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arXiv:quant-ph/0407022 · Physical Review A

Arbitrarily accurate composite pulse sequences

Kenneth R. Brown, Aram W. Harrow, Isaac L. Chuang

Published Nov 19, 2004• 222 citations
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Abstract

Systematic errors in quantum operations can be the dominating source of imperfection in achieving control over quantum systems. This problem, which has been well studied in nuclear magnetic resonance, can be addressed by replacing single operations with composite sequences of pulsed operations, which cause errors to cancel by symmetry. Remarkably, this can be achieved without knowledge of the amount of error $ϵ$. Independent of the initial state of the system, current techniques allow the error to be reduced to $O({ϵ}^{3})$. Here, we extend the composite pulse technique to cancel errors to $O({ϵ}^{n})$, for arbitrary $n$.

DOI
10.1103/physreva.70.052318

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概要 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.

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Systematic errors in quantum operations can be the dominating source of imperfection in achieving control over quantum systems. This problem, which has been well studied in nuclear magnetic resonance, can be addressed by replacing single operations with composite sequences of pulsed operations, which cause errors to cancel by symmetry. Remarkably, this can be achieved without knowledge of the amount of error ϵϵϵ. Independent of the initial state of the system, current techniques allow the error to be reduced to O(ϵ3)O({ϵ}^{3})O(ϵ3). Here, we extend the composite pulse technique to cancel errors to O(ϵn)O({ϵ}^{n})O(ϵn), for arbitrary nnn.
Implementing a single-qubit unitary is often hampered by imperfect control. Systematic amplitude errors $\ensuremath{\epsilon}$, caused by incorrect duration or strength of a pulse, are an especially common problem. But a sequence of imperfect pulses can provide a better implementation of a desired operation, as compared to a single primitive pulse. We find optimal pulse sequences consisting of LLL primitive $\ensuremath{\pi}$ or $2\ensuremath{\pi}$ rotations that suppress such errors to arbitrary order $O({\ensuremath{\epsilon}}^{n})$ on arbitrary initial states. Optimality is demonstrated by proving an L=O(n)L=O(n)L=O(n) lower bound and saturating it with L=2nL=2nL=2n solutions. Closed-form solutions for arbitrary rotation angles are given for n=1,2,3,4n=1,2,3,4n=1,2,3,4. Perturbative solutions for any nnn are proven for small angles, while arbitrary angle solutions are obtained by analytic continuation up to n=12n=12n=12. The derivation proceeds by a novel algebraic and nonrecursive approach, in which finding amplitude error correcting sequences can be reduced to solving polynomial equations.
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.
[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。