Ballistic bosonic noise suppression with hybrid qumode-qubit rotation gates

Abstract

Noise suppression is of paramount importance for reliable quantum information processing and computation. We show that for any single-mode bosonic code (qumode) corrupted by thermal~noise at rate~$η$ and mean \mbox{excitation}~$\bar{n}$, a hybrid continuous-discrete-variable~(CV-DV) interferometer using only a single qubit ancilla~(DV) and two controlled~Fourier~(CF) gates sandwiching the noise channel suppresses its effects to $\mathcal{O}(η^2)$ \emph{without} any active error correction or destructive measurements of the encoded state and with high success probabilities~$>0.5$ if~$η(1+\bar{n})<0.5$. This suppression scheme works by conditionally monitoring the photon-number parities after the interferometer. Bosonic codes with two logical states of the same photon-number parity (like-parity codes) are \emph{completely resilient} to DV amplitude- and phase-damping ancilla noise. For such codes, the interferometer simplifies to the use of a qumode rotation gate and a \emph{single} CF~gate. This presents a clear advantage of our CF-gate-based error suppression scheme over previously-proposed ``bypass'' protocols, where qubit information transferred to the DV mode is readily corrupted by damping~noise. Finally, we present a simple extension to direct communication of qumode states between two parties over a noisy channel using a preshared DV entangled state, by implementing a CF gate in the first laboratory and its inverse in the other. Such a communication protocol achieves a similar fidelity performance at the same success rate as the single-party case, but with greater resilience to the ancilla noise than DV~teleportation. Resource-efficient multi-qubit codes that depend on a few essential long-range interactions can benefit from it.

Figures (7)

• Figure 1: Bosonic noise suppression with (a) conditional rotation (condrot) gates and (b) a series of compositions of conditional displacements and the conditional rotations gates. (c) An optical circuit implementing conditional rotations using the dispersive interactions between a single atom in a cavity and traveling waves Reiserer2015:Controlled with two circulators and a time-dependent optical switch to redirect the beams onto a cavity containing an atom to implement the noise suppression unitaries $U_\mathrm{s}$ and $U_\mathrm{s}^\dagger$. The atomic state is read out (green arrow) Hacker2019:DeterministicNunn2021:Heralding and the CV output $\rho_{\textsc{b},\text{succ}}$ is heralded on measuring the ground state. For bosonic codes with logical states of identical parity, the first hybrid gate can be replaced with a local qumode rotation as shown by the inset in (a).
• Figure 2: Comparison of average suppression performance with conditional Fourier (CF) gates between a like (even)-parity code $\texttt{bin}(2,4)$ and an opposite-parity bosonic code: $\texttt{cat}(6,1.916)$, both of which have similar average Gaussian moments of the photon-number distribution $(\langle n\rangle\cong 4$, $\langle n^2 \rangle\cong 20$, $\langle a^2\rangle=0)$ with respect to the state $C_\textsc{l}$. The former is more resilient to the composite amplitude and phase damping qubit noise of equal strengths $p$. The dashed and dot-dashed lines represent their respective, roughly identical performances. Insets show average success probability.
• Figure 3: The CF-gate-only interferometer requires no information about the noise parameters. Its average suppression performance remains impervious to uncalibrated ancilla damping, contrary to a series of conditional displacement gates and conditional rotation gates ([PQP-condrot]$^L$) [see Fig. \ref{['fig:protocols']} (b)], numerically optimized for known noise parameters for photon loss ($\eta=0.05$) and thermal noise ($\eta=0.05$, $\bar{n}=0.5$). Insets show average success probability.
• Figure 4: Comparison between using conditional Fouriers and the conditional displacements prescribed by the "bypass" protocol of Ref. Park2022:Slowing for two- and four-component cat codes. The latter uses a total of four conditional gates for the $\texttt{cat}(2,\alpha)$ codes. An additional DV ancilla, along with four more conditional displacements and two bosonic rotations by $\pi/4$, were used in "bypass" for $\texttt{cat}(4,\alpha)$. Although the opposite-parity codes are not robust to the DV damping noise in our protocol, the results show that the performance is better for moderately high $|\alpha|\sim 2$ and more robust to unknown DV damping noise. Moreover, our CF-only interferometer uses at most two conditional gates with a single ancilla. The differences in the number of gates become important when the conditional gates are noisy. Here, all noisy entangling gates have additional 1% loss and composite DV damping rates. Insets show average success probability.
• Figure 5: The like parity codes are also resilient to the composite amplitude and phase damping ancilla noise in quantum communication. (a) Circuit for bosonic signal transmission with preshared DV entanglement and classical communications as a resource. (b) An optical circuit to implement the protocol for quantum signal transmission using two remote cavity atom systems. (c) The average performance of various encodings under the noise suppression protocol for both photon loss of rate $\eta=0.05$ and thermal noise of the same rate and $\bar{n}=0.5$. The unmarked dashed, dot-dashed, and dotted lines refer to the average unsuppressed fidelities for the binomial, cat, and GKP codes, while the solid one is the average DV teleportation fidelity. Insets show average success probability. The filled and unfilled markers refer to heralding the bosonic output on the ancillary measurement outcome $00$, and both outcomes $00$ and $11$, respectively.