Quantum Lego Power-up: Designing Transversal Gates with Tensor Networks

TL;DR

This paper introduces a tensor-network framework called the quantum lego formalism to systematically design quantum error-correcting codes with useful transversal gates, including non-Clifford gates, enhancing fault-tolerant quantum computation.

Contribution

It develops a novel tensor-network approach that enables the construction of quantum codes supporting addressable transversal gates, including non-Clifford types, which are difficult with traditional methods.

Findings

Constructed new code families supporting transversal $T$, $CCZ$, $SH$, and $K_3$ gates.

Built holographic and fractal-like codes with addressable transversal gates.

Lowered overhead for universal fault-tolerant quantum computation using these codes.

Abstract

Transversal gates are the simplest form of fault-tolerant gates and are relatively easy to implement in practice. Yet designing codes that support useful transversal operations -- especially non-Clifford or addressable gates -- remains difficult within the stabilizer formalism or CSS constructions alone. We show that these limitations can be overcome using tensor-network frameworks such as the quantum lego formalism, where transversal gates naturally appear as global or localized symmetries. Within the quantum lego formalism, small codes carrying desirable symmetries can be "glued" into larger ones, with operator-flow rules guiding how logical symmetries are preserved. This approach enables the systematic construction of codes with addressable transversal single- and multi-qubit gates targeting specific logical qubits regardless of whether the gate is Clifford or not. As a proof of principle, we build new finite-rate code families that support strongly transversal $T$, $CCZ$, $SH$, and Gottesman's $K_3$ gates, structures that are challenging to realize with conventional methods. We further construct holographic and fractal-like codes that admit addressable transversal inter-, meso-, and intra-block $T$, $CS$, and $C^\ell Z$ gates. As a corollary, we demonstrate that the heterogeneous holographic Steane-Reed-Muller black hole code also supports fully addressable transversal inter- and intra-block $CZ$ gates, significantly lowering the overhead for universal fault-tolerant computation.