Experimental Demonstration of a Brachistochrone Nonadiabatic Holonomic Quantum-Gate Scheme in a Trapped Ion

Abstract: Nonadiabatic holonomic quantum computation (NHQC) offers intrinsic resilience to certain control imperfections. However, conventional nonadiabatic holonomic protocols are constrained by the fixed-pulse-area condition, which limits flexibility and prolongs duration of small-angle gates. Here we experimentally demonstrate a universal brachistochrone nonadiabatic holonomic quantum gate scheme in a trapped 40Ca+ ion, and realized the construction of pX gate under the conventional NHQC, brachistochrone NHQC (BNHQC) and composite BNHQC (CBNHQC) protocols. By characterizing the performance of gate performance in the presence of dissipation, Rabi-frequency errors and detuning errors, we show that BNHQC and CBNHQC outperform conventional NHQC, and BNHQC can offer a favorable balance between operation speed and robustness. It further shows that keeping high fidelity and strong robustness need decrease the accumulated population of excited state in the evolution process. These results highlight nonadiabatic holonomic computation as a practical route toward fast and robust quantum gates in trapped-ion platforms.

• The paper experimentally compares three nonadiabatic holonomic protocols in a single trapped 40Ca+ ion, showing that brachistochrone NHQC shortens the √X gate to 0.866 times the conventional duration without reducing fidelity.
• The measurements show a clear trade-off: composite BNHQC delivers the highest process fidelity of 99.5% and strongest Rabi-error resistance, while BNHQC performs best against detuning and minimizes gate time.
• The paper identifies accumulated population in the lossy auxiliary state as the main factor linking decoherence sensitivity and control robustness, offering a practical principle for designing improved holonomic gates.

Overview

This paper reports an experimental comparison of three nonadiabatic holonomic quantum computation (NHQC) protocols implemented on a single trapped $^{40}\mathrm{Ca}^{+}$ ion: conventional NHQC, brachistochrone NHQC (BNHQC), and composite BNHQC (CBNHQC). The central motivation is that conventional NHQC imposes a fixed-pulse-area constraint, so the gate duration $\tau_N = 2\pi/\Omega$ is independent of the target rotation angle. This is particularly wasteful for small-angle gates, which needlessly prolong exposure to decoherence. BNHQC removes this constraint by replacing the abrupt phase jump of conventional NHQC with a continuous time-optimal phase modulation derived from the quantum brachistochrone equation, while CBNHQC wraps the brachistochrone construction in a composite-pulse structure to suppress systematic control errors at the cost of a longer duration.

The authors benchmark all three protocols using the $\sqrt{X}$ gate, characterizing state and process fidelities as well as robustness against engineered dissipation on the auxiliary level and against detuning and Rabi-frequency errors. The headline finding is a clear trade-off: BNHQC minimizes gate time ($\tau_B = \sqrt{3}\pi/\Omega$ versus $2\pi/\Omega$ for NHQC), CBNHQC achieves the highest fidelity ($\tau_C = \sqrt{7}\pi/\Omega$), and the accumulated population in the auxiliary excited state emerges as the unifying quantity governing both decoherence sensitivity and error robustness.

Control protocols

All three protocols operate on a $\Lambda$-type three-level system in which the qubit is encoded in $\{|g\rangle, |e\rangle\}$ and an auxiliary state $|a\rangle$ couples to both via two laser drives. The interaction Hamiltonian takes the standard form

$$H = \frac{\Omega(t)}{2}e^{i\phi_1(t)}\left[e^{i\phi(t)}\sin\frac{\theta}{2}|g\rangle + \cos\frac{\theta}{2}|e\rangle\right]\langle a| + \text{H.C.},$$

with $\tau_N = 2\pi/\Omega$0. After adiabatic elimination of $\tau_N = 2\pi/\Omega$1, each protocol yields the effective qubit rotation $\tau_N = 2\pi/\Omega$2, so universality follows from the three controllable parameters $\tau_N = 2\pi/\Omega$3. The $\tau_N = 2\pi/\Omega$4 gate corresponds to $\tau_N = 2\pi/\Omega$5.

Conventional NHQC uses constant Rabi frequencies over $\tau_N = 2\pi/\Omega$6 with phases held at $\tau_N = 2\pi/\Omega$7 for the first half and jumped to $\tau_N = 2\pi/\Omega$8 for the second half, producing an unsmooth trajectory on the Bloch sphere.

BNHQC replaces the phase jump with a linear-in-time modulation $\tau_N = 2\pi/\Omega$9, obtained by solving the quantum brachistochrone equation under fixed pulse area. The resulting duration

$\sqrt{X}$0

shrinks toward zero as $\sqrt{X}$1, so small-angle rotations benefit most; for the $\sqrt{X}$2 gate, $\sqrt{X}$3.

CBNHQC splits the rotation into two BNHQC segments of angle $\sqrt{X}$4 with a relative phase offset, giving $\sqrt{X}$5 for $\sqrt{X}$6 — longer than even the conventional protocol — but with composite-pulse symmetry that cancels systematic errors to first order.

A limitation worth noting: the speed advantage of BNHQC is angle-dependent and vanishes near $\sqrt{X}$7, where $\sqrt{X}$8. The paper does not explore gates other than $\sqrt{X}$9 experimentally, so the claimed generality rests on theory rather than measurement across the full parameter space.

Experimental system

The experiment uses a single $\tau_B = \sqrt{3}\pi/\Omega$0 ion in a five-segment linear trap (CIQTEK) with secular frequencies $\tau_B = \sqrt{3}\pi/\Omega$1 MHz. The qubit is encoded in Zeeman sublevels of the $\tau_B = \sqrt{3}\pi/\Omega$2 ground-state manifold ($\tau_B = \sqrt{3}\pi/\Omega$3), split by 26.55 MHz under a 9.485 G bias field, and the auxiliary state is $\tau_B = \sqrt{3}\pi/\Omega$4, addressed by a bichromatic 729 nm laser forming the $\tau_B = \sqrt{3}\pi/\Omega$5 system. State preparation exceeds 99.5% fidelity via optical pumping plus microwave-assisted repumping, and readout uses 729 nm shelving pulses combined with fluorescence detection (~60 kcps bright signal versus <1 kcps dark).

A distinctive feature of the setup is the deliberately controllable dissipation channel: an 854 nm laser couples $\tau_B = \sqrt{3}\pi/\Omega$6 to $\tau_B = \sqrt{3}\pi/\Omega$7, creating effective decay channels $\tau_B = \sqrt{3}\pi/\Omega$8 and $\tau_B = \sqrt{3}\pi/\Omega$9 with branching ratio $2\pi/\Omega$0. This allows the authors to tune the auxiliary-state decay rate $2\pi/\Omega$1 and directly measure how dissipation during the holonomic evolution degrades each protocol — a capability most NHQC demonstrations lack.

Fidelity results

State tomography tracks the full evolution from $2\pi/\Omega$2 to the ideal output $2\pi/\Omega$3, confirming that BNHQC reaches the target state fastest, consistent with its shorter theoretical duration. Final state fidelities are:

Protocol	Measured state fidelity	Predicted	Measured process fidelity	Predicted
NHQC	98.5(4)%	99.0%	98.4(2)%	98.8%
BNHQC	98.6(7)%	99.3%	98.8(3)%	99.2%
CBNHQC	99.2(6)%	99.4%	99.5(2)%	99.4%

Process tomography reconstructs the $2\pi/\Omega$4 matrix from four input states $2\pi/\Omega$5, and the measured process fidelities agree with numerical simulations within roughly 0.5%, attributed mainly to state preparation and measurement (SPAM) errors. Two implications follow directly. First, CBNHQC's fidelity advantage is real but modest (~0.7 percentage points over NHQC), purchased with a ~53% longer gate than BNHQC. Second, the fact that BNHQC matches or slightly exceeds NHQC despite its shorter duration indicates that the time-optimal trajectory does not sacrifice geometric character or introduce additional infidelity in practice.

Robustness characterization

The authors probe three error channels by deliberately increasing the decay rate ($2\pi/\Omega$6 kHz at reduced Rabi frequency $2\pi/\Omega$7 kHz) and scanning detuning errors $2\pi/\Omega$8 and Rabi-frequency errors $2\pi/\Omega$9 over wide ranges.

Dissipation: CBNHQC maintains the highest fidelity under increasing $\tau_C = \sqrt{7}\pi/\Omega$0, followed by BNHQC, with NHQC lowest and deviating significantly from the other two. The explanation offered is mechanistic: NHQC accumulates more population in $\tau_C = \sqrt{7}\pi/\Omega$1 and runs longer, so it couples more strongly to the lossy auxiliary state.

Detuning errors: BNHQC exhibits the best robustness, attributable to its shorter evolution time reducing accumulated phase error. Notably, the robustness is asymmetric — BNHQC and CBNHQC tolerate positive detuning errors better than negative ones, whereas NHQC is approximately symmetric.

Rabi-frequency errors: CBNHQC performs best, consistent with composite-pulse cancellation of amplitude errors. The paper also reports that Rabi errors dominate over detuning errors in magnitude of impact, and that BNHQC/CBNHQC are more tolerant of negative Rabi errors.

The unifying observation — stated explicitly by the authors — is that high fidelity and strong robustness both correlate with minimizing the accumulated population in the auxiliary excited state throughout the evolution. This reframes protocol design: rather than choosing between "geometric" and "dynamical" character per se, one should minimize transient excited-state occupation, which BNHQC achieves through shorter duration and CBNHQC through destructive interference of excitation amplitudes.

Limitations and open questions

Several caveats bound the conclusions. The experimental demonstration covers only the $\tau_C = \sqrt{7}\pi/\Omega$2 gate; the claimed universal applicability and the angle-dependent speedup of BNHQC for small-angle rotations are not verified experimentally, even though small angles are precisely where BNHQC's advantage should be largest. All comparisons are single-qubit; extending the speed–robustness trade-off to entangling holonomic gates in trapped ions remains open. The asymmetry in error tolerance (positive detuning, negative Rabi errors) is observed empirically but not given a quantitative analytical account, leaving open whether it can be exploited in pulse design. Finally, the reported fidelities (~98–99%) remain SPAM-limited and below the thresholds achieved by optimized dynamical trapped-ion gates, so the practical case for holonomic gates would need demonstration at higher baseline fidelity or under realistic noise spectra rather than engineered single-channel dissipation.

Conclusion

This work provides a controlled, like-for-like experimental comparison of conventional, brachistochrone, and composite brachistochrone nonadiabatic holonomic protocols in a trapped ion, using a tunable dissipation channel to isolate decoherence effects. The results establish that BNHQC shortens gate duration without fidelity loss, CBNHQC buys additional systematic-error suppression at the cost of longer duration, and auxiliary-state population is the key resource governing performance across all error channels. The study clarifies the speed–robustness trade-off within holonomic quantum control and identifies a concrete design principle — minimize excited-state occupation — for future holonomic gate engineering on trapped-ion platforms.