arXiv:1307.2211 · Physical Review A

Optimal arbitrarily accurate composite pulse sequences

Guang Hao Low, Theodore J. Yoder, Isaac L. Chuang

Published Feb 28, 2014• 44 citations

Abstract

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 $L$ 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 $L=O(n)$ lower bound and saturating it with $L=2n$ solutions. Closed-form solutions for arbitrary rotation angles are given for $n=1,2,3,4$. Perturbative solutions for any $n$ are proven for small angles, while arbitrary angle solutions are obtained by analytic continuation up to $n=12$. The derivation proceeds by a novel algebraic and nonrecursive approach, in which finding amplitude error correcting sequences can be reduced to solving polynomial equations.

DOI: 10.1103/physreva.89.022341

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arXiv:1307.2211 · Physical Review A

Optimal arbitrarily accurate composite pulse sequences

Guang Hao Low, Theodore J. Yoder, Isaac L. Chuang

Published Feb 28, 2014• 44 citations

arXiv HTML PDF

Abstract

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

L

primitive

\ensuremath{\pi}

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)

lower bound and saturating it with

L=2n

solutions. Closed-form solutions for arbitrary rotation angles are given for

n=1,2,3,4

. Perturbative solutions for any

n

are proven for small angles, while arbitrary angle solutions are obtained by analytic continuation up to

n=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.

DOI: 10.1103/physreva.89.022341

Related papers

Arbitrarily accurate composite pulse sequences

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

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 $n$ .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 $n$.

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Systematic control errors remain a primary obstacle to realizing high-fidelity single-qubit gates. We introduce composite pulse sequences that implement X and Hadamard gates while simultaneously compensating amplitude (Rabi-frequency), detuning (frequency), and duration errors. Our construction uses two complementary strategies: (i) derivative-based cancellation of error terms in the full unitary (not just the transition probability), formulated via the Cayley-Klein parametrization, and (ii) direct minimization of the average gate infidelity over prescribed error ranges. We derive symmetric five-pulse solutions with closed-form phases that cancel all first-order terms (including the mixed derivative), and numerically optimize longer sequences -- up to 15 pulses -- to achieve higher-order suppression. We also show that standard ``universal'' five-pulse sequences (U5a/U5b) emerge as simple phase-shifted instances of our symmetric solutions, yielding broad robustness to both detuning and amplitude errors. Finally, we construct variable-area sequences for $R_x(π/2)$ , which, up to virtual Z rotations, benchmark the Hadamard gate. Across all families we observe the expected trade-off between sequence length and robustness window, with substantial boosts in fidelity over large error domains.Systematic control errors remain a primary obstacle to realizing high-fidelity single-qubit gates. We introduce composite pulse sequences that implement X and Hadamard gates while simultaneously compensating amplitude (Rabi-frequency), detuning (frequency), and duration errors. Our construction uses two complementary strategies: (i) derivative-based cancellation of error terms in the full unitary (not just the transition probability), formulated via the Cayley-Klein parametrization, and (ii) direct minimization of the average gate infidelity over prescribed error ranges. We derive symmetric five-pulse solutions with closed-form phases that cancel all first-order terms (including the mixed derivative), and numerically optimize longer sequences -- up to 15 pulses -- to achieve higher-order suppression. We also show that standard ``universal'' five-pulse sequences (U5a/U5b) emerge as simple phase-shifted instances of our symmetric solutions, yielding broad robustness to both detuning and amplitude errors. Finally, we construct variable-area sequences for $R_x(π/2)$, which, up to virtual Z rotations, benchmark the Hadamard gate. Across all families we observe the expected trade-off between sequence length and robustness window, with substantial boosts in fidelity over large error domains.

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Composite pulse sequences, which produce arbitrary predefined rotations of a two-state system at an angle $\ensuremath{\theta}$ on the Bloch sphere, are presented. The composite sequences can contain arbitrarily many pulses and can compensate experimental errors in the pulse amplitude and duration to any desired order. A special attention is devoted to two classes of $\ensuremath{\pi}/2$ sequences---symmetric and asymmetric---the phases of which are given by simple formulas in terms of rational multiples of $\ensuremath{\pi}$ for any number of constituent pulses. This allows one to construct arbitrarily accurate $\ensuremath{\pi}/2$ composite rotations. These $\ensuremath{\pi}/2$ composite sequences are used to construct three classes of arbitrarily long composite $\ensuremath{\theta}$ sequences by pairing two $\ensuremath{\pi}/2$ composite sequences, one of which is shifted by a phase $\ensuremath{\pi}\ensuremath{-}\ensuremath{\theta}$ with respect to the other one.Composite pulse sequences, which produce arbitrary predefined rotations of a two-state system at an angle $\ensuremath{\theta}$ on the Bloch sphere, are presented. The composite sequences can contain arbitrarily many pulses and can compensate experimental errors in the pulse amplitude and duration to any desired order. A special attention is devoted to two classes of $\ensuremath{\pi}/2$ sequences---symmetric and asymmetric---the phases of which are given by simple formulas in terms of rational multiples of $\ensuremath{\pi}$ for any number of constituent pulses. This allows one to construct arbitrarily accurate $\ensuremath{\pi}/2$ composite rotations. These $\ensuremath{\pi}/2$ composite sequences are used to construct three classes of arbitrarily long composite $\ensuremath{\theta}$ sequences by pairing two $\ensuremath{\pi}/2$ composite sequences, one of which is shifted by a phase $\ensuremath{\pi}\ensuremath{-}\ensuremath{\theta}$ with respect to the other one.

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The Hamiltonian control of n qubits requires precision control of both the strength and timing of interactions. Compensation pulses relax the precision requirements by reducing unknown but systematic errors. Using composite pulse techniques designed for single qubits, we show that systematic errors for n -qubit systems can be corrected to arbitrary accuracy given either two non-commuting control Hamiltonians with identical systematic errors or one error-free control Hamiltonian. We also examine composite pulses in the context of quantum computers controlled by two-qubit interactions. For quantum computers based on the XY interaction, single-qubit composite pulse sequences naturally correct systematic errors. For quantum computers based on the Heisenberg or exchange interaction, the composite pulse sequences reduce the logical single-qubit gate errors but increase the errors for logical two-qubit gates.The Hamiltonian control of n qubits requires precision control of both the strength and timing of interactions. Compensation pulses relax the precision requirements by reducing unknown but systematic errors. Using composite pulse techniques designed for single qubits, we show that systematic errors for n -qubit systems can be corrected to arbitrary accuracy given either two non-commuting control Hamiltonians with identical systematic errors or one error-free control Hamiltonian. We also examine composite pulses in the context of quantum computers controlled by two-qubit interactions. For quantum computers based on the XY interaction, single-qubit composite pulse sequences naturally correct systematic errors. For quantum computers based on the Heisenberg or exchange interaction, the composite pulse sequences reduce the logical single-qubit gate errors but increase the errors for logical two-qubit gates.

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Nested Bloch-Sphere Rotations Achieve Arbitrarily Accurate SU(2) Gates — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Composite pulse sequences on the Bloch sphere achieve arbitrarily accurate SU(2) rotations by nesting error-compensating rotations, with the key structural insight being that any single-qubit gate is a 3D rotation in SU(2) ≅ SO(3), parameterized by Euler angles on the sphere. MATH: - Qubit state: |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩, with θ ∈ [0,π], φ ∈ [0,2π) — the Bloch sphere coordinates. - SU(2) generators: Pauli matrices σ_x, σ_y, σ_z, satisfying [σ_i, σ_j] = 2iε_{ijk}σ_k. - Rotation operator: R_n̂(θ) = exp(−iθ n̂·σ/2) = cos(θ/2)I − i sin(θ/2)(n̂·σ). - Composite pulse: U_total = Π_k R_{n̂_k}(θ_k), designed so that first-order (and higher) error terms in pulse amplitude/duration vanish: ∂U_total/∂ε |_{ε=0} = 0, up to desired order N. - The arXiv paper (1810.03833) explicitly constructs sequences with arbitrary N, using nested commutators of rotation axes to cancel error terms — a Lie-algebraic approach. CONNECTION: - The Bloch sphere is the unit sphere S², and SU(2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Composite pulse sequences on the Bloch sphere achieve arbitrarily accurate SU(2) rotations by nesting error-compensating rotations, with the key structural insight being that any single-qubit gate is a 3D rotation in SU(2) ≅ SO(3), parameterized by Euler angles on the sphere. MATH: - Qubit state: |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩, with θ ∈ [0,π], φ ∈ [0,2π) — the Bloch sphere coordinates. - SU(2) generators: Pauli matrices σ_x, σ_y, σ_z, satisfying [σ_i, σ_j] = 2iε_{ijk}σ_k. - Rotation operator: R_n̂(θ) = exp(−iθ n̂·σ/2) = cos(θ/2)I − i sin(θ/2)(n̂·σ). - Composite pulse: U_total = Π_k R_{n̂_k}(θ_k), designed so that first-order (and higher) error terms in pulse amplitude/duration vanish: ∂U_total/∂ε |_{ε=0} = 0, up to desired order N. - The arXiv paper (1810.03833) explicitly constructs sequences with arbitrary N, using nested commutators of rotation axes to cancel error terms — a Lie-algebraic approach. CONNECTION: - The Bloch sphere is the unit sphere S², and SU(2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 1, 20260 citations

Implementing a single-qubit unitary is often hampered by imperfect control. Systematic amplitude errors

\ensuremath{\epsilon}

L

primitive

\ensuremath{\pi}

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)

lower bound and saturating it with

L=2n

solutions. Closed-form solutions for arbitrary rotation angles are given for

n=1,2,3,4

. Perturbative solutions for any

n

are proven for small angles, while arbitrary angle solutions are obtained by analytic continuation up to

n=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.

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

We propose various composite

π

-pulse sequences for implementing robust z-axis rotation gates widely used in quantum information processing (QIP) scenarios, and discuss their error tolerance of the pulse strength error (PSE) and off-resonance error (ORE), i.e., two typical systematic errors in NMR systems. Compared to existing composite gates, our designed composite pulses exhibit higher gate fidelity and shorter sequence time cost. Furthermore, such short pulse sequences are generalized by simple analytic formulas which can suppress these errors to any desired order simultaneously with linear length scaling, indicating an efficient construction of them with arbitrary accuracy.

R_x(π/2)

, which, up to virtual Z rotations, benchmark the Hadamard gate. Across all families we observe the expected trade-off between sequence length and robustness window, with substantial boosts in fidelity over large error domains.