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arXiv:2405.15609 · PRX Quantum

Quantum Control without Quantum States

Modesto Orozco-Ruiz, Nguyen H. Le, Florian Mintert

Published Dec 20, 2024• 2 citations

Abstract

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

DOI: 10.1103/prxquantum.5.040346

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Robust quantum control can achieve noise resilience of quantum systems and quantum technological devices. While the need for noise resilience grows with the number of fluctuating quantities, and thus typically with the number of qubits, most numerically exact optimal control techniques are limited to systems of few interacting qubits. This paper exploits quantum control that avoids explicit reference to quantum states in exponentially large Hilbert space. Exemplary control protocols for spin chains are discussed in terms of noise-resilient preparation of highly entangled states.

The development of quantum control methods is an essential task for emerging quantum technologies. In general, the process of optimizing quantum controls scales very unfavorably in system size due to the exponential growth of the Hilbert space dimension. Here, I present a scalable subsystem-based method for quantum optimal control on a large quantum system. The basic idea is to cast the original problem as a robust control problem on subsystems with requirement of robustness in respect of inter-subsystem couplings, thus enabling a drastic reduction of problem size. The method is in particular suitable for target quantum operations that are local, e.g., elementary quantum gates. As an illustrative example, I employ it to tackle the demanding task of pulse searching on a 12-spin coupled system, achieving substantial reduction of memory and time costs. This work has significant implications for coherent quantum engineering on near-term intermediate-scale quantum devices.

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.

Quantum optimal control experiments and simulations have successfully manipulated the dynamics of systems ranging from atoms to biomolecules. Surprisingly, these collective works indicate that the effort (i.e., the number of algorithmic iterations) required to find an optimal control field appears to be essentially invariant to the complexity of the system. The present work explores this matter in a series of systematic optimizations of the state-to-state transition probability on model quantum systems with the number of states

N

ranging from 5 through 100. The optimizations occur over a landscape defined by the transition probability as a function of the control field. Previous theoretical studies on the topology of quantum control landscapes established that they should be free of suboptimal traps under reasonable physical conditions. The simulations in this work include nearly 5000 individual optimization test cases, all of which confirm this prediction by fully achieving optimal population transfer of at least 99.9

%

on careful attention to numerical procedures to ensure that the controls are free of constraints. Collectively, the simulation results additionally show invariance of required search effort to system dimension

N

. This behavior is rationalized in terms of the structural features of the underlying control landscape. The very attractive observed scaling with system complexity may be understood by considering the distance traveled on the control landscape during a search and the magnitude of the control landscape slope. Exceptions to this favorable scaling behavior can arise when the initial control field fluence is too large or when the target final state recedes from the initial state as

N

increases.

Digital-analog is a quantum computational paradigm that employs the natural interaction Hamiltonian of a system as the entangling resource, combined with single qubit gates, to implement universal quantum operations. As in the case of its digital gate-based counterpart, designing digital-analog circuits that employ optimal quantum resources often requires an exceedingly large classical computational time. In this work we find a suboptimal solution to this exponentially large problem, showing that it can be solved within polynomial computational time. In particular, we provide an exact solution for the problem of expressing arbitrary two-body Hamiltonians as the sum of local unitary transformations of an arbitrary Ising Hamiltonian, with the total number of required terms being at most quadratic in system size. This allows us to design a digital-analog simulation protocol that avoids employing numerical optimization over a large parameter space at the preprocessing stage, minimizing computational resources and allowing for further scaling.

Quantum optimal control, a toolbox for devising and implementing the shapes of external fields that accomplish given tasks in the operation of a quantum device in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The last few years have seen a rapid evolution and expansion of the field. We review here recent progress in our understanding of the controllability of open quantum systems and in the development and application of quantum control techniques to quantum technologies. We also address key challenges and sketch a roadmap for future developments.