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Optimal control of Gmon qubits using Pontyagin's minimum principle: preparing a maximally entangled state with singular bang-bang protocols

Seraph L. Bao, Silken Kleer, Ruoyu Wang, Armin Rahmani

Published Apr 5, 2017• 0 citations
OpenAlex

Abstract

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.

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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.
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.
Great strides have recently been made in developing stable and coherent quantum systems. One such system is the superconducting gmon qubit architecture developed by Martinis et. al [1]. Martinis et. al presents a simple coherent system with highly tuneable qubit-qubit couplings, making it a great candidate for use in quantum computing. Of specific interest to the field of quantum computing is the controlled evolution of these qubit systems to process information. To achieve this type of evolution we worked within the gmon qubit architecture and found optimal solutions for evolving a three qubit system from an easily achieved product state to a more difficult to achieve maximally entangled state. These solutions were found using a simulated annealing algorithm that minimized the error between the maximally entangled state and the final state of the system with respect to the protocols of the coupling parameters that compose the Hamiltonian of the system. These protocols, confirmed using Pontryagin's minimum principle, turned out to be simple bang-bang protocols that greatly shortcut adiabatic evolution and offer a structure that can be used to help employ quantum gates or regulate the energy of a quantum computer.
Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's minimum principle, we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in Schr\"odinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example of how Pontryagin's minimum principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.
Optimal control theory is applied to analyze the time-optimal solution with a single scalar control knob in a two-level quantum system without quantum decoherence. Emphasis is placed on the dependence on the maximum control strength umax{u}_{\text{max}}umax​. General constraints on the optimal protocol are derived and used to rigorously parametrize the time-optimal solution. Two concrete problems are investigated. For generic state preparation problems, both multiple bang-bang and bang-singular-bang are legitimate and should be considered. Generally, the optimal is bang-bang for small umax{u}_{\text{max}}umax​, and there exists a state-dependent critical amplitude above which singular control emerges. For the X-gate operation of a qubit, the optimal protocol is exclusively multiple bang-bang. The minimum gate time is about 80% of that based on the resonant Rabi $\ensuremath{\pi}$-pulse over a wide range of control strength; in the ${u}_{\text{max}}\ensuremath{\rightarrow}0$ limit, this ratio is derived to be $\ensuremath{\pi}/4$. To develop practically feasible protocols, we present methods to smooth the abrupt changes in the bang-bang control while preserving perfect gate fidelity. The presence of bang-bang segments in the time-optimal protocol indicates that the high-frequency components and a full calculation (instead of the commonly adopted rotating wave approximation) are essential for the ultimate quantum speed limit.
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.
A formalism based on Pontryagin's maximum principle is applied to determine the time-optimal protocol that drives a general initial state to a target state by a Hamiltonian with limited control, i.e., there is a single control field with bounded amplitude. The coupling between the bath and the qubit is modeled by a Lindblad master equation. Dissipation typically drives the system to the maximally mixed state; consequently, there generally exists an optimal evolution time beyond which the decoherence prevents the system from getting closer to the target state. For some specific dissipation channel, however, the optimal control can keep the system from the maximum entropy state for infinitely long. The conditions under which this specific situation arises are discussed in detail. The numerical procedure to construct the time-optimal protocol is described. In particular, the formalism adopted here can efficiently evaluate the time-dependent singular control which turns out to be crucial in controlling either an isolated or a dissipative qubit.