Discrete Adjoints for Accurate Numerical Optimization with Application
to Quantum Control
- URL: http://arxiv.org/abs/2001.01013v2
- Date: Thu, 19 Nov 2020 17:27:45 GMT
- Title: Discrete Adjoints for Accurate Numerical Optimization with Application
to Quantum Control
- Authors: N. Anders Petersson, Fortino M. Garcia, Austin E. Copeland, Ylva L.
Rydin and Jonathan L. DuBois
- Abstract summary: This paper considers the optimal control problem for realizing logical gates in a closed quantum system.
The system is discretized with the Stormer-Verlet scheme, which is a symplectic partitioned Runge-Kutta method.
A parameterization of the control functions based on B-splines with built-in carrier waves is also introduced.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: This paper considers the optimal control problem for realizing logical gates
in a closed quantum system. The quantum state is governed by Schrodinger's
equation, which we formulate as a time-dependent Hamiltonian system in terms of
the real and imaginary parts of the state vector. The system is discretized
with the Stormer-Verlet scheme, which is a symplectic partitioned Runge-Kutta
method. Our main theoretical contribution is the derivation of a compatible
time-discretization of the adjoint state equation, such that the gradient of
the discrete objective function can be calculated exactly, at a computational
cost of solving two Schr\"odinger systems, independently of the number of
parameters in the control functions. A parameterization of the control
functions based on B-splines with built-in carrier waves is also introduced.
The carrier waves are used to specify the frequency spectra of the control
functions, while the B-spline functions specify their envelope and phase. This
approach allows the number of control parameters to be independent of, and
significantly smaller than, the number of time steps for integrating
Schrodinger's equation. We consider Hamiltonians that model the dynamics of a
superconducting multi-level qudit and present numerical examples of how the
proposed technique can be combined with the interior point L-BFGS algorithm
from the IPOPT package for realizing quantum gates. In a set of test cases, the
proposed algorithm is shown to compare favorably with QuTiP/pulse_optim and
Grape-Tensorflow.
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