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arXiv:1010.1622

Manipulating quantum information on the controllable systems or subspaces

Ming Zhang, Zairong Xi, Jiahua Wei

Published Oct 8, 2010• 1 citation
arXivHTMLPDF

Abstract

In this paper, we explore how to constructively manipulate qubits by rotating Bloch spheres. It is revealed that three-rotation and one-rotation Hamiltonian controls can be constructed to steer qubits when two tunable Hamiltonian controls are available. It is demonstrated in this research that local-wave function controls such as Bang-Bang, triangle-function and quadratic function controls can be utilized to manipulate quantum states on the Bloch sphere. A new kind of time-energy performance index is proposed to trade-off time and energy resource cost, in which control magnitudes are optimized in terms of this kind of performance. It is further exemplified that this idea can be generalized to manipulate encoded qubits on the controllable subspace.

DOI
10.48550/arxiv.1010.1622

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In this paper, we explore how to constructively manipulate qubits by rotating Bloch spheres. It is revealed that three-rotation and one-rotation Hamiltonian controls can be constructed to steer qubits when two tunable Hamiltonian controls are available. It is demonstrated in this research that local-wave function controls such as Bang-Bang, triangle-function and quadratic function controls can be utilized to manipulate quantum states on the Bloch sphere. A new kind of time-energy performance index is proposed to trade-off time and energy resource cost, in which control magnitudes are optimized in terms of this kind of performance. It is further exemplified that this idea can be generalized to manipulate encoded qubits on the controllable subspace.
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This paper presents an approach based on constructing control Hamiltonians to steer qubit states. When three tunable control Hamiltonians are available, we can adjust the controls to transform two-level quantum systems from an arbitrary initial pure state to another arbitrary target pure state along the unique geodesic curve on Block sphere. In terms of a kind of time-energy performance J = ∫0tf[λ + (1 - λ)E(u(t))]dt, we discuss how to design control magnitude to minimize this performance. An example indicates the feasibility and efficiency of this approach on optimal control of two-level quantum systems.
Abstract. We present a constructive control scheme for solving quantum state engineering problems based on a parametrization of the state vector in terms of complex hyperspherical coordinates. Unlike many control schemes based on factorization of unitary operators, the scheme gives explicit expressions for all generalized Euler angles in terms of the hyperspherical coordinates of the initial and final states. The factorization, when applicable, has added benefits that phase rotations can be combined and performed concurrently. The control procedure can be realized using simple bang-bang or square-wave-function controls. Optimal time-energy control is considered to find the optimal control amplitude. The extension of the scheme to implement arbitrary unitary operators is also discussed.
Hybrid qubits enable the hybridization of charge and spin degrees of freedom, which provides a way to realize both a relatively long coherence time and rapid qubit manipulation. Here, we use microwave driving to demonstrate the coherent operation of a tunable hybrid qubit, including X-rotation, Z-rotation, and rotation around an arbitrary axis in the X–Y panel of the Bloch sphere. Moreover, the coherence properties of the qubit and its tunability are studied. The measured coherence time of the X-rotation reaches ∼14.3 ns. While for the Z-rotation, the maximum decoherence time is ∼5.8 ns due to the larger sensitivity to noise. By employing the Hahn echo sequence to mitigate the influence of the low-frequency noise, we have improved the qubit coherence time from ∼5.8 ns to ∼15.0 ns. Our results contribute to a further understanding of the hybrid qubit and a step towards achieving high-fidelity qubit gates in the hybrid qubit.
In the control of dissipative quantum systems, the slow relaxation modes usually set the ultimate manipulation timescale. We show that this bottleneck can be bypassed when dissipation itself is used as the engine of the transfer. For a qubit with non-unital, anisotropic Bloch relaxation characterized by the ratio κ=T1/T2>1κ=T_1/T_2>1κ=T1​/T2​>1, coherent control can only rotate the Bloch vector, whereas the bath alone can change its length. The fastest transfer between two thermal states therefore consists in steering the Bloch vector, at every instant, along the direction in which the bath contracts it most rapidly. This direction is obtained in closed form and lies slightly beyond the equatorial plane. With strong pulses, the resulting minimum transfer time is a fraction of T2T_2T2​, instead of several T1T_1T1​ for passive relaxation. With bounded fields, the Pontryagin maximum principle yields a bang--singular--bang synthesis whose singular arc is precisely this fastest-contraction direction, and the minimum time exceeds its strong-pulse limit by a correction inversely proportional to the field amplitude. We also determine the full time--energy Pareto front: moderate accelerations require little energy, and the time optimum lies at its steep, high-energy end. Off-axis targets are treated within the same framework; their only additional benefit is a shorter final rotation. These results apply directly to magnetic-resonance spins, nitrogen-vacancy centers and superconducting qubits, all of which operate deep in the strong-pulse regime.
We present a constructive control scheme for solving quantum state engineering problems based on the parameterization of the state vector in terms of complex hyperspherical coordinates. Unlike many control schemes based on factorization of unitary operators, the scheme gives explicit expressions for all generalized Euler angles in terms of the hyperspherical coordinates of the initial and final states. The factorization, when applicable, has added benefits that phase rotations can be combined and performed concurrently. The control procedure can be realized using simple bang–bang or square-wavefunction controls. Optimal time–energy control is considered to find the optimal control amplitude. The extension of the scheme to implement arbitrary unitary operators is also discussed.