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 the states 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. See the Optimization of an X-Gate for a Transmon Qubit for an example. For more advanced gate optimizations, also see How to optimize towards a two-qubit gate up to single-qubit corrections, How to optimize towards an arbitrary perfect entangler, How to optimize in a dissipative system, and How to optimize for robust pulses.

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 given by

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

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

See the Optimization of a state-to-state transfer in a lambda system with RWA for an example.

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 krotov.shapes.zero_shape() in the pulse_options that you pass to optimize_pulses().

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 the control target is reached when the Objective.initial_state evolves to a specific target state under the optimal control fields, that target state should be included 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. This function should calculate the states \(\ket{\chi_k}\) based on the forward-propagated states \(\ket{\phi_k(T)}\) and the list of objectives. For convenience, 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() which returns the value \(J_T\). This is not required to run an optimization since 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())

See krotov.functionals for some standard functionals. An example for a more advanced functional is the Optimization towards a Perfect Entangler.

How to penalize population in a forbidden subspace

In principle, 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 levels in the forbidden subspace. A non-Hermitian Hamiltonian is usually a good way to realize this. See the Optimization of a dissipative state-to-state transfer in a Lambda system for an example.

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]. This optimizes towards a point in the Weyl chamber.

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

How to optimize towards an arbitrary perfect entangler

Closely releated to an optimization towards a point in the Weyl chamber is the optimizatin towards an arbitrary perfectly entangling two-qubit gate. Geometrically, this means optimizing towards the polyhedron of perfect entanglers in the Weyl chamber.

Use krotov.objectives.gate_objectives() with gate='PE' in order to construct a list of objectives suitable for an optimization using a “perfect entanglers” functional [WattsPRA2015][GoerzPRA2015]. This is illustrated in the Optimization towards a Perfect Entangler.

Again, the chi_constructor is available in the weylchamber package.

How to optimize in a dissipative system

To optimize a dissipative system, it is sufficient to set an Objective with a density matrix for the initial_state and target, and a Liouvillian in Objective.H. See the Optimization of Dissipative Qubit Reset for an example.

Instead of a Liouvillian, it is also possible to set Objective.H to the system Hamiltonian, and Objective.c_ops to the appropriate Lindblad operators. However, it is generally much more efficient to use krotov.objectives.liouvillian() to convert a time-dependent Hamiltonian and a list of Lindblad operators into a time-dependent Liouvillian. In either case, the propagate routine passed to optimize_pulses() must be aware of and compatible with the convention for the objectives.

Specifically for gate optimization, the routine gate_objectives() can be used to automatically set appropriate objectives for an optimization in Liouville space. The parameter liouville_states_set indicates that the system dynamics are in Liouville space and sets an appropriate choice of matrices that track the optimization according to Ref. [GoerzNJP2014]. See the Optimization of a Dissipative Quantum Gate for an example.

For weak dissipation, it may also be possible to avoid the use of density matrices altogether, and to instead use a non-Hermitian Hamiltonian. For example, you may use the effective Hamiltonian from the MCWF method [PlenioRMP1998],

\[\Op{H}_{\text{eff}} = \Op{H} - \frac{i}{2} \sum_k \Op{L}_k^\dagger \Op{L}_k\,,\]

for the Hermitian Hamiltonian \(\Op{H}\) and the Lindblad operators \(\Op{L}_k\). Propagating \(\Op{H}_{\text{eff}}\) (without quantum jumps) will lead to a decay in the norm of the state corresponding to how much dissipation the state is subjected to. Numerically, this will usually increase the value of the optimization functional (that is, the error). Thus the optimization can be pushed towards avoiding decoherence, without explicitly performing the optimization in Liouville space. See the Optimization of a dissipative state-to-state transfer in a Lambda system for an example.

How to optimize for robust pulses

Control pulses can be made robust with respect to variations in the system by doing an ensemble optimization, as proposed in Ref. [GoerzPRA2014]. The idea if to sample a representative selection of possible system Hamiltonians, and to optimize over an average of the entire ensemble.

An appropriate set of objectives can be generated with the ensemble_objectives() function.

How to parallelize the optimization

Krotov’s method is inherently parallel accross different objectives. See krotov.parallelization, and the Optimization of an X-Gate for a Transmon Qubit for an example.

How to maximize numerical efficiency

For systems of non-trivial size, the main numerical effort should be in the simulation of the system dynamics. Every iteration of Krotov’s method requires a full backward propagation and a full forward propagation of the states associated with each objective. Therefore, the best numerical efficiency can be achieved by optimizing the performance of the propagator that is passed to optimize_pulses().

One possibility is to implement problem-specific propagators, such as krotov.propagators.DensityMatrixODEPropagator. Going further, you might consider implementing the propagator with the help of lower-level instructions, e.g., by using Cython.

How to deal with the optimization running out of memory

Krotov’s method requires the storage of at least one set of propagated state over the entire time grid, for each objective. For the second-order update equation, up to three sets of stored states per objective may be required. In particular for larger systems and dynamics in Liouville space, the memory required for storing these states may be prohibitively expensive.

The optimize_pulses() accepts a storage parameter to which a constructor for an array-like container can be passed wherein the propagated states will be stored. It is possible to pass custom out-of-memory storage objects, such as Dask arrays. This may carry a significant penalty in runtime, however, as states will have to be read from disk, or across the network.