Letting the tiger out of its cage: bosonic coding without concatenation

TL;DR

This work introduces tiger codes, a broad class of intrinsically multimode bosonic quantum error-correcting codes defined by integer matrices $G$ and $H$ that satisfy a CSS-type constraint $HG^T=0$. Codewords are projected coherent states whose supports are constrained by $H$ and whose stabilizers (Z-type) and dissipators (X-type) define a joint, continuously-rotated codespace linked to GKZ hypergeometric functions. The framework unifies many existing bosonic encodings (cat, pair-cat, binomial, dual-rail, etc.) and yields new constructions, including tiger and liger surface codes with increasing $X$- and $Z$-distances and topological protection, as well as finite-support variants. Dephasing is shown to be exponentially suppressed with energy density for infinite-support codes, with $d_Z$ acting as a key geometric distance between codeword constellations, and loss errors are detectable and correctable up to $d_X-1$ losses under Knill-Laflamme conditions. The results provide a practical, experimentally accessible route to multimode bosonic QEC and point toward topological bosonic codes and GKZ-driven code design avenues.

Abstract

Continuous-variable cat codes are encodings into a single photonic or phononic mode that offer a promising avenue for hardware-efficient fault-tolerant quantum computation. Protecting information in a cat code requires measuring the mode's occupation number modulo two, but this can be relaxed to a linear occupation-number constraint using the alternative two-mode pair-cat encoding. We construct multimode codes with similar linear constraints using any two integer matrices satisfying a CSS-like homological condition of a quantum rotor code. Just like the pair-cat code, syndrome extraction can be performed in tandem with stabilizing dissipation using current superconducting-circuit designs. The framework includes codes with various finite- or infinite-dimensional codespaces, and codes with finite or infinite Fock-state support. It encompasses two-component cat, pair-cat, dual-rail, two-mode binomial, various bosonic repetition codes, and aspects of chi-squared encodings while also yielding codes from homological products, lattices, generalized coherent states, and algebraic varieties. Among our examples are analogues of repetition codes, the Shor code, and a surface-like code that is not a concatenation of a known cat code with the qubit surface code. Codewords are coherent states projected into a Fock-state subspace defined by an integer matrix, and their overlaps are governed by Gelfand-Kapranov-Zelevinsky hypergeometric functions.

Figures (5)

• Figure 1: The tiger codewords of this work are superpositions of continuous sets of coherent-state "stripes" --- circles or, more generally, tori. In contrast, quantum spherical codewords are superpositions of a discrete set of coherent states. The use of continuous superpositions allows for linear Fock-state constraints to define the code, while discrete superpositions correspond to modular and, more generally, representation-theoretic constraints. Two-mode coherent states of values $\boldsymbol{\alpha}=\alpha(e^{i\phi_1},e^{i\phi_2})$ of fixed coordinate amplitude $\alpha$ but varying phases $\phi_{1}$ and $\phi_2$ form a torus, which is depicted either as itself or as a square with opposite boundaries identified. Panels (a) and (b) depict the subset of coherent states that form the logical codewords of a tiger code and a quantum spherical code, respectively. The logical $+$ and $-$ states are defined as equal-weight superpositions of coherent states located at the black and white points. Therefore, the $Z$-distance of tiger code is given by the minimum squared Euclidean distance between points belonging to the constellations $\mathcal{C}_+=\{\alpha(e^{i\phi}, e^{-i\phi})|\forall \phi \in \mathbb{T}\}$ and $\mathcal{C}_{-}=\{\alpha(e^{i\phi}, e^{-i(\phi+\pi)})|\forall \phi \in \mathbb{T}\}$. The tiger code example is the pair-cat codewords defined by matrices $G =22$ and $H =1-1$, while the quantum spherical code constellations are$\mathcal{C}_+=\{(\alpha,\alpha), (i\alpha ,- \alpha), (- \alpha, \alpha), (-i \alpha,- \alpha)\}$ and $\mathcal{C}_-=\{(\alpha ,-\alpha), (i\alpha, \alpha), (- \alpha,- \alpha), (-i \alpha,\alpha)\}$.
• Figure 2: (a) The structure of tiger surface code lattice and its stabilizers, where each white circle represents a physical mode, a pink square represents an $X$-check acting on the physical modes it connects, and a blue square represents a $Z$-check acting on the physical modes it connects. Each row contains $r$ physical modes, and there are $2m-1$ rows in total, forming a 2D lattice with $2mr-r$ physical modes. The undetectable $X$-error $\mathbf{\hat{a}}^{\dagger \mathbf{q}} \mathbf{\hat{a}}^{\mathbf{p}}$, with minimal 1-norm, and the logical $Z$ operator are string operators connecting the top-bottom and left-right boundaries, respectively. (b) The form of the $X$ and $Z$-checks. (c) The liger surface code with $d_X=2$ and $d_Z=4r$, encoding a logical qubit into a chain involving $3r$ physical modes.
• Figure 3: In panel (a), the vertical string operator in the light orange region represents a lowest-weight undetectable $X$-error $\mathbf{\hat{a}}^{\dagger \mathbf{q}} \mathbf{\hat{a}}^{ \mathbf{p}}$. Thus, the tiger surface code can correct loss errors $\mathbf{\hat{a}}^{\mathbf{v}}$ whose 1-norm is at least $\lfloor \frac{m-1}{2} \rfloor$. In panel (b), the diagonal string operator in the dark orange region represents the lowest-weight logical $X$ operator. It implies that tiger surface code can detect loss error $\mathbf{\hat{a}}^{\mathbf{p}}$ one any $(2m-2)$ modes. Panel (c) represents the error pattern that gives upper bound of the squared Euclidean distance between two codewords in Eq. \ref{['eq:z-distance']}, which corresponds to a "smeared" version of the defining logical $Z$ operator of tiger surface code. In this error pattern, each physical mode inside the green horizontal strips undergoes a phase-space rotation by an angle of $\pm \frac{\pi}{m}$. The physical modes inside the green vertical strip in the first column undergo configuration-space rotations by angles $\frac{-2(m-1)\pi}{m},\frac{2(m-2)\pi}{m},\cdots , \frac{(-1)^{m-1} 2\pi}{m}$, from top to bottom.
• Figure 4: Pictorial comparison between infinite-support pair-cat code and finite-support two-mode binomial code in the two-mode Fock space $(n_1,n_2)$. Panel (a) represents the Fock-state structure of the pair-cat code for different choices of occupation-number difference, $\Delta = n_2-n_1$. Each choice specifies a 1D semi-infinite lattice (dashed line with $\Delta$) extending in the upper-right direction in the two-mode Fock space, and the codewords are superpositions of the Fock states in the semi-infinite lattice. Inside each lattice, the black (white) circles are in the Fock-state support of codeword $\mathopen{}\mathclose{\left\vert {\overline{0}} \right\rangle_\Delta$ ($\mathopen{}\mathclose{\left\vert {\overline{1}}} \right\rangle_\Delta$). Since the 1D lattice with $\Delta$ involves infinite numbers of Fock states, the codewords are superpositions of Fock states involving infinitely many terms. Panel (b) represents the Fock-state structure of the two-mode binomial code for different choices of the total occupation number, $\Delta = n_1+n_2$. Each choice specifies a finite 1D lattice in the two-mode Fock space, and the codewords are superpositions of Fock states in the finite lattice. The finite lattices from left to right correspond to $\Delta=1,2,3,4$. Inside each lattice, the black (white) circles are in the Fock-state support of codeword $\mathopen{}\mathclose{\left\vert {\overline{0}} \right\rangle_\Delta$ ($\mathopen{}\mathclose{\left\vert {\overline{1}}} \right\rangle_\Delta$). The supports for both $\mathopen{}\mathclose{\left\vert {\overline{0}}} \right\rangle_\Delta$ and $\mathopen{}\mathclose{\left\vert {\overline{1}}} \right\rangle_\Delta$ are finite.
• Figure 5: Potential implementation of four-mode tiger codes is analogous to the realization of the pair-cat code in Ref. albert2019pair. A line with a cross-sign represents a Josephson junction ancilla, pink rounded rectangles represent low-Q cavities for entropy extraction, and the blue rounded rectangle at the top represents the readout cavity for $Z$-syndrome extraction.

Theorems & Definitions (1)

• Definition : tiger code