Complementary 3D color codes for transversal quantum logic

• URL: http://arxiv.org/abs/2607.05107v1
• Date: Mon, 06 Jul 2026 14:01:07 GMT
• Title: Complementary 3D color codes for transversal quantum logic
• Abstract summary: Transversal logical gates provide a direct route to fault-tolerant quantum computation, but the Eastin-Knill theorem forbids a universal gate set within a single quantum error-correcting code.<n>We propose a hybrid architecture based on the tetrahedral three-dimensional color code and its Hadamard-bitsed counterpart, which we call the H-tetrahedral code.<n>We complete a universal gate set through a pieceably fault-tolerant round-leaved construction of a logical controlled-$Z$ gate between two H-tetrahedral codes.
• Score: 4.568256986294634
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Transversal logical gates provide a direct route to fault-tolerant quantum computation, but the Eastin-Knill theorem forbids a universal transversal gate set within a single quantum error-correcting code. We propose a hybrid architecture based on the tetrahedral three-dimensional color code and its Hadamard-transformed counterpart, which we call the H-tetrahedral code. The two encodings support complementary transversal non-Clifford operations. Combined with bitwise Hadamard transformations that switch between the two encodings and a one-way transversal logical CNOT from the tetrahedral code to the H-tetrahedral code, these operations realize an almost-universal transversal logical gate set that enables both the creation of entanglement and logical states with magic. We complete a universal gate set through a pieceably fault-tolerant round-robin construction of a logical controlled-$Z$ gate between two H-tetrahedral codes. This logical entangling gate is interleaved with reduced-overhead Steane-type syndrome extraction using logical two-dimensional color-code auxiliary qubits. Our construction provides a new route toward implementing classically hard-to-simulate quantum algorithms where magic and most entangling operations are transversal while the resource overhead is concentrated in a small number of non-transversal Clifford entangling operations.

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