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arXiv:2607.28783

Quantum Optimal Control at Intermediate Times: Controlling Revivals in Spin Chains

Juan J. Omiste, Ignacio R. Solá

quant-phphysics.comp-phSubmitted Jul 30, 2026• Announced Aug 3, 2026• 0 citations
arXivHTMLPDF

Abstract

Standard Quantum Optimal Control (QOC) protocols typically maximize a physical objective just at a final target time. However, tracking or measuring the quantum state during the time evolution requires the control at intermediate stages of their evolution. In this work, we extend QOC to accommodate the simultaneous optimization of observables at arbitrary intermediate times. Using a variational approach, we show that intermediate observations induce discontinuities in the costate trajectory, which can be handled with a Krotov algorithm leading to monotonic optimization and convergent results in the limit of zero temporal measurement windows. We apply this multi-time formulation to a Heisenberg XXX spin chain to control the propagation, including field-free revivals, of a Dicke state excitation. Our results demonstrate that simultaneous optimization of the same observable reshapes the driving field to balance intermediate targets with final populations. Finally, we show how this framework enables dynamic tracking of spin excitations and the active manipulation of post-pulse quantum state revivals.

Comments
9 pages, 7 figures

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Standard Quantum Optimal Control (QOC) protocols typically maximize a physical objective just at a final target time. However, tracking or measuring the quantum state during the time evolution requires the control at intermediate stages of their evolution. In this work, we extend QOC to accommodate the simultaneous optimization of observables at arbitrary intermediate times. Using a variational approach, we show that intermediate observations induce discontinuities in the costate trajectory, which can be handled with a Krotov algorithm leading to monotonic optimization and convergent results in the limit of zero temporal measurement windows. We apply this multi-time formulation to a Heisenberg XXX spin chain to control the propagation, including field-free revivals, of a Dicke state excitation. Our results demonstrate that simultaneous optimization of the same observable reshapes the driving field to balance intermediate targets with final populations. Finally, we show how this framework enables dynamic tracking of spin excitations and the active manipulation of post-pulse quantum state revivals.
We present a novel derivation of the Quantum Optimal Control Theory (QOCT). First, we identify the QOCT equations as the Euler-Lagrange equations of the functional associated to the control scheme considering a general window function that accounts for the measurement process. In this framework, we prove that the extremal functions found by Peirce et al. [Phys. Rev. A 37, 4950 (1988)], can be obtained using the well-established theory of variations with appropriately chosen boundary conditions. Second, we derive the QOCT canonical equations to maximize an observable at a given time T by taking the limit of an infinite narrow window function, that is, a delta distribution. We show that the costate is discontinuous and vanishes after the measurement time. Finally, we illustrate the convergence to the limiting case with numerical computations on a spin-chain Heisenberg model.
Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring the properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time.
Abstract. Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time. Submitted to: New J. Phys. † Author to whom any correspondence should be addressed. ar
We show that combining ideas from the fields of quantum invariants and optimal control can be used to design optimal quantum control solutions without explicit reference to quantum states. The states are specified only implicitly in terms of operators to which they are eigenstates. The scaling in numerical effort of the resultant approach is not given by the typically exponentially growing effort required for the specification of a time-evolved quantum state, but it is given by the effort required for the specification of a time-evolved operator. For certain Hamiltonians, this effort can be polynomial in the system size. We describe how control problems for state preparation and the realization of propagators can be formulated in this approach, and we provide explicit control solutions for a spin chain with an extended Ising Hamiltonian. The states considered for state-preparation protocols include eigenstates of Hamiltonians with more than pairwise interactions and these Hamiltonians are also used for the definition of target propagators. The cost of describing suitable time-evolving operators grows only quadratically with the system size, allowing us to construct explicit control solutions for up to 50 spins. While subexponential scaling is obtained only in special cases, we provide several examples that demonstrate favorable scaling beyond the extended Ising model. Published by the American Physical Society 2024
The control of quantum dynamics is often concerned with finding time-dependent optimal control fields that can take a system from an initial state to a final state to attain the desired value of an observable. This paper presents a general method for formulating monotonically convergent algorithms to iteratively improve control fields. The formulation is based on a two-point boundary-value quantum control paradigm (TBQCP) expressed as a nonlinear integral equation of the first kind arising from dynamical invariant tracking control. TBQCP is shown to be related to various existing techniques, including local control theory, the Krotov method, and optimal control theory. Several accelerated monotonic convergence schemes for iteratively computing control fields are derived based on TBQCP. Numerical simulations are compared with the Krotov method showing that the new TBQCP schemes are efficient and remain monotonically convergent over a wide range of the iteration step parameters and the control pulse lengths, which is attributable to the trap-free character of the transition probability quantum dynamics control landscape.
The control of quantum dynamics via specially tailored laser pulses is a long-standing goal in physics and chemistry. Partly, this dream has come true, as sophisticated pulse-shaping experiments allow us to coherently control product ratios of chemical reactions. The theoretical design of the laser pulse to transfer an initial state to a given final state can be achieved with the help of quantum optimal control theory (QOCT). This tutorial provides an introduction to QOCT. It shows how the control equations defining such an optimal pulse follow from the variation of a properly defined functional. We explain the most successful schemes to solve these control equations and show how to incorporate additional constraints in the pulse design. The algorithms are then applied to simple quantum systems and the obtained pulses are analysed. Besides the traditional final-time control methods, the tutorial also presents an algorithm and an example to handle time-dependent control targets.