Transversal dimension jump for product qLDPC codes

TL;DR

The paper introduces transversal dimension jump for lifted-product qLDPC codes, enabling universal fault-tolerant quantum computation by bridging 3D LP codes with their 2D component codes via homomorphic, one-way transversal CNOTs and teleportation-based code switching. It formalizes the homomorphic CNOT using chain maps and injectivity conditions on the induced homology maps, then demonstrates this mechanism across several code families: hypergraph product, univariate lifted-product, and multivariate bicycle/tricycle codes. Case studies show practical, high-rate 3D-2D code pairs with depth-2 cup-product CCZ gates, explicit finite-size examples with favorable distances and thresholds, and efficient magic-state preparation via a single CCZ layer with postselection, all under realistic, circuit-level noise assumptions. Collectively, these results offer a scalable, low-overhead path to universal quantum computation with qLDPC codes, reducing reliance on distillation and enabling broader hardware implementations and code-design options.

Abstract

We introduce transversal dimension jump, a code-switching protocol for lifted product (LP) quantum low-density parity-check (qLDPC) codes across different chain-complex dimensions, enabling universal fault-tolerant quantum computation with low overhead. The construction leverages the product structure of LP codes to implement one-way transversal CNOTs between a 3D code and its 2D component codes, enabling teleportation-based switching. Combined with constant-depth CCZ gates in 3D LP codes and low-overhead transversal Clifford gates in 2D LP codes, this yields universal, high-rate quantum logical computation with high thresholds and low space-time costs. Beyond asymptotic schemes, we identify explicit 3D-2D LP code pairs supporting cup-product CCZ gates, including bivariate tricycle-bicycle families such as the $[[81,3,5]]$-$[[54,2,6]]$ pair, where the 3D tricycle codes admit depth-2 CCZ, weight-6 stabilizers, and pseudo-thresholds $\gtrsim 0.4\%$. As a byproduct, we show that the 3D codes enable highly efficient magic-state preparation: a single round of stabilizer measurements followed by depth-2 CCZ and postselection produces states with error $<10^{-9}$ and success probability $\sim 35\%$. Our results establish a native integration of qLDPC codes with complementary transversal gates-covering nearly all practically relevant families known so far-and open a broad design space for scalable, low-overhead universal quantum computation.

Figures (2)

• Figure 1: Transversal dimension jump. (a) A 3D LP code $\mathcal{Q}_{\mathrm{ABC}}$, defined as the lifted product of base classical codes $\mathcal{C}_A$, $\mathcal{C}_B$, and $\mathcal{C}_C$, can be visualized on a cube. Any of its 2D component codes, such as $\mathcal{Q}_{\mathrm{AB}}$ (the product of two base codes), corresponds to a 2D slice of $\mathcal{Q}_{\mathrm{ABC}}$. A logical CNOT circuit can be realized via transversal physical CNOTs between $\mathcal{Q}_{\mathrm{AB}}$ and the slice of $\mathcal{Q}_{\mathrm{ABC}}$. To see the logical action, the logical $Z$ operators of the 2D codes, indicated by the string-like operators, are mapped through the CNOT circuit to the corresponding $Z$ operators of the 3D code on the associated slice. (b-c) Consequently, a subset of logical qubits of $\mathcal{Q}_{\mathrm{ABC}}$ can be teleported from (panel (b)) or to (panel (c)) one or more copies of the 2D codes in parallel, provided the CNOT circuits are also logically transversal (with one-to-one couplings between the logical qubits). The logical teleportation circuits are one-bit teleportation circuits that involve the transversal CNOTs, transversal logical initialization in the $X$ (resp. $Z$) basis, transversal measurements in the $Z$ (resp. $X$) basis --- denoted by $M_Z$ (resp. $M_X$) --- and corresponding Pauli $X$ (resp. $Z$) feedback corrections.
• Figure 2: Logical error rates of 3D codes and the transversal dimension jump circuit. (a) Memory logical error rate per logical qubit per code cycle for the BT codes in Table \ref{['table:codes']}, compared against standard 3D toric codes. (b) Logical error rate per logical qubit for the transversal teleportation circuit in Fig. \ref{['fig:overview']}(b) using the BB/BT code pairs in Table \ref{['table:codes']}. (c) Logical error rate per logical qubit for single-shot $|+\rangle$ state preparation using the BT codes, compared against 2D surface codes.

Theorems & Definitions (2)

• Lemma 1: Homomorphic CNOT
• Theorem 2: Transversal dimension jump