dq.jsmesolve

jsmesolve(
    H: QArrayLike | TimeQArray,
    jump_ops: list[QArrayLike | TimeQArray],
    thetas: ArrayLike,
    etas: ArrayLike,
    rho0: QArrayLike,
    tsave: ArrayLike,
    keys: PRNGKeyArray,
    *,
    exp_ops: list[QArrayLike] | None = None,
    solver: Solver | None = None,
    gradient: Gradient | None = None,
    options: Options = Options()
) -> JSMESolveResult

Solve the jump stochastic master equation (SME).

Warning

This function has not been implemented yet. The following API is indicative of the planned implementation.

The jump SME describes the evolution of a quantum system measured by a jump detector (for example photodetection in quantum optics). This function computes the evolution of the density matrix $\rho(t)$ at time $t$, starting from an initial state $\rho_0$, according to the jump SME ($\hbar=1$, time is implicit(1)) $$ \begin{split} \dd\rho =&~ -i[H, \rho]\,\dt + \sum_{k=1}^N \left( L_k \rho L_k^\dag - \frac{1}{2} L_k^\dag L_k \rho - \frac{1}{2} \rho L_k^\dag L_k \right)\dt \\ &+ \sum_{k=1}^N \left( \frac{\theta_k \rho+ \eta_k L_k\rho L_k^\dag}{\theta_k + \eta_k \tr{L_k\rho L_k^\dag}} - \rho \right)\left(\dd N_k - \left(\theta_k + \eta_k \tr{L_k\rho L_k^\dag}\right) \dt\right), \end{split} $$ where $H$ is the system's Hamiltonian, $\{L_k\}$ is a collection of jump operators, each continuously measured with dark count rate $\theta_k\geq0$ and efficiency $0\leq\eta_k\leq1$ ($\eta_k=0$ for purely dissipative loss channels) and $\dd N_k$ are independent point processes with law $$ \begin{split} \mathbb{P}[\dd N_k = 0] &= 1 - \mathbb{P}[\dd N_k = 1], \\ \mathbb{P}[\dd N_k = 1] &= \left(\theta_k + \eta_k \tr{L_k\rho L_k^\dag}\right) \dt. \end{split} $$

With explicit time dependence:
- $\rho\to\rho(t)$
- $H\to H(t)$
- $L_k\to L_k(t)$
- $\dd N_k\to \dd N_k(t)$

The continuous-time measurements are defined by the point processes $\dd N_k$. The solver returns the times at which the detector clicked, $I_k = \{t \in [t_0, t_\text{end}[ \,|\, \dd N_k(t)=1\}$.

Warning

For now, jsmesolve() only supports linearly spaced tsave with values that are exact multiples of the solver fixed step size dt.

Parameters

H (qarray-like or time-qarray of shape (...H, n, n)) –
Hamiltonian.
jump_ops (list of qarray-like or time-qarray, each of shape (n, n)) –
List of jump operators.
thetas (array-like of shape (len(jump_ops),)) –
Dark count rate for each loss channel.
etas (array-like of shape (len(jump_ops),)) –
Measurement efficiency for each loss channel with values between 0 (purely dissipative) and 1 (perfectly measured). No measurement is returned for purely dissipative loss channels.
rho0 (qarray-like of shape (...rho0, n, 1) or (...rho0, n, n)) –
Initial state.
tsave (array-like of shape (ntsave,)) –
Times at which the states and expectation values are saved. The equation is solved from tsave[0] to tsave[-1].
keys (list of PRNG keys) –
PRNG keys used to sample the point processes. The number of elements defines the number of sampled stochastic trajectories.
exp_ops (list of array-like, each of shape (n, n), optional) –
List of operators for which the expectation value is computed.
solver –
Solver for the integration. No defaults for now, you have to specify a solver (supported: EulerMaruyama).
gradient –
Algorithm used to compute the gradient. The default is solver-dependent, refer to the documentation of the chosen solver for more details.
options –
Generic options (supported: save_states, cartesian_batching, save_extra, nmaxclick).
Detailed options API
```
dq.Options(
    save_states: bool = True,
    cartesian_batching: bool = True,
    save_extra: callable[[Array], PyTree] | None = None,
    nmaxclick: int = 10_000,
)
```
Parameters
- save_states - If True, the state is saved at every time in tsave, otherwise only the final state is returned.
- cartesian_batching - If True, batched arguments are treated as separated batch dimensions, otherwise the batching is performed over a single shared batched dimension.
- save_extra (function, optional) - A function with signature f(QArray) -> PyTree that takes a state as input and returns a PyTree. This can be used to save additional arbitrary data during the integration, accessible in result.extra.
- nmaxclick - Maximum buffer size for result.clicktimes, should be set higher than the expected maximum number of jump event.

Returns

dq.JSMESolveResult object holding the result of the jump SME integration. Use result.states to access the saved states, result.expects to access the saved expectation values and result.clicktimes to access the detector click times.

Detailed result API

dq.JSMESolveResult

For the shape indications we define ntrajs as the number of trajectories (ntrajs = len(keys)) and nLm as the number of measured loss channels (those for which the measurement efficiency is not zero).

Attributes

states (qarray of shape (..., ntrajs, nsave, n, n)) - Saved states with nsave = ntsave, or nsave = 1 if options.save_states=False.
final_state (qarray of shape (..., ntrajs, n, n)) - Saved final state.
expects (array of shape (..., ntrajs, len(exp_ops), ntsave) or None) - Saved expectation values, if specified by exp_ops.
clicktimes (array of shape (..., ntrajs, nLm, nmaxclick)) - Times at which the detectors clicked. Variable-length array padded with None up to nmaxclick.
extra (PyTree or None) - Extra data saved with save_extra() if specified in options.
keys (PRNG key array of shape (ntrajs,)) - PRNG keys used to sample the Wiener processes.
infos (PyTree or None) - Solver-dependent information on the resolution.
tsave (array of shape (ntsave,)) - Times for which results were saved.
solver (Solver) - Solver used.
gradient (Gradient) - Gradient used.
options (Options) - Options used.

Advanced use-cases

Defining a time-dependent Hamiltonian or jump operator

If the Hamiltonian or the jump operators depend on time, they can be converted to time-arrays using dq.pwc(), dq.modulated(), or dq.timecallable(). See the Time-dependent operators tutorial for more details.

Running multiple simulations concurrently

The Hamiltonian H and the initial density matrix rho0 can be batched to solve multiple SMEs concurrently. All other arguments (including the PRNG key) are common to every batch. The resulting states, click times and expectation values are batched according to the leading dimensions of H and rho0. The behaviour depends on the value of the cartesian_batching option.

If cartesian_batching = True (default value)If cartesian_batching = False

The results leading dimensions are

... = ...H, ...rho0

For example if:

H has shape (2, 3, n, n),
rho0 has shape (4, n, n),

then result.states has shape (2, 3, 4, ntrajs, ntsave, n, n).

The results leading dimensions are

... = ...H = ...rho0  # (once broadcasted)

For example if:

H has shape (2, 3, n, n),
rho0 has shape (3, n, n),

then result.states has shape (2, 3, ntrajs, ntsave, n, n).

See the Batching simulations tutorial for more details.

Warning

Batching on jump_ops, thetas and etas is not yet supported, if this is needed don't hesitate to open an issue on GitHub.