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\[\gdef\op#1{\hat{#1}} \gdef\Liouvillian{\mathcal{L}} \gdef\Re{\operatorname{Re}} \gdef\Im{\operatorname{Im}}\]

The QuantumPropagators packages provides solvers for the dynamic equations of quantum mechanics, most importantly the Schrödinger and Liouville equations. We refer to the numerical evaluation of a quantum state $|Ψ(t + dt)⟩$ from a state $|Ψ(t)\rangle$ under a given equation of motion as "time propagation".

Getting started

As a simple "Hello World" example, we use the propagate function to simulate a π/2 Rabi flip in a two level system:

In a two-level system with ground state $|0⟩$ and excited state $|1⟩$, a constant driving field between the two levels with a pulse area of π/2 results in a population inversion, transforming the initial state $|0⟩$ into $-i |1⟩$,

using QuantumPropagators: propagate, ExpProp

Ψ₀ = ComplexF64[1, 0]  #  = |0⟩
H = ComplexF64[0 1; 1 0]
tlist = collect(range(0, π/2, length=101))

Ψ = propagate(Ψ₀, H, tlist; method=ExpProp)

print("Ψ = $(round.(Ψ; digits=3))\n")

# output

Ψ = ComplexF64[0.0 + 0.0im, 0.0 - 1.0im]

We've used a simple exponential propagator (ExpProp) here, which directly calculates U = exp(-1im * H * dt) in every time step and applies it to the current state Ψ.

Instead of just returning the final state, we can use storage=true to return an array with all the states at every point in the time grid (tlist). This allows us to plot the Rabi oscillation over time:

using Plots
states = propagate(Ψ₀, H, tlist; method=ExpProp, storage=true)
plot(tlist./π, abs.(states').^2; label=["ground" "excited"],
     xlabel="pulse area / π", ylabel="population", legend=:right)
Example block output

The storage parameter provides a powerful way to obtain arbitrary dynamic quantities from the propagation:

  • If given as true, return a storage array with the propagated states at each point in time instead of just the final state.
  • If given a pre-allocated storage array, fill it with the propagated states at each point in time, and return the final state.
  • If given in combination with observables, put arbitrary "observable" data derived from the propagated states in the storage array.

See the discussion of Expectation Values for details.

Approaches to time propagation

We are primarily interested in the time propagation of a quantum state under the Schrödinger equation (ħ = 1)

\[i \frac{\partial}{\partial t} |\Psi(t)⟩ = \op{H}(t) |\Psi(t)⟩\,,\]

which describes the dynamics of a closed quantum system. For open quantum systems, the equivalent equation is the Liouville equation, which we write as

\[i \frac{\partial}{\partial t} \hat{\rho}(t) = \Liouvillian(t) \hat{\rho}(t)\,.\]

This form differs from most textbooks by a factor of $i$, but has the benefit that it is structurally identical to the Schrödinger equation, so that the propagation methods do not actually need to know whether they are propagating a Hilbert space vector or a (vectorized) density matrix. See liouvillian with convention=:LvN for how to construct an appropriate $\Liouvillian$.

There are two fundamental approaches to solving the Schrödinger equation (or any equation of motion):

  1. We can analytically solve the Schrödinger equation and then numerically evaluate the solution. Mathematically, this is the application of the time evolution operator $\op{H}$ as $|Ψ(t+dt)⟩ = \op{U}(t) |Ψ(t)⟩$. For a piecewise-constant $\op{H}(t)$where there is a time-independent $\op{H}$ in the interval $[t, t+dt]$, the time evolution operator is well-known to be

    \[\op{U} = \exp[-i \op{H} dt]\,.\]

    The propagated state $\op{U} |Ψ(t)⟩$ would then be obtained, e.g., by expanding the exponentiation of the operator $\op{H}$ into a polynomial series. This can be done to arbitrary precision by truncating the series at an appropriate point.

  2. We can use a general ODE solver, e.g., using some kind of Runge-Kutta scheme. These work by following the derivative between $t$ and $t+dt$ with some adaptive internal step size to achieve a given precision.

The first case of a piecewise-constant time evolution operator is particularly relevant to quantum control, since the two most venerable methods of quantum control (GRAPE and Krotov's method) are inherently piecewise-constant. Hence, the QuantumPropagators package implements two efficient polynomial propagators that have a long history in quantum control, using Chebychev polynomials for closed quantum systems [1, 2] and Newton polynomials for open quantum systems [3–5].

For propagation via an ODE Solver, QuantumPropagators delegates to the DifferentialEquations.jl, respectively the OrdinaryDiffEq.jl package.

Propagation methods

The propagate function has a mandatory method keyword argument. It should be passed a module that implements the method. For the built-in methods, this would be one of the following submodule of QuantumPropagators:

using QuantumPropagators: ExpProp, Cheby, Newton

The equation of motion is implicit in the propagation method. The above methods target the Schrödinger or Liouville equations for a piecewise-constant Hamiltonian that is evaluated on the midpoints of the propagation time grid.

The two core method are:

  • method=Cheby: Evaluate the application of the unitary time-evolution operator via an expansion into Chebychev polynomials.
  • method=Newton: Evaluate the application of the non-unitary time-evolution operator via an expansion into Newton polynomials, for a Liouvillian or a non-Hermitian Hamiltonian.

Furthermore, as a fallback for very small system or for debugging,

  • method=ExpProp: Explicitly construct the time evolution operator by matrix exponentiation and apply it to the state.

If the OrdinaryDiffEq.jl or (equivalently) th DifferentialEquations.jl package is loaded, QuantumPropagators can delegate to it:

using OrdinaryDiffEq

allows to pass

Unlike any of the built-in methods, OrdinaryDiffEq is able to propagate for time-continuous generators. This is the default for that propagator (pwc=false). By setting pwc=true or piecewise=true) the ODE solvers can also be used for piecewise-constant Hamiltonians or Liouvillians, providing an alternative to the built-in method=Cheby and method=Newton.

See the more extended discussion of Propagation Methods for more details.

Dynamical generators

In the initial example, the "generator" H that is the second argument to propagate was a simple static operator. In general, we will want time-dependent Hamiltonians or Liouvillians. The standard way to initialize a time-dependent Hamiltonian is via the hamiltonian function, e.g., as hamiltonian(Ĥ₀, (Ĥ₁, ϵ₁), (Ĥ₂, ϵ₂)). The Ĥ₀, Ĥ₁, and Ĥ₂ are static operators, and ϵ₁ and ϵ₂ are control fields, typically functions (or function-like objects) of time t. For piecewise-constant propagators, ϵ₁ and ϵ₂ may also be an array of amplitude values appropriate to the time grid tlist. The tuple-syntax for the time-dependent terms is inspired by QuTiP.

Generally, the generator, or the operators/controls inside the tuples can be a arbitrary objects, as long as some relevant methods are implemented for these objects, see the full section on Dynamical Generators.

Open quantum systems are handled identically to closed quantum system, except that Hamiltonian operator are replaced by Liouvillian super-operators. For any system of non-trivial Hilbert space dimension, all (super-)operators should be sparse matrices.

The Propagator interface

As a lower-level interface than propagate, the QuantumPropagators package defines an interface for "propagator" objects. These are initialized via init_prop as, e.g.,

using QuantumPropagators: init_prop

propagator = init_prop(Ψ₀, H, tlist; method)

with a mandatory method keyword argument.

The propagator is a propagation-method-dependent object with the interface described by AbstractPropagator and QuantumPropagators.Interfaces.check_propagator.

The prop_step! function can then be used to advance the propagator:

using QuantumPropagators: prop_step!

Ψ = prop_step!(propagator)  # single step

while !isnothing(prop_step!(propagator)); end  # go to end
Ψ = propagator.state

print("Ψ = $(round.(Ψ; digits=3)))\n")
print("t = $(round(propagator.t / π; digits=3))π\n")

# output

Ψ = ComplexF64[0.0 + 0.0im, 0.0 - 1.0im])
t = 0.5π

Backward propagation

When propagate or init_prop are called with backward=true, the propagation is initialized to run backward. The initial state is then defined at propagator.t == tlist[end] and each prop_step! moves to the previous point in tlist. The equation of motion is the Schrödinger or Liouville equation with a negative $dt$. For a Hermitian generator, doing a forward propagation followed by a backward propagation will recover the initial state. For a non-Hermitian generator, this no longer holds. Note that in optimal control methods such as GRAPE or Krotov's method, obtaining gradients involves a "backward propagation with the adjoint generator" (when the generator is non-Hermitian and adjoint/non-adjoint makes a difference). The propagate routine with backward=true will not automatically take this adjoint of the generator; instead, the adjoint generator must be passed explicitly.

Parameterized controls

Controls may depend on a list of tunable parameters. Such controls must be especially defined and should be subtypes of QuantumPropagators.Controls.ParameterizedFunction, or more generally "functors" of a single float t for which QuantumPropagators.Controls.get_parameters is implemented.

Independently, all propagators have a field parameters that is a dict of controls to propagation parameters for that control. For continuous-time propagators, such as a propagator initialized with method=OrdinaryDiffEq and pwc=false, these propagation parameters are exactly the analytic parameters of the control as obtained by QuantumPropagators.Controls.get_parameters.

For piecewise-constant propagators (all the default built-in propagators), the propagation parameters are always the values of the control evaluated on the mid points of the time grid, see QuantumPropagators.Controls.discretize_on_midpoints. Specifically, the analytic parameters from get_parameters(control) are not used as propagation parameters for piecewise propagators.

In any case, mutating propagator.parameters affects the propagation in subsequent calls to prop_step!.

Connection to DifferentialEquations.jl

The QuantumPropagators API is structured similarly to the DifferentialEquations.jl

Note that the equation of motion for QuantumPropagators is implicit in the propagation method (usually the Schrödinger/Liouville equation), so the initialization of a Propagator via the initial state and the "generator" is more specialized than DifferentialEquations' Problem Interface.

The propagator returned by using DifferentialEquations; init_prop(Ψ₀, H, tlist; method=DifferentialEquations) is a thin wrapper around DifferentialEquations' integrator. That propagator uses an un-exported function QuantumPropagators.ode_function to wrap around the evaluation of a time-dependent generator. The ode_function wrapper could also be used directly to enable working with the data structures defined in QuantumPropagators in the context of the DifferentialEquations package.