Build three-qubit quantum codes enabling transversal CCZ gates
Building codes with transversal CCZ using projective geometry and SAT solvers
Information Theory
Summary
Quantum computers need special error-correcting codes to protect information and perform operations reliably. This paper shows how to create codes with three logical qubits that allow a key quantum gate called CCZ to be applied easily across all physical bits at once. The authors use mathematical objects from projective geometry and computer solvers to find codes with good error protection and this special property. They also prove that smaller codes with similar features don’t exist below a certain size, giving limits on how compact these codes can be.
What this means in practice
- •For quantum hardware developers: Design quantum memory and processors supporting efficient fault-tolerant three-qubit CCZ gates using codes constructed from projective geometry and SAT solver techniques.
- •For quantum compiler engineers: Implement logical gate sets including transversal CCZ operations in error-corrected quantum circuits enabled by these newly constructed three-qubit codes.
Authors
Bohan Lu, Kenneth R. Brown
Abstract
We construct CSS codes with three logical qubits whose logical CCZ gate is implemented by transversal physical $T/T^{\dagger}$ gates and whose distance is at least $3$. These codes start with $8$-divisible $X$-stabilizers defined by projective geometry, which guarantees $d_Z\geq 3$. A SAT solver then finds three $X$-logical operators compatible with the transversal CCZ. We build codes at thirteen block lengths from $n=48$ to $n=496$, with $d_Z=3$ or $4$, and illustrate the method with the $[[48,3,3]]$ code $Q_{48}$. Additionally, we prove that no CSS code with three logical qubits, $Z$-distance at least $3$, and a quasi-transversal CCZ gate (transversal $T$ with a diagonal Clifford correction) exists below $n=39$. The $[[47,3,3]]$ code of Jacinto et al. serves as an upper bound, leaving $39\leq n\leq46$ open.