Glossary

In the context of the JuliaQuantumControl ecosystem, we apply the following nomenclature.

Generator

Dynamical generator (Hamiltonian / Liouvillian) for the time evolution of a state, i.e., the right-hand-side of the equation of motion (up to a factor of $i$) such that $|Ψ(t+dt)⟩ = e^{-i Ĥ dt} |Ψ(t)⟩$ in the infinitesimal limit. We use the symbols $G$, $Ĥ$, or $L$, depending on the context (general, Hamiltonian, Liouvillian).

Examples for supported forms a Hamiltonian are the following, from the most general case to simplest and most common case of linear controls,

The $Ĥ_0$ is the Drift Term and each term under the sum over $l$ is a Control Term. In the most general case, Eq. (G1), the control term is a Hamiltonian that depends on a set of control amplitudes. More commonly, the control term is separable into the Control Amplitude $a_l(t)$ and the Control Operator $Ĥ_l$. The control amplitude $a_l(t)$ depends in turn on the Control Function (or simply "control") $ϵ_l(t)$, which is the function we can control directly. The control may further depend on a Pulse Parametrization, $ϵ_l(t) = ϵ_l(u_l(t))$ or a set of Control Parameters, $ϵ_l(t) = ϵ_l({u_n})$.

In an open quantum system, the structure of Eqs. (G1–G3) is the same, but with Liouvillian (super-)operators acting on density matrices instead of Hamiltonians acting on state vectors. See liouvillian with convention=:TDSE.

Drift Term

A term in the dynamical generator that does not depend on any controls.

Control Term

A term in the dynamical generator that depends on one or more controls.

Control Function

(aka "Control") A function $ϵ_l(t)$ in the Generator that is directly controllable, typically corresponding to a physical Control Field. Conceptually a function, but may be specified in terms of Control Parameters.

Control Field

A function that corresponds directly to some kind of physical drive (laser amplitude, microwave pulse, etc.). The term can be ambiguous in that it usually corresponds to the Control Function $ϵ(t)$, but depending on how the control problem is formulated, it can also correspond to the Control Amplitude $a(t)$.

Control Operator

(aka "control Hamiltonian/Liouvillian"). The operator $Ĥ_l$ in Eqs. (G2, G3). This is a static operator which forms the Control Term together with a Control Amplitude. The control generator is not a well-defined concept in the most general case of non-separable controls terms, Eq. (G1)

Control Amplitude

The time-dependent coefficient for the Control Operator in Eq. (G2), or, in the most general case of Eq. (G1), a function on which the control term depends directly. The mapping from a Control Function to an Control Amplitude can encompass a variety of different concepts:

• Non-linear coupling of a control field to the operator, e.g., the quadratic coupling of the laser field to a Stark shift operator
• Transfer functions, e.g., to model the response of an electronic device to the optimal control field $ϵ(t)$.
• Noise in the amplitude of the control field
• Non-controllable aspects of the control amplitude, e.g. a "guided" control amplitude $a_l(t) = R(t) + ϵ_l(t)$ or a non-controllable envelope $S(t)$ in $a_l(t) = S(t) ϵ(t)$ that ensures switch-on- and switch-off in a CRAB pulse ϵ(t).

In Qiskit Dynamics, the "control amplitude" is called "Signal", see Connecting Qiskit Pulse with Qiskit Dynamics, where a Qiskit "pulse" corresponds roughly to our Control Function.

Control Parameters

Non-time-dependent parameters that a Control Function depends on, $ϵ(t) = ϵ(\{u_n\}, t)$. One common parametrization of a control field is as a Pulse, where the control parameters are the amplitude of the field at discrete points of a time grid. Parametrization as a "pulse" is implicit in Krotov's method and standard GRAPE.

More generally, the control parameters could also be spectral coefficients (CRAB) or simple parameters for an analytic pulse shape (e.g., position, width, and amplitude of a Gaussian shape). All optimal control methods find optimized control fields by varying the control parameters.

Pulse

(aka "control pulse") A control field discretized to a time grid, usually on the midpoints of the time grid, in a piecewise-constant approximation. Stored as a vector of floating point values. The parametrization of a control field as a "pulse" is implicit for Krotov's method and standard GRAPE. One might think of these methods to optimize the control fields directly, but a conceptually cleaner understanding is to think of the discretized "pulse" as a vector of control parameters for the time-continuous control field.

Pulse Parametrization

The use of a function $u(t)$ such that $ϵ(t) = ϵ(u(t))$ for the purpose of constraining the amplitude of the control field $ϵ(t)$. See e.g. SquareParametrization, where $ϵ(t) = u^2(t)$ to ensure that $ϵ(t)$ is positive. Since Krotov's method inherently has no constraints on the optimized control fields, pulse parameterization is a method of imposing constraints on the amplitude in this context. This is different from, albeit related to, the Control Amplitude, e.g. $a(ϵ(t)) = ϵ^2(t)$ in that the amplitude parameterization does not reflect how the control field physically couples to the control Hamiltonian. Note that "parameterization" here has nothing to do with the "parametrization" in terms of Control Parameters: the pulse parametrization is a parametrization with a function, whereas the control parameters are values.

Control Derivative

The derivative of the dynamical Generator with respect to the control $ϵ(t)$. In the case of linear controls terms in Eq. (G3), the control derivative is the Control Operator coupling to $ϵ(t)$. In general, however, for non-linear control terms, the control derivatives still depends on the control fields and is thus time dependent. We commonly use the symbol $μ$ for the control derivative (reminiscent of the dipole operator)

Parameter Derivative

The derivative of a control with respect to a single control parameter. The derivative of the dynamical Generator with respect to that control parameter is then the product of the Control Derivative and the parameter derivative.

Gradient

The derivative of the optimization functional with respect to all Control Parameters, i.e. the vector of all parameter derivatives.