The highest-level API of the QuantumPropagators.jl package (apart from some convenience functions) consists of a single function:

  • propagate — Propagate a quantum state over an entire time grid under a given generator

At a slightly lower level, propagation of quantum states in encapsulated by The Propagator interface:

  • init_prop — Initialize a propagator object, which is of some concrete (method-dependent) sub-type of AbstractPropagator
  • reinit_prop! — Re-initialize the propagator
  • prop_step! — Advance the propagator by a single time step forward or backward

In some cases, the ability to mutate the propagator after each propagation step can be useful. This can be achieved with the following private methods:

  • set_state! — Mutate the current quantum state of the propagator (not exported)
  • set_t! — Mutate the current time of the propagator (not exported)

The dynamics of a quantum state are determined by a time-dependent dynamical generator (a Hamiltonian or Liouvillian). The QuantumPropagators package re-exports the two main initialization routines for generators from QuantumPropagators.Generators:

  • hamiltonian — Construct a time-dependent generator for a propagation in Hilbert space under the Schrödinger equation
  • liouvillian — Construct a time-dependent generator for a propagation in Liouville space under the master equation in Lindblad form

To set up the time-dependent control fields in a Hamiltonian, methods from the submodules QuantumPropagators.Controls, QuantumPropagators.Shapes, and QuantumPropagators.Amplitudes can be used.

The above constitutes the main interface of QuantumPropagators. At the lowest level, further functionality is provided by sub-modules like QuantumPropagators.Cheby, which defines a standalone API specifically for the Chebychev propagation method. The full list of sub-modules and their public members is:

Convenience functions

There are some "convenience functions" that wrap around propagate for common tasks:

Reference for top-level QuantumPropagators module

Public Members:

Re-exported Members:

Private Members:

Public members


Wrapper around the parameters of a call to propagate.

    generator, tlist;
    pre_propagation=nothing, post_propagation=nothing,

    pre_propagation=nothing, post_propagation=nothing,

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.


Initialize a Propagator.

propagator = init_prop(
    state, generator, tlist;
    method,  # mandatory keyword argument

initializes a propagator for the time propagation of the given state over a time grid tlist under the time-dependent generator (Hamiltonian/Liouvillian) generator.


  • 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 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.


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

using QuantumPropagators: Cheby

cheby_propagator = init_prop(

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(
    func=(z -> exp(-1im * z)),

initializes a NewtonPropagator.

Method-specific keyword arguments

  • m_max: maximum Krylov dimension, cf. NewtonWrk
  • func, norm_min, relerr, max_restarts: parameter to pass to newton!
using QuantumPropagators: ExpProp

exp_propagator = init_prop(
    func=(H_dt -> exp(-1im * H_dt))

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(
    method=OrdinaryDiffEq,  # or: `method=DifferentialEquations`

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.

Advance 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.


Propagate a state over an entire time grid.

state = propagate(
    method,  # mandatory keyword argument
    observables=<store state>,

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:

  1. If check=true (default), check that state, generator, and tlist are consistent with the required interface.

  2. Initialize a propagator via init_prop:

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

  3. Call and return the result of

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


  • 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.
  • 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.


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(
    observables=<store state>,

re-initializes the given propagator with state (see reinit_prop!) and then calls the lower-level propagate(propagator; ...).

state = propagate(
    observables=<store state>,

propagates a freshly initialized propagator (immediately after init_prop). Used in the higher-level propagate(state, generator, tlist; kwargs...).


Propagate a state through a sequence of generators.

states = propagate_sequence(

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).


  • 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.


Re-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.

    transform_control_ranges=((c, ϵ_min, ϵ_max, check) => (ϵ_min, ϵ_max)),

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))
            return (min(ϵ_min, 5 * ϵ_min), max(ϵ_max, 5 * ϵ_max))

    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.


Private members


Abstract base type for all Propagator objects.

All Propagator objects must be instantiated via init_prop and implement the following interface.


  • 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.


  • 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


Determine 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.


  • 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

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.


Set 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
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.


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

set_state!(propagator, propagator.state)

See also


Enable the collection of TimerOutputs data.


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.


Wrap 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.


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.


Set 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



The following routines define dynamical generators (Hamiltonians, Liouvillians). This includes the initialization of generators and the methods that define how these generators contain controls and control amplitudes. These methods must be defined for any custom generator or control amplitude.


Public Members:

Public members


A 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

  • Operator for static generators, which may be obtained from a Generator via evaluate.

A 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 (@ref) via evaluate.


Initialize 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
  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]

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):

# 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
  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.


Construct a Liouvillian Generator.

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

calculates the sparse Liouvillian super-operator from the Hamiltonian and a list c_ops of Lindblad operators.

With convention=:LvN, applying the resulting 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 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 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 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 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.



The following routines define useful function $S(t)$ that can be used for control functions or amplitudes.


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Blackman window shape.

blackman(t, t₀, T; a=0.16)


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

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.


Box 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.


Flat 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.



The following routines define method that must be defined for any control function, or for generators with respect to control functions.


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Abstract 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.


Evaluate 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.


Evaluate 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

Evaluate 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.


Update 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.


Extract 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).


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


Obtain 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 [7], cf. the example of a ParameterizedFunction, or the amplitudes associated with spectral components in a random truncated basis [11].

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.


Shift 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.

Substitute 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.


Midpoint 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



The following types define control amplitudes $a(\{ϵ_l(t)\}, t)$ that depend on one or more control functions $ϵ_l(t)$.


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A 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.


Product 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.



The following routines allow to manage and extend storage arrays storage parameter in propagate. See the discussion of Expectation Values for more details.


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Obtain 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.


Create 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.


Apply 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.


Obtain "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.


Place 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.



The following routines implement time propagation via expansion in Chebychev polynomials.


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Workspace 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.


Evaluate Ψ = exp(-𝕚 * H * dt) Ψ in-place.

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


  • 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 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.


Calculate Chebychev coefficients.

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

return an array of coefficiencts larger than limit.


  • Δ: the spectral radius of the underlying operator
  • dt: the time step

See also cheby_coeffs! for an in-place version.


Calculate 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.



The following routines implement time propagation via expansion in Newton polynomials using a restarted Arnoldi scheme to determine evaluation points.


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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.

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.


  • 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.


Private members

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).


  • 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
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.


  • 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


The following routines implement time propagation via explicit exponentiation of the dynamical generator.


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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.


Private members


Chebychev propagation relies on estimating the spectral range of the Hamiltonian, which in turn may be done via Arnoldi iteration.


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Calculate 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;
    state=random_state(H; rng),

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.


Private members


Calculate 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.


Random 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.



Arnoldi iteration is an approximate method to find extremal eigenvalues of a dynamical generator by projecting it into a Krylov subspace. It is used to estimate spectral ranges for Chebychev Propagation and to find appropriate Leja points to support Newton Propagation


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Extend 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.

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.

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+1st 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+1st 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.


  • 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


The following routines allow to test whether custom data structures match the interface requirements of QuantumPropagators.


Public Members:

Public members


Check 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:

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.


Check that control can be evaluated on a time grid.

@test check_control(
    for_time_continuous=(control isa Function),

verifies the given control (one of the elements of the tuple returned by get_controls):

If for_time_continuous:

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.


Check the dynamical generator for propagating state over tlist.

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

verifies the given generator:

If for_pwc (default):

If for_time_continuous:

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.


Check that op is a valid operator that can be applied to state.

@test check_operator(op; state, tlist=[0.0, 1.0],
                     for_mutable_state=true, for_immutable_state=true,
                     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:

If for_immutable_state (e.g., for use in propagators with inplace=false):

  • op * state must be defined

If for_mutable_state (e.g., for use in propagators with inplace=true):

  • 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.


Check a ParameterizedFunction instance.

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

verifies that the given f:

See also


Check 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), with support for in-place operations (for_mutable_state=true) if propagator.inplace is 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.


Check that state is a valid element of a Hilbert space.

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

verifies the following requirements:

  • The inner product (LinearAlgebra.dot) of two states must return a Complex number type.
  • 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.

If for_immutable_state:

  • state + state and state - state must be defined
  • copy(state) must be defined
  • c * state for a scalar c must be defined
  • norm(state + state) must fulfill the triangle inequality
  • 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 for_mutable_state:

  • similar(state) must be defined and return a valid state
  • copyto!(other, state) 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 with for_immutable_state.

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.


Check 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.