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arXiv:2307.13168 · AVS Quantum Science

A practical approach to determine minimal quantum gate durations using amplitude-bounded quantum controls

Stefanie Günther, N. Anders Petersson

Published Dec 1, 2023• 1 citation
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Abstract

We present an iterative scheme to estimate the minimal duration in which a quantum gate can be realized while satisfying hardware constraints on the control pulse amplitudes. The scheme performs a sequence of unconstrained numerical optimal control cycles that each minimize the gate fidelity for a given gate duration alongside an additional penalty term for the control pulse amplitudes. After each cycle, the gate duration is adjusted based on the inverse of the resulting maximum control pulse amplitudes by re-scaling the dynamics to a new duration where control pulses satisfy the amplitude constraints. Those scaled controls then serve as an initial guess for the next unconstrained optimal control cycle, using the adjusted gate duration. We provide multiple numerical examples that each demonstrate fast convergence of the scheme toward a gate duration that is close to the quantum speed limit, given the control pulse amplitude bound. The proposed technique is agnostic to the underlying system and control Hamiltonian models, as well as the target unitary gate operation, making the time-scaling iteration an easy to implement and practically useful scheme for reducing the durations of quantum gate operations.

DOI
10.1116/5.0173373

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We present an iterative scheme to estimate the minimal duration in which a quantum gate can be realized while satisfying hardware constraints on the control pulse amplitudes. The scheme performs a sequence of unconstrained numerical optimal control cycles that each minimize the gate fidelity for a given gate duration alongside an additional penalty term for the control pulse amplitudes. After each cycle, the gate duration is adjusted based on the inverse of the resulting maximum control pulse amplitudes by re-scaling the dynamics to a new duration where control pulses satisfy the amplitude constraints. Those scaled controls then serve as an initial guess for the next unconstrained optimal control cycle, using the adjusted gate duration. We provide multiple numerical examples that each demonstrate fast convergence of the scheme toward a gate duration that is close to the quantum speed limit, given the control pulse amplitude bound. The proposed technique is agnostic to the underlying system and control Hamiltonian models, as well as the target unitary gate operation, making the time-scaling iteration an easy to implement and practically useful scheme for reducing the durations of quantum gate operations.
Current efforts to build quantum computers focus mainly on the two-state qubit, which often involves suppressing readily-available higher states. In this work, we break this abstraction and synthesize short-duration control pulses for gates on generalized d-state qudits. We present Incremental Pulse Reseeding, a practical scheme to guide optimal control software to the lowest-duration pulse by iteratively seeding the optimizer with previous results. We find a near-linear relationship between Hilbert space dimension and gate duration through explicit pulse optimization for one- and two-qudit gates on transmons. Our results suggest that qudit operations are much more efficient than previously expected in the practical regime of interest and have the potential to significantly increase the computational power of current hardware.
In this work we derive a lower bound for the minimum time required to implement a target unitary transformation through a classical time-dependent field in a closed quantum system. The bound depends on the target gate, the strength of the internal Hamiltonian and the highest permitted control field amplitude. These findings reveal some properties of the reachable set of operations, explicitly analyzed for a single qubit. Moreover, for fully controllable systems, we identify a lower bound for the time at which all unitary gates become reachable. We use numerical gate optimization in order to study the tightness of the obtained bounds. It is shown that in the single qubit case our analytical findings describe the relationship between the highest control field amplitude and the minimum evolution time remarkably well. Finally, we discuss both challenges and ways forward for obtaining tighter bounds for higher dimensional systems, offering a discussion about the mathematical form and the physical meaning of the bound.
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Abstract Quantum optimal control plays a vital role in many quantum technologies, including quantum computation. One of the most important control parameters to optimise for is the evolution time (pulse duration). However, most existing works focus on finding the shortest evolution time theoretically possible without offering explicit pulse constructions under practical constraints like noise in the system. This paper addresses these limitations by introducing an efficient method to perform the chopped random basis optimisation in the presence of noise, specifically when the noise commutes with the gate Hamiltonian, i.e. when the unitary and dissipative parts of the Liouvillian commute. This noise-aware approach allows for direct optimisation of the evolution time alongside other control parameters, significantly reducing the computational cost compared to full noisy simulations. The protocol is demonstrated through numerical simulations on state-to-state transfer and gate compilation problems under several noise models. Results show that the optimised fidelity has a strong dependence on evolution time due to noise, drift Hamiltonian, and local traps in optimisation, highlighting the necessity of optimising evolution time in practical settings that can lead to a substantial gain in the fidelity. Our pulse optimisation protocol can consistently escape from many local minima in all of our examples. We hope that our protocol can be the start of many more works on the crucial topic of control pulse time optimisation in practical settings.
We present a general framework for finding the time-optimal quantum evolution inspired by the classical problem called the brachistochrone problem.At first, the time-optimal control problem for achieving a pure state evolution from a given initial state to a desired final state is formulated.We minimize the duration time for this transition by finding the optimal control for the Hamiltonian in an available set subject to a certain constraint dictated by physical conditions.Then, it will be extended to the unitary evolution which has greater relevance for quantum computation.This provides a systematic way to make an efficient unitary gate on a certain physical system.To demonstrate this framework, we choose the anisotropic Heisenberg model and find a general solution.Since it is not always possible to find a general solution for the fundamental equation for the time-optimal control, we introduce the simulated annealing method as a simple numerical method to show that our framework can be used for the system whose general solution cannot be found.The numerical analysis based on the Krotov iteration method is introduced and applied to the time-optimal quantum control problem to solve the problem efficiently.The optimal realization of the quantum Fourier transform for two or more qubit systems are derived.We consider two generalizations of the quantum brachistochrone for mixed state evolutions.One is the unitary evolution with ancilla system ended up with the final measurement on the ancilla.The other is the repetition of measurement or decoherence model which can be represented by the evolution with the master equation in the Lindblad form.Also, the relation between the fidelity to the target and the duration time is considerd.A.4 Quantum speed limit . . . . . . . . .
Quantum optimal control for gate optimization aims to provide accurate, robust, and fast pulse sequences to achieve gate fidelities on quantum systems below the error correction threshold. Many methods have been developed and successfully applied in simulation and on quantum hardware. In this paper, we establish a connection between the iterative linear quadratic regulator and quantum optimal control by adapting it to gate optimization of quantum systems. We include constraints on the controls and their derivatives to enable smoother pulses. We achieve high-fidelity simulation results forX\mathbf{X}Xand cross-resonance gates on one- and two-qubit fixed-frequency transmons simulated with two and three levels.