How to optimize towards a quantum gate

To optimize towards a quantum gate \(\Op{O}\) in a closed quantum system, set one Objective for each state in the logical basis, with the basis state \(\ket{\phi_k}\) as the initial_state and \(\Op{O} \ket{\phi_k}\) 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-valued 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 in the RWA for an example.

How to use args in time-dependent control fields

QuTiP requires that the functions that are used to express time-dependencies have the signature func(t, args) where t is a scalar value for the time and args is a dict containing values for static parameters, see QuTiP’s documentation on using the args variable. Most of the krotov package’s Examples use closures or hardcoded values instead of args. For example, in the Optimization of a State-to-State Transfer in a Two-Level-System, the Hamiltonian is defined as:

def hamiltonian(omega=1.0, ampl0=0.2):
   """Two-level-system Hamiltonian

      omega (float): energy separation of the qubit levels
      ampl0 (float): constant amplitude of the driving field
   H0 = -0.5 * omega * qutip.operators.sigmaz()
   H1 = qutip.operators.sigmax()

   def guess_control(t, args):
      return ampl0 * krotov.shapes.flattop(
            t, t_start=0, t_stop=5, t_rise=0.3, func="blackman"

   return [H0, [H1, guess_control]]

Note how ampl0 is used in guess_control as a closure from the surrounding hamiltonian scope, t_stop and t_rise are hardcoded, and args is not used at all. The function could be rewritten as:

def guess_control(t, args):
   """Initial control amplitude.

      t (float): Time value at which to evaluate the control.
      args (dict): Dictionary containing the value "ampl0" with the
        amplitude of the driving field, "t_stop" with the time at which the
        control shape ends, and "t_rise" for the duration of the
        switch-on/switch-off time.
    return args['ampl0'] * krotov.shapes.flattop(

def hamiltonian(omega=1.0):
    """Two-level-system Hamiltonian

        omega (float): energy separation of the qubit levels
    H0 = -0.5 * omega * qutip.operators.sigmaz()
    H1 = qutip.operators.sigmax()

    return [H0, [H1, guess_control]]

ARGS = dict(ampl0=0.2, t_stop=5, t_rise=0.3)

The ARGS must be passed to optimize_pulses() via the pulse_options parameter:

pulse_options = {
   guess_control: dict(lambda_a=5, update_shape=S, args=ARGS)

Both Objective.mesolve() and Objective.propagate() take an optional args dict also.

The args in pulse_options are used automatically when evaluating the respective initial guess. Note that the use of args does not extend to update_shape, which is always a function of t only. Any other parameters in the update_shape are best set via functools.partial(), see the Optimization of a Dissipative State-to-State Transfer in a Lambda System.

Compare that example to the Optimization of a State-to-State Transfer in a Lambda System in the RWA. In the latter, the values for the parameters in the control fields and the Hamiltonian are hardcoded, while in the former, all parameters are centrally defined in a dict which is passed to the optimization and propagation routines.

How to stop the optimization when the error crosses some threshold

By default, an optimization stops after a predefined number of iterations (iter_stop parameter in optimize_pulses()). However, through the interplay of the info_hook and the check_convergence routine passed to optimize_pulses(), the optimization can be stopped based on the optimization success or the rate of convergence: The info_hook routine should return the value of the optimization functional or error, which is accessible to check_convergence via the Result.info_vals attribute, see krotov.convergence for details.

Generally, you should use the krotov.info_hooks.print_table() function as an info_hook, which receives a function to evaluate the optimization functional \(J_T\) as a parameter. Then, use krotov.convergence.value_below() as a check_convergence routine to stop the optimization when \(J_T\) falls below some given threshold.

See the thee Optimization of a State-to-State Transfer in a Lambda System in the 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 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 \(\tau_k = \Braket{\phi_k^{\tgt}}{\phi_k(T)}\). 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

On many quantum computing platforms, applying arbitrary single-qubit gates is easy compared to entangling two-qubit gates. A specific entangling gate like CNOT is combined with single-qubit gates to form a universal set of gates. For a given physical system, it can be hard to know a-priori which entangling gates are easy or even possible to realize. For example, trapped neutral atoms only allow for the realization of diagonal two-qubit gates [54][27] like CPHASE. However, the CPHASE gate is “locally equivalent” to CNOT: only additional single-qubit operations are required to obtain one from the other. A “local-invariants functional” [55] defines an optimization with respect to a such a local equivalence class, and thus is free to find the specific realization of a two-qubit gate that is easiest to realize.

Use krotov.objectives.gate_objectives() with local_invariants=True in order to construct a list of objectives suitable for an optimization using the local-invariant functional [55]. This optimizes towards a point in the Weyl chamber.

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

The optimization towards a local equivalence class may require use of the second-order update equation, see Second order update.

How to optimize towards an arbitrary perfect entangler

The relevant property of a gate is often its entangling power, and the requirement for a two-qubit gate in a universal set of gates is that it is a “perfect entangler”. A perfect entangler can produce a maximally entangled state from a separable input state. Since 85% of all two-qubit gates are perfect entanglers [56][57], a functional that targets an arbitrary perfect entangler [28][29] solves the control problem with the least constraints.

The optimization towards an arbitrary perfect entangler is closely related to an optimization towards a point in the Weyl chamber (How to optimize towards a two-qubit gate up to single-qubit corrections): It turns out that in the geometric representation of the Weyl chamber, all the perfect entanglers lie within a polyhedron, and we can simply minimize the geometric distance to the surface of this polyhedron.

Use krotov.objectives.gate_objectives() with gate='PE' in order to construct a list of objectives suitable for an optimization using the perfect entanglers functional [28][29]. This is illustrated in the Optimization towards a Perfect Entangler.

Again, the chi_constructor is available in the weylchamber package.

Both the optimization towards a local equivalence class and an arbitrary perfect entangler may require use of the second-order update equation, see Second order update.

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. [27]. 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 [58],

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

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 fields can be made robust with respect to variations in the system by performing an “ensemble optimization” [26]. The idea is to sample a representative selection of possible system Hamiltonians, and to optimize over an average of the entire ensemble. In the functional, Eq. (1), respectively the update Eq. (12), the index \(k\) now numbers not only the states, but also different ensemble Hamiltonians: \(\Op{H}(\{\epsilon_l(t)\}) \rightarrow \{\Op{H}_k(\{\epsilon_l(t)\})\}\).

The example considered in Ref. [26] is that of a CPHASE two-qubit gate on trapped Rydberg atoms. Two classical fluctuations contribute significantly to the gate error: deviations in the pulse amplitude (\(\Omega = 1\) ideally), and fluctuations in the energy of the Rydberg level (\(\Delta_{\text{ryd}} = 0\) ideally). Starting from a set of objectives for the unperturbed system, see How to optimize towards a quantum gate, ensemble_objectives() creates an extended set of objectives that duplicates the original objectives once for each Hamiltonian from a set perturbed Hamiltonian \(\Op{H}(\Omega \neq 1, \Delta_{\text{ryd}} \neq 0)\). As shown in Ref. [27], an optimization over the average of all these objectives results in controls that are robust over a wide range of system perturbations.

A simpler example of an ensemble optimization is Ensemble Optimization for Robust Pulses, which considers a state-to-state transition in a Lamba-System with a dissipative intermediary state.

How to apply spectral constraints

In principle, Krotov’s method can include spectral constraints while maintaining the guarantee for monotonic convergence [59] . However, the calculation of the pulse update with such spectral constraints requires solving a Fredholm equation of the second kind, which has not yet been implemented numerically. Thus, the krotov package does not support this approach (and no such support is planned).

A “cheap” alternative that usually yields good results is to apply a spectral filter to the optimized pulses after each iteration. The optimize_pulses() function allows this via the modify_params_after_iter argument.

For example, the following function restricts the spectrum of each pulse to a given range:

def apply_spectral_filter(tlist, w0, w1):
   """Spectral filter for real-valued pulses.

   The resulting filter function performs a Fast-Fourier-Transform (FFT) of
   each optimized pulse, and sets spectral components for angular
   frequencies below `w0` or above `w1` to zero. The filtered pulse is then
   the result of the inverse FFT, and multiplying again with the update
   shape for the pulse, to ensure that the filtered pulse still fulfills
   the required boundary conditions.

       tlist (numpy.ndarray): Array of time grid values. All pulses must be
           defined on the intervals of this time grid
       w0 (float): The lowest allowed (angular) frequency
       w1 (float): The highest allowed (angular) frequency

       callable: A function that can be passed to
       `modify_params_after_iter` to apply the spectral filter.

    dt = tlist[1] - tlist[0]  # assume equi-distant time grid

    n = len(tlist) - 1  # = len(pulse)
    # remember that pulses are defined on intervals of tlist

    w = np.abs(np.fft.fftfreq(n, d=dt / (2.0 * np.pi)))
    # the normalization factor 2π means that w0 and w1 are angular
    # frequencies, corresponding directly to energies in the Hamiltonian
    # (ħ = 1).

    flt = (w0 <= w) * (w <= w1)
    # flt is the (boolean) filter array, equivalent to an array of values 0
    # and 1

    def _filter(**kwargs):
        # same interface as an `info_hook` function
        pulses = kwargs['optimized_pulses']
        shape_arrays = kwargs['shape_arrays']
        for (pulse, shape) in zip(pulses, shape_arrays):
            spectrum = np.fft.fft(pulse)
            # apply the filter by element-wise multiplication
            spectrum[:] *= flt[:]
            # after the inverse fft, we should also multiply with the
            # update shape function. Otherwise, there is no guarantee that
            # the filtered pulse will be zero at t=0 and t=T (assuming that
            # is what the update shape is supposed to enforce). Also, it is
            # important that we overwrite `pulse` in-place (pulse[:] = ...)
            pulse[:] = np.fft.ifft(spectrum).real * shape

    return _filter

This function is passed to optimize_pulses() as e.g.

modify_params_after_iter=apply_spectral_filter(tlist, 0, 7)

to constrain the spectrum of the pulse to angular frequencies \(\omega \in [0, 7]\). You may want to explore how such a filter behaves in the example of the Optimization of an X-Gate for a Transmon Qubit.

Modifying the optimized pulses “manually” through a modify_params_after_iter function means that we lose all guarantees of monotonic convergence. If the optimization with a spectral filter does not converge, you should increase the value of \(\lambda_a\) in the pulse_options that are passed to optimize_pulses(). A larger value of \(\lambda_a\) results in smaller updates in each iteration. This should also translate into the filter pulses being closer to the unfiltered pulses, increasing the probability that the changes due to the filter do not undo the monotonic convergence. You may also find that the optimization fails if the control problem physically cannot be solved with controls in the desired spectral range. Without a good physical intuition, trial and error may be required.

How to limit the amplitude of the controls

Amplitude constraints on the control can be realized indirectly through parametrization [60]. For example, consider the physical Hamiltonian \(\Op{H} = \Op{H}_0 + \epsilon(t) \Op{H}_1\).

There are several possible parametrizations of \(\epsilon(t)\) in terms of an unconstrained function \(u(t)\):

  • For \(\epsilon(t) \ge 0\):

    \[\epsilon(t) = u^2(t)\]
  • For \(0 \le \epsilon(t) < \epsilon_{\max}\):

    \[\epsilon(t) = \epsilon_{\max} \tanh^2\left(u(t)\right)\]
  • For \(\epsilon_{\min} < \epsilon(t) < \epsilon_{\max}\):

    \[\epsilon(t) = \frac{\epsilon_{\max} - \epsilon_{\min}}{2} \tanh\left(u(t)\right) + \frac{\epsilon_{\max} + \epsilon_{\min}}{2}\]

Krotov’s method can now calculate the update \(\Delta u(t)\) in each iteration, and then \(\Delta \epsilon(t)\) via the above equations.

There is a caveat: In the update equation (12), we now have the term

\[\begin{split}\Bigg( \left.\frac{\partial \Op{H}}{\partial u}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\u^{(i+1)}(t)\end{matrix}}} \Bigg) = \Bigg( \left.\frac{\partial \epsilon}{\partial u}\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\u^{(i+1)}(t)\end{matrix}}} \Bigg)\end{split}\]

on the right hand side. As the dependendence of \(\epsilon(t)\) on \(u(t)\) is non-linear, we are left with a dependency on the unknown updated parametrization \(u^{(i+1)}(t)\). We resolve this by approximating \(u^{(i+1)}(t) \approx u^{(i)}(t)\), or equivalently \(\Delta u(t) \ll u(t)\), which can be enforced by choosing a sufficiently large value of \(\lambda_a\) in the pulse_options that are passed to optimize_pulses().

Currently, the krotov package does not yet support parametrizations in the above form, although this is a planned feature. In the meantime, you could modify the control to fit within the desired amplitude constaints in the same way as applying spectral constaints, see How to apply spectral constraints.

How to parallelize the optimization

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

How to prevent losing an optimization result

Optimizations usually take several hundred to several thousand iterations to fully converge. Thus, the optimize_pulses() routine may require significant runtime (often multiple days for large problems). Once an optimization has completed, you are strongly encouraged to store the result to disk, using Result.dump(). You may also consider using dump_result() during the check_convergence step to dump the current state of the optimization to disk at regular intervals. This protects you from losing work if the optimization is interrupted in any way, like an unexpected crash.

In order to continue after such a crash, you can restore a Result object containing the recent state of the optimization using Result.load() (with the original objectives and finalize=True if the dump file originates from dump_result()). You may then call optimize_pulses() and pass the loaded Result object as continue_from. The new optimization will start from the most recent optimized controls as a guess, and continue to count iterations from the previous result. See How to continue from a previous optimization for further details.

How to continue from a previous optimization

See How to prevent losing an optimization result for how to continue from an optimization that ended (crashed) prematurely. Even when an optimization has completed normally, you may still want to continue with further iterations – either because you find that the original iter_stop was insufficient to reach full convergence, or because you would like to modify some parameters, like the λₐ values for each control. In this case, you can again call optimize_pulses() and pass the Result object from the previous optimization as continue_from. Note that while you are free to change the pulse_options between the two optimization, the objectives must remain the same. The functional (chi_constructor) and the info_hook should also remain the same (otherwise, you may and up with inconsistencies in your Result). The Result object returned by the second optimization will include all the data from the first optimization.

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, see krotov.propagators. 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.

How to avoid the overhead of QuTiP objects

If you know what you are doing, it is possible to set up an Objective without any qutip.Qobj instances, using arbitrary low-level objects instead. See the Optimization with numpy Arrays for an example.