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QuantumCollocation.jl

Direct Collocation for Quantum Optimal Control (arXiv)

Motivation

In quantum optimal control, we are interested in finding a pulse sequence $a_{1:T-1}$ to drive a quantum system and realize a target gate $U_{\text{goal}}$. We formulate this problem as a nonlinear program (NLP) of the form

\[\begin{aligned} \underset{U_{1:T}, a_{1:T-1}, \Delta t_{1:T-1}}{\text{minimize}} & \quad \ell(U_T, U_{\text{goal}})\\ \text{ subject to } & \quad f(U_{t+1}, U_t, a_t, \Delta t_t) = 0 \\ \end{aligned}\]

where $f$ defines the dynamics, implicitly, as constraints on the states and controls, $U_{1:T}$ and $a_{1:T-1}$, which are both free variables in the solver. This optimization framework is called direct collocation. For details of our implementation please see our award-winning paper Direct Collocation for Quantum Optimal Control.

The gist of the method is that the dynamics are given by the solution to the Schrodinger equation, which results in unitary evolution given by $\exp(-i \Delta t H(a_t))$, where $H(a_t)$ is the Hamiltonian of the system and $\Delta t$ is the timestep. We can approximate this evolution using Pade approximants:

\[\begin{aligned} f(U_{t+1}, U_t, a_t, \Delta t_t) &= U_{t+1} - \exp(-i \Delta t_t H(a_t)) U_t \\ &\approx U_{t+1} - B^{-1}(a_t, \Delta t_t) F(a_t, \Delta t_t) U_t \\ &= B(a_t, \Delta t_t) U_{t+1} - F(a_t, \Delta t_t) U_t \\ \end{aligned}\]

where $B(a_t)$ and $F(a_t)$ are the backward and forward Pade operators and are just polynomials in $H(a_t)$.

This implementation is possible because direct collocation allows for the dynamics to be implicit. Since numerically calculating matrix exponentials inherently requires an approximation – the Padé approximant is commonly used – utilizing this formulation significantly improves performance, as, at least here, no matrix inversion is required.

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