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Physical Review A

Reduced-dynamics approach for optimally designing unitary transformations

Kenta Arai, Yukiyoshi Ohtsuki

Published Dec 1, 2015• 7 citations
Publisher

Abstract

Quantum optimal control is a powerful tool for designing a pulse that executes a specified gate, i.e., a unitary transformation in quantum computation. If we consider qubits characterized by different spectroscopic properties, we may design a pulse that executes a specified gate on a specified qubit ($S$ qubit) with the inclusion of neighboring qubits ($B$ qubits) but does not explicitly include the other qubits. In this situation, first we introduce sets of reduced time-evolution operators associated with the $S$ and $B$ qubits. Then, we apply quantum optimal control to these sets to design optimal pulses that could selectively execute the specified gate on the $S$ qubit while actively removing the effects of the pulse on the $B$ qubits, which would lead to a scalable pulse-design procedure. The present pulse design is applied to the quantum computation proposed by DeMille with the aim of executing the Hadamard and controlled-not gates on the $S$ qubits while actively suppressing the pulse-induced population transitions of the $B$ qubit states. The numerical results illustrate high fidelity even after introducing the extra qubits that are not included in the pulse design.

DOI
10.1103/physreva.92.062302

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Quantum optimal control is a powerful tool for designing a pulse that executes a specified gate, i.e., a unitary transformation in quantum computation. If we consider qubits characterized by different spectroscopic properties, we may design a pulse that executes a specified gate on a specified qubit (SSS qubit) with the inclusion of neighboring qubits (BBB qubits) but does not explicitly include the other qubits. In this situation, first we introduce sets of reduced time-evolution operators associated with the SSS and BBB qubits. Then, we apply quantum optimal control to these sets to design optimal pulses that could selectively execute the specified gate on the SSS qubit while actively removing the effects of the pulse on the BBB qubits, which would lead to a scalable pulse-design procedure. The present pulse design is applied to the quantum computation proposed by DeMille with the aim of executing the Hadamard and controlled-not gates on the SSS qubits while actively suppressing the pulse-induced population transitions of the BBB qubit states. The numerical results illustrate high fidelity even after introducing the extra qubits that are not included in the pulse design.
Quantum computing algorithms can be decomposed into a universal set of elementary one- and two-qubit gates. Different physical implementations of quantum computing, however, employ interactions that permit direct conditional dynamics on multiple qubits in a single step. In this work, we leverage quantum optimal control techniques to design single continuous laser pulses that implement multi-qubit controlled-phase, -NOT and -swap (Fredkin) gates on Rydberg atom quantum processors. The identification of robust multi-qubit operations leads to reduced operation time and less decoherence, and the control field provides continuous protection of the atoms from environmental noise. Notably, we find that the controlled-swap (Fredkin) gate, implemented using this approach achieves 99.74\% fidelity while accounting for imperfections such as spontaneous emission, laser fluctuations, and Doppler dephasing.
A quantum gate is realized by specific unitary transformations operating on states representing qubits. Considering a quantum system employed as an element in a quantum computing scheme, the task is therefore to enforce the pre-specified unitary transformation. This task is carried out by an external time dependent field. Optimal control theory has been suggested as a method to compute the external field which alters the evolution of the system such that it performs the desire unitary transformation. This study compares two recent implementations of optimal control theory to find the field that induces a quantum gate. The first approach is based on the equation of motion of the unitary transformation. The second approach generalizes the state to state formulation of optimal control theory. This work highlight the formal relation between the two approaches.
Constructing high-fidelity control fields that are robust to control, system, and/or surrounding environment uncertainties is a crucial objective for quantum information processing. Using the two-state Landau-Zener model for illustrative simulations of a controlled qubit, we generate optimal controls for $\ensuremath{\pi}/2$ and $\ensuremath{\pi}$ pulses and investigate their inherent robustness to uncertainty in the magnitude of the drift Hamiltonian. Next, we construct a quantum-control protocol to improve system-drift robustness by combining environment-decoupling pulse criteria and optimal control theory for unitary operations. By perturbatively expanding the unitary time-evolution operator for an open quantum system, previous analysis of environment-decoupling control pulses has calculated explicit control-field criteria to suppress environment-induced errors up to (but not including) third order from $\ensuremath{\pi}/2$ and $\ensuremath{\pi}$ pulses. We systematically integrate this criteria with optimal control theory, incorporating an estimate of the uncertain parameter to produce improvements in gate fidelity and robustness, demonstrated via a numerical example based on double quantum dot qubits. For the qubit model used in this work, postfacto analysis of the resulting controls suggests that realistic control-field fluctuations and noise may contribute just as significantly to gate errors as system and environment fluctuations.
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.
Quantum computing algorithms can be decomposed into a universal set of elementary one- and two-qubit gates. Different physical implementations of quantum computing, however, employ interactions that permit direct conditional dynamics on multiple qubits in a single step. In this work, we leverage quantum optimal control techniques to design single continuous laser pulses that implement multi-qubit controlled-phase and -swap (Fredkin) gates on Rydberg atom quantum processors. The identification of robust multi-qubit operations leads to reduced operation time and less decoherence, and the control field provides continuous protection of the atoms from environmental noise. Notably, we find that the controlled-SWAP (Fredkin) gate, implemented using this approach, achieves 99.88% fidelity while accounting for imperfections such as spontaneous emission, laser fluctuations, and Doppler dephasing.
Abstract Optimal control techniques provide a means to tailor the control pulses required to generate customized quantum gates, which helps to improve the resilience of quantum simulations to gate errors and device noise. However, the significant amount of (classical) computation required to generate customized gates can quickly undermine the effectiveness of this approach, especially when pulse optimization needs to be iterated. We propose a method to reduce the computational time required to generate the control pulse for a Hamiltonian that is parametrically dependent on a time-varying quantity. We use simple interpolation schemes to accurately reconstruct the control pulses from a set of pulses obtained in advance for a discrete set of predetermined parameter values. We obtain a reconstruction with very high fidelity and a significant reduction in computational effort. We report the results of the application of the proposed method to device-level quantum simulations of the unitary (real) time evolution of two interacting neutrons based on superconducting qubits.