Achievable rates for concatenated square Gottesman-Kitaev-Preskill codes

TL;DR

This work shows that concatenating square GKP codes with quantum polar codes yields explicit, scalable schemes to achieve the coherent-information rate of the Gaussian displacement noise channel for all noise strengths, by exploiting analog information from GKP syndrome. It also establishes the capacity of bosonic pure-loss and amplification channels via a random-coding existence proof based on averaging over self-orthogonal GF$(d^2)$ codes, using a δ-good lattice framework to approximate good sphere packing. The combination provides both constructive, efficient encoders/decoders for displacement noise and an existence guarantee for capacity-achieving sequences on loss/amplification channels, highlighting the deep link between lattice structure and stabilizer codes in a concatenated GKP setting. Taken together, the results offer new, explicit methods for good GKP lattices and broaden the practical potential of GKP-based quantum communication.

Abstract

The Gottesman-Kitaev-Preskill (GKP) codes are known to achieve optimal rates under displacement noise and pure loss channels, which establishes theoretical foundations for its optimality. However, such optimal rates are only known to be achieved at a discrete set of noise strength with the current self-dual symplectic lattice construction. In this work, we develop a new coding strategy using concatenated continuous variable - discrete variable encodings to go beyond past results and establish GKP's optimal rate over all noise strengths. In particular, for displacement noise, the rate is obtained through a constructive approach by concatenating GKP codes with a quantum polar code and analog decoding. For pure loss channel, we prove the existence of capacity-achieving GKP codes through a random coding approach. These results highlight the capability of concatenation-based GKP codes and provides new methods for constructing good GKP lattices.

Figures (10)

• Figure 1: (a) Depiction of the GKP lattice for a square GKP qudit of $d$ levels. Displacements in the stabilizer group lie in the square lattice spaced by $\sqrt{2\pi d}$ (black dots) and the displacements giving logical operations are in the square lattice spaced apart by $\sqrt{2\pi/d}$ (pink dots). (b) Figure depicting relation between the obtained syndrome and logical errors for the case of $d=3$. Obtaining a particular syndrome $s_1$ implies a possible set of errors (see top right line graph) for the shift along $\hat{q}\sqrt{\frac{3}{2\pi}}$. These on correction by the smallest possible displacement for getting back to the logical codespace may result in logical errors (red arrows) or result in no logical error at all (green arrows). (c) By noting that the value of $p_1(u,s_1)$ is only a function of $u+s_1$, we compare these distributions for different values of $d$ for the value of $\sigma=0.5$. In our convention, $(u+s_1)/d$ will always lie in $[-1/2,1/2)$ and we can see that over this normalized range, there is a strong concentration near $0$ for $p_1(u,s_1)$ as $d$ is increased. Further, we plot the difference $p_1(u,s_1)-p_{\mathrm{lim}}(u,s_1)$ (which is also purely a function of $(u+s_1)/d$) and note that increasing the value of $d$ shows a clear exponential suppression in this difference.
• Figure 2: (a) Achievable rates for the square GKP qudit with analog information ($I^{\mathrm{sq}}_{d,\mathrm{analog}}$) and without analog information ($I^{\mathrm{sq}}_{d,\text{no analog}}$) plotted in solid line and dashed lines respectively for different values of $d$. The dashed line uses the distribution $p(u)$ obtained by averaging over the syndrome hence resulting in not using any analog information. (b) A rough depiction of what the concentration of $p_1(u,s_1)$ looks like in the limit of large $d$. While there is contribution from terms that are centered around $\pm l$ for $l=1,2\dots$ (see plotted Gaussians with dashed lines), their contribution is exponentially small in magnitude. Additionally, the spread of $p_{\mathrm{lim}}(u,s_1)$ being $\mathcal{O}(\sigma/\sqrt{d})$ over the normalized variable $(u-s_1)/d$ results in the size of typical error set growing much smaller than the actual dimension of Hilbert space. (c) Numerical evaluation of $I^{\mathrm{sq}}_{d,\mathrm{analog}}$ (for values of $d=2$ to $d=17$ shown in solid lines) showing a clear convergence to the coherent information of the Gaussian displacement noise channel (dotted line). Notably for finite and reasonably small $d$ ($<10$), we already see the value of $I^{\mathrm{sq}}_{d,\mathrm{analog}}$ get numerically close to coherent information while also crossing the value of rates achieved by rescaled self-dual lattices (dashed-dotted line).
• Figure 3: A heuristic circuit diagram explaining the structure of the concatenation. Here we treat each single mode square GKP as its own qudit which undergoes an encoding operation $V$ which creates the codewords for the concatenated outer polar code. The operations in $V$ are logical qudit operations (in this case purely composed of CSUM gates in the case of the polar code). The $K$ qudits are used for encoding information and the $N-K$ qudits are frozen to some chosen state. The logical noise operation $\mathcal{N}_{\mathrm{logical}}$ acts on all the encoded qudits and also outputs classical information in the form of the analog GKP syndrome of each individual mode. Following this the decoding operation proceeds, which involves inverting the encoding operation and then measuring the previously frozen $N-K$ qudits which gives the syndrome information of the outer polar code. This along with all the individual GKP syndromes is then used in a classical decoder (the successive cancellation decoder) giving a correction to be applied on the $K$ qudits carrying information.
• Figure 4: Effects of channel polarization on increasing block length $N$. Here we simulate the values of $Z(W_1^{(i)})$ and $Z(W_2^{(i)})$ for displacement spread $\sigma=0.4$ using kernel with $\alpha=2$ (blue dots) and $\alpha=1$ (orange dots) for $d=5$. Information will be encoded in dits that have both $Z(W_1^{(i)})$ and $Z(W_2^{(i)})$ being small hence we plot $\max\{Z(W_1^{(i)}),Z(W_2^{(i)})\}$. As can be seen for small block lengths, not enough of the indices have both the values being small as can be seen for $N=512$, and as can be seen in increasing value of $N$ till finally reaching $N=4096$. Notably, the polarization phenomenon is observed to be quicker using the kernel of $\alpha=2$ compared to $\alpha=1$ as can be inferred by comparing the fraction of indices satisfying $\max\{Z(W_1^{(i)}),Z(W_2^{(i)})\}<\delta$ for some small enough $\delta$.
• Figure 5: Performance of polar codes concatenated to square GKP qudits of $d=2$, $d=5$, and $d=7$. Here we find an explicit sequence of codes with increasing $N$ which satisfies $P_{e,1}\leq c_e N^{-\beta}$ and $P_{e,2}\leq c_e N^{-\beta}$ at a given value of $\sigma$ as the value of $N$ is increased. For the above plot we use $c_e=0.5$ and $\beta=-2/9$. The kernel choice for $d=2$ is $\alpha=1$, $d=5$ is $\alpha=2$, and for $d=7$ is $\alpha=3$. We find that we are clearly able to approach the solid line representing $I_{d,\mathrm{analog}}^{\mathrm{sq}}$ for different values of $d$ which in turn is able to exceed the rate achievable using rescaled self-dual lattices (dashed-dotted line).

Theorems & Definitions (22)

• Definition 5.1: $\delta$-good lattice
• Theorem 5.1
• Definition A.1: Polar transform
• Definition A.2: Polar code $(N,K,\mathcal{A},u_{\mathcal{A}_c})_d$
• Definition A.3: Discrete memoryless channel
• Theorem A.1: Classical polar coding
• proof
• Definition B.1: Quantum polar transform
• Theorem B.1: Quantum polar code achievable rate for Pauli noise
• proof