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

TL;DR

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, and significantly lowering the overhead for universal fault-tolerant computation.

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.

Figures (21)

• Figure 1: (a) A tensor as a quantum lego block. A tensor can represent a state or a map depending on how the legs are assigned. In-going arrows mark input/logical legs while out-going arrows mark output/physical legs. (b) Unitary symmetry of a tensor. (c) Symmetries of the tensor network is generated by operator matching.
• Figure 2: (a,b) Unitary symmetry of a tensor $\mathbf{V}$ (purple) can manifest as different types of multi-qubit FT gates based on the different groupings of the tensor legs. (c) $CZ$s acting on pairs of qubits connected by the dotted lines performs a locally transversal logical $CZ$ gate, but the $\mathbf{V}$ encoding tensor also admits unitary symmetry when $CZ$s act on all vertices of any colored cubes. (d) As an explicit example of (b), a unitary symmetry of a single Steane encoding tensor is $CZ^{\otimes 4}$. Taking just the purple (P) leg as logical produces the Steane code whereas taking both purple and green (P+G) produces a $[[6,2,2]]$ code with intrablock transversal $CZ$. The green leg can be chosen to be the physical qubit on any of the 3 vertices of the triangle.
• Figure 3: $[[14,2,2]]$ codes obtained from the encoding tensor of the punctured $[[15,1,3]]$ QRM encoding tensor admit addressable $CZ$ gates. Weight 8 X and Z checks act on the vertices of the plaquettes with the same color. Additional Z checks connect the two layers, which are not shown in the figure.
• Figure 4: Transversal $S$ and $H$ gates emerge from unitary symmetries that are not unitary product such as $CZ$ and $SWAP$.
• Figure 5: Local deformation followed by gluing is equivalent to projecting the relevant sites to unitary-deformed Bell states $U\otimes I|\Phi^+\rangle$ represented by the green edges.

Theorems & Definitions (68)

• Definition 2.1
• Lemma 2.2
• Proposition 2.3
• Definition 2.4
• Definition 2.5: Cleanability
• Lemma 2.6
• Proposition 2.7
• proof
• Corollary 2.8
• Proposition 2.9