Engineering Non-Gaussian Bosonic Gates through Quantum Signal Processing

TL;DR

This paper addresses the bottleneck of implementing non-Gaussian gates in bosonic quantum technologies by introducing a quantum signal processing (QSP) framework tailored to hybrid qumode-qubit systems. By mapping joint qumode-qubit dynamics to dressed subspaces, the authors construct exact, finite-time non-Gaussian gates whose boson-number-dependent phases are controlled via polynomials (e.g., $F(z),G(z)$ or $P(z),Q(z)$) evaluated at $z=e^{i\Phi_n}$, with $\Phi_n$ set by the interaction (dispersive or JC). A key advancement is the mod-$k$ gate, a generalization of parity, which enables deterministic multi-component cat states, qudit entangling gates, and logic gates for rotation-symmetric codes, all with time $T_{total}=4\pi/\chi$ (dispersive) or similar JC-compatible times, independent of the Hilbert-space size. The framework also extends to non-unitary operations, including noiseless linear amplification and generalized-parity measurements, via controlled entanglement and projective readouts. Overall, the QSP approach provides exact, fast, and versatile non-Gaussian gate engineering for a wide range of hybrid platforms, with strong implications for bosonic codes and CV quantum information processing.

Abstract

Non-Gaussian operations are essential for most bosonic quantum technologies. Yet, realizable non-Gaussian gates are rather limited in type and generally suffer from accuracy-duration trade-offs. In this work, we propose to use quantum signal processing (QSP) techniques to engineer non-Gaussian gates on hybrid qumode-qubit systems. For systems with dispersive coupling, our scheme can generate a new non-Gaussian gate that produces a phase shift depending on the modulus of the boson number. This gate reproduces the selective number-dependent arbitrary phase (SNAP) gates under certain parameter choices, but with higher accuracy within a short, fixed and excitation-independent interaction time. The gate unlocks new applications, for example, in entangling logical qudits and deterministically generating multi-component cat states. Additionally, our versatile QSP formalism can be extended to systems with other interactions, and also engineer non-unitary operations, such as noiseless linear amplification and generalized-parity measurement.

Figures (6)

• Figure 1: Illustration of QSP circuits. (a) QSP is a sequence of operations that alternatively applies fixed phase-shifts $\hat{Z}_\Phi$ and qubit rotations $\hat{R}_m$ for $M$ rounds. (b) To engineer a qumode gate, qubit driving $\hat{T}_m$ and hybrid qumode-qubit interactions are applied alternatively for $M$ rounds. (c) The hybrid interaction induces excitation-number-dependent phase-shifts, such that each dressed subspace effectively undergoes QSP. After the sequence, the qubit returns to its ground state and is disentangled from the qumode.
• Figure 2: Gate infidelities of both the multi-tone driving (blue dots) and QSP (red dots) implementations of SNAP gates with $N_{\text{max}}=4$. The details of the fidelities and simulations can be found in Appendix \ref{['appendix_gate_fidelity']} and \ref{['appendix_multi']}, respectively. The presented data consists of $50$ samples of SNAP gates with randomly chosen target phases. For the multi-tone driving method, the infidelities decrease gradually with implementation time, because a weaker drive implements the gate more slowly but reduces the off-resonant effects. In contrast, the QSP method implements the same gates in a fixed time of $T_{\text{total}} = 4\pi/\chi$, but their infidelities are orders of magnitude lower.
• Figure 3: Quantum circuit of the two-mode mod-$k$ gate. Both qumodes A and B are dispersively coupled to the qubit and undergo controlled-phase-shifts. By applying well-designed qubit rotations between each round of controlled-phase-shifts, the two-mode mod-$k$ gate is applied to the qumodes.
• Figure 4: Wigner functions of (left) a coherent state with amplitude $\alpha =4$ and (right) an equal-amplitude $5$-component cat state generated by using the mod-$5$ gate with the phases in Eq. \ref{['relation']}.
• Figure 5: Gate infidelities of both the multi-tone driving (blue curve and dots) and QSP (red dots) implementations of SNAP gates with $N_{\text{max}}=3$ in a JC system. The details of the fidelities and the simulations can be found in Appendix \ref{['appendix_gate_fidelity']} and \ref{['appendix_multi']} respectively. (a) Infidelity of the hybridization process that is described in Sec. \ref{['hybridization']}. The QSP implementation achieves an infidelity that is eleven orders of magnitude lower under the same total implementation time, $T_{\text{total}} = 92\pi/(\sqrt{5}\lambda)$. (b) Infidelity of the phase modification process that is described in Sec. \ref{['sec_JC_kernel']}. The presented data consists of $50$ samples of the phase modification for SNAP gates with randomly chosen target phases. Our QSP method can generate gates with a fixed implementation time $T_{\text{total}} = 46\pi/(\sqrt{5}\lambda)$. Their infidelities are orders of magnitude lower than those for multi-tone driving.