Logical channel for heralded and pure loss with the Gottesman-Kitaev-Preskill code

TL;DR

This paper addresses how photon loss shapes the logical channel of Gottesman-Kitaev-Preskill (GKP) qubits by deriving analytic expressions for damping and pure-loss channels, both with and without error correction. It develops two loss representations—photon counting (discrete) and heterodyne (continuous)—and shows how they connect, enabling exact expressions for heralded and unheralded loss effects on the GKP logical qubit. A key result is that the loss-induced logical channel is not a stochastic Pauli channel, with non-Pauli features emerging as loss increases, while optimal damping can mitigate deterioration. The work also demonstrates that photon-subtraction heralding can yield highly non-Pauli, potentially magic-state-generating channels, offering a route to engineer non-Pauli dynamics in GKP-based fault-tolerance. Overall, the analytic framework clarifies the interplay of damping, loss, and EC in realistic optical quantum computing with GKP codes and informs strategies for optimizing logical-channel performance and resource states.

Abstract

Photon loss is the dominant source of noise in optical quantum systems. The Gottesman-Kitaev-Preskill (GKP) bosonic code provides significant protection; however, even low levels of loss can generate uncorrectable errors that another concatenated code must handle. In this work, we characterize these errors by deriving analytic expressions for the logical channel that arises from pure loss acting on approximate GKP qubits. Unlike random displacement noise, we find that the loss-induced logical channel is not a stochastic Pauli channel. We also provide analytic expressions for the logical channel for "heralded loss," when the light scattered out of the signal mode is measured either by photon number counting -- i.e., photon subtraction -- or heterodyne detection. These offer a pathway to intentionally inducing non-Pauli channels for, e.g., magic-state production.

Figures (7)

• Figure 1: Effects of loss on a GKP state. Solid dots indicate spikes in the Wigner function of a high-quality GKP $\lvert\bar{0}\rangle$ state. Larger circles show the evolution of each spike under pure loss---it is drawn towards the origin with added noise.
• Figure 2: (a) We consider the following logical channel that maps ideal GKP states to ideal GKP states: An ideal GKP state undergoes damping of strength $0 \leq \beta \leq \infty$ followed by pure loss at rate $0 \leq \gamma \leq 1$. Then, ideal GKP error correction projects the damaged state back into the GKP subspace. (b) A realization of the composed channels, with damping and loss each described by a dilation using an ancilla mode prepared in vacuum coupled to the data mode with a beam splitter. The circuit runs right-to-left. Tracing over the ancilla mode for loss gives the pure loss channel. Further, we consider both single-syndrome error correction and error correction averaged over syndromes. We produce the $4 \times 4$ single-qubit process matrix describing the logical GKP qubit map for all cases. (c) Table summarizing our results. We give either the Bloch vector components of the Kraus operator (from which the process matrix follows) or the process matrix elements themselves. Figures include depictions of the process matrices and Bloch vector trajectories given ideal GKP Pauli eigenstates. ${}^*$ We treat the trace as projective measurements of the loss mode in two relevant bases corresponding to heterodyne and number detection, respectively, and then discarding the outcome. By retaining the outcome, this additionally allows us to find conditional qubit maps that depend on the measurement basis and are heralded by the outcome. $^\dagger$ We focus on heralded photon-counting outcomes, since number-resolving detection is a key non-Gaussian operation employed in photon subtraction and Gaussian boson sampling.
• Figure 3: Process matrices and Bloch vectors for no loss ($\gamma = 0$) and various damping parameters. Top row: each subplot is a plot of the process matrix element $\chi_{\bm{a} \bm{a}'}(\bm{m})$ conditioned on syndrome outcomes $\{m_q\}$ and $\{m_p\}$. See Fig. \ref{['fig:legend']} for a guide to reading these plots and a color legend. Bottom row: syndrome-averaged process matrices, normalized to their trace, for each value of $\beta$. To the right are trajectories in the XZ plane of the qubit Bloch sphere as a function of $\beta$ for GKP Pauli $X/Z$ eigenstates with damping. The top right plot shows Bloch vectors conditional on syndrome $(0,0)$. The bottom right plot shows Bloch vectors averaged over syndromes. This plot reproduces curves found in Shaw et al.shaw_stabilizer_2024. Highlighted on each are low damping (solid circles), medium damping (open circles), and high damping (crossed circles). In each plot, a star indicates the fixed point of maximum damping. For the syndrome $(0,0)$, this state is $\lvert+H\rangle$; for the syndrome-averaged case, this state is given by Eq. \ref{['eq:avgECvacstate']}.
• Figure 4: Guide to read the process-matrix figures. (a) We represent the matrix elements of a single-qubit process matrix $\bm{\mathrm{\chi}}$ as subplots. The plots are laid out in an array just as if they were elements of a matrix. For syndrome-averaged process matrices, the values in each subplot are summed and then normalized by the trace of the syndrome-averaged process matrix, $\chi_{\bm{a} \bm{a}'} \rightarrow \chi_{\bm{a} \bm{a}'}/\sum_{\bm{a}} \chi_{\bm{a} \bm{a}}$, and then plotted. The matrix-element values are overlaid as a final step. (b) For post-selected syndromes, each subplot is a plot of $\chi_{\bm{a}\bm{a}'}(\bm{m})$ over the region of fractional syndromes $[-\frac{\sqrt{\pi}}{2}, \frac{\sqrt{\pi}}{2} )\times [-\frac{\sqrt{\pi}}{2}, \frac{\sqrt{\pi}}{2} )$. For visual comparison, these plots are also normalized to the trace of the syndrome-averaged process matrix. (c) In all plots, phase is given by color, and amplitude is given by brightness according to the color wheel.
• Figure 5: Process matrices and Bloch vectors for fixed damping ($\beta = 0.1$) and various loss rates. See Fig. \ref{['fig:legend']} for a guide to reading these plots and a color legend. Top row: Process matrices conditioned on syndrome outcomes $\{m_q\}$ and $\{m_p\}$. Bottom row: syndrome-averaged process matrices for each value of loss parameter $\gamma$. To the right are trajectories in the XZ plane of the qubit Bloch sphere as a function of $\gamma$ for GKP Pauli $X/Z$ eigenstates. Top: Bloch vectors conditional on syndrome $(0,0)$. Bottom: syndrome-averaged Bloch vectors. Highlighted on each are no loss (solid circles), high loss (open circles), and the fixed point of complete loss (star), where the channel is described by Eq. \ref{['eq:qubitchannel_maxloss']}.