Towards Non-Abelian Quantum Signal Processing: Efficient Control of Hybrid Continuous- and Discrete-Variable Architectures

TL;DR

This paper extends quantum signal processing to a non-Abelian setting (NA-QSP) for hybrid continuous-variable/discrete-variable quantum systems by introducing the Gaussian-Controlled-Rotation (GCR) pulse that uses non-commuting oscillator operators to robustly entangle and disentangle oscillators and qubits. GCR achieves a substantial reduction in circuit depth (\geq$4.5\times$) and improved error resilience compared with Abelian BB1, and can be concatenated into BB1(GCR) to sharpen square-wave responses for modular readout. The authors develop fully analytical, deterministic schemes for preparing Gaussian and non-Gaussian oscillator states (squeezed, cat, and GKP), including high-fidelity GKP state preparation and measurement-free, error-corrected gate teleportation, enabling fault-tolerant universal control of GKP qudits. They also demonstrate NA-QSP's applicability to phase estimation and outline a path toward a general theory of non-Abelian QSP with potential extensions to multi-mode and higher-dimensional codes, offering a principled framework for robust CV-DV quantum control with practical hardware fault-tolerance implications.

Abstract

Robust quantum control is crucial for achieving operations below the quantum error correction threshold. Quantum Signal Processing (QSP) transforms a unitary parameterized by $θ$ into one governed by a polynomial function $f(θ)$, a feature that underpins key quantum algorithms. Originating from composite pulse techniques in NMR, QSP enhances robustness against systematic control errors. We extend QSP to a new class, non-abelian QSP, which utilizes non-commuting control parameters, $\hatθ_1, \hatθ_2, \dots$, representing quantum harmonic oscillator positions and momenta. We introduce a fundamental non-abelian composite pulse sequence, the Gaussian-Controlled-Rotation (GCR), for entangling and disentangling a qubit from an oscillator. This sequence achieves at least a $4.5\times$ speedup compared to the state-of-the-art abelian QSP pulse BB1, while maintaining performance. Though quantum fluctuations in the control parameters are unavoidable, the richer commutator algebra of non-abelian QSP enhances its power and efficiency. Non-abelian QSP represents the highest tier of QSP variants tailored for hybrid oscillator-qubit architectures, unlocking new possibilities for such systems. We demonstrate the utility of GCR in high-fidelity preparation of continuous-variable oscillator states, including squeezed, Fock, cat, and GKP states, using fully analytical schemes that match numerically optimized methods in fidelity and depth while enabling mid-circuit error detection. Furthermore, we propose a high-fidelity QSP-based end-of-the-line GKP readout and a measurement-free, error-corrected gate teleportation protocol for logical operations on GKP bosonic qudits, bridging the gap between idealized theoretical and experimentally realistic versions of the GKP code. Finally, we showcase a GCR-based phase estimation algorithm for oscillator-based quantum computing.

Figures (16)

• Figure 1: Framework of composite pulses in phase space and its applications. Blue curves show the probability distribution of the oscillator position $|\braket{x|\psi}|^2$, with green arrows indicating the ancilla qubit’s spin orientation for the state $\ket{g}\otimes\ket{\alpha_\Delta}$ (Eq. (\ref{['eq:Gaussian']})). The goal is to rotate the spin from $\ket{g}$ to $\ket{\mp i}$ using only oscillator-controlled rotations, based on the sign of $\braket{\hat{x}}=\pm\alpha$, independent of position uncertainty $\Delta$. Black dashed lines plot the response function $\braket{\sigma_\mathrm{y}}$; flatter curves indicate improved performance post QSP. Gray panel: Depicts the hybrid CV-DV system in a product state, controlled via the phase space instruction set (Eq. (\ref{['eq:phasespaceIS']})). Conditional displacements $\mathrm{CD}(\gamma,\sigma_\phi)$ implement oscillator-controlled qubit rotations $\mathrm{R}_\phi(\gamma \hat{x})$. Blue panel: Introduces our key idea—applying QSP-based composite pulses in oscillator phase space to suppress rotation errors from wavefunction uncertainty. We introduce the non-Abelian composite pulse sequence, $\mathrm{GCR}$, and compare it to the standard $\mathrm{BB1}$ pulse wimperis1994broadband, including a concatenated BB1(GCR) construction (Sec. \ref{['sec:GCR']}). The dashed black lines show how $\braket{\sigma_\mathrm{y}}$ varies as a function of $x$, illustrating the response function before and after QSP sequences are applied. The panel summarizes key applications covered in this work, due to the enhanced performance shown in the right figure.
• Figure 2: Performance of non-Abelian composite pulse sequence $\mathrm{GCR}$ in quantum phase space for $\theta=\pi/2$ and $\Delta=1$.(a) Comparison against $\mathrm{BB1}(\theta)$ for a special case of $\chi=\frac{\pi}{4|\alpha|}$. The colored lines denote the various merits of correctness (failure probability: solid, infidelity: dashed) obtained from simulations using QuTiP Johansson2013 and the black lines denote the corresponding analytical expressions quoted in Eqs. (\ref{['eq:fail_gcr']}-\ref{['eq:reset_fid_gcr']}). See App. \ref{['app:comp_err']} for derivation. The infidelities of $\mathrm{GCR}(\theta)$ scale as $\chi^4$ while the failure probability of both schemes scales as $\chi^6$. (b) Performance of $\mathrm{GCR}(\theta)$ for the coherent basis $\{\ket{\alpha_\Delta+i\beta_\Delta}\}$ where $\alpha\neq0,\beta\neq 0$. (Left) For varying $|\alpha|$ and fixed $|\beta|(=5)$, the simulated failure probability (solid) and infidelity (dashed) show that this variation of $\mathrm{GCR}(\theta)$ also improves upon the rotation errors with the same efficiency as confirmed by the black lines, again plotting Eqs. (\ref{['eq:fail_gcr']}-\ref{['eq:reset_fid_gcr']}). (Right) For varying $|\beta|$ and a fixed $|\alpha|=5$, we show that this improvement does not depend on $|\beta|$ as suggested by Eq. (\ref{['eq:pnot0']}) since it just requires a simple rotation to keep the anomaly coming from this state with center at $\braket{\hat{x}}\neq 0$ and $\braket{\hat{p}}\neq 0$ in check.
• Figure 3: Readout binning using Gaussian-controlled-$\mathrm{BB1}$ pulse sequence, $\mathrm{BB1}(\mathrm{GCR})$. All plots follow the same legend, as given on the right. (a) Readout binning for the case of $|\alpha|=\sqrt{\pi}/2$. The plot gives the probability to measure a $+1$ outcome upon $\sigma_\textrm{y}$ measurement, post the $\mathrm{BB1}(\mathrm{GCR})$ and $\mathrm{BB1}$ in red and blue, respectively. The x-axis represents the initial oscillator state with mean modular position value $\braket{\hat{x}}/(2|\alpha|)$ on which this pulse was applied. Note that a binning readout tells us whether the oscillator is in a specific bin (of size $2|\alpha|$) or not, using qubit measurement outcome. Overall, $\mathrm{BB1}(\mathrm{GCR})$ yields a flatter response function compared to $\mathrm{BB1}$. (b) Logarithmic scale for plot (a) to quantify the advantage of $\mathrm{BB1}(\mathrm{GCR})$ precisely, for bins which support opposite qubit measurement outcomes. BB1(GCR) shows an order of magnitude improvement in comparison to BB1 at the peaks of the target square wave response. (c) The hybrid fidelity $F_\mathrm{H}$ after each pulse. Interestingly, the $\mathrm{BB1}(\mathrm{GCR}(90))$ pulse has better fidelity as well.
• Figure 4: Deterministic preparation of squeezed states.(a) Deterministic squeezing protocol with incremental $\mathrm{GCR}$. (b) Narration of $\mathrm{GCR}$ as a squeezing gadget $\mathcal{S}(\Delta,\Delta')$. The plots show how this sequence introduces a small amount of squeezing while unentangling the qubit from the final state for $\Delta=1,|\alpha|=0.25$. (c) Variation in fidelity and circuit duration with varying squeezing rate $|\alpha|_{k+1}=a\Delta_{k}^c$ with 10 points in the range $c\in[-2,0]$, for three different protocols with $a\in\{0.06,0.13,0.27\}$ aimed at a target squeezing of $11.2 \ \mathrm{dB}$. (d) Squeezing (maroon) and anti-squeezing (red) are shown as a function of the circuit duration for the faster protocol with $c=2$. See App. \ref{['app:squeezing']} for definitions of $S_x,S_p$ in terms of $\Delta$. (e) Fisher information for the faster protocol. The empty circles in (d,e) represent a plot of the results for the case when post-selection is activated. The $\mu$s is the total length of conditional displacements; conversion to actual runtime can be found in App. \ref{['app:squeezing']}.
• Figure 5: Deterministic preparation of two-legged cat states.(a) Deterministic cat state preparation requires an un-entangling sequence given by $\mathcal{U}$. (b) Entangling-unentangling gadgets using $\mathrm{GCR}$. (c) We show numerical results with options of no correction ($\mathcal{U}=\mathrm{R}_\textrm{y}(\theta'\hat{x}/|\alpha|)$ in yellow), univariate or traditional QSP correction ($\mathcal{U}=\mathrm{BB1}$ in cyan), bivariate non-Abelian QSP correction ($\mathcal{U}=\mathrm{GCR}$ in red). (Left) Success probability of ancilla ending in-state $\ket{g}$. (Right) Fidelity of output oscillator state with the desired cat state upon success.