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Quantum optimal control for Pauli operators based on spin-1/2 system

Jinfang Li, Zixuan Xin, Jie‐Ru Hu, Dongshan He

Published Oct 14, 2022• 0 citations
Publisher

Abstract

Abstract Quantum control is an important basis for quantum computing, quantum com-munication and quantum simulation. The key of quantum control is to realize quantum logicoperators with high fidelity. In this paper, based on spin-1/2 system, the optimal simulationof three Pauli logic operators is carried out by using quantum optimal control theory. Under Pauli z spin-presentation, the results show that under the given quantum initial state, the Pauli operators achieve the expected target quantum state with a high fidelity of 0.9999. When the control pulse is applied on x-axis, the number of iterations required to optimize Pauli x operator to achieve the target state is the least, and the number of iterations requiredto optimize Pauli z operator is the most. In addition, the comparison shows that when thefidelity reach 0.9999, the population of the quantum final state can reach the ideal theoreti-cal expectation. Besides, the optimization fidelity of Hadamard gate can also reaches 0.9999 based on spin-1/2 system. Finally, the study on the phase evolution of quantum states showsthat the phase difference between the initial and final quantum states of optimized Pauli xand Pauli z logic operator is π/2, and there is no phase difference between the final quantumstate and the initial quantum state after the evolution of optimized Pauli y operator.

DOI
10.21203/rs.3.rs-2151303/v1

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Abstract Quantum control is an important basis for quantum computing, quantum com-munication and quantum simulation. The key of quantum control is to realize quantum logicoperators with high fidelity. In this paper, based on spin-1/2 system, the optimal simulationof three Pauli logic operators is carried out by using quantum optimal control theory. Under Pauli z spin-presentation, the results show that under the given quantum initial state, the Pauli operators achieve the expected target quantum state with a high fidelity of 0.9999. When the control pulse is applied on x-axis, the number of iterations required to optimize Pauli x operator to achieve the target state is the least, and the number of iterations requiredto optimize Pauli z operator is the most. In addition, the comparison shows that when thefidelity reach 0.9999, the population of the quantum final state can reach the ideal theoreti-cal expectation. Besides, the optimization fidelity of Hadamard gate can also reaches 0.9999 based on spin-1/2 system. Finally, the study on the phase evolution of quantum states showsthat the phase difference between the initial and final quantum states of optimized Pauli xand Pauli z logic operator is π/2, and there is no phase difference between the final quantumstate and the initial quantum state after the evolution of optimized Pauli y operator.
We study the manipulation of a two level quantum mechanical system which is the basic unit of representing data in quantum information science, a qubit. This research is motivated by the design of the quantum Hadamard gate for solid state spin qubits. We consider the problem of driving the initial state of a qubit into a desired final state in a specified time, while minimizing the energies of the control fields. Optimal control theory is employed to define optimum control fields which satisfy both dynamical constraints and optimality conditions. Time evolution of the system is calculated within the Schrödinger equation and a gradient method is used for tailoring optimal control fields.
Quantum Fourier transform (QFT) is a quantum analogue of the classical discrete Fourier transform. It is a fundamental quantum gate in quantum algorithms which has an exponential advantage over the classical computation and has been excessively studied. Normally, an n-qubit quantum Fourier transform could be resolved into the tensor product of n single-qubit operations, and each operation could be implemented by a Hadamard gate and a controlled phase gate. Then the complexity of an n-qubit QFT is of order O(n2). To reduce the complexity of quantum operations, optimal control (OC) method has recently been used successfully to find the minimum time for implementing a quantum operation. Up to now, two types of quantum optimal control methods have been presented, i.e. analytical and numerical methods. The analytical approach is to change the problem of efficient synthesis of unitary transformations into the geometrical one of finding the shortest paths. Numerical optimal control procedures are based on the gradient methods (GRAPE, Gradient Ascent Pulse Engineering) and Krotov methods. Notable application mainly focus on nuclear magnetic resonance fields, including imaging, liquid-state NMR, solid-state NMR, and NMR quantum computation. One obvious advantage of optimal control NMR quantum computation is that the OC unitary evolution transformation pulse sequences are normally shorter than the conventional corresponding ones. Here we use the optimal control method to find the minimum duration for implementing QFT quantum gate. A linear spin chain with nearest-neighbor Ising interaction is used to find the optimization. And the optimized pulse sequence is experimentally demonstrated on an NMR quantum information processor. By using optimal control method with numerical calculation, a three-qubit QFT in an indirect-linear-coupling chain system is optimized. The duration of the OC QFT is obviously shorter than that of conventional approaches. The OC pulse sequence has been experimentally implemented on a liquid-state NMR spectrometer. To verify the optimally controlled pulse sequence for the three-qubit QFT, different initial states are assumed. The accuracy of the OC pulse sequence could be demonstrated by the consistency of theoretical simulation spectra with the experimental results. The good consistency between the simulation and the experimental spectra demonstrates that the OC QFT is of high fidelity.
Implementation of quantum gates are important for quantum computations in physical system made of polar molecules. We investigate the feasibility of implementing gates based on pendular states of the molecular system by two different quantum optical control methods. Firstly, the Multi-Target optimal control theory and the Multi-Constraint optimal control theory are described for optimizing control fields and accomplish the optimization of quantum gates. Numerical results show that the controlled NOT gate (CNOT) can be realized under the control of above methods with high fidelities (0.975 and 0.999) respectively. In addition, in order to examine the dependence of the fidelity on energy difference in the same molecular system, the SWAP gate in the molecular system is also optimized with high fidelity (0.999) by the Multi-Constraint optimal control theory with the zero-area and constant-fluence constraints.
Quantum Lyapunov control was developed in order to transform a quantum system from arbitrary initial states to a target state. The idea is to find control fields that steer the Lyapunov function to zero as $t\ensuremath{\rightarrow}\ensuremath{\infty}$, meanwhile the quantum system is driven to the target state. In order to shorten the time required to reach the target state, we propose two designs to optimize Lyapunov control in this paper. The first design makes the Lyapunov function decrease as fast as possible with a constraint on the total power of control fields, and the second design has the same purpose but with a constraint on each control field. Examples of a three-level system demonstrate that the evolution time for Lyapunov control can be significantly shortened, especially when high control fidelity is required. Besides, this optimal Lyapunov-based quantum control is robust against uncertainties in the free Hamiltonian and decoherence in the system compared to conventional Lyapunov control. We apply our optimal design to cool a nanomechanical resonator where a shorter cooling time is found with respect to the cooling time by the conventional Lyapunov design.
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.
Coherent quantum control shows promise in preparing and manipulating internal states within a quantum system coherently. Specifically, it aims to smoothly transition the system from an initial state to a desired target state using external electromagnetic fields. This transfer of population between ground states is crucial in various quantum processes, including applications in photochemistry for controlling reactions with fast laser fields, enhancing Nuclear Magnetic Resonance images, quantum optics and quantum information science. Quantum information processing manipulates data with unitary operators called quantum logic gates, which control state transitions in qubits. These gates are vital for future quantum computers, requiring high efficiency and robustness despite experimental limitations and errors, like inconsistent control fields. The primary goal in quantum computing is to transfer population from the ground state to the target state with a high fidelity , good robustness and minimum interaction time between the electromagnetic fields and the system. Currently, creating a control method that meets all three criteria is a significant challenge in quantum information. However, by merging reverse engineering using the Lewis-Riesenfeld dynamic invariant with optimal control theory, we've shown that designing such a control field can be simplified into a Lagrange optimization problem with integral constraints. This hybrid method, called RIO (Robust Inverse Optimization), is adaptable and applicable to diverse quantum control challenges.