Logical channels in approximate Gottesman-Kitaev-Preskill error correction

TL;DR

This work tackles the challenge of finite-energy GKP states leaking from the code space by developing a logical-channel description of GKP error correction under damping. It presents two channel-twirl strategies—stabilizer-group twirling (passive) and a minimal Pauli-twirling (active)—to produce completely positive, trace-preserving (CPTP) logical channels, enabling consistent qubit-level analysis of repeated GKP teleportation. The authors show that stabilizer twirling reduces the damping to a Gaussian random-noise channel in the high-quality limit, while minimal Pauli twirling yields a broadly applicable CPTP channel without relying on stabilizer symmetry, and both approaches support decoding optimization. Their numerical comparisons indicate that for small damping, standard binning decoding suffices and GRN approximations are effective, with differences diminishing as code energy increases. The results reinforce the usefulness of stochastic GKP state models and SB decoding, and lay groundwork for optimizing decoding in GKP-based fault tolerance and scalable quantum computation.

Abstract

The GKP encoding is a top contender among bosonic codes for fault-tolerant quantum computation. Analysis of the GKP code is complicated by the fact that finite-energy code states leak out of the ideal GKP code space and are not orthogonal. We analyze a variant of the GKP stabilizer measurement circuit using damped, approximate GKP states that virtually project onto the ideal GKP code space between rounds of error correction even when finite-energy GKP states are used. This allows us to identify logical maps between projectors; however, due to finite-energy effects, these maps fail to resolve completely positive, trace-preserving channels on the logical code space. We present two solutions to this problem based on channel twirling the damping operator. The first twirls over the full stabilizer group motivated by standard binning (SB) decoding that converts small amounts of damping into Gaussian random noise. The second twirls over a set of representative Pauli shifts that keeps the energy in the code finite and allow for arbitrary decoding. This approach is not limited to small damping, can be applied when logical GKP unitaries or other sources of CV noise are present, and allows us to study general decoding, which can be optimized to the noise in the circuit. Focusing on damping, we compare decoding strategies tailored to different levels of effective squeezing. While our results indicate that SB decoding is suboptimal for finite-energy GKP states, the advantage of optimized decoding over SB decoding shrinks as the energy in the code increases, and moreover the performance of both strategies converges to that of the stabilizer-twirled logical channel. These studies provide stronger arguments for commonplace procedures in the analysis of GKP error correction:(i) using stochastically shifted GKP states in place of coherently damped ones, and(ii) the use of SB decoding.

Figures (7)

• Figure 1: (a) Circuit showing the composition of state preparation, $N$ rounds of teleportation, and auxiliary state-assisted readout. (b) Associated global map given an input state of the form in Eq. \ref{['eq:physical-state']}. Above the global map, we label the CV Kraus operators, Eq. \ref{['eq:telKrausOp_shifted']}. Below the global map, we label the quasi-local qubit maps that appear between GKP projectors, Eq. \ref{['eq:correlatedqubitop']}. The corrections commute with GKP projectors, so we reorder them here to better indicate the qubit maps, and we use Eq. \ref{['eq:Pauliconnections']} to write the corrections as GKP subspace Paulis, noting that $\bm{\mathrm{a}}(\mu_t) = \bm{\mathrm{c}}(\mu_t) \text{ mod } 2$. Note that normalizations have been excluded throughout.
• Figure 2: Wigner function and marginal probability distributions for the GKP state $\ket{\bar{0}}$ in a mixture over envelopes, Eq. \ref{['eq:unbiasedencoding']}, for three values of $\beta = \{0.4, 0.2, 0.1\}$. Asymmetries across the $q=0$ and $p=0$ axes arise from the choice of envelope centers, $S = \{ (0,0), (0,1), (1,0), (1,1)) \}$ in units of $\sqrt{\pi}$, which are marked on the Wigner functions with a white 'x'. The mixture probabilities in Eq. \ref{['eq:unbiasedencoding_physical']} for $\bm{\mathrm{b}} \in S$ are (a) $\{0.31, 0.31, 0.19, 0.19\}$, (b) $\{0.26, 0.26, 0.24, 0.24\}$, and (c) $\{0.25, 0.25, 0.25, 0.25\}$.
• Figure 3: Graphical illustration of the twirl-aware recovery. Each subplot shows locations of the final envelope given particular bit values and corrective shifts chosen from Table \ref{['eq:shiftdecoder']}. The pink circle is centered at the location of the final envelope when no correction is applied --- i.e. the location of the bit vector $\bm{\mathrm{b}}_t$. Arrows depict the shifts that implement $\bar{X}$ (blue), $\bar{Y}$ (green), and $\bar{Z}$ (red) logical corrections. The dotted circles are centered at locations of the final envelope after corrections. The size of the circle has no meaning, it was chosen simply for illustrative purposes.
• Figure 4: (a) Circuit showing a CPTP channel $\mathcal{E} = \mathop{\mathrm{\ooalign{$∑$\cr\hidewidth$∫$\hidewidth\cr} }}\limits_k \hat{E}_k \cdot \hat{E}_k^\dagger$ acting between two rounds of teleportation with bit-shifted, damped auxiliary states. (b) The associated chain of operators using the noise channel's Kraus operators $\hat{E}_k$. (c) By twirling over the local bits $\bm{\mathrm{b}}$ and using the twirl-aware decoder, we can replace the CV Kraus operators with their twirl equivalents, Eq. \ref{['eq:telKrausOp_shiftednew']}. This allows us to identify the qubit Kraus operators in Eq. \ref{['eq:newQubitops_extranoise']}.
• Figure 5: Comparison of the standard binning (SB) and optimal decoders. Representative patch of the SB decoder indicating which logical GKP Pauli to apply for a given syndrome $\{m_q, m_p\}$. A similar patch is shown for the numerically optimized decoder when $\beta = 0.4$. The key difference is that the decision boundaries have been shifted---the SB boundaries are shown with dashed lines for reference. The decision boundaries for optimal-decoder patches further from the origin are shifted more.