Complementary 3D color codes for transversal quantum logic

arXiv (Cornell University)

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.

Authors 6

1. Friederike Butt Aachen Institute for Quantum Information

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information , RWTH Aachen University , Aachen , Germany

   Institute for Theoretical Nanoelectronics (PGI-2) , Forschungszentrum Jülich , Jülich , Germany

2. Luis Colmenárez Aachen Institute for Quantum Information

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information , RWTH Aachen University , Aachen , Germany

   Institute for Theoretical Nanoelectronics (PGI-2) , Forschungszentrum Jülich , Jülich , Germany

3. Erik Weilandt

   Technical University of Munich

   Affiliation as printed

   Technische Universität München , Munich , Germany

4. Tom Peham

   Technical University of Munich

   Affiliation as printed

   Technische Universität München , Munich , Germany

5. Robert Wille

   Technical University of Munich

   Affiliation as printed

   Technische Universität München , Munich , Germany

6. Markus Müller Aachen Institute for Quantum Information

   Forschungszentrum Jülich · RWTH Aachen University

   Affiliation as printed

   Institute for Quantum Information , RWTH Aachen University , Aachen , Germany

   Institute for Theoretical Nanoelectronics (PGI-2) , Forschungszentrum Jülich , Jülich , Germany