Continuous-Variable Quantum MacWilliams Identities

TL;DR

This work introduces continuous-variable weight distributions A(r) and B(r) and establishes a CV quantum MacWilliams identity that links them for any pair of trace-class operators. Using these tools, the authors derive a quantum analogue of the Cohn-Elkies sphere-packing bound and a CV Levenshtein bound, bounding code size K in terms of distance d and number of modes N under a displacement-noise model. They show that ideal GKP codes based on the E8 and Leech lattices achieve maximal distances under natural assumptions, highlighting a form of optimality for CV quantum codes in high-dimensional sphere-packings. The framework also connects weight distributions to entanglement fidelities and phase-space representations, offering a versatile set of methods for CV quantum error correction and potentially enabling SDP-based refinements in the future.

Abstract

We derive bounds on general quantum error correcting codes against the displacement noise channel. The bounds limit the distances attainable by codes and also apply in an approximate setting. Our main result is a quantum analogue of the classical Cohn-Elkies bound on sphere packing densities attainable in Euclidean space. We further derive a quantum version of Levenshtein's sphere packing bound and argue that Gottesman--Kitaev--Preskill (GKP) codes based on the $E_8$ and Leech lattices achieve optimal distances. The main technical tool is a continuous-variable version of the quantum MacWilliams identities, which we introduce. The identities relate a pair of weight distributions which can be obtained for any two trace-class operators. General properties of these weight distributions are discussed, along with several examples.

Figures (4)

• Figure 1: Weight distributions of example pure states. In the pure-state case the primary and the dual weight distributions coincide, i.e. $\mathbf{A} = \mathbf{B}$, and only the primary weight distribution is shown. Left: The weight distributions of a coherent states possess a single peak close to the origin, reflecting the localization of the state in phase space. The weight distribution is identical for all coherent states $|{\alpha} \rangle$. Right: The weight distributions of the first Fock state $|{1} \rangle$ and the third Fock state $|{3} \rangle$ are shown. In general, the weight distribution of the $n$-th Fock state oscillates $n+1$ times before decaying rapidly.
• Figure 2: Weight distributions of example error correcting codes encoding a logical qubit. Left: Weight distributions of a cat code with well-separated coherent states ($\lVert {\boldsymbol{\alpha}} \rVert = 4$). The distance between the coherent states is reflected in the dual distribution $\mathbf{B}$, which has a second peak at $2 \lVert {\boldsymbol{\alpha}} \rVert$, the length of a displacement required to shift a coherent state $|{\pm\alpha} \rangle$ onto the opposite coherent state $|{\mp\alpha} \rangle$. Right: Weight distributions of the single-mode square GKP code. Both primary and dual weight distributions are sums of integer multiples of delta functions, displayed as vertical lines. The primary weight distribution $\mathbf{A}$ is the distribution of lengths of vectors within the lattice $\mathcal{L}$ underlying the GKP stabilizer $\mathcal{S}$. For the square GKP code the this lattice is $\mathcal{L} = 2\sqrt{\pi}\mathbb{Z}^{2}$ and contains a single vector of length $0$, as well as $4$ vectors of length $2\sqrt{\pi}$ and $\sqrt{8 \pi}$, among longer ones. This is reflected in the heights of peaks of the primary distribution $\mathbf{A}$ at the respective lengths. The dual distribution $\mathbf{B}$ contains equivalent information about the dual lattice; $\mathcal{L}^{\perp} = \frac{1}{2}\mathcal{L}$ in the case of a square GKP code with $K=2$. As GKP states are unnormalizable, both distributions are also unnormalizable and have peaks at arbitrarily large arguments $r$. The distance of the square GKP code is $\sqrt{\pi}$, indicated by the smallest argument $r$ where $\mathbf{B}(r) > \mathbf{A}(r).$
• Figure 3: Left: Plot of the quotient $f_8(\sqrt{2}x ) / \hat{f}_8(d^2 x /\sqrt{2})$ over the unit interval for various values of $d$. The quotient visibly achieves a maximum value of $(2\pi)^4$ as long as $d \leq d_{8}^{\textnormal{(max)}} \approx 3.4286$. For higher values of $d$ the quotient achieves values exceeding $(2\pi)^4$. Right: Plot of the quotient $f_{24}(2 x) / \hat{f}_{24}(d^2 x /2)$ over the unit interval for various values of $d$. The quotient visibly achieves a maximum value of $(2\pi)^{12}$ as long as $d \leq d_{24}^{\textnormal{(max)}} \approx 4.9193$. For higher values of $d$ the quotient achieves values exceeding $(2\pi)^{12}$.
• Figure 4: Main steps in lower bounding the convolution product $g * \chi_{j_{N}}(y)$. a) the product is obtained given by integrating $g$ over the displaced Ball $B_{j_{N}}(y\hat{e}_1)$. As $g$ itself has support only within the radius $j_{N}$ Ball around the origin the total integration region is the intersection of two Balls (light gray). b) the integral of $g$ over $\mathbb{R}^{2N}$ is $1$, hence we can instead consider the integral over the complementary crescent shaped region (dark gray). c) to obtain an upper bound on the integral over the crescent shaped region we add some extra region on the top and bottom and then integrate an upper bound on $g$ over the enlarged region (dark gray). Some angles and points from the main text are also shown. d) the enlarged region can be deformed to the product of a $2N-1$-ball and the interval $[0, y]$ as it has constant width $y$ in direction $\hat{e}_{1}$.

Theorems & Definitions (13)

• Theorem 1: Quantum Cohn-Elkies bound
• Theorem 2: Quantum Levenshtein bound
• Theorem 3: Optimality of $E_8$ and Leech GKP codes
• Theorem 4: Continuous-variable quantum MacWilliams identity
• Definition 1: Approximate quantum error detection code
• Lemma 1
• proof
• Theorem 4: Quantum Cohn-Elkies bound
• Lemma 2
• Theorem 4: Quantum Levenshtein bound