A small qubit leak becomes a big problem after many pulses

The vulnerability of quantum sensors to even minor qubit leaks is amplified by the thousands of pulses now routinely applied to extend sensing duration. A new analysis reveals that certain dynamical decoupling protocols, commonly used to achieve long qubit coherence, can degrade sensing performance far more than they affect coherence itself, challenging the assumption that long coherence guarantees good sensing.

Researchers found that differing pulse protocols cause coherence times to vary by orders of magnitude due to qubit leakage, where pulses couple to states outside the two-level qubit subspace. Even a very small leak can become problematic after a large number of pulses.

Carr-Purcell Sequence Sensitivity Reveals Pulse Error Impact

Pulse imperfections during quantum sensing can dramatically alter results, with coherence times differing by orders of magnitude depending on the specific pulse protocol employed, according to new simulations. The work focuses on the Carr-Purcell (CP) pulse sequence and its variants, revealing that seemingly minor deviations from ideal pulses accumulate over the thousands of pulses now common in sensing applications. Researchers tested these sequences through both experimental measurements and simulations of nuclear magnetic resonance spectroscopy using unpolarized nuclear spins.

The simulations modeled a single spin-1/2 quantum sensor detecting a single spin-1/2 target, beginning with the sensor in a pure spin-up state and the target in a mixed state. Results demonstrate that certain pulse protocols are far more susceptible to off-resonant coupling than others, leading to substantial variations in coherence.

A simulation of a 256 π pulse sequence revealed that the probability of measuring spin-up at the sequence’s end is highly sensitive to both rotation error and pulse detuning, ranging from a probability of one to zero depending on these factors. These effects were observed even when the interaction between the sensing and target spins was absent. Further analysis showed that finite-width pulses exacerbate the problem, as detuning impacts both phase evolution between pulses and the pulses themselves, eliminating the periodic behavior seen with idealized, delta-function pulses.

The simulations used a Rabi frequency where a π pulse duration equaled 2/3τ, highlighting the sensitivity to pulse timing. “We find that different pulse protocols are affected by this off-resonant coupling very differently, resulting in coherence times that differ by orders of magnitude,” the paper reports, emphasizing the critical need for pulse sequences robust to errors.

CPMG Sequence Performance with Sensing and Pulse Imperfections

CPMG sequences, while often favored for their resilience to qubit leaks, exhibit a surprising vulnerability when employed for sensing; the peak signal splits when the target spin’s precession aligns with the sequence, compromising performance in a manner similar to the Carr-Purcell (CP) protocol. This splitting occurs because the sensing qubit evolves away from its initially protected state, becoming susceptible to rotational errors induced by imperfect pulses.

Simulations reveal that at a detuning of 0.5, the phases within a CPMG sequence align with those of an APCP sequence at zero detuning, demonstrating a complex interplay between detuning and sequence performance. A simulation of 256 π pulses across CP, CPMG, APCP, XY16, and MLEV32Y protocols highlights these differences, visually demonstrating how the probability of measuring a specific spin state varies with detuning.

While systems with minimal leaks and significant pulse errors would ideally benefit from protocols like APCP or CPMG, offering leak insensitivity and XY16 or MLEV32Y, offering pulse error insensitivity, no single protocol currently delivers both advantages simultaneously. However, MLEV32Y appears to offer the best compromise for systems contending with both issues. Further details regarding these simulations and related findings are available in reference, offering a deeper dive into the nuances of dynamical decoupling and its impact on quantum sensing fidelity.

XY Family Struggles with Off-Resonant Qubit Leakage

The XY family of pulse sequences, often described as demonstrating resilience to pulse errors during sensing and in the absence of a target spin, yet exhibits heightened sensitivity to off-resonant coupling compared to both CPMG and APCP protocols. Simulations focused on XY16, finding it outperformed shorter sequences like XY4 and XY8, but showed negligible differences when compared to XY32, suggesting diminishing returns with increased pulse complexity.

Experimental results mirrored this trend, with XY16 consistently exceeding XY8 in performance, while XY32 and XY64 offered no further improvement. Similarly, the MLEV family of sequences also provides robustness against pulse errors, but also suffers from off-resonant coupling. Simulations identified MLEV8, MLEV8Y and MLEV32Y as the least susceptible to this effect. The source of this leakage stems from the drive coupling the sensing qubit levels, which can also couple to a third, “leak” state, with the strength of the coupling dependent on the detuning between levels.

This three-level system, where states |1〉 and |2〉 form the sensing qubit and |3〉 represents the leak state, highlights the vulnerability of these protocols to unintended transitions, even when decoherence and environmental noise are absent. The simulations assume equal transition strengths between |1〉↔|2〉 and |2〉↔|3〉, providing a controlled environment to assess the impact of off-resonant coupling.

MLEV Sequences Balance Error Robustness and Leakage

While CP demonstrates extreme sensitivity to pulse errors, the MLEV family, though offering robustness against those errors, also suffers from leakage, a coupling to states outside the primary qubit subspace, that impacts performance. Results show strong differences between the protocols, with the probability distributions diverging significantly depending on the sequence employed. Error bars reflect both measurement error and day-to-day variations in pulse imperfections, providing a comprehensive assessment of sequence reliability.

The researchers report, highlighting the critical need to select protocols based on specific experimental parameters and error profiles. The data emphasise that minimizing leakage is as important as maximizing coherence for effective quantum sensing.

Simulating NMR Spectroscopy with a Quantum Sensor

Simulations demonstrate that a π pulse duration of 2/3τ represents the longest possible pulse before overlap occurs between π/2 and π pulses, establishing a practical limit for pulse design. This constraint is critical because the cumulative effect of imperfect pulses becomes increasingly problematic as the number of pulses in a sensing sequence grows, a common feature in extending qubit lifetimes and performing NMR spectroscopy. Specifically, the simulations considered rotation fraction and detuning parameters to model pulse errors, with detuning representing the drive pulse’s deviation from the sensor spin’s resonant frequency.

A rotation fraction of one indicates a perfect pulse, while lower values simulate partially executed pulses, such as a nominal π pulse becoming a π/2 pulse. These simulations reveal that the probability of the sensor spin remaining in its initial state after the sensing sequence is heavily influenced by both the chosen pulse protocol and the magnitude of these errors.

The choice of pulse sequence is therefore not solely dictated by achieving long coherence times; certain protocols exhibit greater robustness against pulse imperfections than others. The data show that the performance of the sensing sequence is plotted as a function of the Larmor precession frequency of the target spin, providing a visual representation of how errors distort the NMR signal. This detailed analysis provides a pathway for optimizing pulse sequences to enhance the accuracy and reliability of quantum sensors used in NMR spectroscopy and beyond.

Dipole-Dipole Interaction Impacts Target Spin Frequency

The measured precession frequency of a target spin remains unaltered by interaction with a sensing spin, even when both are subject to a static bias magnetic field, according to new simulations. Researchers focused on a spin-1/2 sensing particle in a superposition of states to explore this interaction, simplifying the analysis by assuming equal magnitude for the $S_z{S}_z$ and $S_z{S}_x$ interaction terms.

Simulations considered a scenario where the interaction strength and duration of the sensing sequence are perfectly matched, allowing for conditional flipping of the sensing spin in the presence of the target spin under ideal conditions. This analysis demonstrates that the sensor’s ability to detect the target spin’s frequency is sensitive to the precise timing and shape of the applied pulses, even in the absence of decoherence or qubit leakage.

Pulse Imperfections Modeled via Rotation and Detuning

Researchers modeled these imperfections using “rotation” and “detuning” parameters, quantifying how far pulses deviate from their intended strength and frequency, respectively. The analysis demonstrates that inhomogeneous distributions of these errors cause signal broadening, but the combined effect of both rotation and detuning errors is more complex. These calculations assumed a sequence of 256 π pulses, with results plotted as a function of rotation error and pulse detuning, revealing a stark contrast between scenarios with and without a target spin interaction.

The simulations used delta-function pulses, allowing researchers to isolate the effects of detuning on phase evolution between pulses. The behavior differs significantly depending on the pulse shape.

For true delta-function pulses, detuning affects only the phase evolution, creating repeating patterns in the simulation results. “For the sequences shown here, the results for detunings from 0.5 to 1 are a mirror image of those from 0 to 0.5, and the results from rotation fractions from 1 to 2 are a mirror image of those from 0 to 1,” the researchers note, highlighting the predictable, yet sensitive, response of the system. This sensitivity highlights the need for precise pulse calibration as quantum sensing protocols become more complex and rely on increasingly long sequences of operations.

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