We apply the theory of optimal control to the dynamics of two Gmon qubits, with the goal of preparing a desired entangled ground state from an initial unentangled one. Given an initial state, a target state, and a Hamiltonian with a set of permissible controls, can we reach the target state with coherent quantum evolution and, in that case, what is the minimum time required? The adiabatic theorem provides a far from optimal solution in the presence of a spectral gap. Optimal control yields the fastest possible way of reaching the target state and helps identify unreachable states. In the context of a simple quantum system, we provide examples of both reachable and unreachable target ground states and show that the unreachability is due to a symmetry. We find the optimal protocol in the reachable case using three different approaches: (i) a brute-force numerical minimization (ii) an efficient numerical minimization using the bang-bang ansatz expected from the Pontryagin minimum principle, and (iii) direct solution of the Pontryagin boundary value problem, which yields an analytical understanding of the numerically obtained optimal protocols. Interestingly, our system provides an example of singular control, where the Pontryagin theorem does not guarantee bang-bang protocols. Nevertheless, all three approaches give the same bang-bang protocol.
Grover's algorithm is a cornerstone of quantum algorithms and is strictly optimal in oracle-query complexity. While the full search problem admits no further improvement, one may trade accuracy for speed in the partial search problem, where the task is to identify only the block containing the target item. The best known quantum algorithm for the partial search problem is the Grover-Radhakrishnan-Korepin (GRK) algorithm, whose optimality has long been conjectured but not proved. In this work, we prove the optimality of GRK in the large-block limit. We formulate partial search as a time-optimal control problem and apply the Pontryagin maximum principle to derive the switching-function dynamics, establish the bang-bang structure of regular extremals, and exclude non-optimal switching patterns. As a result, we show that the optimal regular extremal has the global-local-global form, which yields a control-theoretic proof of the asymptotic optimality of the GRK algorithm in oracle-query complexity.
We apply the theory of optimal control to the dynamics of two ``gmon'' qubits, with the goal of preparing a desired entangled ground state from an initial unentangled one. Given an initial state, a target state, and a Hamiltonian with a set of permissible controls, can we reach the target state with coherent quantum evolution, and, in that case, what is the minimum time required? The adiabatic theorem provides a far from optimal solution in the presence of a spectral gap. Optimal control yields the fastest possible way of reaching the target state and helps identify unreachable states. In the context of a simple quantum system, we provide examples of both reachable and unreachable target ground states, and show that the unreachability is due to a symmetry. We find the optimal protocol in the reachable case using three different approaches: (i) a brute-force numerical minimization, (ii) an efficient numerical minimization using the bang-bang ansatz expected from the Pontryagin minimum principle, and (iii) direct solution of the Pontryagin boundary value problem, which yields an analytical understanding of the numerically obtained optimal protocols. Interestingly, our system provides an example of singular control, where the Pontryagin theorem does not guarantee bang-bang protocols. Nevertheless, all three approaches give the same bang-bang protocol.
Optimization is one of the key applications of quantum computing where a quantum speedup has been an eagerly anticipated outcome. A promising approach to optimization using quantum dynamics is to consider a linear combination $s(t)B+[1\ensuremath{-}s(t)]C$ of two noncommuting Hamiltonians
B and
C, where
C encodes the solution to the optimization problem in its ground state,
B is a Hamiltonian whose ground state is easy to prepare, and $s(t)\ensuremath{\in}[0,1]$ is the bounded ``switching schedule'' or ``path,'' with $t\ensuremath{\in}[0,{t}_{f}]$. This approach encompasses two of the most widely studied quantum-optimization algorithms: quantum annealing [QA; continuous
s(t)] and the quantum approximate optimization algorithm [QAOA; piecewise constant
s(t)]. While it is notoriously difficult to prove a quantum advantage for either algorithm, it is possible to compare and contrast them by finding the optimal
s(t). Here we provide a rigorous analysis of this quantum optimal control problem, entirely within the geometric framework of Pontryagin's maximum principle of optimal control theory. We extend earlier results, derived in a purely closed-system setting, to open systems. This is the natural setting for experimental realizations of QA and QAOA. In the closed-system setting it was shown that the optimal solution is a ``bang-anneal-bang'' schedule, with the bangs characterized by
s(t)=0 and
s(t)=1 in finite subintervals of
[0,tf], in particular,
s(0)=0 and
s(tf)=1, in contrast to the standard prescription
s(0)=1 and
s(tf)=0 of QA. As an example, we prove that for a single spin-
1/2, the optimal solution in the closed-system setting is the bang-bang schedule, switching midway from $s\ensuremath{\equiv}0$ to $s\ensuremath{\equiv}1$. For finite-dimensional environments and without any approximations we identify sufficient conditions ensuring that either the bang-anneal, anneal-bang, or bang-anneal-bang schedules are optimal, and recover the optimality of
s(0)=0 and
s(tf)=1. However, for infinite-dimensional environments and a system described by an adiabatic Redfield master equation we do not recover the bang-type optimal solution. In fact we can only identify conditions under which
s(tf)=1, and even this result is not recovered in the fully Markovian limit, suggesting that the pure anneal-type schedule is optimal. Our open-system results have implications for the use of experimental quantum-information processors, which are by necessity noisy, and suggest that in this practical sense the optimal schedules for quantum optimization are likely to be continuous.
Reliable high-fidelity quantum state transformation has always been considered as an inseparable part of quantum information processing. In this regard, Pontryagin Minimum (or maximum) Principle (PMP) has proved to play an important role to achieve the maximum fidelity in an optimum time or energy. Motivated by this, in this work, we formulate a control constrained optimal control problem where we aim to minimize time and also energy subjected to a quantum system satisfying the bilinear Schrödinger equation. We derive the first order optimality conditions through the application of PMP resulting in a boundary value problem. Next, in order to obtain efficient numerical results, we exploit a particular family of physics-informed neural networks that are specifically designed to tackle the indirect method based on the PMP. We show that this method can significantly speed up the process by first obtaining a set of relations which finally let us compute the optimal control strategy to determine the time- and energy-optimal protocol driving a general initial state to a target state by a quantum Hamiltonian with bounded control. We make use of the so-called “qutip” package in python, and the newly developed “tfc” python package.
We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity.The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.