How-Tos¶

How to optimize towards a quantum gate¶

To optimize towards a quantum gate \(\Op{O}\) in a closed quantum system, set one Objective for state in the logical basis, with the basis state \(\ket{\Psi_k}\) as the initial_state and \(\Op{O} \ket{\Psi_k}\) as the target.

You may use krotov.gate_objectives() to construct the appropriate list of objectives.

How to optimize complex control fields¶

This implementation of Krotov’s method requires real-valued control fields. You must rewrite your Hamiltonian to contain the real part and the imaginary part of the field as two independent controls. This is always possible. For example, for a driven harmonic oscillator in the rotating wave approximation, the interaction Hamiltonian is

\[\Op{H}_\text{int} = \epsilon^*(t) \Op{a} + \epsilon(t) \Op{a}^\dagger = \epsilon_{\text{re}}(t) (\Op{a} + \Op{a}^\dagger) + \epsilon_{\text{im}}(t) (i \Op{a} - i \Op{a}^\dagger)\]

where \(\epsilon_{\text{re}}(t)= \Re[\epsilon(t)]\) and \(\epsilon_{\text{im}}(t) = \Im[\epsilon(t)]\) are considered as two independent (real-valued) controls.

How to exclude a control from the optimization¶

In order to force the optimization to leave any particular control field unchanged, set its update shape to zero_shape:

krotov.PulseOptions(lambda_a=1, shape=krotov.shapes.zero_shape)

How to define a new optimization functional¶

In order to define a new optimization functional \(J_T\):

Decide on what should go in Objective.target to best describe the physical control target. If it the control target is fulfilled when the Objective.initial_state evolves to a specific target state under the optimal control fields, that target state should to in target.
Define a function chi_constructor that calculates the boundary condition for the backward-propagation in Krotov’s method

\[\ket{\chi_k(T)} \equiv - \left. \frac{\partial J_T}{\partial \bra{\phi_k(T)}} \right\vert_{\ket{\phi_k(T)}}\]

or the equivalent experession in Liouville space. The function calculates the states \(\ket{\chi_k}\) based on the forward-propagated states \(\ket{\phi_k(T)}\) and the list of objectives. For convenience, for when target contains a target state, chi_constructor will also receive tau_vals containing the overlaps \(\tau_k = \Braket{\phi_k(T)}{\phi_k^{\tgt}}\). See chis_re() for an example.
Optionally, define a function that can be used as an info_hook in optimize_pulses() and that returns the value \(J_T\). This is not required to run an optimization, as the functional is entirely implicit in chi_constructor. However, calculating the value of the functional is useful for convergence analysis (check_convergence in optimize_pulses())

How to optimize towards a two-qubit gate up to single-qubit corrections¶

Use krotov.objectives.gate_objectives() with local_invariants=True in order to construct a list of objectives suitable for an optimization using a “local-invariant functional” [MullerPRA11].

The weylchamber package contains the suitable chi_constructor routines to pass to optimize_pulses().

How to penalize population in a forbidden subspace¶

In principal, optimize_pulses() has a state_dependent_constraint. However, this has some caveats. Most notably, it results in an inhomogeneous equation of motion, which is currently not implemented.

The recommended “workaround” is to place artificially high dissipation on the the levels in the forbidden subspace. A non-Hermitian Hamiltonian is usually a good way to realize this.

How to optimize towards an arbitrary perfect entangler¶

Use krotov.objectives.gate_objectives() with gate=PE in order to construct a list of objectives suitable for an optimization using a “perfect entanglers” [WattsPRA2015][GoerzPRA2015].

The weylchamber package contains the suitable chi_constructor routines to pass to optimize_pulses().