A remarkably simple result is derived for the minimal time Tmin required to drive a general initial state to a final target state by a Landau-Zener-type Hamiltonian or, equivalently, by time-dependent laser driving. The associated protocol is also derived. A surprise arises for some states when the interaction strength is bounded by a constant c. Then, for large c, the optimal driving is of type bang-off-bang and for increasing c one recovers the unconstrained result. However, for smaller c the optimal driving can suddenly switch to bang-bang type. We discuss the notion of quantum speed limit time.
Nonlinear two-level Landau-Zener type equations for systems with relevance for Bose-Einstein condensates and nonlinear optics are considered and the minimal time
Tmin to drive an initial state to a given target state is investigated. Surprisingly, the nonlinearity may be canceled by a time-optimal unconstrained driving and
Tmin becomes independent of the nonlinearity. For constrained and unconstrained driving explicit expressions are derived for
Tmin, the optimal driving, and the protocol.
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. 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, 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 determine the optimal quantum manipulation protocols for implementing high-fidelity, fast single-qubit gates. We demonstrate that the time-optimal pulse sequence is a ``bang-bang'' sequence: discrete pulses of either positive or negative maximum amplitude or zero. The non-adiabatic bang-bang pulse sequence minimizes gate duration providing a speedup for low-frequency architectures. We first derive the protocol for a transversally driven two-level system, extracting exact analytic expressions for minimum gate times. We then extend this framework to general multilevel architectures, identifying conditions that enable the coherent suppression of leakage errors. Using the fluxonium circuit as a representative case study, we optimize
X/2 and
Y/2 gates through a combination of discrete bang sequences and continuous waveform smoothing. This approach preserves near-optimal execution speeds while mitigating transitions outside the computational subspace. Open-system simulations demonstrate that these sequences outperform commensurate and resonant pulse schemes across different fluxonium regimes, achieving low-error manipulation significantly faster than standard resonant control, even in the presence of
1/f flux noise and dissipation.
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.
A central problem in quantum computing is the finding of an unknown target state that encodes the solution of a certain computational task. To accomplish this goal, the evolution from a given initial state is performed with an associated total Hamiltonian, which is a time-dependent combination of two time-independent Hamiltonians: the problem-Hamiltonian, whose ground state is the unknown target, and a driving-Hamiltonian, whose ground state is the initial state. Here, we analyze this computational problem in the light of optimal control theory, considering each Hamiltonian modulated by an independent control function. For bounded controls, there is a minimum total evolution time beyond which the target state can be exactly prepared. We show that below this minimum time a possible optimal solution consists of both controls constantly tuned at their upper bound, provided the gradients of the associated control Hamiltonian with respect to the controls (the switching functions) are positive. We refer to this type of solution as the double-bang solution. We show that the double-bang solution is optimal for a teleportation protocol up to the minimum time. Additionally, we combine the double-bang solution and the adiabatic gate teleportation protocol to implement universal quantum computing. This approach to quantum computing is very appealing because of its simplicity and experimental feasibility. To corroborate our analytical results, we propose the use of a numerical quantum optimal control technique adapted to limit the amplitude of the controls, which converges to the double-bang solution when the final evolution time is shorter than the minimum time. We compare the fidelity of the teleported state obtained for the numerically optimized two-control functions with the usual one-control function scheme and with the quantum approximate optimization algorithm (QAOA). We find that the two-control approach has a better performance than the other approaches. Moreover, we investigate the energetic cost and the robustness against systematic errors in the teleportation protocol, considering different schemes.