Optimization towards a Perfect Entangler¶

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2\vphantom{#1}\right\rangle}¶

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This example demonstrates the optimization with an "unconventional" optimization target. Instead of a state-to-state transition, or the realization of a specific quantum gate, we optimize for an arbitrary perfectly entangling gate. See

• P. Watts, et al., Phys. Rev. A 91, 062306 (2015)

• M. H. Goerz, et al., Phys. Rev. A 91, 062307 (2015)

for details.

Hamiltonian¶

We consider a generic two-qubit Hamiltonian (motivated from the example of two superconducting transmon qubits, truncated to the logical subspace),

$$ \begin{equation} \op{H}(t) = - \frac{\omega_1}{2} \op{\sigma}_{z}^{(1)} - \frac{\omega_2}{2} \op{\sigma}_{z}^{(2)} + 2 J \left( \op{\sigma}_{x}^{(1)} \op{\sigma}_{x}^{(2)} + \op{\sigma}_{y}^{(1)} \op{\sigma}_{y}^{(2)} \right) + u(t) \left( \op{\sigma}_{x}^{(1)} + \lambda \op{\sigma}_{x}^{(2)} \right), \end{equation} $$

where $\omega_1$ and $\omega_2$ are the energy level splitting of the respective qubit, $J$ is the effective coupling strength and $u(t)$ is the control field. $\lambda$ defines the strength of the qubit-control coupling for qubit 2, relative to qubit 1.

We use the following parameters:

These are for illustrative purposes only, and do not correspond to any particular physical system.

The initial guess is defined as

We instantiate the Hamiltonian with this guess pulse

As well as the canonical two-qubit logical basis,

with the corresponding projectors to calculate population dynamics below.

Objectives for a perfect entangler¶

Our optimization target is the closest perfectly entangling gate, quantified by the perfect-entangler functional

$$ \begin{equation} F_{PE} = g_3 \sqrt{g_1^2 + g_2^2} - g_1, \end{equation} $$

where $g_1, g_2, g_3$ are the local invariants of the implemented gate that uniquely identify its non-local content. The local invariants are closely related to the Weyl coordinates $c_1, c_2, c_3$, which provide a useful geometric visualization in the Weyl chamber. The perfectly entangling gates lie within a polyhedron in the Weyl chamber and $F_{PE}$ becomes zero at its boundaries. We define $F_{PE} \equiv 0$ for all perfect entanglers (inside the polyhedron)

A list of four objectives that encode the minimization of $F_{PE}$ are generated by calling the gate_objectives function with the canonical basis, and "PE" as target "gate".

The initial states in these objectives are not the canonical basis states, but a Bell basis,

Since we don't know which perfect entangler the optimization result will implement, we cannot associate any "target state" with each objective, and the target attribute is set to the string 'PE'.

We can treat the above objectives as a "black box"; the only important consideration is that the chi_constructor that we will pass to optimize_pulses to calculating the boundary condition for the backwards propagation,

$$ \begin{equation} \ket{\chi_{k}} = \frac{\partial F_{PE}}{\partial \bra{\phi_k}} \Bigg|_{\ket{\phi_{k}(T)}}\,, \end{equation} $$

must be consistent with how the objectives are set up. For the perfect entanglers functional, the calculation of the $\ket{\chi_{k}}$ is relatively complicated. The weylchamber package (https://github.com/qucontrol/weylchamber) contains a suitable routine that works on the objectives exactly as defined above (specifically, under the assumption that the $\ket{\phi_k}$ are the appropriate Bell states):

Again, the key point to take from this is that when defining a new or unusual functional, the chi_constructor must be congruent with the way the objectives are defined. As a user, you can choose whatever definition of objectives and implementation of chi_constructor is most suitable, as long they are compatible.

Second Order Update Equation¶

As the perfect-entangler functional $F_{PE}$ is non-linear in the states, Krotov's method requires the second-order contribution in order to guarantee monotonic convergence (see D. M. Reich, et al., J. Chem. Phys. 136, 104103 (2012) for details). The second order update equation reads

$$ \begin{align} \epsilon^{(i+1)}(t) & = \epsilon^{ref}(t) + \frac{S(t)}{\lambda_a} \Im \Bigg\{ \sum_{k=1}^{N} \Bigg\langle \chi_k^{(i)}(t) \Bigg\vert \left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t) \\\epsilon^{(i+1)}(t)\end{matrix}}} \Bigg\vert \phi_k^{(i+1)}(t) \Bigg\rangle \\ & \qquad + \frac{1}{2} \sigma(t) \Bigg\langle \Delta\phi_k(t) \Bigg\vert \left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\\epsilon^{(i+1)}(t)\end{matrix}}} \Bigg\vert \phi_k^{(i+1)}(t) \Bigg\rangle \Bigg\}\,, \end{align} $$

where the term proportional to $\sigma(t)$ defines the second-order contribution. In order to use the second-order term, we need to pass a function to evaluate this $\sigma(t)$ as sigma to optimize_pulses. We use the equation

$$ \begin{equation} \sigma(t) = -\max\left(\varepsilon_A,2A+\varepsilon_A\right) \end{equation} $$

with $\varepsilon_A$ a small non-negative number, and $A$ a parameter that can be recalculated numerically after each iteration (see D. M. Reich, et al., J. Chem. Phys. 136, 104103 (2012) for details).

Generally, $\sigma(t)$ has parametric dependencies like $A$ in this example, which should be refreshed for each iteration. Thus, since sigma holds internal state, it must be implemented as an object subclassing from krotov.second_order.Sigma:

This combines the evaluation of the function, sigma(t), with the recalculation of $A$ (or whatever parametrizations another $\sigma(t)$ function might contain) in sigma.refresh, which optimize_pulses invokes automatically at the end of each iteration.

Optimization¶

Before running the optimization, we define the shape function $S(t)$ to maintain the smooth switch-on and switch-off, and the $\lambda_a$ parameter that determines the overall magnitude of the pulse update in each iteration:

In previous examples, we have used info_hook routines that display and store the value of the functional $J_T$. Here, we will also want to analyze the optimization in terms of the Weyl chamber coordinates $(c_1, c_2, c_3)$. We therefore write a custom print_fidelity routine that prints $F_{PE}$ as well as the gate concurrence (as an alternative measure for the entangling power of quantum gates), and results in the storage of a nested tuple (F_PE, (c1, c2, c3)) for each iteration, in Result.info_vals.

This structure must be taken into account in a check_convergence routine. This would affect routines like krotov.convergence.value_below that assume that Result.info_vals contains the values of $J_T$ only. Here, we define a check that stops the optimization as soon as we reach a perfect entangler:

We can visualize how each iteration of the optimization brings the dynamics closer to the polyhedron of perfect entanglers (using the Weyl chamber coordinates that we calculated in the info_hook routine print_fidelity, and that were stored in Result.info_vals).

The final optimized control field looks like this: