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Physical Review A

Employing typicality in optimal control theory: Addressing large Hilbert spaces

Aviv Aroch, Ronnie Kosloff, Shimshon Kallush

Published Feb 6, 2023• 2 citations
Publisher

Abstract

Controlling the dynamics of quantum systems is a crucial task in quantum science and technology. Obtaining the driving field that transforms the quantum systems into its objective is a typical control task. This task is hard, scaling unfavorably with the size of the quantum map. To tackle this issue we employ typicality to assist in finding the control field for such transformation. To demonstrate the method we choose the control task of cooling the fine-structure states of the aluminum monofluoride molecule, at relatively high temperature. As a result high-rotational states are occupied, meaning a high effective Hilbert space. Using quantum typicality, we demonstrate that we can simulate an ensemble of states, enabling a control task addressing simultaneously many states. We employ this method to find a control field for cooling molecules with a large number of internal sates, corresponding to high initial temperatures.

DOI
10.1103/physreva.107.022603

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Controlling the dynamics of quantum systems is a crucial task in quantum science and technology. Obtaining the driving field that transforms the quantum systems into its objective is a typical control task. This task is hard, scaling unfavorably with the size of the quantum map. To tackle this issue we employ typicality to assist in finding the control field for such transformation. To demonstrate the method we choose the control task of cooling the fine-structure states of the aluminum monofluoride molecule, at relatively high temperature. As a result high-rotational states are occupied, meaning a high effective Hilbert space. Using quantum typicality, we demonstrate that we can simulate an ensemble of states, enabling a control task addressing simultaneously many states. We employ this method to find a control field for cooling molecules with a large number of internal sates, corresponding to high initial temperatures.
Controlling the dynamics of quantum systems is a crucial task in quantum science and technology. Obtaining the driving field that transforms the quantum systems to its objective is a typical control task. This task is hard, scaling unfavorably with the size of Hilbert space. To tackle this issue we employ typicality to assist in finding the control field for such systems. To demonstrate the method we choose the control task of cooling the fine structure states of the AlF molecule, from relatively high temperatures which results in large Hilbert space. Using quantum typicality, we demonstrate that we can simulate an ensemble of states, enabling a control task addressing simultaneously many states. We employ this method to find a control field for cooling molecules with large number of internal sates, corresponding to high initial temperatures.
Abstract. The robust manipulation of ensembles of quantum systems exhibiting variations or uncertainties in the system parameters and dynamics has been a long-standing problem in the field of quantum control. Existing methods for addressing quantum ensemble control primarily rely on the use of state-space models that describe the time evolution of quantum states. In this paper, we introduce a distinctive quantum control paradigm by mapping the quantum ensemble control problem to a dual problem defined on a coordinate system established by a moment transform. This transformation introduces a new type of quantization in quantum systems, enabling tractable systems-theoretic analysis and effective optimal control design, which are typically opaque. Subsequently, we develop an iterative computational algorithm to solve the transformed optimal moment control problems. We conduct a convergence analysis for the proposed algorithm and demonstrate its applicability and effectiveness on various challenging pulse design problems in quantum control.
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.
The major task in quantum control theory is to find an external field that transforms the system from one state to another or executes a predetermined unitary transformation. We investigate the difficulty of computing the control field as the size of the Hilbert space is increased. In the models studied the controls form a small closed subalgebra of operators. Complete controllability is obtained by the commutators of the controls with the stationary Hamiltonian. We investigate the scaling of the computation effort required to converge a solution for the quantum control task with respect to the size of the Hilbert space. The models studied include the double-well Bose Hubbard model with the SU(2) control subalgebra and the Morse oscillator with the Heisenberg-Weil algebra. We find that for initial and target states that are classified as generalized coherent states (GCSs) of the control subalgebra the control field is easily found independent of the size of the Hilbert space. For such problems, a control field generated for a small system can serve as a pilot for finding the field for larger systems. Attempting to employ pilot fields that generate superpositions of GCSs or cat states failed. No relation was found between control solutions of different Hilbert space sizes. In addition the task of finding such a field scales unfavorably with Hilbert space sizes. We demonstrate the use of symmetry to obtain quantum transitions between states without phase information. Implications to quantum computing are discussed.
The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms, but (ii) the required time for convergence to optimal controls can scale exponentially with the Hilbert space dimension. Depending on the control-system Hamiltonian, the landscape structure and scaling may vary. This work introduces methods for quantifying Hamiltonian-dependent and kinematic effects on control optimization dynamics in order to classify quantum systems according to the search effort and control resources required to implement arbitrary unitary transformations.
The dynamics of a quantum system driven by an external field is well described by a unitary transformation generated by a time-dependent Hamiltonian. The inverse problem of finding the field that generates a specific unitary transformation is the subject of study. The unitary transformation which can represent an algorithm in a quantum computation is imposed on a subset of quantum states embedded in a larger Hilbert space. Optimal control theory is used to solve the inversion problem irrespective of the initial input state. A unified formalism based on the Krotov method is developed leading to a different scheme. The schemes are compared for the inversion of a two-qubit Fourier transform using as registers the vibrational levels of the $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ electronic state of Na2.{\mathrm{Na}}_{2}.Na2​. Raman-like transitions through the $A{}^{1}{\ensuremath{\Sigma}}_{u}^{+}$ electronic state induce the transitions. Light fields are found that are able to implement the Fourier transform within a picosecond time scale. Such fields can be obtained by pulse-shaping techniques of a femtosecond pulse. Of the schemes studied, the square modulus scheme converges fastest. A study of the implementation of the Q qubit Fourier transform in the Na2{\mathrm{Na}}_{2}Na2​ molecule was carried out for up to five qubits. The classical computation effort required to obtain the algorithm with a given fidelity is estimated to scale exponentially with the number of levels. The observed moderate scaling of the pulse intensity with the number of qubits in the transformation is rationalized.