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arXiv:1204.2437 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Geometric aspects of composite pulses

Tsubasa Ichikawa, Masamitsu Bando, Yasushi Kondo, Mikio Nakahara

Published Sep 3, 2012• 30 citations
arXivHTMLPDF

Abstract

Unitary operations acting on a quantum system must be robust against systematic errors in control parameters for reliable quantum computing. Composite pulse technique in nuclear magnetic resonance realizes such a robust operation by employing a sequence of possibly poor-quality pulses. In this study, we demonstrate that two kinds of composite pulses-one compensates for a pulse length error in a one-qubit system and the other compensates for a J-coupling error in a two-qubit system-have a vanishing dynamical phase and thereby can be seen as geometric quantum gates, which implement unitary gates by the holonomy associated with dynamics of cyclic vectors defined in the text.

DOI
10.1098/rsta.2011.0358

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[Abstract] Quantum operations robust against random noise and systematic errors are essential ingredients in the quantum control. We have theoretically studied such robust quantum operations against the errors, that consist of several elementary operations and are called composite quantum gates. In this article, we experimentally discuss the robustness of various composite quantum gates by using NMR (Nuclear Magnetic Resonance) that is one of the most familiar quantumly controlled systems. [概要] 量子制御には乱雑なノイズや系統的なエラーの影響を受け難い量子操作が必要である。我々は今までエラー耐性を持つ量子操作（複数の基本的な量子操作を組み合わせた複合量子ゲート）の理論的な研究を主に行ってきた。本稿では、最も身近な量子制御であるNMR（核磁気共鳴）を用いて、様々な複合量子ゲートのエラー耐性を実験的に調べる。

•Feb 1, 2014•0 citations
Unitary operations acting on a quantum system must be robust against systematic errors in control parameters for reliable quantum computing. Composite pulse technique in nuclear magnetic resonance realizes such a robust operation by employing a sequence of possibly poor-quality pulses. In this study, we demonstrate that two kinds of composite pulses-one compensates for a pulse length error in a one-qubit system and the other compensates for a J-coupling error in a two-qubit system-have a vanishing dynamical phase and thereby can be seen as geometric quantum gates, which implement unitary gates by the holonomy associated with dynamics of cyclic vectors defined in the text.
We propose a simple formalism to design unitary gates robust against given systematic errors. This formalism generalizes our previous observation [Y. Kondo and M. Bando, J. Phys. Soc. Jpn. 80, 054002 (2011)] that vanishing dynamical phase in some composite gates is essential to suppress pulse-length errors. By employing our formalism, we derive a composite unitary gate which can be seen as a concatenation of two known composite unitary operations. The obtained unitary gate has high fidelity over a wider range of error strengths compared to existing composite gates.
Composite pulse sequences designed for nuclear magnetic resonance experiments are currently being applied in many quantum information processing technologies. We present an analysis of a family of composite pulse sequences used to address systematic pulse-length errors in the execution of quantum gates. It has been demonstrated by Cummins et al. [Phys. Rev. A 67, 042308 (2003)] that for this family of composite pulse sequences, the fidelity of the resulting unitary operation compared with the ideal unitary operation is $1\ensuremath{-}C{ϵ}^{6}$, where ϵϵϵ is the fractional error in the length of the pulse. We derive an exact expression for the sixth-order coefficient CCC and from this deduce conditions under which this sixth-order dependence is observed. We also present pulse sequences which achieve the same fidelity.
Composite pulses---sequences of pulses with well-defined relative phases---are an efficient, robust, and flexible technique for coherent control of quantum systems. Composite sequences can compensate for a variety of experimental errors in the driving field (e.g., in the pulse amplitude, duration, detuning, chirp, etc.) or in the quantum system and its environment (e.g., inhomogeneous broadening, stray electric or magnetic fields, unwanted couplings, etc.). The control parameters are the relative phases between the constituent pulses in the composite sequence, an accurate control over which is required in all composite sequences reported hitherto. In this paper, we introduce two types of composite pulse sequences which, in addition to error compensation in the basic experimental parameters, compensate for systematic errors in the composite phases. In the first type of such composite sequences, which compensate for pulse area errors, relative phase errors of over 10% can be tolerated with reasonably short sequences while maintaining the fidelity above the 99.99% quantum computing benchmark. In the second type of composite sequences, which compensate for simultaneous pulse area and detuning errors, relative phase errors of over 5% can be compensated.
The precision of quantum operations is affected by unavoidable systematic errors. A composite pulse (CP), which has been well investigated in nuclear magnetic resonance (NMR), is a technique that suppresses the influence of systematic errors by replacing a single operation with a sequence of operations. In one-qubit operations, there are two typical systematic errors, Pulse Length Error (PLE) and Off Resonance Error (ORE). Recently, it was found that PLE robust CPs have a clear geometric property. In this study, we show that ORE robust CPs also have a simple geometric property, which is associated with trajectories on the Bloch sphere of the corresponding operations. We discuss the geometric property of ORE robust CPs using two examples.
The control of qubit states is often impeded by systematic control errors. Compensating pulse sequences have emerged as a resource efficient method for quantum error reduction. In this review, we discuss compensating composite pulse methods, and introduce a unifying control-theoretic framework using a dynamic interaction picture. This admits a novel geometric picture where sequences are interpreted as vector paths on the dynamical Lie algebra. Sequences for single-qubit and multi-qubit operations are described with this method.
[Abstract] Quantum operations robust against random noise and systematic errors are essential ingredients in the quantum control. We have theoretically studied such robust quantum operations against the errors, that consist of several elementary operations and are called composite quantum gates. In this article, we experimentally discuss the robustness of various composite quantum gates by using NMR (Nuclear Magnetic Resonance) that is one of the most familiar quantumly controlled systems. [概要] 量子制御には乱雑なノイズや系統的なエラーの影響を受け難い量子操作が必要である。我々は今までエラー耐性を持つ量子操作（複数の基本的な量子操作を組み合わせた複合量子ゲート）の理論的な研究を主に行ってきた。本稿では、最も身近な量子制御であるNMR（核磁気共鳴）を用いて、様々な複合量子ゲートのエラー耐性を実験的に調べる。