Decoding Multimode Gottesman-Kitaev-Preskill Codes with Noisy Auxiliary States

TL;DR

This work investigates decoding multimode GKP codes under realistic noise where auxiliary GKP states used in Steane-type QEC are imperfect. It introduces a noise-correlated MED (COR-MED) decoder that exploits correlations between storage errors and auxiliary-state errors, maintaining CVP-like classical complexity while significantly reducing logical error rates. Across square, hexagonal, tesseract, and D4 multimode GKP encodings, COR-MED achieves at least an order-of-magnitude improvement in p_L over standard MED, with up to ~25× gains in some regimes. The approach suggests that certain multimode GKP codes (notably D4) become more practical for fault-tolerant quantum computing, motivating future concatenation with qubit codes and experimental exploration of Steane-type QEC with correlated-noise decoders.

Abstract

In order to achieve fault-tolerant quantum computing, we make use of quantum error correction schemes designed to protect the logical information of the system from decoherence. A promising way to preserve such information is to use the multimode Gottesman-Kitaev-Preskill (GKP) encoding, which encodes logical qubits into several harmonic oscillators. In this work, we focus on decoding the measurements obtained from Steane-type quantum error correction protocols for multimode GKP codes. We propose a decoder that considers the noise present on the auxiliary states, more specifically by tracking the correlations between errors on different modes spreading throughout the error-correction circuit. We show that leveraging the correlations between measurement results and the actual error affecting the multimode GKP state can decrease the logical error probability by at least an order of magnitude, yielding more robust quantum computation.

Figures (10)

• Figure 1: Logical Voronoi cell $\mathcal{V}^\perp$ (grey hexagon) for a GKP code defined on a hexagonal lattice. The black dots represent the logical lattice $\Lambda^\perp$ and the teal arrows represent all three Voronoi-relevant vectors $\boldsymbol{\lambda}_{\mathcal{V}, j}^\perp$ and their inverse. The beige arrow represents a vector $\mathbf{v}$ that has its projections $\mathbf{v}_{\boldsymbol{\lambda}_{\mathcal{V}, j}^\perp}$ on the Voronoi-relevant vectors smaller than half of their length. These projections are represented by the green vectors. As we can see, the condition \ref{['eq: gen_voronoi_cell']} is respected for all Voronoi-relevant vectors, implying that $\mathbf{v}$ lies in the logical Voronoi cell. All points located in the grey hexagon also respect that condition, justifying the form of $\mathcal{V}^\perp$. Finally, we represent the distance $d$ by the radius of the dotted circle.
• Figure 2: (a) General form of a Steane-type QEC circuit for multimode GKP codes. The errors acting on the auxiliary states are shown in red to emphasize the distinction between the analysis where they are present (\ref{['subsec: qec_perf_aux']}) and when they are not (\ref{['subsec: qec_noisy_aux']}). Each $\hat{U}_{T_j}$ acts only on $|\psi\rangle$ and its corresponding auxiliary qunaught $|\varnothing_{\eta_j} \rangle$. (b) An example of a Steane-type QEC circuit for the square GKP code, measuring operators $\hat{h}'_{\square, j}$ defined by \ref{['eq: unity_stab_quad_measurement']}. This circuit is the one that was originally proposed in Ref. originalGKP, adapted with our notation. For a more detailed comparison, we refer the reader to \ref{['fig: square_GKP_steane_type_QEC_original']} of \ref{['subsec: appendix_gen_steane_type_mm_circuit_square_gkp']}. Notably, we remark that here we measure both the $\hat{q}$ and $\hat{p}$ quadratures of the auxiliaries.
• Figure 3: (a) Example scenario of the noiseless auxiliary decoding process for the square GKP that underwent a random translation $\boldsymbol{\xi}_\square$ (red arrow underneath the green arrow). The top figure shows the measurement result, and the bottom figure shows the decoding result, using the MED decoder defined in section \ref{['subsec: qec_perf_aux', 'subsec: qec_noisy_aux']}. The grey square delimits the logical Voronoi cell of the square GKP storage lattice $\Lambda_\square^\perp$ (black dots). If the final displacement of the storage is located in the light grey colored areas, it leads to a logical error. For the measurement process, we identify two different quantities in the figure, $\boldsymbol{\chi}(\mathbf{z})$ and $\boldsymbol{\chi}_{[\Lambda_\square^\perp]}(\mathbf{z})$, both represented by the same green arrow. The first one represents the raw value of the measurement result and the second the measurement result modulo the logical lattice, $\boldsymbol{\chi}_{[\Lambda_\square^\perp]}(\mathbf{z}) = \boldsymbol{\chi}(\mathbf{z}) + \boldsymbol{\lambda}_\square^\perp = \boldsymbol{\chi}(\mathbf{z}) ~\text{mod}~ \Lambda^{\perp}_{\square}$. In this case, both quantities are equal since the error is inside the Voronoi cell. Thus, the measurement completely recovers the error, hence the green arrow directly on top of the red one. We also show two of the three different errors, $\boldsymbol{\xi}_\square'$ and $\boldsymbol{\xi}_\square"$ (red dotted arrow), that would yield the same measurement syndrome $\boldsymbol{\chi}(\mathbf{z})$. For the decoding process, solving the CVP, we obtain a correction given by the teal arrow. In this case, we correctly identified the error, and the total displacement after the correction (orange point) brings back the storage to the center of phase space. (b) Same example scenario with the same error, but with noisy auxiliary states. This noise shifts the measurement result and the storage by $\boldsymbol{\chi}(\ell \boldsymbol{\delta}_\text{m})$ and $\boldsymbol{\delta}_\square$, respectively (purple arrows). This is represented by sampling translations from normal distributions with covariance $\Sigma_{\boldsymbol{\delta}_{\text{m}}} = \text{Cov}\{-\boldsymbol{\chi}(\ell \boldsymbol{\delta}_\text{m})\}$ and $\Sigma_{\boldsymbol{\delta}_{\square}} = \text{Cov}\{\boldsymbol{\delta}_\square\}$, where the notation $\text{Cov}\{\mathbf{y}\}$ means the covariance matrix associated with the vector of random variables $\mathbf{y}$. These are represented by the purple circles, with the same color indicating that they are correlated. Here, the circles are arbitrarily drawn for visual purposes. Looking at the bottom figure, since $\boldsymbol{\xi}_\square-\bar{\boldsymbol{\xi}}_\square$ is outside the Voronoi cell, we see that the correction now yields a logical error. Thus, the noise present on the auxiliaries can send correctable errors to uncorrectable ones. Importantly, the shifts $\boldsymbol{\chi}(\ell \boldsymbol{\delta}_\text{m})$ and $\boldsymbol{\delta}_\square$ are correlated, which is taken into account in the new decoder presented in \ref{['subsec: qec_noisy_aux']}.
• Figure 4: Performance comparison of the MED decoder \ref{['eq: med_cvp_decoder']} (orange dotted curve) with our COR-MED decoder \ref{['eq: solved_noise_corr_mld_decoder']} (teal dotted curve). We show the probability of having a logical error $p_L$ as a function of the noise variance $\sigma^2/2\pi$ in the system for the (a) Square GKP, (b) hexagonal GKP, (c) Tesseract GKP, and (d) D$_4$ GKP. For the D$_4$ GKP, we show the value of $\sigma^2/2\pi$ with light grey lines, which corresponds to a $\Delta_{(\text{dB})} \approx 11$dB squeezed GKP state. Each figure is simulated over a range of $\sigma^2/2\pi = [0.002, 0.010]$. We also show the variance $\zeta^*_\text{G}$ at which $p_L = 10^{-6}$ for each storage lattice.
• Figure 5: Steane-type QEC circuit for the square GKP code proposed measuring $\hat{\mathbf{h}}_\square$, where $\ell_\square = 2\sqrt{\pi}$. The main differences between this circuit and the one from \ref{['fig: square_GKP_steane_type_QEC_original']} are in the squeezing of the auxiliaries, the squeezing of the $\hat{U}_{T_j}$ operators, and the measurements, which are all made on the $\hat{q}$ quadratures.