Memory-optimised Cubic Splines for High-fidelity Quantum Operations

TL;DR

Memory-efficient quantum pulse generation is achieved by combining cubic spline interpolation with symmetry-based compression and a two-stage, quantisation-aware fixed-point fitting strategy implemented on FPGA via recursive coefficient generation. A floating-point spline fit is refined with hardware-aware optimisation to minimize quantisation error, enabling high-fidelity operations at substantially reduced memory footprints. Benchmark results on neutral-atom gates and atom transport demonstrate large memory Savings (up to ~8×10^2–6×10^2) with fidelity close to AWG limits, validating the approach for scalable quantum control. The method is platform-agnostic and has potential applications across qubit platforms, offering a practical path to memory-constrained, high-bandwidth quantum control in large-scale systems.

Abstract

Radio-frequency pulses are widespread for the control of quantum bits and the execution of operations in quantum computers. The ability to tune key pulse parameters such as time-dependent amplitude, phase, and frequency is essential to achieve maximal gate fidelity and mitigate errors. As systems scale, a larger fraction of the control electronic processing will move closer to the qubits, to enhance integration and minimise latency in operations requiring fast feedback. This will constrain the space available in the memory of the control electronics to load time-resolved pulse parameters at high sampling rates. Cubic spline interpolation is a powerful and widespread technique that divides the pulse into segments of cubic polynomials. We show an optimised implementation of this strategy, using a two-stage curve fitting process and additional symmetry operations to load a high-sampling pulse output on an FPGA. This results in a favourable accuracy versus memory footprint trade-off. By simulating single-qubit population transfer and atom transport on a neutral atom device, we show that we can achieve high fidelities with low memory requirements. This is instrumental for scaling up the number of qubits and gate operations in environments where memory is a limited resource.

Figures (8)

• Figure 1: Illustrative diagram of Direct Digital Synthesis (DDS) using cubic splines for amplitude modulation.
• Figure 2: Pulse shaper block diagram as implemented on FPGA.
• Figure 3: We show the effect of the quantisation-aware fitting method for a Gaussian pulse of length $30000$ partitioned into $7$ segments and multiplied with an NCO frequency of $f=2$MHz for illustration purposes. Top-left: Comparison of the ideal floating point fit of the desired Gaussian amplitude envelope ($y_{ref}(t)=\sin(2 \pi ft)e^{-(t-15000)^2/(1.6e^7)}$) to the quantised 36pt initial fit. Top-right: Absolute percentage error of the quantised fits with respect to the desired output. The optimised fitting method reduces the error by an order of magnitude. Bottom-left: Fourier Transformation of the desired signal and the initial fit (where x=0 corresponds to the centre frequency of $2$ MHz). Bottom-right: Absolute error in Fourier Transform between quantised fits and the desired signal. The optimised fit again improves the error by an order of magnitude. The cubic polynomial coefficients for two example segments are shown for reference in App. \ref{['sec:examples']}
• Figure 4: We consider two fixed length 20 $\mu$s Blackman pulses used to implement an $X(\pi)$ gate and compare the fidelity (top) as well as the memory usage to store the pulse (bottom) between the optimised fit, the floating point fit and the AWG implementation. The top plot shows that as the number of segments increases, the optimised fitting fidelity no longer differs from the fidelity obtained with the initial fit, however the point where the optimised fit reaches a comparable fidelity to the AWG is already at $\approx6$ segments (compared to $\approx20$) which implies a significantly lower memory overhead. This is exhibited in the bottom plot where one can see that we obtain almost three orders of magnitude of compression for $6$ segments when comparing the cubic spline and AWG implementations.
• Figure 5: We compare fidelities (top) across the floating point fit, the optimised fit, and the AWG implementation for an atom transport with a piecewise quadratic pulse (cf. Eq. \ref{['eq:pw_qu']} with a length of $31 \mu$s and a trap depth of $20 \mu$K) and also compare their respective memory usage (bottom). The top plot shows that for a small number of segments the cubic polynomial representations are insufficient for high-fidelity transport even with the optimised fitting because small pulse kinks lead to immediate loss of the atom from the trap. At larger segment numbers we can simulate atom transport with extremely high fidelities and much more modest memory costs. The improvement in fidelity from floating point to quantisation-aware fit can be as large as two orders of magnitude. The fidelity in fact can even exceed that attained with the AWG implementation, as the piecewise quadratic pulse used here is not the optimal waveform for atom transport pagano2024optimalcontroltransportneutral. The bottom plot shows similar results to those obtained in Fig. \ref{['fig:2mu_pulse_segs']} and we obtain almost three orders of magnitude of compression at 6 segments when comparing the optimised fit to the AWG implementation.