement for a Lindblad model.

Current function signatures

The important QuTiP 5.3 call boundaries are:

sesolve(H, psi0, tlist, *, e_ops=None, args=None, options=None)
mesolve(H, rho0, tlist, c_ops=None, *, e_ops=None, args=None, options=None)
mcsolve(H, state, tlist, c_ops=(), *, e_ops=None, ntraj=500,
        args=None, options=None, seeds=None, target_tol=None, timeout=None)
brmesolve(H, psi0, tlist, a_ops=None, sec_cutoff=0.1, *,
          c_ops=None, e_ops=None, args=None, options=None)
ssesolve(H, psi0, tlist, sc_ops=(), heterodyne=False, *,
         e_ops=None, args=None, ntraj=500, options=None, seeds=None,
         target_tol=None, timeout=None)
smesolve(H, rho0, tlist, c_ops=(), sc_ops=(), heterodyne=False, *,
         e_ops=None, args=None, ntraj=500, options=None, seeds=None,
         target_tol=None, timeout=None)

In 5.3, e_ops, args, and options are keyword-only. Passing solver options as arbitrary solver keyword arguments was removed. The old qutip.Options export is no longer present; pass an ordinary dictionary.

sesolve: closed unitary evolution

import numpy as np
from qutip import basis, sesolve, sigmax, sigmaz

H = 0.5 * sigmax()
psi0 = basis(2, 0)
tlist = np.linspace(0.0, 10.0, 201)

result = sesolve(
    H,
    psi0,
    tlist,
    e_ops={"z": sigmaz()},
    options={"atol": 1e-10, "rtol": 1e-8, "progress_bar": ""},
)
z = np.asarray(result.e_data["z"])

Audit Hamiltonian Hermiticity and state norm. If norm drift is material, tighten tolerances or investigate the model; do not normalize away unexplained error.

mesolve: Lindblad or explicit Liouvillian evolution

import numpy as np
from qutip import basis, mesolve, sigmam, sigmaz

gamma = 0.2
excited = basis(2, 0)
tlist = np.linspace(0.0, 12.0, 241)

result = mesolve(
    0.5 * sigmaz(),
    excited,
    tlist,
    c_ops=[np.sqrt(gamma) * sigmam()],
    e_ops={"p_excited": excited.proj()},
    options={
        "method": "adams",
        "atol": 1e-10,
        "rtol": 1e-8,
        "store_final_state": True,
        "progress_bar": "",
    },
)

When no collapse operators are supplied and H is not a superoperator, mesolve may defer to sesolve. If H is an explicit Liouvillian, document whether it is in valid Lindblad form or represents a deliberate approximation.

mcsolve: quantum jumps

result = mcsolve(
    0.5 * sigmaz(),
    excited,
    tlist,
    [np.sqrt(gamma) * sigmam()],
    e_ops=[excited.proj()],
    ntraj=500,
    seeds=20260723,
    options={
        "keep_runs_results": False,
        "progress_bar": "",
    },
)

mean_population = result.expect[0]
trajectory_spread = result.std_expect[0]
seed_manifest = result.seeds

seeds may be one integer/SeedSequence used to spawn trajectory seeds, or a list with one seed per trajectory. QuTiP stores the realized seeds in the result. Reusing them enables a paired comparison, but paired runs are not independent replicates.

Trajectory rigor:

• increase ntraj and report stabilization or a target uncertainty;
• distinguish trajectory standard deviation from standard error of the mean;
• set options={"keep_runs_results": True} only when individual trajectories are actually required, because memory scales with trajectories and time points;
• collapse records are available per trajectory through result attributes such as collapse; do not print unbounded records by default;
• target_tol may stop before ntraj, so report the number actually run;
• report timeout termination and solver stats.

For mixed initial conditions, current QuTiP can sample the initial mixture; the result then includes initial-state accounting. State this extra source of sampling explicitly.

brmesolve: Bloch-Redfield evolution

import numpy as np
from qutip import basis, brmesolve, sigmax, sigmaz

def one_sided_spectrum(w):
    return 0.03 * w if w > 0.0 else 0.0

tlist = np.linspace(0.0, 30.0, 601)
result = brmesolve(
    0.5 * sigmaz(),
    basis(2, 0),
    tlist,
    a_ops=[(sigmax(), one_sided_spectrum)],
    e_ops={"z": sigmaz()},
    sec_cutoff=0.1,
    options={"atol": 1e-10, "rtol": 1e-8, "progress_bar": ""},
)

The coupling operator in each a_ops pair is normally Hermitian, and the spectrum is a function of angular frequency. Prefer a current QuTiP Environment object when it represents the bath more clearly.

Required checks:

• weak system-bath coupling (Born approximation);
• short bath memory compared with system dynamics (Markov approximation);
• stationary bath and correct positive/negative-frequency convention;
• secular choice: sec_cutoff=-1 disables secularization and can worsen positivity;
• density-matrix trace, Hermiticity, and minimum eigenvalue over time.

See advanced.md for the physical boundaries.

Stochastic Schrödinger and master equations

Use these for diffusive continuous-measurement unravellings, not jump trajectories:

result = smesolve(
    H,
    rho0,
    tlist,
    c_ops=unmonitored_channels,
    sc_ops=monitored_channels,
    heterodyne=False,  # homodyne
    e_ops={"signal": measured_quadrature},
    ntraj=200,
    seeds=20260723,
    options={"dt": 0.001, "store_measurement": True, "progress_bar": ""},
)

• c_ops are unmonitored deterministic dissipation channels.
• sc_ops are monitored stochastic channels.
• heterodyne=False is homodyne; True is heterodyne.
• Legacy integer noise selectors are not current API.
• Measurement records can be much larger than expectation summaries; store them only when needed.
• Converge the stochastic integration step dt separately from ntraj.

Time-dependent systems and QobjEvo

Pythonic callable coefficients

import numpy as np
from qutip import QobjEvo, sigmax, sigmaz

def pulse(t, amplitude, center, width):
    return amplitude * np.exp(-0.5 * ((t - center) / width) ** 2)

H = QobjEvo(
    [0.5 * sigmaz(), [sigmax(), pulse]],
    args={"amplitude": 0.2, "center": 5.0, "width": 1.0},
)
H_at_t = H(5.0)
H.arguments(amplitude=0.15)

The Pythonic form f(t, parameter, ...) is current. The legacy f(t, args_dictionary) form is deprecated in 5.3 and scheduled for removal in 5.5.

Sampled coefficients

coefficient_times = np.linspace(0.0, 10.0, 501)
samples = np.cos(coefficient_times)
H = QobjEvo(
    [0.5 * sigmaz(), [sigmax(), samples]],
    tlist=coefficient_times,
    order=3,
)

Sample times must be sorted and match coefficient length. order=0 gives a left/previous-value step function; the default cubic spline can overshoot. Converge the coefficient sampling independently of solver output times.

QuTiP also accepts expression strings and may compile them, but this skill does not construct such expressions from configuration or user text. Use trusted callables or numeric arrays in generated workflows.

Option dictionaries and integrators

Common solver options:

options = {
    "method": "adams",
    "atol": 1e-10,
    "rtol": 1e-8,
    "nsteps": 10000,
    "store_states": False,
    "store_final_state": True,
    "progress_bar": "",
}

Options are solver- and integrator-specific. Consult the selected solver's .options documentation before using a key.

Common integration choices:

• adams: non-stiff default for many deterministic solvers;
• bdf: stiff systems;
• lsoda: switches between non-stiff and stiff methods;
• explicit high-order methods such as dop853, vern7, or vern9;
• diag: diagonalization-based constant-system evolution;
• krylov: suitable cases where Krylov evolution is beneficial.

An integrator label is not an accuracy certificate. Compare at least two tolerance levels, inspect warnings/stats, and compare a second method for a representative stiff or difficult case.

QuTiP 5.3 adds matrix_form for mesolve/MESolver:

options = {"matrix_form": True, "atol": 1e-10, "rtol": 1e-8}

It uses matrix-matrix products and may reduce memory in suitable models. Benchmark both time and verified observables before adopting it.

Time grids

tlist is the requested output grid, not necessarily the integrator's internal step sequence.

Checks:

1. finite, strictly increasing values;
2. includes the physically intended initial time;
3. resolves the fastest Hamiltonian, decay, drive-envelope, and measurement scales in the returned output;
4. long enough to capture transients or steady behavior;
5. converged under a denser output grid where downstream interpolation, FFT, or peak detection depends on sampling.

For FFT spectra, a uniform taulist is required and its spacing/window set Nyquist range and frequency resolution.

Result semantics

Current solver results can contain:

Attribute	Meaning
times	returned times
states	stored states; may be empty depending on options and e_ops
final_state	final state when requested
expect	list aligned with list-form e_ops
e_data	dictionary keyed like dict-form e_ops
options	effective result/solver options
solver	solver name
stats	timing and diagnostic statistics

Multi-trajectory results add trajectory counts, seeds, standard deviations, average states, measurements, and optionally individual run results. Never assume .states means individual trajectories; inspect keep_runs_results and the concrete result type.

QuTiP 5.3 adds Result.plot_expect() and MultiTrajResult.plot_expect(). Prefer explicit axes in reusable code.

Repeated systems

Solver classes (SESolver, MESolver, MCSolver, BRSolver, SMESolver, and others) can reuse a constructed right-hand side across runs:

from qutip import MESolver

solver = MESolver(H, c_ops, options={"atol": 1e-10, "rtol": 1e-8})
first = solver.run(rho_a, tlist, e_ops=e_ops)
second = solver.run(rho_b, tlist, e_ops=e_ops)

Use the class interface when repeated setup is material, and verify that updated arguments and initial states are the intended ones.

Portable output

Do not load untrusted serialized Python or QuTiP objects. For exchange and audit, save:

• scalar configuration and assumptions;
• real/imaginary numeric arrays;
• dimensions, versions, tolerances, seeds, and solver stats;
• checksums and schema versions where long-term provenance matters.

The bundled CLIs use bounded strict JSON and never pickle results.

Local tools

• ../scripts/two_level_simulation.py: bounded mesolve/mcsolve example.
• ../scripts/solver_config_planner.py: solver and option checklist.
• ../scripts/convergence_sweep.py: deterministic or trajectory convergence.
• ../scripts/result_audit.py: portable JSON audit.

Sources (verified 2026-07-23)

• Dynamics API
• QobjEvo and coefficients API
• Dynamics user guide
• Monte Carlo guide
• Stochastic solver guide
• Bloch-Redfield guide
• QuTiP 5.3.0 release
• QuTiP 5 migration changelog