Structural Conditions for Native CCZ Magic-State Fountains in qLDPC Codes

TL;DR

This main theorem shows that if a CSS code family on $n$ qubits admits $\Omega(n^{1+\gamma})$ magic-friendly triples whose supports have bounded per-qubit participation, then there exists a constant-depth circuit of physical $\CCZ$ gates implementing $\Omega(n^{\gamma})$ logical $\CCZ$ gates in parallel while preserving distance up to a constant factor.

Abstract

Quantum low-density parity-check (qLDPC) codes promise constant-rate, linear-distance families with bounded-weight checks, and recent work has realized transversal or constant-depth non-Clifford gates on various (often non-LDPC) codes. However, no explicit \emph{qubit} qLDPC family is known that simultaneously has constant rate, linear distance, bounded stabilizer weight, and a native \emph{magic-state fountain} that prepares many non-Clifford resource states in constant depth. We take a structural approach and identify coding-theoretic conditions under which a CSS qLDPC family necessarily supports a constant-depth $\CCZ$ magic-state fountain. The key ingredients are: (i) an algebraic notion of \emph{magic-friendly triples} of $X$-type logical operators, defined by pairwise orthogonality and a triple-overlap form controlling diagonal $\CCZ$ phases, and (ii) a 3-uniform hypergraph model of physical $\CCZ$ circuits combined with a packing lemma that turns large collections of such triples with bounded overlaps into bounded-degree hypergraphs. Our main theorem shows that if a CSS code family on $n$ qubits admits $Ω(n^{1+γ})$ magic-friendly triples whose supports have bounded per-qubit participation, then there exists a constant-depth circuit of physical $\CCZ$ gates implementing $Ω(n^γ)$ logical $\CCZ$ gates in parallel while preserving distance up to a constant factor. For asymptotically good qLDPC families such as quantum Tanner codes, this reduces the existence of a native $\CCZ$ magic-state fountain to a concrete combinatorial problem about counting and distributing magic-friendly triples in the logical $X$ space.

Figures (3)

• Figure 1: Toy 3-uniform hypergraph with three hyperedges (triangles), distinguished by three line styles (solid, dashed, dotted). Gates in each style class can be executed in one layer of disjoint $\mathsf{CCZ}$ gates.
• Figure 2: Schematic for the packing lemma. Qubits $1,\dots,9$ are shown as nodes on a line. Each candidate support set $S_t$ (grey dots in a row above) marks the qubits in the union of the supports of one triple $(x_t,y_t,z_t)$ of logical $X$ operators. The greedy algorithm selects a subcollection of supports $T_t$ (black dots in rows below) so that no two selected supports share a qubit, yielding bounded per-qubit participation.
• Figure 3: Supports of a magic-friendly triple $(x,y,z)$ on four qubits for $x=(0,1,1,1)$, $y=(1,0,1,1)$, $z=(1,1,0,1)$. Solid, dashed, and dotted rings mark the qubits in the supports of $x$, $y$, and $z$, respectively. Qubit 4 lies in all three supports and is the unique triple overlap.

Theorems & Definitions (13)

• Definition 1: Magic-friendly triple
• Remark 1: Logical $Z$ corrections
• Definition 2: 3-uniform hypergraph and degree
• Lemma 1: Edge coloring and constant depth
• Lemma 2: Packing lemma
• Theorem 1: Native constant-depth $\mathsf{CCZ}$ from many magic-friendly triples
• Lemma 3
• proof
• proof : Proof of Lemma \ref{['lem:coloring-main']}
• proof : Proof of Lemma \ref{['lem:packing-main']}