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Tackling Systematic Errors in Quantum Logic Gates with Composite Rotations
H. K. Cummins, G. Llewellyn, J. A. Jones
TL;DR
Systematic errors in quantum logic gates, including off-resonance and pulse-length errors, must be addressed because they affect NMR quantum computations. The paper uses composite rotations and quaternion-based analysis to construct and compare three families of fully compensating single-qubit pulse sequences. The corpse family removes the f^2 fidelity term and outperforms a simple pulse for |f| ≤0.663; W1 cancels all error terms below sixth order, while BB1 retains off-resonance tolerance.
Problem
Systematic errors in quantum logic gates, including off-resonance and pulse-length errors, must be addressed because they affect NMR quantum computations.
Method
The paper uses composite rotations and quaternion-based analysis to construct and compare three families of fully compensating single-qubit pulse sequences.
Results
The corpse family removes the f^2 fidelity term and outperforms a simple pulse for |f| ≤0.663; W1 cancels all error terms below sixth order, while BB1 retains off-resonance tolerance.
Takeaways & Limitations
Composite pulses show promise for reducing data errors in NMR quantum computers and may apply more generally to quantum-computing implementations.
Takeaways & Limitations
Simultaneous compensation for off-resonance and pulse-length errors remains complicated and unresolved in this work.
Abstract
from arXiv · showhide
We describe the use of composite rotations to combat systematic errors in single qubit quantum logic gates and discuss three families of composite rotations which can be used to correct off-resonance and pulse length errors. Although developed and described within the context of NMR quantum computing these sequences should be applicable to any implementation of quantum computation.
I. INTRODUCTION
Systematic errors matter in quantum computation because they accumulate across long sequences of quantum logic gates. Composite rotations reduce the resulting sensitivity by replacing single Bloch-sphere rotations with composed rotations.
- I. INTRODUCTION: Although random errors from decoherence have received substantial attention, systematic errors in quantum gates also require consideration.The paper distinguishes these systematic effects from random errors arising from decoherence processes.
- I. INTRODUCTION: Systematic pulse errors can accumulate significantly across the long gate cascades required by complex NMR quantum algorithms.Errors that are negligible in conventional NMR experiments can become important when many pulses are combined.
- I. INTRODUCTION: Composite rotations reduce sensitivity to rotational imperfections by replacing single rotations with composed rotations on the Bloch sphere.The approach targets systematic errors through modified rotational trajectories rather than treating only random decoherence errors.
II. SYSTEMATIC ERRORS IN NMR QUANTUM COMPUTERS
NMR quantum computers use RF pulses for single-qubit gates, making those gates vulnerable to pulse-length and off-resonance errors. Composite pulses can reduce this sensitivity, but quantum computation requires fully compensating sequences that work for arbitrary input states.
- II. SYSTEMATIC ERRORS IN NMR QUANTUM COMPUTERS: NMR single-qubit gates are implemented with RF pulses, while two-qubit gates use sculpted scalar spin–spin coupling.The platform encodes qubits in spin-1/2 nuclear states within a magnetic field.
- II. SYSTEMATIC ERRORS IN NMR QUANTUM COMPUTERS: Pulse-length errors change the achieved rotation angle, whereas off-resonance errors tilt the rotation away from the intended axis.Pulse-length errors can arise from incorrect duration or RF-field strength, while off-resonance effects reflect detuning from the transition.
- II. SYSTEMATIC ERRORS IN NMR QUANTUM COMPUTERS: Conventional composite pulses are often unsuitable for quantum computation because they assume known initial spin states.Fully compensating type A sequences are required for pulses occurring inside computations, where the starting state is not known.
III. OFF-RESONANCE ERRORS
The paper derives off-resonance-compensating composite pulses with quaternion algebra and identifies the corpse family as the strongest member of its solution group. The corpse sequence removes the second-order fidelity error and outperforms a simple pulse over substantial off-resonance ranges.
- III. OFF-RESONANCE ERRORS: Quaternion multiplication represents the composite pulse, and quaternion fidelity compares it with the ideal rotation while accounting for equivalent quaternion signs.The off-resonance fraction is f = δ/ν1, and the sequence quaternion is obtained by multiplying the individual pulse quaternions.
- III. OFF-RESONANCE ERRORS: The composite sequence uses three pulses along x, −x, and x, with nominal angles constrained to reproduce the target rotation and cancel unwanted quaternion components.The angles satisfy relations enforcing the target scalar component and zero y and z components, subject to positive physical pulse angles.
- III. OFF-RESONANCE ERRORS: The corpse family appears to be the best member of the derived solution group because its added 2π rotations give the smallest fourth-order error term.All solutions remove the f^2 term; the fourth-order term is minimized when n = n1 − n2 + n3 = 0, including n1 = 1, n2 = 1, n3 = 0.
- III. OFF-RESONANCE ERRORS: Shortcorpse is somewhat shorter than corpse but performs less well, while Table I lists corpse pulse rotation angles for different target rotations.The corpse family uses phases +x, −x, +x.
- III. OFF-RESONANCE ERRORS: For a 180° target, corpse outperforms a simple pulse whenever |f| ≤ 0.663; for a 30° target, it outperforms one whenever |f| ≤ 0.297.Figure 1 compares simple and corpse-pulse fidelity as a function of off-resonance fraction f.
IV. PULSE LENGTH ERRORS
The pulse-length-error family uses three-pulse composite rotations with adjustable phases and angles, simplified through time symmetry and first-order error cancellation. The resulting scrofulous sequences are designed to be insensitive to pulse-length errors and are compared with plain pulses.
- Pulse-length-error compensation: Three-pulse sequences with arbitrary rotation and phase angles are developed to correct pulse-length errors.The six initial parameters are reduced to three during the derivation.
- Pulse-length-error compensation: Time symmetry, with θ1 = θ3 and φ1 = φ3, removes the composite quaternion’s z component and simplifies the analysis.The symmetric sequence remains a rotation about an axis in the xy plane.
- Pulse-length-error compensation: The approach builds on a previously described 180° sequence that compensates pulse-length errors, while selecting θ2 = π as the more productive family for further development.The earlier sequence is cited as.
- Pulse-length-error compensation: The design cancels the first-order term in the fractional pulse-power error g, making the sequences insensitive to pulse-length errors.The target rotation and phase are then adjusted through suitable values of θ1 and φ1.
- Pulse-length-error compensation: The resulting sequences are called scrofulous, and their pulse rotation and phase angles are tabulated for target angles before comparison with plain 180° pulses.Performance comparisons are reported in Figure 2, while Table II provides numerical sequence parameters.
V. THE BB1 FAMILY
The BB1 family appends an error-correcting W1 sequence to a target rotation, with phases chosen to cancel pulse-length errors. Its higher-order cancellation makes BB1 outperform simple pulses over the stated error range, while related Wn sequences offer limited gains beyond W1.
- BB1 construction: BB1 uses a W1 error-correction sequence with constrained phases, including φ2 = 3φ1, to cancel first-order pulse-length error components.The correction sequence is combined with a θx pulse, and the phase choices eliminate the remaining first-order components.
- BB1 performance: The positive BB1 solution also completely removes second-order error terms, reflecting a broader property of W1 and related sequences.The sequence is identical to the earlier construction by Wimperis.
- BB1 performance: BB1 performance is unchanged when W1 is placed before, after, or within the target pulse; all error terms below sixth order are cancelled.The sixth-order term still depends on the target rotation angle θ.
- BB1 performance: For target angles below 180°, BB1 outperforms a simple pulse when |g| < 1.Figure 3 compares simple and BB1 pulse fidelity for a 180° target rotation as a function of fractional pulse-length error g.
- Wn extensions: Using multiple Wn correction sequences cancels second- and fourth-order errors in every case, but n = 2 gives only a small sixth-order improvement over W1.The authors therefore identify simpler W1-based pulses as likely most effective in practice.
- Scope: The same error-correcting-sequence approach has not succeeded for off-resonance effects, despite its success against pulse-length errors.This marks a scope boundary for the approach discussed here.
VI. SIMULTANEOUS ERRORS
The paper analyzes how its composite pulses respond when off-resonance and pulse-length errors occur simultaneously, but does not yet design sequences that compensate for both. For 180° pulses, scrofulous is more off-resonance-sensitive than plain, whereas BB1 retains near-plain off-resonance sensitivity while tolerating pulse-length errors.
- Analysis scope: The analysis evaluates each sequence's sensitivity to the other error type rather than solving the unresolved problem of simultaneous compensation.The authors calculate quaternion fidelities and examine Maclaurin expansions, while treating one error at a time in the analytic procedure.
- Corpse response: For corpse, pulse-length-error behavior without off-resonance effects is identical to that of a simple pulse because its +x, −x, +x sequence reduces accordingly.The simultaneous-error behavior of a 180° pulse is presented in Figure 4.
- Scrofulous response: For 180° pulses, scrofulous has fidelity F ≈ 1 − 2f^2 versus F ≈ 1 − f^2/2 for plain pulses, making it considerably more off-resonance-sensitive.The general-angle analysis is difficult because θ1 depends on the arcsinc function, so the comparison concentrates on 180° pulses.
- BB1 response: BB1 achieves strong pulse-length-error tolerance with little or no added off-resonance sensitivity, including under simultaneous errors.The time-symmetrised BB1 sequence matches a plain pulse through second order for off-resonance effects, and Figure 4 confirms this for simultaneous errors.
VII. CONCLUSIONS
The paper concludes that composite pulses show promise for reducing data errors in NMR quantum computers.
- Composite pulses show great promise for reducing data errors in NMR quantum computers.