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arXiv:1810.03833 · Physical Review A

Arbitrarily accurate variable rotations on the Bloch sphere by composite pulse sequences

Boyan T. Torosov, Nikolay V. Vitanov

Published Jan 3, 2019• 33 citations
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Abstract

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.

DOI
10.1103/physreva.99.013402

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Composite pulse sequences, which produce arbitrary predefined rotations of a two-state system at an angle \ensuremathθ\ensuremath{\theta}\ensuremathθ 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 \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2 sequences---symmetric and asymmetric---the phases of which are given by simple formulas in terms of rational multiples of \ensuremathπ\ensuremath{\pi}\ensuremathπ for any number of constituent pulses. This allows one to construct arbitrarily accurate \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2 composite rotations. These \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2 composite sequences are used to construct three classes of arbitrarily long composite \ensuremathθ\ensuremath{\theta}\ensuremathθ sequences by pairing two \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2 composite sequences, one of which is shifted by a phase \ensuremathπ\ensuremath−\ensuremathθ\ensuremath{\pi}\ensuremath{-}\ensuremath{\theta}\ensuremathπ\ensuremath−\ensuremathθ with respect to the other one.
Composite pulse sequences, which produce arbitrary predefined rotations of a qubit on the Bloch sphere, are presented. The composite sequences contain up to 17 pulses and can compensate up to 8 orders of experimental errors in the pulse amplitude and the pulse duration. Composite sequences for three basic quantum gates, X (not), Hadamard and arbitrary rotation, are derived. Three classes of composite sequences are presented: one symmetric and two asymmetric. They contain as their lowest members two well-known composite sequences: the three-pulse symmetric SCROFULOUS pulse and the four-pulse asymmetric BB1 pulse, which compensate first- and second-order errors, respectively. The shorter sequences are derived analytically, and the longer ones numerically (instead by nesting and concatenation, as usually done hitherto). Consequently, the composite sequences derived here match or outperform the existing ones in terms of either speed or accuracy, or both. For example, we derive a second-order composite sequence, which is faster (by about 13%) than the famous BB1 sequence. For higher-order sequences, the speedup becomes much more pronounced. This is important for quantum information processing as the sequences derived here provide more options for finding the sweet spot between ultrahigh fidelity and high speed.
We present three classes of symmetric broadband composite pulse sequences. The composite phases are given by analytic formulas (rational fractions of $\ensuremath{\pi}$) valid for any number of constituent pulses. The transition probability is expressed by simple analytic formulas and the order of pulse area error compensation grows linearly with the number of pulses. Therefore, any desired compensation order can be produced by an appropriate composite sequence; in this sense, they are arbitrarily accurate. These composite pulses perform equally well as or better than previously published ones. Moreover, the current sequences are more flexible as they allow total pulse areas of arbitrary integer multiples of $\ensuremath{\pi}$.
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.com
Narrowband composite pulse sequences containing an arbitrary number NNN of identical pulses are presented. The composite phases are given by a very simple analytic formula and the transition probability is merely sin2N(A/2){\mathrm{sin}}^{2N}(A/2)sin2N(A/2), where AAA is the pulse area. These narrowband sequences can be made accurate to any order with respect to variations in AAA for sufficiently many constituent pulses, i.e., excitation can be suppressed below any desired value for any pulse area but $\ensuremath{\pi}$.
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.
We propose various composite πππ-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.