# QuantumPropagators Package

## Package Index

$\gdef\tgt{\text{tgt}}$ $\gdef\tr{\operatorname{tr}}$ $\gdef\Re{\operatorname{Re}}$ $\gdef\Im{\operatorname{Im}}$

`QuantumPropagators.AbstractPropagator`

`QuantumPropagators.Amplitudes.LockedAmplitude`

`QuantumPropagators.Amplitudes.ShapedAmplitude`

`QuantumPropagators.Cheby.ChebyWrk`

`QuantumPropagators.ChebyPropagator`

`QuantumPropagators.Controls.ParameterizedFunction`

`QuantumPropagators.ExpProp.ExpPropWrk`

`QuantumPropagators.ExpPropagator`

`QuantumPropagators.Generators.Generator`

`QuantumPropagators.Generators.Operator`

`QuantumPropagators.Generators.ScaledOperator`

`QuantumPropagators.Newton.NewtonWrk`

`QuantumPropagators.NewtonPropagator`

`QuantumPropagators.PWCPropagator`

`QuantumPropagators.PiecewisePropagator`

`QuantumPropagators.Propagation`

`QuantumPropagators.Arnoldi.arnoldi!`

`QuantumPropagators.Arnoldi.diagonalize_hessenberg_matrix`

`QuantumPropagators.Arnoldi.extend_arnoldi!`

`QuantumPropagators.Cheby.cheby`

`QuantumPropagators.Cheby.cheby!`

`QuantumPropagators.Cheby.cheby_coeffs`

`QuantumPropagators.Cheby.cheby_coeffs!`

`QuantumPropagators.Controls.discretize`

`QuantumPropagators.Controls.discretize_on_midpoints`

`QuantumPropagators.Controls.evaluate`

`QuantumPropagators.Controls.evaluate!`

`QuantumPropagators.Controls.get_controls`

`QuantumPropagators.Controls.get_parameters`

`QuantumPropagators.Controls.get_tlist_midpoints`

`QuantumPropagators.Controls.substitute`

`QuantumPropagators.Controls.t_mid`

`QuantumPropagators.ExpProp.expprop`

`QuantumPropagators.ExpProp.expprop!`

`QuantumPropagators.Generators.hamiltonian`

`QuantumPropagators.Generators.liouvillian`

`QuantumPropagators.Interfaces.check_amplitude`

`QuantumPropagators.Interfaces.check_control`

`QuantumPropagators.Interfaces.check_generator`

`QuantumPropagators.Interfaces.check_operator`

`QuantumPropagators.Interfaces.check_parameterized`

`QuantumPropagators.Interfaces.check_parameterized_function`

`QuantumPropagators.Interfaces.check_propagator`

`QuantumPropagators.Interfaces.check_state`

`QuantumPropagators.Interfaces.check_tlist`

`QuantumPropagators.Interfaces.supports_inplace`

`QuantumPropagators.Newton.extend_leja!`

`QuantumPropagators.Newton.extend_newton_coeffs!`

`QuantumPropagators.Newton.newton!`

`QuantumPropagators.Shapes.blackman`

`QuantumPropagators.Shapes.box`

`QuantumPropagators.Shapes.flattop`

`QuantumPropagators.SpectralRange.random_state`

`QuantumPropagators.SpectralRange.ritzvals`

`QuantumPropagators.SpectralRange.specrange`

`QuantumPropagators.Storage.get_from_storage`

`QuantumPropagators.Storage.get_from_storage!`

`QuantumPropagators.Storage.init_storage`

`QuantumPropagators.Storage.map_observable`

`QuantumPropagators.Storage.map_observables`

`QuantumPropagators.Storage.write_to_storage!`

`QuantumPropagators.cheby_get_spectral_envelope`

`QuantumPropagators.disable_timings`

`QuantumPropagators.enable_timings`

`QuantumPropagators.init_prop`

`QuantumPropagators.ode_function`

`QuantumPropagators.prop_step!`

`QuantumPropagators.propagate`

`QuantumPropagators.propagate_sequence`

`QuantumPropagators.reinit_prop!`

`QuantumPropagators.set_state!`

`QuantumPropagators.set_t!`

`QuantumPropagators.timings_enabled`

`QuantumPropagators`

Public Symbols:

Private Symbols:

`ExpPropagator`

`timings_enabled`

`set_state!`

`enable_timings`

`ode_function`

`NewtonPropagator`

`disable_timings`

`set_t!`

`PiecewisePropagator`

`ChebyPropagator`

`AbstractPropagator`

`cheby_get_spectral_envelope`

`PWCPropagator`

Submodules:

`QuantumPropagators.Amplitudes`

`QuantumPropagators.Arnoldi`

`QuantumPropagators.Cheby`

`QuantumPropagators.Controls`

`QuantumPropagators.ExpProp`

`QuantumPropagators.Generators`

`QuantumPropagators.Interfaces`

`QuantumPropagators.Newton`

`QuantumPropagators.Shapes`

`QuantumPropagators.SpectralRange`

`QuantumPropagators.Storage`

### Reference

`QuantumPropagators.Propagation`

— TypeWrapper around the parameters of a call to `propagate`

.

```
Propagation(
generator, tlist;
pre_propagation=nothing, post_propagation=nothing,
kwargs...
)
Propagation(
propagator;
pre_propagation=nothing, post_propagation=nothing,
kwargs...
)
```

is a wrapper around the arguments for `propagate`

/ `init_prop`

, for use within `propagate_sequence`

.

The positional and keyword arguments are those accepted by the above propagation routines, excluding the initial state. A `Propagation`

may in addition include the `pre_propagation`

and `post_propagation`

keyword arguments recognized by `propagate_sequence`

.

`QuantumPropagators.init_prop`

— FunctionInitialize a `Propagator`

.

```
propagator = init_prop(
state, generator, tlist;
method, # mandatory keyword argument
backward=false,
inplace=QuantumPropagators.Interfaces.supports_inplace(state),
piecewise=nothing,
pwc=nothing,
kwargs...
)
```

initializes a propagator for the time propagation of the given `state`

over a time grid `tlist`

under the time-dependent generator (Hamiltonian/Liouvillian) `generator`

.

**Arguments**

`state`

: The "initial" state for the propagation. For`backward=false`

, this state is taken to be at initial time (`tlist[begin]`

); and for`backward=true`

, at the final time (`tlist[end]`

)`generator`

: The time-dependent generator of the dynamics`tlist`

: The time grid over which which the propagation is defined. This may or may not be equidistant.

**Mandatory keyword arguments**

`method`

: The propagation method to use. May be given as a name (`Symbol`

), but the recommended usage is to pass a module implementing the propagation method, e.g.,`using QuantumPropagators: Cheby; method = Cheby`

. Passing a module ensures that the code implementing the method is correctly loaded. This is particularly important for propagators using third-party backends, like with`method=OrdinaryDiffEq`

.

**Optional keyword arguments**

`backward`

: If`true`

, initialize the propagator for a backward propagation. The resulting`propagator.t`

will be`tlist[end]`

, and subsequent calls to`prop_step!`

will move backward on`tlist`

.`inplace`

: If`true`

, the`state`

property of the resulting propagator will be changed in-place by any call to`prop_step!`

. If`false`

, each call to`prop_step!`

changes the reference for`propagator.state`

, and the propagation will not use any in-place operations. Not all propagation methods may support both in-place and not-in-place propagation. In-place propagation is generally more efficient for larger Hilbert space dimensions, but may not be compatible, e.g., with automatic differentiation.`piecewise`

: If given as a boolean,`true`

enforces that the resulting propagator is a`PiecewisePropagator`

, and`false`

enforces that it not a`PiecewisePropagator`

. For the default`piecewise=nothing`

, whatever type of propagation is the default for the given`method`

will be used. Throw an error if the given`method`

does not support the required type of propagation.`pwc`

: Like`piecewise`

, but for the stronger`PWCPropagator`

.

All other `kwargs`

are method-dependent and are ignored for methods that do not support them.

The type of the returned `propagator`

is a sub-type of `AbstractPropagator`

, respectively a sub-type of `PiecewisePropagator`

if `piecewise=true`

or a sub-type of `PWCPropagator`

if `pwc=true`

.

**Internals**

Internally, the (mandatory) keyword `method`

is converted into a fourth positional argument. This allows propagation methods to define their own implementation of `init_prop`

via multiple dispatch. However, when *calling* `init_prop`

in high-level code, `method`

must always be given as a keyword argument.

**See also**

`reinit_prop!`

— Re-initialize a propagator`propagate`

— Higher-level propagation interface- `check_propagator — a function to verify the interface described above.

```
using QuantumPropagators: Cheby
cheby_propagator = init_prop(
state,
generator,
tlist;
method=Cheby,
inplace=QuantumPropagators.Interfaces.supports_inplace(state),
backward=false,
verbose=false,
parameters=nothing,
control_ranges=nothing,
specrange_method=:auto,
specrange_buffer=0.01,
cheby_coeffs_limit=1e-12,
check_normalization=false,
specrange_kwargs...
)
```

initializes a `ChebyPropagator`

.

**Method-specific keyword arguments**

`control_ranges`

: a dict the maps the controls in`generator`

(see`get_controls`

) to a tuple of min/max values. The Chebychev coefficients will be calculated based on a spectral envelope that assumes that each control can take arbitrary values within the min/max range. If not given, the ranges are determined automatically. Specifying manual control ranges can be useful when the the control amplitudes (`parameters`

) may change during the propagation, e.g. in a sequential-update control scheme.`specrange_method`

: Method to pass to the`specrange`

function`specrange_buffer`

: An additional factor by which to enlarge the estimated spectral range returned by`specrange`

, in order to ensure that Chebychev coefficients are based on an overestimation of the spectral range.`cheby_coeffs_limit`

: The maximum magnitude of Chebychev coefficients that should be treated as non-zero`check_normalization`

: Check whether the Hamiltonian has been properly normalized, i.e., that the spectral range of`generator`

has not been underestimated. This slowes down the propagation, but is advisable for novel`generators`

.`uniform_dt_tolerance=1e-12`

: How much the intervals of`tlist`

are allowed to vary while still being considered constant.`specrange_kwargs`

: All further keyword arguments are passed to the`specrange`

function. Most notably, with the default`specrange_method=:auto`

(or`specrange_method=:manual`

), passing`E_min`

and`E_max`

allows to manually specify the spectral range of`generator`

.

```
using QuantumPropagators: Newton
newton_propagator = init_prop(
state,
generator,
tlist;
method=Newton,
inplace=QuantumPropagators.Interfaces.supports_inplace(state),
backward=false,
verbose=false,
parameters=nothing,
m_max=10,
func=(z -> exp(-1im * z)),
norm_min=1e-14,
relerr=1e-12,
max_restarts=50,
_...
)
```

initializes a `NewtonPropagator`

.

**Method-specific keyword arguments**

```
using QuantumPropagators: ExpProp
exp_propagator = init_prop(
state,
generator,
tlist;
method=ExpProp,
inplace=QuantumPropagators.Interfaces.supports_inplace(state),
backward=false,
verbose=false,
parameters=nothing,
func=(H_dt -> exp(-1im * H_dt))
convert_state=_exp_prop_convert_state(state),
convert_operator=_exp_prop_convert_operator(generator),
_...
)
```

initializes an `ExpPropagator`

.

**Method-specific keyword arguments**

`func`

: The function to evaluate. The argument`H_dt`

is obtained by constructing an operator`H`

from`generator`

via the`evaluate`

function and the multiplied with the time step`dt`

for the current time interval. The propagation then simply multiplies the return value of`func`

with the current state`convert_state`

: Type to which to temporarily convert states before multiplying the return value of`func`

.`convert_operator`

: Type to which to convert the operator`H`

before multiplying it with`dt`

and plugging the result into`func`

The `convert_state`

and `convert_operator`

parameters are useful for when the `generator`

and or `state`

are unusual data structures for which the relevant methods to calculate `func`

are not defined. Often, it is easier to temporarily convert them to standard complex matrices and vectors than to implement the missing methods.

```
using OrdinaryDiffEq # or: `using DifferentialEquations`
ode_propagator = init_prop(
state,
generator,
tlist;
method=OrdinaryDiffEq, # or: `method=DifferentialEquations`
inplace=QuantumPropagators.Interfaces.supports_inplace(state),
backward=false,
verbose=false,
parameters=nothing,
piecewise=false,
pwc=false,
alg=OrdinaryDiffEq.Tsit5(),
solver_options...
)
```

initializes a propagator that uses an ODE solver from the OrdinaryDiffEq.jl package as a backend.

By default, the resulting propagator is for time-continuous controls that can be evaluated with `evaluate(control, t)`

for any `t`

in the range of `tlist[begin]`

to `tlist[end]`

. The controls may be parametrized, see `get_parameters`

. Any parameters will be available in the `parameters`

attribute of the resulting `ode_propagator`

, as a dictionary mapping controls to a vector of parameter values. Mutating `ode_propagator.parameters[control]`

will be reflected in any subsequent call to `prop_step!`

.

If `pwc=true`

(or, equivalently `piecewise=true`

), all controls will be discretized with `discretize_on_midpoints`

and the propagation will be for piecewise constant dynamics. The resulting `ode_propagator`

will be an instance of `PWCPropagator`

, with the corresponding semantics. In particular, the `ode_propatator.parameters`

will be a mapping of controls to discretized pulse values, *not* the analytical parameters obtained with `get_parameters(control)`

as in the default case.

Internally, the `generator`

will be wrapped with `QuantumPropagators.ode_function`

. The resulting function `f`

will be called internally as `f(ϕ, Ψ, vals_dict, t)`

or `f(Ψ, vals_dict, t)`

depending on the `inplace`

keyword argument.

**Method-specific keyword arguments**

`pwc`

: Whether to propagate for piecewise-constant controls or, with the default`pwc=false`

, for time-continuous controls.`piecewise`

: Currently equivalent to`pwc`

, but future version may change this to allow for other piecewise (e.g., piecewise-linear) controls.`parameters`

: If given, a mapping of controls to parameter values (`pwc=false`

) or pulse values on the intervals of the time grid (`pwc=true`

). By default, the`parameters`

are determined automatically using`get_parameters`

, respectively`discretize_on_midpoints`

if`pwc=true`

. If they are given manually, they must follow the exact same semantics. In particular, for`pwc=false`

, any parameters must alias the parameters in the controls, such that mutating`parameters`

is automatically reflected in`evaluate`

. The`parameters`

will be available as an attribute of the`ode_propagator`

.`alg`

: The algorithm to use for the ODE Solver, see the list of solvers in the DifferentialEquations manual. The default`Tsit5()`

method is the recommended choice for non-stiff problems.`solver_options`

: All other keyword arguments are passed to the ODE solver, see the list of Solve Keyword Arguments in the DifferentialEquations manual. Note that the options for "Default Algorithm Hinting" do not apply, since`alg`

must be specified manually. Also, the "Output Control" is managed by the`ode_propagator`

, so these options should not be used.

`QuantumPropagators.prop_step!`

— FunctionAdvance the `propagator`

by a single time step.

`state = prop_step!(propagator)`

returns the state obtained from propagating to the next point on the time grid from `propagator.t`

, respectively the previous point if `propagator.backward`

is true.

When the propagation would lead out of the time grid, `prop_step!`

leaves `propagator`

unchanged and returns `nothing`

. Thus, a return value of `nothing`

may be used to signal that a propagation has completed.

`QuantumPropagators.propagate`

— FunctionPropagate a state over an entire time grid.

```
state = propagate(
state,
generator,
tlist;
method, # mandatory keyword argument
check=true,
backward=false,
inplace=QuantumPropagators.Interfaces.supports_inplace(state),
verbose=false,
piecewise=nothing,
pwc=nothing,
storage=nothing,
observables=<store state>,
callback=nothing,
show_progress=false,
init_prop_kwargs...)
```

propagates `state`

of the entire time grid and returns the propagated states, or a storage array of data collected during the propagation. This high-level routine performs the following three steps:

If

`check=true`

(default), check that`state`

,`generator`

, and`tlist`

are consistent with the required interface.Initialize a

`propagator`

via`init_prop`

:`init_prop(state, generator, tlist; method, inplace, init_prop_kwargs...)`

Call and return the result of

`propagate(propagator; storage, observables, show_progress, callback)`

**Arguments**

`state`

: The "initial" state for the propagation. For`backward=false`

, this state is taken to be at initial time (`tlist[begin]`

); and for`backward=true`

, at the final time (`tlist[end]`

)`generator`

: The time-dependent generator of the dynamics`tlist`

: The time grid over which which the propagation is defined. This may or may not be equidistant.

**Mandatory keyword arguments**

`method`

: The propagation method to use. May be given as a name (`Symbol`

), but the recommended usage is to pass a module implementing the propagation method, cf.`init_prop`

.

**Optional keyword arguments**

`check`

: if`true`

, check that`state`

,`generator`

, and`tlist`

pass`check_state`

,`check_generator`

and`check_tlist`

, respectively.`backward`

: If`true`

, propagate backward in time`inplace`

: If`true`

, propagate using in-place operations. If`false`

, avoid in-place operations. Not all propagation methods support both in-place and not-in-place propagation. Note that`inplace=true`

requires that`QuantumPropagators.Interfaces.supports_inplace`

for`state`

is`true`

.`piecewise`

: If given as a boolean, ensure that the internal`propagator`

is an instance of`PiecewisePropagator`

, cf.`init_prop`

.`pwc`

: If given a a boolean, do a piecewise constant propagation where the generator in each interval is constant (the internal`propagator`

is a`PWCPropagator`

, cf.`init_prop`

)`storage`

: Flag whether to store and return the propagated states / observables, or pre-allocated storage array. See Notes below.`observables`

: Converters for data to be stored in`storage`

. See Notes below.`callback`

: Function to call after each propagation step. See Notes below.`show_progress`

: Whether to show a progress bar. See Notes below.

All remaining keyword arguments are passed to `init_prop`

to initialize the `Propagator`

that is used internally to drive the optimization. Unknown keyword arguments will be ignored.

**Notes**

In general, there is no requirement that `tlist`

has a constant time step, although some propagation methods (most notably `Cheby`

) only support a uniform time grid.

If `storage`

is given as a container pre-allocated via `init_storage`

, it will be filled with data determined by the `observables`

. Specifically, after each propagation step,

```
data = map_observables(observables, tlist, i, state)
write_to_storage!(storage, i, data)
```

is executed, where `state`

is defined at time `tlist[i]`

. See `map_observables`

and `write_to_storage!`

for details. The default values for `observables`

results simply in the propagated states at every point in time being stored.

The `storage`

parameter may also be given as `true`

, and a new storage array will be created internally with `init_storage`

and returned instead of the propagated state:

```
data = propagate(
state, generator, tlist; method,
backward=false; storage=true, observables=observables,
callback=nothing, show_progress=false, init_prop_kwargs...)
```

If `backward`

is `true`

, the input state is assumed to be at time `tlist[end]`

, and the propagation progresses backward in time (with a negative time step `dt`

). If `storage`

is given, it will be filled back-to-front during the backward propagation.

If `callback`

is given as a callable, it will be called after each propagation step, as `callback(propagator, observables)`

where `propagator`

is `Propagator`

object driving the propagation. The `callback`

is called before calculating any observables. Example usage includes writing data to file, or modifying `state`

via `set_state!`

, e.g., removing amplitude from the lowest and highest level to mitigate "truncation error".

If `show_progress`

is given as `true`

, a progress bar will be shown for long-running propagation. In order to customize the progress bar, `show_progress`

may also be a function that receives `length(tlist)`

and returns a `ProgressMeter.Progress`

instance.

If `in_place=false`

is given, the propagation avoids in-place operations. This is slower than `inplace=true`

, but is often required in the context of automatic differentiation (AD), e.g., with Zygote. That is, use `in_place=false`

if `propagate`

is called inside a function to be passed to `Zygote.gradient`

, `Zygote.pullback`

, or a similar function. In an AD context, `storage`

and `show_progress`

should not be used.

The `propagate`

routine returns the propagated state at `tlist[end]`

, respectively `tlist[1]`

if `backward=true`

, or a storage array with the stored states / observable data if `storage=true`

.

```
state = propagate(
state,
propagator;
storage=nothing,
observables=<store state>,
show_progress=false,
callback=nothing,
reinit_prop_kwargs...
)
```

re-initializes the given `propagator`

with `state`

(see `reinit_prop!`

) and then calls the lower-level `propagate(propagator; ...)`

.

```
state = propagate(
propagator;
storage=nothing,
observables=<store state>,
show_progress=false,
callback=nothing,
)
```

propagates a freshly initialized `propagator`

(immediately after `init_prop`

). Used in the higher-level `propagate(state, generator, tlist; kwargs...)`

.

`QuantumPropagators.propagate_sequence`

— FunctionPropagate a state through a sequence of generators.

```
states = propagate_sequence(
state,
propagations;
storage=nothing,
pre_propagation=nothing,
post_propagation=nothing,
kwargs...
)
```

takes an `initial`

state and performs a sequence of `propagate`

calls using the parameters in `propagations`

. The initial state for each step in the sequence is the state resulting from the previous step. Optionally, before and after each step, a `pre_propagation`

and `post_propagation`

function may modify the state instantaneously, e.g., to perform a frame transformation. Return the vector of states at the end of each step (after any `post_propagation`

, before any next `pre_propagation`

of the next step).

**Arguments**

`state`

: The initial state`propagations`

: A vector of`Propagation`

instances, one per step in the sequence, each containing the arguments for the call to`propagate`

for that step. The`Propagation`

contains the generator and time grid for each step as positional parameters, or alternatively a pre-initialized`Propagator`

, and any keyword arguments for`propagate`

that are specific to that step. Note that`propagate`

keyword arguments that are common to all steps can be given directly to`propagate_sequence`

.`storage`

: If`storage=true`

, return a vector of storage objects as returned by`propagate(…, storage=true)`

for each propagation step, instead of the state after each step. To use a pre-initialized`storage`

, each`Propagation`

in`propagations`

should have a`storage`

keyword argument instead.`pre_propagation`

: If not`nothing`

, must be a function that receives the same arguments as`propagate`

and returns a state. Called immediately before the`propagate`

of each step, and the state returned by`pre_propagation`

will become the initial state for the subsequent call to`propagate`

. Generally,`pre_propagation`

would be different in each step of the sequence, and should be given as a keyword argument in a particular`Propagation`

.`post_propagation`

: If not`nothing`

, a function that receives the same arguments as`propagate`

and returns a state, see`pre_propagation`

. The returned state becomes the initial state for the next step in the sequence (and may be further processed by the following`pre_propagation`

). Like`pre_propagation`

, this will generally be set as a keyword argument for a particular`Propagation`

, not as a global keyword argument to`propagate_sequence`

.

All other keyword arguments are forwarded to `propagate`

. Thus, keyword arguments that are common to all steps in the sequence should be given as keyword arguments to `propagate_sequence`

directly.

`QuantumPropagators.reinit_prop!`

— FunctionRe-initialize a propagator.

`reinit_prop!(propagator, state; kwargs...)`

resets the `propagator`

to `state`

at the beginning of the time grid, respectively the end of the time grid if `propagator.backward`

is true.

At a minimum, this is equivalent to a call to `set_state!`

follow by a call to `set_t!`

, but some propagators may have additional requirements on re-initialization, such as refreshing expansion coefficients for `ChebyPropagator`

. In this case, the `kwargs`

may be additional keyword arguments specific to the concrete type of propagator.

```
reinit_prop!(
propagator::ChebyPropagator,
state;
transform_control_ranges=((c, ϵ_min, ϵ_max, check) => (ϵ_min, ϵ_max)),
kwargs...
)
```

re-initializes an existing `ChebyPropagator`

. This may or may not involve recalculating the Chebychev coefficients based on the current control amplitudes in `propagator.parameters`

.

**Method-specific keyword arguments**

`transform_control_ranges`

: a function`(c, ϵ_min, ϵ_max, check) => (ϵ_min′, ϵ_max′)`

. For each control`c`

, the function is called with`check=true`

and`ϵ_min`

(`ϵ_max`

) the current minimum (maximum) values for the control from`propagator.parameters`

). The Chebychev coefficients will be recalculated if the existing coefficients were obtained assuming a range for`c`

outside the*returned*`ϵ_min′, ϵ_max′`

.If the coefficients do need to be recalculated,

`transform_control_ranges`

is called a second time with`check=false`

, and the returned`(ϵ_min′, ϵ_max′)`

are used for estimating the new spectral range.For example,

`function transform_control_ranges(c, ϵ_min, ϵ_max, check) if check return (min(ϵ_min, 2 * ϵ_min), max(ϵ_max, 2 * ϵ_max)) else return (min(ϵ_min, 5 * ϵ_min), max(ϵ_max, 5 * ϵ_max)) end end`

will re-calculate the Chebychev coefficients only if the current amplitudes differ by more than a factor of two from the ranges that were used when initializing the propagator (

`control_ranges`

parameter in`init_prop`

, which would have had to overestimate the actual amplitudes by at least a factor of two). When re-calculating, the`control_ranges`

will overestimate the amplitudes by a factor of five. With this`transform_control_ranges`

, the propagation will be stable as long as the amplitudes do not change dynamically by more than a factor of 2.5 from their original range, while also not re-calculating coefficients unnecessarily in each pass because of modest changes in the amplitudes.The

`transform_control_ranges`

argument is only relevant in the context of optimal control, where the same`propagator`

will be used for many iterations with changing control field amplitudes.

All other keyword arguments are ignored.

`QuantumPropagators.ExpPropagator`

— TypePropagator for propagation via direct exponentiation (`method=QuantumPropagators.ExpProp`

)

This is a `PWCPropagator`

.

`QuantumPropagators.timings_enabled`

— FunctionCheck whether the collection of `TimerOutputs`

data is active.

`QuantumPropagators.timings_enabled()`

returns `true`

if `QuantumPropagators.enable_timings()`

was called, and `false`

otherwise or after `QuantumPropagators.disable_timings()`

.

`QuantumPropagators.set_state!`

— FunctionSet the current `state`

of the `propagator`

.

`set_state!(propagator, state)`

sets the `propagator.state`

property and returns `propagator.state`

. In order to mutate the current state after a call to `prop_step!`

, the following pattern is recommended:

```
Ψ = propagator.state
foo_mutate!(Ψ)
set_state!(propagator, Ψ)
```

where `foo_mutate!`

is some function that mutates `Ψ`

. This is guaranteed to work efficiently both for in-place and not-in-place propagators, without incurring unnecessary copies.

`foo_mutate!(propagator.state)`

by itself is not a safe operation. Always follow it by

`set_state!(propagator, propagator.state)`

**See also**

`set_t!`

— set`propagator.t`

.

`QuantumPropagators.enable_timings`

— FunctionEnable the collection of `TimerOutputs`

data.

`QuantumPropagators.enable_timings()`

enables certain portions of the package to collect `TimerOutputs`

internally. This aids in profiling and benchmarking propagation methods.

Specifically, after `enable_timings()`

, for any `ChebyPropagator`

or `NewtonPropagator`

, timing data will become available in `propagator.wrk.timing_data`

(as a `TimerOutput`

instance). This data is reset when the propagator is re-instantiated with `init_prop`

or re-initialized with `reinit_prop!`

. This makes the data local to any call of `propagate`

.

Note that `enable_timings()`

triggers recompilation, so `propagate`

should be called at least twice to avoid compilation overhead in the timing data. There is still a small overhead for collecting the timing data.

The collection of timing data can be disabled again with `disable_timings`

.

Returns `QuantumPropagators.timings_enabled()`

, i.e., `true`

if successful.

`QuantumPropagators.ode_function`

— FunctionWrap around a `Generator`

, for use as an ODE function.

`f = ode_function(generator, tlist; c=-1im)`

creates a function suitable to be passed to `ODEProblem`

.

\[\gdef\op#1{\hat{#1}} \gdef\ket#1{\vert{#1}\rangle}\]

With `generator`

corresponding to $\op{H}(t)$, this implicitly encodes the ODE

\[\frac{\partial}{\partial t} \ket{\Psi(t)} = c \op{H}(t) \ket{\Psi(t)}\]

for the state $\ket{\Psi(t)}$. With the default $c = -i$, this corresponds to the Schrödinger equation, or the Liouville equation with `convention=:LvN`

.

The resulting `f`

works both in-place and not-in-place, as

```
f(ϕ, Ψ, vals_dict, t) # in-place `f(du, u, p, t)`
ϕ = f(Ψ, vals_dict, t) # not-in-place `f(u, p, t)`
```

Calling `f`

as above is functionally equivalent to calling `evaluate`

to obtain an operator `H`

from the original time-dependent `generator`

, and then applying `H`

to the current quantum state `Ψ`

:

```
H = evaluate(f.generator, t; vals_dict=vals_dict)
ϕ = c * H * Ψ
```

where `vals_dict`

may be a dictionary mapping controls to values (set as the parameters `p`

of the underlying ODE solver).

If `QuantumPropagators.enable_timings()`

has been called, profiling data is collected in `f.timing_data`

.

`QuantumPropagators.NewtonPropagator`

— TypePropagator for Newton propagation (`method=QuantumPropagators.Newton`

).

This is a `PWCPropagator`

.

`QuantumPropagators.disable_timings`

— FunctionDisable the collection of `TimerOutputs`

data.

`QuantumPropagators.disable_timings()`

disables the collection of timing data previously enabled with `enable_timings`

. This triggers recompilation to completely remove profiling from the code. That is, there is zero cost when the collection of timing data is disabled.

Returns `QuantumPropagators.timings_enabled()`

, i.e., `false`

if successful.

`QuantumPropagators.set_t!`

— FunctionSet the current time for the propagation.

`set_t!(propagator, t)`

Sets `propagator.t`

to the given value of `t`

, where `t`

must be an element of `propagator.tlist`

.

**See also**

`set_state!`

— set`propagator.state`

`QuantumPropagators.PiecewisePropagator`

— Type`AbstractPropagator`

sub-type for piecewise propagators.

A piecewise propagator is determined by a single parameter per control and time grid interval. Consequently, the `propagator.parameters`

are a dictionary mapping the controls found in the `generator`

via `get_controls`

to a vector of values defined on the intervals of the time grid, see `discretize_on_midpoints`

. This does not necessarily imply that these values are the piecewise-constant amplitudes for the intervals. A general piecewise propagator might use interpolation to obtain actual amplitudes within any given time interval.

When the amplitudes *are* piecewise-constant, the propagator should be a concrete instantiation of a `PWCPropagator`

.

`QuantumPropagators.ChebyPropagator`

— TypePropagator for Chebychev propagation (`method=QuantumPropagators.Cheby`

).

This is a `PWCPropagator`

.

`QuantumPropagators.AbstractPropagator`

— TypeAbstract base type for all `Propagator`

objects.

All `Propagator`

objects must be instantiated via `init_prop`

and implement the following interface.

**Properties**

`state`

(read-only): The current quantum state in the propagation`tlist`

(read-only): The time grid for the propagation`t`

(read-only): The time at which`state`

is defined. An element of`tlist`

.`parameters`

: parameters that determine the dynamics. The structure of the parameters depends on the concrete`Propagator`

type (i.e., the propagation method). Mutating the`parameters`

affects subsequent propagation steps.`backward`

: Boolean flag to indicate whether the propagation moves forward or backward in time`inplace`

: Boolean flag to indicate whether`propagator.state`

is modified in-place or is recreated by every call of`prop_step!`

or`set_state!`

. With`inplace=false`

, the propagator should generally avoid in-place operations, such as calls to`QuantumPropagators.Controls.evaluate!`

.

Concrete `Propagator`

types may have additional properties or fields, but these should be considered private.

**Methods**

`reinit_prop!`

— reset the propagator to a new initial state at the beginning of the time grid (or the end, for backward propagation)`prop_step!`

– advance the propagator by one step forward (or backward) on the time grid.`set_state!`

— safely mutate the current quantum`state`

of the propagation. Note that directly mutating the`state`

property is not safe. However,`Ψ = propagator.state; foo_mutate!(Ψ), set_state!(propagator, Ψ)`

for some mutating function`foo_mutate!`

is guaranteed to be safe and efficient for both in-place and not-in-place propagators.`set_t!`

— safely mutate the current time (`propagator.t`

), snapping to the values of`tlist`

.

**See also**

`PiecewisePropagator`

— specialization of`AbstractPropagator`

for piecewise propagation methods.`PWCPropagator`

— specialization of`PiecewisePropagator`

for piecewise-constant propagation methods.

`QuantumPropagators.cheby_get_spectral_envelope`

— FunctionDetermine the spectral envelope of a `generator`

.

```
E_min, E_max = cheby_get_spectral_envelope(
generator, tlist, control_ranges, method; kwargs...
)
```

estimates a lower bound `E_min`

the lowest eigenvalue of the generator for any values of the controls specified by `control_ranges`

, and an upper bound `E_max`

for the highest eigenvalue.

This is done by constructing operators from the extremal values for the controls as specified in `control_ranges`

and taking the smallest/largest return values from `specrange`

for those operators.

**Arguments**

`generator`

: dynamical generator, e.g. a time-dependent`tlist`

: The time grid for the propagation`control_ranges`

: a dict that maps controls that occur in`generator`

(cf.`get_controls`

to a tuple of minimum and maximum amplitude for that control`method`

: method name to pass to`specrange`

`kwargs`

: Any remaining keyword arguments are passed to`specrange`

`QuantumPropagators.PWCPropagator`

— Type`PiecewisePropagator`

sub-type for piecewise-constant propagators.

Like the more general `PiecewisePropagator`

, this is characterized by `propagator.parameters`

mapping the controls in the `generator`

to a vector of amplitude value on the midpoints of the time grid intervals.

The propagation will use these values as constant within each interval.

`QuantumPropagators.Amplitudes`

Public:

### Reference

`QuantumPropagators.Amplitudes.LockedAmplitude`

— TypeA time-dependent amplitude that is not a control.

`ampl = LockedAmplitude(shape)`

wraps around `shape`

, which must be either a vector of values defined on the midpoints of a time grid or a callable `shape(t)`

.

`ampl = LockedAmplitude(shape, tlist)`

discretizes `shape`

to the midpoints of `tlist`

.

`QuantumPropagators.Amplitudes.ShapedAmplitude`

— TypeProduct of a fixed shape and a control.

`ampl = ShapedAmplitude(control; shape=shape)`

produces an amplitude $a(t) = S(t) ϵ(t)$, where $S(t)$ corresponds to `shape`

and $ϵ(t)$ corresponds to `control`

. Both `control`

and `shape`

should be either a vector of values defined on the midpoints of a time grid or a callable `control(t)`

, respectively `shape(t)`

. In the latter case, `ampl`

will also be callable.

`ampl = ShapedAmplitude(control, tlist; shape=shape)`

discretizes `control`

and `shape`

to the midpoints of `tlist`

.

`QuantumPropagators.Arnoldi`

Private:

### Reference

`QuantumPropagators.Arnoldi.extend_arnoldi!`

— FunctionExtend dimension of Hessenberg matrix by one.

`extend_arnoldi!(Hess, q, m, H, dt; norm_min=1e-15)`

extends the entries in `Hess`

from size (m-1)×(m-1) to size m×m, and the list `q`

of Arnoldi vectors from m to (m+1). It is assumed that the input `Hess`

was created by a call to `arnoldi!`

with `extended=false`

or a previous call to `extend_arnoldi!`

. Note that `Hess`

itself is not resized, so it must be allocated to size m×m or greater on input.

`QuantumPropagators.Arnoldi.diagonalize_hessenberg_matrix`

— Function`diagonalize_hessenberg_matrix(Hess, m; accumulate=false)`

Diagonalize the m × m top left submatrix of the given Hessenberg matrix.

If `accumulate`

is `true`

, return the concatenated eigenvalues for `Hess[1:1,1:1]`

to `Hess[1:m,1:m]`

, that is, all sumatrices of size 1 through `m`

.

`QuantumPropagators.Arnoldi.arnoldi!`

— Function`m = arnoldi!(Hess, q, m, Ψ, H, dt=1.0; extended=true, norm_min=1e-15)`

Calculate the Hessenberg matrix and Arnoldi vectors of `H dt`

, from `Ψ`

.

For a given order `m`

, the `m×m`

Hessemberg matrix is calculated and stored in in the pre-allocated `Hess`

. Further an array of `m`

normalized Arnoldi vectors is stored in in the pre-allocated `q`

, plus one additional unnormalized Arnoldi vector. The unnormalized `m+1`

st vector could be used to easily extend a given `m×m`

Hessenberg matrix to a `(m+1)×(m+1)`

matrix.

If the extended Hessenberg matrix is requested (`extended=true`

, default), the `m+1`

st Arnoldi vector is also normalized, and it's norm will be stored in `m+1, m`

entry of the (extended) Hessenberg matrix, which is an `(m+1)×(m+1)`

matrix.

Return the size `m`

of the calculated Hessenberg matrix. This will usually be the input `m`

, except when the Krylov dimension of `H`

starting from `Ψ`

is less then `m`

. E.g., if `Ψ`

is an eigenstate of `H`

, the returned `m`

will be 1.

See https://en.wikipedia.org/wiki/Arnoldi_iteration for a description of the algorithm.

**Arguments**

`Hess::Matrix{ComplexF64}`

: Pre-allocated storage for the Hessemberg matrix. Can be uninitialized on input. The matrix must be at least of size`m×m`

, or`(m+1)×(m+1)`

if`extended=true`

. On output, the`m×m`

sub-matrix of`Hess`

(with the returned output`m`

) will contain the Hessenberg matrix, and all other elements of`Hess`

be be set to zero.`q`

: Pre-allocated array of states similar to`Ψ`

, as storage for the calculated Arnoldi vectors. These may be un-initialized on input. Must be at least of length`m+1`

`m`

: The requested dimensions of the output Hessenberg matrix.`Ψ`

: The starting vector for the Arnoldi procedure. This can be of any type, as long as`Φ = H * Ψ`

results in a vector similar to`Ψ`

, there is an inner products of`Φ`

and`Ψ`

(`Ψ⋅Φ`

is defined), and`norm(Ψ)`

is defined.`H`

: The operator (up to`dt`

) for which to calculate the Arnoldi procedure. Can be of any type, as long as`H * Ψ`

is defined.`dt`

: The implicit time step; the total operator for which to calculate the Arnoldi procedure is`H * dt`

`extended`

: If`true`

(default), calculate the extended Hessenberg matrix, and normalized the final Arnoldi vector`norm_min`

: the minimum value of the norm of`Ψ`

at which`Ψ`

should be considered the zero vector

`QuantumPropagators.Cheby`

Public:

### Reference

`QuantumPropagators.Cheby.ChebyWrk`

— TypeWorkspace for the Chebychev propagation routine.

`ChebyWrk(Ψ, Δ, E_min, dt; limit=1e-12)`

initializes the workspace for the propagation of a state similar to `Ψ`

under a Hamiltonian with eigenvalues between `E_min`

and `E_min + Δ`

, and a time step `dt`

. Chebychev coefficients smaller than the given `limit`

are discarded.

`QuantumPropagators.Cheby.cheby`

— FunctionEvaluate `Ψ = exp(-𝕚 * H * dt) Ψ`

.

`Ψ_out = cheby(Ψ, H, dt, wrk; E_min=nothing, check_normalization=false)`

acts like `cheby!`

but does not modify `Ψ`

in-place.

`QuantumPropagators.Cheby.cheby!`

— FunctionEvaluate `Ψ = exp(-𝕚 * H * dt) Ψ`

in-place.

`cheby!(Ψ, H, dt, wrk; E_min=nothing, check_normalization=false)`

**Arguments**

`Ψ`

: on input, initial vector. Will be overwritten with result.`H`

: Hermitian operator`dt`

: time step`wrk`

: internal workspace`E_min`

: minimum eigenvalue of H, to be used instead of the`E_min`

from the initialization of`wrk`

. The same`wrk`

may be used for different values`E_min`

, as long as the spectra radius`Δ`

and the time step`dt`

are the same as those used for the initialization of`wrk`

.`check_normalizataion`

: perform checks that the H does not exceed the spectral radius for which the workspace was initialized.

The routine will not allocate any internal storage. This implementation requires `copyto!`

`lmul!`

, and `axpy!`

to be implemented for `Ψ`

, and the three-argument `mul!`

for `Ψ`

and `H`

.

`QuantumPropagators.Cheby.cheby_coeffs`

— FunctionCalculate Chebychev coefficients.

`a::Vector{Float64} = cheby_coeffs(Δ, dt; limit=1e-12)`

return an array of coefficients larger than `limit`

.

**Arguments**

`Δ`

: the spectral radius of the underlying operator`dt`

: the time step

See also `cheby_coeffs!`

for an in-place version.

`QuantumPropagators.Cheby.cheby_coeffs!`

— FunctionCalculate Chebychev coefficients in-place.

`n::Int = cheby_coeffs!(coeffs, Δ, dt, limit=1e-12)`

overwrites the first `n`

values in `coeffs`

with new coefficients larger than `limit`

for the given new spectral radius `Δ`

and time step `dt`

. The `coeffs`

array will be resized if necessary, and may have a length > `n`

on exit.

See also `cheby_coeffs`

for an non-in-place version.

`QuantumPropagators.Controls`

Public:

`ParameterizedFunction`

`discretize`

`discretize_on_midpoints`

`evaluate`

`evaluate!`

`get_controls`

`get_parameters`

`get_tlist_midpoints`

`substitute`

`t_mid`

### Reference

`QuantumPropagators.Controls.ParameterizedFunction`

— TypeAbstract type for function-like objects with `parameters`

.

A struct that is an implementation of a `ParameterizedFunction`

:

- must have a
`parameters`

field that is an`AbstractVector`

of floats (e.g., a`ComponentArrays.ComponentVector`

) - must be callable with a single float argument
`t`

, - may define getters and setters for referencing the values in
`parameters`

with convenient names.

The `parameters`

field of any `ParameterizedFunction`

can be accessed via `get_parameters`

.

See How to define a parameterized control for an example. You may use the `QuantumPropagators.Interfaces.check_parameterized_function`

to check the implementation of a `ParameterizedFunction`

subtype.

`QuantumPropagators.Controls.discretize`

— FunctionEvaluate `control`

at every point of `tlist`

.

`values = discretize(control, tlist; via_midpoints=true)`

discretizes the given `control`

to a Vector of values defined on the points of `tlist`

.

If `control`

is a function, it is first evaluated at the midpoint of `tlist`

, see `discretize_on_midpoints`

, and then the values on the midpoints are converted to values on `tlist`

. This discretization is more stable than directly evaluating the control function at the values of `tlist`

, and ensures that repeated round-trips between `discretize`

and `discretize_on_midpoints`

can be done safely, see the note in the documentation of `discretize_on_midpoints`

.

The latter can still be achieved by passing `via_midpoints=false`

. While such a direct discretization is suitable e.g. for plotting, but it is unsuitable for round-trips between `discretize`

and `discretize_on_midpoints`

(constant controls on `tlist`

may result in a zig-zag on the intervals of `tlist`

).

If `control`

is a vector, a copy of `control`

will be returned if it is of the same length as `tlist`

. Otherwise, `control`

must have one less value than `tlist`

, and is assumed to be defined on the midpoints of `tlist`

. In that case, `discretize`

acts as the inverse of `discretize_on_midpoints`

. See `discretize_on_midpoints`

for how control values on `tlist`

and control values on the intervals of `tlist`

are related.

`QuantumPropagators.Controls.discretize_on_midpoints`

— FunctionEvaluate `control`

at the midpoints of `tlist`

.

`values = discretize_on_midpoints(control, tlist)`

discretizes the given `control`

to a Vector of values on the midpoints of `tlist`

. Hence, the resulting `values`

will contain one less value than `tlist`

.

If `control`

is a vector of values defined on `tlist`

(i.e., of the same length as `tlist`

), it will be converted to a vector of values on the intervals of `tlist`

. The value for the first and last "midpoint" will remain the original values at the beginning and end of `tlist`

, in order to ensure exact boundary conditions. For all other midpoints, the value for that midpoint will be calculated by "un-averaging".

For example, for a `control`

and `tlist`

of length 5, consider the following diagram:

```
tlist index: 1 2 3 4 5
tlist: ⋅ ⋅ ⋅ ⋅ ⋅ input values cᵢ (i ∈ 1..5)
|̂/ ̄ ̄ ̂\ / ̂\ / ̂ ̄ ̄\|̂
midpoints: x x x x output values pᵢ (i ∈ 1..4)
midpoints index: 1 2 3 4
```

We will have $p₁=c₁$ for the first value, $p₄=c₅$ for the last value. For all other points, the control values $cᵢ = \frac{p_{i-1} + p_{i}}{2}$ are the average of the values on the midpoints. This implies the "un-averaging" for the midpoint values $pᵢ = 2 c_{i} - p_{i-1}$.

An arbitrary input `control`

array may not be compatible with the above averaging formula. In this case, the conversion will be "lossy" (`discretize`

will not recover the original `control`

array; the difference should be considered a "discretization error"). However, any *further* round-trip conversions between points and intervals are bijective and preserve the boundary conditions. In this case, the `discretize_on_midpoints`

and `discretize`

methods are each other's inverse. This also implies that for an optimal control procedure, it is safe to modify *midpoint* values. Modifying the the values on the time grid directly on the other hand may accumulate discretization errors.

If `control`

is a vector of one less length than `tlist`

, a copy of `control`

will be returned, under the assumption that the input is already properly discretized.

If `control`

is a function, the function will be directly evaluated at the midpoints marked as `x`

in the above diagram..

**See also**

`get_tlist_midpoints`

– get all the midpoints on which the control will be discretized.`t_mid`

– get a particular midpoint.`discretize`

– discretize directly on`tlist`

instead of on the midpoints

`QuantumPropagators.Controls.evaluate`

— FunctionEvaluate all controls.

In general, `evaluate(object, args...; vals_dict=IdDict())`

evaluates the `object`

for a specific point in time indicated by the positional `args`

. Any control in `object`

is evaluated at the specified point in time. Alternatively, the `vals_dict`

maps a controls to value ("plug in this value for the given control")

For example,

`op = evaluate(generator, t)`

evaluates `generator`

at time `t`

. This requires that any control in `generator`

is a callable that takes `t`

as a single argument.

`op = evaluate(generator, tlist, n)`

evaluates `generator`

for the n'th interval of `tlist`

. This uses the definitions for the midpoints in `discretize_on_midpoints`

. The controls in `generator`

may be vectors (see `discretize`

, `discretize_on_midpoints`

) or callables of `t`

.

```
op = evaluate(generator, t; vals_dict)
op = evaluate(generator, tlist, n; vals_dict)
```

resolves any explicit time dependencies in `generator`

at the specified point in time, but uses the value in the given `vals_dict`

for any control in `vals_dict`

.

```
a = evaluate(ampl, tlist, n; vals_dict=IdDict())
a = evaluate(ampl, t; vals_dict=IdDict())
```

evaluates a control amplitude to a scalar by evaluating any explicit time dependency, and by replacing each control with the corresponding value in `vals_dict`

.

Calling `evaluate`

for an object with no implicit or explicit time dependence should return the object unchanged.

For generators without any explicit time dependence,

`op = evaluate(generator; vals_dict)`

can be used. The `vals_dict`

in this case must contain values for all controls in `generator`

.

**See also:**

`evaluate!`

— update an existing operator with a re-evaluation of a

generator at a different point in time.

`QuantumPropagators.Controls.evaluate!`

— FunctionUpdate an existing evaluation of a `generator`

.

`evaluate!(op, generator, args..; vals_dict=IdDict())`

performs an in-place update on an `op`

the was obtained from a previous call to `evaluate`

with the same `generator`

, but for a different point in time and/or different values in `vals_dict`

.

`QuantumPropagators.Controls.get_controls`

— FunctionExtract a Tuple of controls.

`controls = get_controls(generator)`

extracts the controls from a single dynamical generator.

For example, if `generator = hamiltonian(H0, (H1, ϵ1), (H2, ϵ2))`

, extracts `(ϵ1, ϵ2)`

.

`get_controls(operator)`

for a static operator (matrix) returns an empty tuple.

`controls = get_controls(trajectories)`

extracts the controls from a list of trajectories (i.e., from each trajectory's `generator`

). Controls that occur multiple times in the different trajectories will occur only once in the result.

`controls = get_controls(problem)`

extracts the controls from `problem.trajectories`

.

`QuantumPropagators.Controls.get_parameters`

— FunctionObtain analytic parameters of the given `control`

.

`parameters = get_parameters(control)`

obtains `parameters`

as an `AbstractVector{Float64}`

containing any tunable analytic parameters associated with the `control`

. The specific type of `parameters`

depends on how `control`

is defined, but a `ComponentArrays.ComponentVector`

should be a common array type.

Mutating the resulting vector must directly affect the control in any subsequent call to `evaluate`

. That is, the values in `parameters`

must alias values inside the `control`

.

Note that the `control`

must be an object specifically designed to have analytic parameters. Typically, it should be implemented as a subtype of `ParameterizedFunction`

. For a simple function `ϵ(t)`

or a vector of pulse values, which are the default types of controls discussed in the documentation of `hamiltonian`

, the `get_parameters`

function will return an empty vector.

More generally,

`parameters = get_parameters(object)`

collects and combines all unique parameter arrays from the controls inside the `object`

. The `object`

may be a `Generator`

, `Trajectory`

, `ControlProblem`

, or any other object for which `get_controls(object)`

is defined. If there are multiple controls with different parameter arrays, these are combined in a `RecursiveArrayTools.ArrayPartition`

. This requires the `RecursiveArrayTools`

package to be loaded. Again, mutating `parameters`

directly affects the underlying controls.

The `parameters`

may be used as part of the `parameters`

attribute of a propagator for time-continuous dynamics, like a general ODE solver, or in an optimization that tunes analytic control parameters, e.g., with a Nelder-Mead method. Examples might include the widths, peak amplitudes, and times of a superposition of Gaussians [20], cf. the example of a `ParameterizedFunction`

, or the amplitudes associated with spectral components in a random truncated basis [21].

The `parameters`

are not intended for optimization methods such as GRAPE or Krotov that fundamentally use a piecewise-constant control ansatz. In the context of such methods, the "control parameters" are always the amplitudes of the `control`

at the mid-points of the time grid, as obtained by `discretize_on_midpoints`

, and `get_parameters`

is ignored.

`parameters = get_parameters(trajectories)`

collects and combines get parameter arrays from all the generators in `trajectories`

. Note that this allows any custom generator type to define a custom `get_parameters`

method to override the default of obtaining the parameters recursively from the controls inside the generator.

`parameters = get_parameters(problem)`

extracts the `parameters`

from `problem.trajectories`

.

`QuantumPropagators.Controls.get_tlist_midpoints`

— FunctionShift time grid values to the interval midpoints

```
tlist_midpoints = get_tlist_midpoints(
tlist; preserve_start=true, preserve_end=true
)
```

takes a vector `tlist`

of length $n$ and returns a `Vector{Float64}`

of length $n-1$ containing the midpoint values of each interval. The intervals in `tlist`

are not required to be uniform.

By default, the first and last point of `tlist`

is preserved, see `discretize_on_midpoints`

. This behavior can be disabled by passing `preserve_start`

and `preserve_end`

as `false`

in order to use the midpoints of the first and last interval, respectively.

**See also**

`t_mid`

– get a particular midpoint.

`QuantumPropagators.Controls.substitute`

— FunctionSubstitute inside the given object.

`object = substitute(object, replacements)`

returns a modified object with the replacements defined in the given `replacements`

dictionary. Things that can be replaced include operators, controls, and amplitudes. For example,

```
generator = substitute(generator::Generator, replacements)
operator = substitute(operator::Operator, replacements)
amplitude = substitute(amplitude, controls_replacements)
```

Note that `substitute`

cannot be used to replace dynamic quantities, e.g. controls, with static value. Use `evaluate`

instead for that purpose.

```
trajectory = substitute(trajectory::Trajectory, replacements)
trajectories = substitute(trajectories::Vector{<:Trajectory}, replacements)
```

recursively substitutes the `initial_state`

, `generator`

, and `target_state`

.

`problem = substitute(problem::ControlProblem, replacements)`

substitutes in `problem.trajectories`

`QuantumPropagators.Controls.t_mid`

— FunctionMidpoint of n'th interval of tlist.

`t = t_mid(tlist, n)`

returns the `t`

that is the midpoint between points `tlist[n+1]`

and `tlist[n]`

, but snapping to the beginning/end to follow the convention explained in `discretize_on_midpoints`

(to preserve exact boundary conditions at the edges of the time grid.)

**See also**

`get_tlist_midpoints`

– get all the midpoints in one go.

`QuantumPropagators.ExpProp`

Public:

Private:

### Reference

`QuantumPropagators.ExpProp.ExpPropWrk`

— Type`ExpPropWrk(v0)`

Workspace for propagation via direct matrix exponentiation.

Initializes the workspace for the propagation of a vector `v0`

`QuantumPropagators.ExpProp.expprop!`

— Function`expprop!(Ψ, H, dt, wrk; func=(H_dt -> exp(-1im * H_dt)), _...)`

Evaluate `Ψ = func(H*dt) Ψ`

by directly evaluating `U = func(H*dt)`

, i.e. by matrix exponentiation for the default `func`

, and then multiplying `U`

and `Ψ`

in-place with `mul!`

.

The workspace `wrk`

must be initialized with `ExpPropWrk`

to provide storage for a temporary state.

Keyword arguments besides `func`

are ignored.

`QuantumPropagators.ExpProp.expprop`

— Function`Ψ_out = expprop(Ψ, H, dt, wrk; func=(H_dt -> exp(-1im * H_dt)), _...)`

evaluates `Ψ_out = func(H*dt) Ψ`

as in `expprop!`

, but not acting in-place.

`QuantumPropagators.Generators`

Public:

### Reference

`QuantumPropagators.Generators.Generator`

— TypeA time-dependent generator.

`Generator(ops::Vector{OT}, amplitudes::Vector{AT})`

produces an object of type `Generator{OT,AT}`

that represents

\[Ĥ(t)= Ĥ_0 + \sum_l a_l(\{ϵ_{l'}(t)\}, t) \, Ĥ_l\,,\]

where $Ĥ_l$ are the `ops`

and $a_l(t)$ are the `amplitudes`

. $Ĥ(t)$ and $Ĥ_l$ may represent operators in Hilbert space or super-operators in Liouville space. If the number of `amplitudes`

is less than the number of `ops`

, the first `ops`

are considered as drift terms ($Ĥ_0$, respectively subsequent terms with $a_l ≡ 1$). At least one time-dependent amplitude is required. Each amplitude may depend on one or more control functions $ϵ_{l'}(t)$, although most typically $a_l(t) ≡ ϵ_l(t)$, that is, the `amplitudes`

are simply a vector of the controls. See `hamiltonian`

for details.

A `Generator`

object should generally not be instantiated directly, but via `hamiltonian`

or `liouvillian`

.

The list of `ops`

and `amplitudes`

are properties of the `Generator`

. They should not be mutated.

**See also**

`QuantumPropagators.Generators.Operator`

— TypeA static operator in Hilbert or Liouville space.

`Operator(ops::Vector{OT}, coeffs::Vector{CT})`

produces an object of type `Operator{OT,CT}`

that encapsulates the "lazy" sum

\[Ĥ = \sum_l c_l Ĥ_l\,,\]

where $Ĥ_l$ are the `ops`

and $c_l$ are the `coeffs`

, which each must be a constant `Number`

. If the number of coefficients is less than the number of operators, the first `ops`

are considered to have $c_l = 1$.

An `Operator`

object would generally not be instantiated directly, but be obtained from a `Generator`

via `evaluate`

.

The $Ĥ_l$ in the sum are considered immutable. This implies that an `Operator`

can be updated in-place with `evaluate!`

by only changing the `coeffs`

.

`QuantumPropagators.Generators.ScaledOperator`

— TypeA static operator with a scalar pre-factor.

`op = ScaledOperator(α, Ĥ)`

represents the "lazy" product $α Ĥ$ where $Ĥ$ is an operator (typically an `Operator`

instance) and $α$ is a scalar.

`QuantumPropagators.Generators.hamiltonian`

— FunctionInitialize a (usually time-dependent) Hamiltonian.

The most common usage is, e.g.,

```
using QuantumPropagators
H₀ = ComplexF64[0 0; 0 1];
H₁ = ComplexF64[0 1; 1 0];
ϵ₁(t) = 1.0;
hamiltonian(H₀, (H₀, ϵ₁))
# output
Generator with 2 ops and 1 amplitudes
ops::Vector{Matrix{ComplexF64}}:
ComplexF64[0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im]
ComplexF64[0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im]
amplitudes::Vector{typeof(ϵ₁)}:
ϵ₁
```

In general,

`H = hamiltonian(terms...; check=true)`

constructs a Hamiltonian based on the given `terms`

. Each term must be an operator or a tuple `(op, ampl)`

of an operator and a control amplitude. Single operators are considered "drift" terms.

In most cases, each control amplitude will simply be a control function or vector of pulse values. In general, `ampl`

can be an arbitrary object that depends on one or more controls, which must be obtainable via `get_controls(ampl)`

. See `QuantumPropagators.Interfaces.check_amplitude`

for the required interface.

The `hamiltonian`

function will generally return a `Generator`

instance. However, if none of the given terms are time-dependent, it may return a static operator (e.g., an `AbstractMatrix`

or `Operator`

):

```
hamiltonian(H₀)
# output
2×2 Matrix{ComplexF64}:
0.0+0.0im 0.0+0.0im
0.0+0.0im 1.0+0.0im
```

```
hamiltonian(H₀, (H₁, 2.0))
# output
Operator with 2 ops and 1 coeffs
ops::Vector{Matrix{ComplexF64}}:
ComplexF64[0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im]
ComplexF64[0.0 + 0.0im 1.0 + 0.0im; 1.0 + 0.0im 0.0 + 0.0im]
coeffs: [2.0]
```

The `hamiltonian`

function may generate warnings if the `terms`

are of an unexpected type or structure. These can be suppressed with `check=false`

.

`QuantumPropagators.Generators.liouvillian`

— FunctionConstruct a Liouvillian `Generator`

.

`ℒ = liouvillian(Ĥ, c_ops=(); convention=:LvN, check=true)`

calculates the sparse Liouvillian super-operator `ℒ`

from the Hamiltonian `Ĥ`

and a list `c_ops`

of Lindblad operators.

With `convention=:LvN`

, applying the resulting `ℒ`

to a vectorized density matrix `ρ⃗`

calculates $\frac{d}{dt} \vec{\rho}(t) = ℒ \vec{\rho}(t)$ equivalent to the Liouville-von-Neumann equation for the density matrix $ρ̂$,

\[\frac{d}{dt} ρ̂(t) = -i [Ĥ, ρ̂(t)] + \sum_k\left( Â_k ρ̂ Â_k^\dagger - \frac{1}{2} A_k^\dagger Â_k ρ̂ - \frac{1}{2} ρ̂ Â_k^\dagger Â_k \right)\,,\]

where the Lindblad operators $Â_k$ are the elements of `c_ops`

.

The Hamiltonian $Ĥ$ will generally be time-dependent. For example, it may be a `Generator`

as returned by `hamiltonian`

. For example, for a Hamiltonian with the terms `(Ĥ₀, (Ĥ₁, ϵ₁), (Ĥ₂, ϵ₂))`

, where `Ĥ₀`

, `Ĥ₁`

, `Ĥ₂`

are matrices, and `ϵ₁`

and `ϵ₂`

are functions of time, the resulting `ℒ`

will be a `Generator`

corresponding to terms `(ℒ₀, (ℒ₁, ϵ₁), (ℒ₂, ϵ₂))`

, where the initial terms is the superoperator `ℒ₀`

for the static component of the Liouvillian, i.e., the commutator with the drift Hamiltonian `Ĥ₀`

, plus the dissipator (sum over $k$), as a sparse matrix. Time-dependent Lindblad operators are not currently supported. The remaining elements are tuples `(ℒ₁, ϵ₁)`

and `(ℒ₂, ϵ₂)`

corresponding to the commutators with the two control Hamiltonians, where `ℒ₁`

and `ℒ₂`

again are sparse matrices.

If $Ĥ$ is not time-dependent, the resulting `ℒ`

will likewise be a static operator. Passing `H=nothing`

with non-empty `c_ops`

initializes a pure dissipator.

With `convention=:TDSE`

, the Liouvillian will be constructed for the equation of motion $i \hbar \frac{d}{dt} \vec{\rho}(t) = ℒ \vec{\rho}(t)$ to match exactly the form of the time-dependent Schrödinger equation. While this notation is not standard in the literature of open quantum systems, it has the benefit that the resulting `ℒ`

can be used in a numerical propagator for a (non-Hermitian) Schrödinger equation without any change. Thus, for numerical applications, `convention=:TDSE`

is generally preferred. The returned `ℒ`

between the two conventions differs only by a factor of $i$, since we generally assume $\hbar=1$.

The `convention`

keyword argument is mandatory, to force a conscious choice.

See Goerz et. al. "Optimal control theory for a unitary operation under dissipative evolution", arXiv 1312.0111v2, Appendix B.2 for the explicit construction of the Liouvillian superoperator as a sparse matrix.

Passing `check=false`

, suppresses warnings and errors about unexpected types or the structure of the arguments, cf. `hamiltonian`

.

`QuantumPropagators.Interfaces`

Public:

`check_amplitude`

`check_control`

`check_generator`

`check_operator`

`check_parameterized`

`check_parameterized_function`

`check_propagator`

`check_state`

`check_tlist`

`supports_inplace`

### Reference

`QuantumPropagators.Interfaces.check_amplitude`

— FunctionCheck amplitude appearing in `Generator`

.

`@test check_amplitude(ampl; tlist, quiet=false)`

verifies that the given `ampl`

is a valid element in the list of `amplitudes`

of a `Generator`

object. Specifically:

`get_controls(ampl)`

must be defined and return a tuple- all controls in
`ampl`

must pass`check_control`

`substitute(ampl, controls_replacements)`

must be defined`evaluate(ampl, tlist, n)`

must be defined and return a Number`evaluate(ampl, tlist, n; vals_dict)`

must be defined and return a Number

If `for_parameterization`

(may require the `RecursiveArrayTools`

package to be loaded):

`get_parameters(ampl)`

must be defined and return a vector of floats. Mutating that vector must mutate the controls inside the`ampl`

.

The function returns `true`

for a valid amplitude and `false`

for an invalid amplitude. Unless `quiet=true`

, it will log an error to indicate which of the conditions failed.

`QuantumPropagators.Interfaces.check_control`

— FunctionCheck that `control`

can be evaluated on a time grid.

```
@test check_control(
control;
tlist,
for_parameterization=true,
for_time_continuous=(control isa Function),
quiet=false
)
```

verifies the given `control`

(one of the elements of the tuple returned by `get_controls`

):

`evaluate(control, tlist, n)`

must be defined and return a`Float64`

`evaluate(control, tlist, n; vals_dict=IdDict(control => v))`

must be defined and return`v`

`discretize(control, tlist)`

must be defined and return a vector of floats of the same size as`tlist`

. Only if`length(tlist) > 2`

.- all values in
`discretize(control, tlist)`

must be finite (`isfinite`

). `discretize_on_midpoints(control, tlist)`

must be defined and return a vector of floats with one element less than`tlist`

. Only if`length(tlist) > 2`

.- all values in
`discretize_on_midpoints(control, tlist)`

must be finite (`isfinite`

)

If `for_time_continuous`

:

`evaluate(control, t)`

must be defined and return a`Float64`

`evaluate(control, t; vals_dict=IdDict(control => v))`

must be defined and return`v`

If `for_parameterization`

:

`get_parameters(control)`

must be defined and return a vector of floats. Mutating that vector must mutate the control.

The function returns `true`

for a valid control and `false`

for an invalid control. Unless `quiet=true`

, it will log an error to indicate which of the conditions failed.

`QuantumPropagators.Interfaces.check_generator`

— FunctionCheck the dynamical `generator`

for propagating `state`

over `tlist`

.

```
@test check_generator(
generator; state, tlist,
for_pwc=true, for_time_continuous=false,
for_expval=true, for_parameterization=false,
atol=1e-14, quiet=false)
```

verifies the given `generator`

:

`get_controls(generator)`

must be defined and return a tuple- all controls returned by
`get_controls(generator)`

must pass`check_control`

`substitute(generator, replacements)`

must be defined- If
`generator`

is a`Generator`

instance, all elements of`generator.amplitudes`

must pass`check_amplitude`

with`for_parameterization`

.

If `for_pwc`

(default):

`op = evaluate(generator, tlist, n)`

must return a valid operator (`check_operator`

), with forwarded keyword arguments (including`for_expval`

)- If
`QuantumPropagators.Interfaces.supports_inplace(op)`

is`true`

,`evaluate!(op, generator, tlist, n)`

must be defined

If `for_time_continuous`

:

`evaluate(generator, t)`

must return a valid operator (`check_operator`

), with forwarded keyword arguments (including`for_expval`

)If

`QuantumPropagators.Interfaces.supports_inplace(op)`

is`true`

,`evaluate!(op, generator, t)`

must be defined

If `for_parameterization`

(may require the `RecursiveArrayTools`

package to be loaded):

`get_parameters(generator)`

must be defined and return a vector of floats. Mutating that vector must mutate the controls inside the`generator`

.

The function returns `true`

for a valid generator and `false`

for an invalid generator. Unless `quiet=true`

, it will log an error to indicate which of the conditions failed.

`QuantumPropagators.Interfaces.check_operator`

— FunctionCheck that `op`

is a valid operator that can be applied to `state`

.

```
@test check_operator(op; state, tlist=[0.0, 1.0],
for_expval=true, atol=1e-14, quiet=false)
```

verifies the given `op`

relative to `state`

. The `state`

must pass `check_state`

.

An "operator" is any object that `evaluate`

returns when evaluating a time-dependent dynamic generator. The specific requirements for `op`

are:

`op`

must not be time-dependent:`evaluate(op, tlist, 1) ≡ op`

`op`

must not contain any controls:`length(get_controls(op)) == 0`

`op * state`

must be defined- The
`QuantumPropagators.Interfaces.supports_inplace`

method must be defined for`op`

. If it returns`true`

, it must be possible to evaluate a generator in-place into the existing`op`

. See`check_generator`

.

If `QuantumPropagators.Interfaces.supports_inplace(state)`

:

- The 3-argument
`LinearAlgebra.mul!`

must apply`op`

to the given`state`

- The 5-argument
`LinearAlgebra.mul!`

must apply`op`

to the given`state`

`LinearAlgebra.mul!`

must match`*`

, if applicable`LinearAlgebra.mul!`

must return the resulting state

If `for_expval`

(typically required for optimal control):

`LinearAlgebra.dot(state, op, state)`

must return return a number`dot(state, op, state)`

must match`dot(state, op * state)`

, if applicable

The function returns `true`

for a valid operator and `false`

for an invalid operator. Unless `quiet=true`

, it will log an error to indicate which of the conditions failed.

`QuantumPropagators.Interfaces.check_parameterized`

— FunctionCheck that that the object supports the parameterization interface.

`@test check_parameterized(object; name="::$typeof(object))", quiet=false)`

verifies that the given `object`

:

- can be passed to
`get_parameters`

, which must return an`AbstractVector`

of`Float64`

- is mutated by mutating the
`parameters`

obtained by`get_parameters`

**See also**

`check_parameterized_function`

is`object`

is a`ParameterizedFunction`

`QuantumPropagators.Interfaces.check_parameterized_function`

— FunctionCheck a `ParameterizedFunction`

instance.

`@test check_parameterized_function(f; tlist; quiet=false)`

verifies that the given `f`

:

- is an instance of
`ParameterizedFunction`

. - has a field
`parameters`

that is an`AbstractVector{Float64}`

. - is a callable as
`f(t)`

for values of`t`

in`tlist`

, returning a`Float64`

. `get_parameters`

provides access to the`parameters`

field.- passes
`check_parameterized`

**See also**

`check_parameterized`

for objects that have parameters (`get_parameters`

), but are not instances of`ParameterizedFunction`

`QuantumPropagators.Interfaces.check_propagator`

— FunctionCheck that the given `propagator`

implements the required interface.

`@test check_propagator(propagator; atol=1e-14, quiet=false)`

verifies that the `propagator`

matches the interface described for an `AbstractPropagator`

. The `propagator`

must have been freshly initialized with `init_prop`

.

`propagator`

must have the properties`state`

,`tlist`

,`t`

,`parameters`

,`backward`

, and`inplace`

`propagator.state`

must be a valid state (see`check_state`

)- If
`propagator.inplace`

is true,`supports_inplace`

for`propagator.state`

must also be true `propagator.tlist`

must be monotonically increasing.`propagator.t`

must be the first or last element of`propagator.tlist`

, depending on`propagator.backward`

`prop_step!(propagator)`

must be defined and return a valid state until the time grid is exhausted- For an in-place propagator, the state returned by
`prop_step!`

must be the`propagator.state`

object - For a not-in-place propagator, the state returned by
`prop_step!`

must be a new object `prop_step!`

must advance`propagator.t`

forward or backward one step on the time grid`prop_step!`

must return`nothing`

when going beyond the time grid`set_t!(propagator, t)`

must be defined and set`propagator.t`

`set_state!(propagator, state)`

must be defined and set`propagator.state`

.`set_state!(propagator, state)`

for an in-place propagator must overwrite`propagator.state`

in-place.`set_state!`

must return the set`propagator.state`

- In a
`PiecewisePropagator`

,`propagator.parameters`

must be a dict mapping controls to a vector of values, one for each interval on`propagator.tlist`

`reinit_prop!`

must be defined and re-initialize the propagator`reinit_prop!(propagator, state)`

must be idempotent. That is, repeated calls to`reinit_prop!`

leave the`propagator`

unchanged.

The function returns `true`

for a valid propagator and `false`

for an invalid propagator. Unless `quiet=true`

, it will log an error to indicate which of the conditions failed.

`QuantumPropagators.Interfaces.check_state`

— FunctionCheck that `state`

is a valid element of a Hilbert space.

`@test check_state(state; normalized=false, atol=1e-15, quiet=false)`

verifies the following requirements:

- The inner product (
`LinearAlgebra.dot`

) of two states must return a Complex number. - The
`LinearAlgebra.norm`

of`state`

must be defined via the inner product. This is the*definition*of a Hilbert space, a.k.a a "complete inner product space" or more precisely a "Banach space (normed vector space) where the norm is induced by an inner product". - The `QuantumPropagators.Interfaces.supports_inplace method must be defined for
`state`

Any `state`

must support the following not-in-place operations:

`state + state`

and`state - state`

must be defined`copy(state)`

must be defined and return an object of the same type as`state`

`c * state`

for a scalar`c`

must be defined`norm(state + state)`

must fulfill the triangle inequality`zero(state)`

must be defined and produce a state with norm 0`0.0 * state`

must produce a state with norm 0`copy(state) - state`

must have norm 0`norm(state)`

must have absolute homogeneity:`norm(s * state) = s * norm(state)`

If `supports_inplace(state)`

is `true`

, the `state`

must also support the following:

`similar(state)`

must be defined and return a valid state of the same type a`state`

`copyto!(other, state)`

must be defined`fill!(state, c)`

must be defined`LinearAlgebra.lmul!(c, state)`

for a scalar`c`

must be defined`LinearAlgebra.axpy!(c, state, other)`

must be defined`norm(state)`

must fulfill the same general mathematical norm properties as for the non-in-place norm.

If `normalized`

(not required by default):

`LinearAlgebra.norm(state)`

must be 1

It is strongly recommended to always support immutable operations (also for mutable states)

The function returns `true`

for a valid state and `false`

for an invalid state. Unless `quiet=true`

, it will log an error to indicate which of the conditions failed.

`QuantumPropagators.Interfaces.check_tlist`

— FunctionCheck that the given `tlist`

is valid.

`@test check_tlist(tlist; quiet=false)`

verifies the given time grid. A valid time grid must

- be a
`Vector{Float64}`

, - contain at least two points (beginning and end),
- be monotonically increasing

The function returns `true`

for a valid time grid and `false`

for an invalid time grid. Unless `quiet=true`

, it will log an error to indicated which of the conditions failed.

`QuantumPropagators.Interfaces.supports_inplace`

— FunctionIndicate whether a given state or operator supports in-place operations

`supports_inplace(state)`

Indicates that propagators can assume that the in-place requirements defined in `QuantumPropagators.Interfaces.check_state`

hold. States with in-place support must also fulfill specific properties when interacting with operators, see `QuantumPropagators.Interfaces.check_operator`

.

`supports_inplace(op)`

Indicates that the operator can be evaluated in-place with `evaluate!`

, see `QuantumPropagators.Interfaces.check_generator`

Note that `supports_inplace`

is not quite the same as `Base.ismutable`

: When using custom structs for states or operators, even if those structs are not defined as `mutable`

, they may still define the in-place interface (typically because their *components* are mutable).

`QuantumPropagators.Newton`

Public:

Private:

### Reference

`QuantumPropagators.Newton.NewtonWrk`

— Type`NewtonWrk(v0, m_max=10)`

Workspace for the Newton-with-restarted-Arnoldi propagation routine.

Initializes the workspace for the propagation of a vector `v0`

, using a maximum Krylov dimension of `m_max`

in each restart iteration. Note that `m_max`

should be smaller than the length of `v0`

.

`QuantumPropagators.Newton.newton!`

— Function```
newton!(Ψ, H, dt, wrk; func=(z -> exp(-1im*z)), norm_min=1e-14, relerr=1e-12,
max_restarts=50, _...)
```

Evaluate `Ψ = func(H*dt) Ψ`

using a Newton-with-restarted-Arnoldi scheme.

**Arguments**

`Ψ`

: The state to propagate, will be overwritten in-place with the propagated state`H`

: Operator acting on`Ψ`

. Together with`dt`

, this is the argument to`func`

`dt`

: Implicit time step. Together with`H`

, this is the argument to`func`

`wkr`

: Work array, initialized with`NewtonWrk`

`func`

: The function to apply to`H dt`

, taking a single (scalar) complex-valued argument`z`

in place of`H dt`

. The default`func`

is to evaluate the time evaluations operator for the Schrödinger equation`norm_min`

: the minimum norm at which to consider a state similar to`Ψ`

as zero`relerr`

: The relative error defining the convergence condition for the restart iteration. Propagation stops when the norm of the accumulated`Ψ`

is stable up to the given relative error`max_restarts`

: The maximum number of restart iterations. Exceeding`max_restarts`

will throw an`AssertionError`

.

All other keyword arguments are ignored.

`QuantumPropagators.Newton.extend_newton_coeffs!`

— Function`extend_newton_coeffs!(a, n_a, leja, func, n_leja, radius)`

Extend the array `a`

of existing Newton coefficients for the expansion of the `func`

from `n_a`

coefficients to `n_leja`

coefficients. Return a new value `n_a=n_a+n_leja`

with the total number of Newton coefficients in the updated `a`

.

**Arguments**

`a`

: On input, a zero-based array of length`n_a`

or greater, containing Newton coefficients. On output, array containing a total`n_leja`

coefficients. The array`a`

will be resized if necessary, and may have a length greater than`n_leja`

on output`n_a`

: The number of Newton coefficients in`a`

, on input. Elements of`a`

beyond the first`n_a`

elements will be overwritten.`leja`

: Array of normalized Leja points, containing at least`n_leja`

elements.`func`

: Function for which to calculate Newton coefficients`n_leja`

: The number of elements in`leja`

to use for calculating new coefficients, and the total number of Newton coefficients on output`radius`

: Normalization radius for divided differences

`QuantumPropagators.Newton.extend_leja!`

— Function`extend_leja!(leja, n, newpoints, n_use)`

Given an array of `n`

(ordered) Leja points, extract `n_use`

points from `newpoints`

, and append them to the existing Leja points. The array `leja`

should be sufficiently large to hold the new Leja points, which are appended after index `n_old`

. It will be re-allocated if necessary and may have a size of up to `2*(n+n_use)`

.

**Arguments**

`leja`

: Array of leja values. Must contain the "old" leja values to be kept in`leja(0:n-1)`

. On output,`n_use`

new leja points will be in`leja(n+:n+n_use-1)`

, for the original value of`n`

. The`leja`

array must use zero-based indexing.`n`

: On input, number of "old" leja points in`leja`

. On output, total number of leja points (i.e.`n=n+n_use`

)`newpoints`

: On input, candidate points for new leja points. The`n_use`

best values will be chosen and added to`leja`

. On output, the values of`new_points`

are undefined.`n_use`

: Number of points that should be added to`leja`

`QuantumPropagators.Shapes`

Public:

### Reference

`QuantumPropagators.Shapes.blackman`

— FunctionBlackman window shape.

`blackman(t, t₀, T; a=0.16)`

calculates

\[B(t; t_0, T) = \frac{1}{2}\left( 1 - a - \cos\left(2π \frac{t - t_0}{T - t_0}\right) + a \cos\left(4π \frac{t - t_0}{T - t_0}\right) \right)\,,\]

for a scalar `t`

, with $a$ = 0.16.

See https://en.wikipedia.org/wiki/Window_function#Blackman_windows

A Blackman shape looks nearly identical to a Gaussian with a 6-sigma interval between `t₀`

and `T`

. Unlike the Gaussian, however, it will go exactly to zero at the edges. Thus, Blackman pulses are often preferable to Gaussians.

`QuantumPropagators.Shapes.box`

— FunctionBox shape (Theta-function).

`box(t, t₀, T)`

evaluates the Heaviside (Theta-) function $\Theta(t) = 1$ for $t_0 \le t \le T$; and $\Theta(t) = 0$ otherwise.

`QuantumPropagators.Shapes.flattop`

— FunctionFlat shape (amplitude 1.0) with a switch-on/switch-off from zero.

`flattop(t; T, t_rise, t₀=0.0, t_fall=t_rise, func=:blackman)`

evaluates a shape function that starts at 0 at $t=t₀$, and ramps to to 1 during the `t_rise`

interval. The function then remains at value 1, before ramping down to 0 again during the interval `t_fall`

before `T`

. For $t < t₀$ and $t > T$, the shape is zero.

The default switch-on/-off shape is half of a Blackman window (see `blackman`

).

For `func=:sinsq`

, the switch-on/-off shape is a sine-squared curve.

`QuantumPropagators.SpectralRange`

Public:

Private:

### Reference

`QuantumPropagators.SpectralRange.specrange`

— FunctionCalculate the spectral range of a Hamiltonian `H`

on the real axis.

`E_min, E_max = specrange(H; method=:auto, kwargs...)`

calculates the approximate lowest and highest eigenvalues of `H`

. Any imaginary part in the eigenvalues is ignored: the routine is intended for (although not strictly limited to) a Hermitian `H`

.

This delegates to

`specrange(H, method; kwargs...)`

for the different methods.

The default `method=:auto`

chooses the best method for the given `H`

. This is `:diag`

for small matrices, and `:arnoldi`

otherwise. If both `E_min`

and `E_max`

are given in the `kwargs`

, those will be returned directly (`method=:manual`

).

Keyword arguments not relevant to the underlying implementation will be ignored.

```
E_min, E_max = specrange(
H, :arnoldi;
rng=Random.GLOBAL_RNG,
state=random_state(H; rng),
m_min=20,
m_max=60,
prec=1e-3,
norm_min=1e-15,
enlarge=true
)
```

uses Arnoldi iteration with `state`

as the starting vector. It approximates the eigenvalues of `H`

with between `m_min`

and `m_max`

Ritz values, until the lowest and highest eigenvalue are stable to a relative precision of `prec`

. The `norm_min`

parameter is passed to the underlying `arnoldi!`

.

If `enlarge=true`

(default) the returned `E_min`

and `E_max`

will be enlarged via a heuristic to slightly over-estimate the spectral radius instead of under-estimating it.

`E_min, E_max = specrange(H, :diag)`

uses exact diagonization via the standard `eigvals`

function to obtain the smallest and largest eigenvalue. This should only be used for relatively small matrices.

`E_min, E_max = specrange(H, :manual; E_min, E_max)`

directly returns the given `E_min`

and `E_max`

without considering `H`

.

`QuantumPropagators.SpectralRange.ritzvals`

— FunctionCalculate a vector for Ritz values converged to a given precision.

`R = ritzvals(G, state, m_min, m_max=2*m_min; prec=1e-5, norm_min=1e-15)`

calculates a complex vector `R`

of at least `m_min`

(assuming a sufficient Krylov dimension) and at most `m_max`

Ritz values.

`QuantumPropagators.SpectralRange.random_state`

— FunctionRandom normalized quantum state.

` Ψ = random_state(H; rng=Random.GLOBAL_RNG)`

returns a random normalized state compatible with the Hamiltonian `H`

. This is intended to provide a starting vector for estimating the spectral radius of `H`

via an Arnoldi method.

`QuantumPropagators.Storage`

Public:

### Reference

`QuantumPropagators.Storage.get_from_storage`

— Function`QuantumPropagators.Storage.get_from_storage!`

— FunctionObtain data from storage.

`get_from_storage!(data, storage, i)`

extracts data from the `storage`

for the i'th time slot. Inverse of `write_to_storage!`

. This modifies `data`

in-place. If `get_from_storage!`

is implemented for arbitrary `observables`

, it is the developer's responsibility that `init_storage`

, `write_to_storage!`

, and `get_from_storage!`

are compatible.

To extract immutable `data`

, the non-in-place version

`data = get_from_storage(storage, i)`

can be used.

`QuantumPropagators.Storage.init_storage`

— FunctionCreate a `storage`

array for propagation.

`storage = init_storage(state, tlist)`

creates a storage array suitable for storing a `state`

for each point in `tlist`

.

`storage = init_storage(state, tlist, observables)`

creates a storage array suitable for the data generated by the `observables`

applied to `state`

, see `map_observables`

, for each point in `tlist`

.

`storage = init_storage(data, nt)`

creates a storage arrays suitable for storing `data`

nt times, where `nt=length(tlist)`

. By default, this will be a vector of `typeof(data)`

and length `nt`

, or a `n × nt`

Matrix with the same `eltype`

as `data`

if `data`

is a Vector of length `n`

.

`QuantumPropagators.Storage.map_observable`

— FunctionApply a single `observable`

to `state`

.

`data = map_observable(observable, tlist, i, state)`

By default, `observable`

can be one of the following:

- A function taking the three arguments
`state`

,`tlist`

,`i`

, where`state`

is defined at time`tlist[i]`

. - A function taking a single argument
`state`

, under the assumption that the observable is time-independent - A matrix for which to calculate the expectation value with respect to the vector
`state`

.

The default `map_observables`

delegates to this function.

`QuantumPropagators.Storage.map_observables`

— FunctionObtain "observable" data from `state`

.

`data = map_observables(observables, tlist, i, state)`

calculates the data for a tuple of `observables`

applied to `state`

defined at time `tlist[i]`

. For a single observable (tuple of length 1), simply return the result of `map_observable`

.

For multiple observables, return the tuple resulting from applying `map_observable`

for each observable. If the tuple is "uniform" (all elements are of the same type, e.g. if each observable calculates the expectation value of a Hermitian operator), it is converted to a Vector. This allows for compact storage in a storage array, see `init_storage`

.

`QuantumPropagators.Storage.write_to_storage!`

— FunctionPlace data into `storage`

for time slot `i`

.

`write_to_storage!(storage, i, data)`

for a `storage`

array created by `init_storage`

stores the `data`

obtained from `map_observables`

at time slot `i`

.

Conceptually, this corresponds roughly to `storage[i] = data`

, but `storage`

may have its own idea on how to store data for a specific time slot. For example, with the default `init_storage`

Vector data will be stored in a matrix, and `write_to_storage!`

will in this case write data to the i'th column of the matrix.

For a given type of `storage`

and `data`

, it is the developer's responsibility that `init_storage`

and `write_to_storage!`

are compatible.