Low Overhead Universal Quantum Computation with Triorthogonal Codes

TL;DR

This work develops two resource-efficient routes to universal fault-tolerant quantum computation using triorthogonal codes: a CZ-transversal Hadamard gate and a transversal code-switching scheme to pair a triorthogonal code with a symmetric CSS code enabling transversal Clifford and non-Clifford operations via state teleportation. Both approaches are designed to integrate with Steane error correction and are illustrated with the $[[15,1,3]]$ code, where gate overhead is shown to be significantly reduced relative to existing methods. The results highlight how combining distinct code structures can realize low-overhead universal computation and may inform scalable architectures and alternative fault-tolerant frameworks.

Abstract

We study the use of triorthogonal codes for universal fault-tolerant quantum computation and propose two methods to circumvent the Eastin-Knill theorem, which prohibits any single quantum error-correcting code from supporting both universality and a transversal gate set. We show that our methods reduce the resource overhead compared with existing fault-tolerant protocols. We develop a simple fault-tolerant implementation of the logical Hadamard gate for triorthogonal codes by exploiting the fact that they have transversal controlled-Z (CZ) gates, resulting in a circuit with reduced overhead. We also introduce a procedure for generating a symmetric Calderbank-Shor-Steane code paired with a triorthogonal code, which allows CNOT and CZ gate transversality across the pair of codes. In addition, we present logical state teleportation circuits that transfer encoded states between the two codes, allowing all logical operations to be performed transversally. Our methods can be integrated into the Steane error correction framework without incurring additional resource cost. Finally, using the 15-qubit code as an example, we demonstrate that our protocols significantly reduce the gate overhead compared with other existing methods. These results highlight the potential of combining distinct code structures to achieve low-overhead, universal fault-tolerant quantum computation.

Figures (8)

• Figure 1: The Steane error correction syndrome extraction circuit. Two ancilla blocks (the second and sixth lines) encoded in logical $\lvert+\rangle_L$ and $\lvert0\rangle_L$ states to load the physical errors from data qubits. Error syndromes can be obtained by measuring ancillas. Additional six ancilla blocks are used to prevent high-weight physical errors on ancilla qubits from propagating to data qubits.
• Figure 2: The triorthogonal code logical Hadamard gate circuit with an ancilla block. The logical CZ gate can be applied transversally since $\mathcal{Q}^T$ satisfies \ref{['Eq:CZsuffcond1']}.
• Figure 3: The state teleportation circuits between $\mathcal{Q}^T$ and $\mathcal{Q}^{\rm Sym}$. (a) The state teleportation from $\mathcal{Q}^T$ to $\mathcal{Q}^{\rm Sym}$. (b) Since the CNOT gate from $\mathcal{Q}^{\rm Sym}$ to $\mathcal{Q}^T$ is not transversal, we can use CZ gate instead. The teleportation circuit is similar to Fig.\ref{['fig:CZHadamard']}, but with additional logical Hadamard gate at beginning, and this logical Hadamard gate can be applied transversally on $\mathcal{Q}^{\rm Sym}$.
• Figure 4: Integration of T-triorthogonal logical Hadamard gate circuit to Steane syndrome extraction. The ancilla blocks are prepared in logical $\lvert+\rangle_L$ states and go through a verification procedure to eliminate high-weight errors.
• Figure 5: The state teleportation circuit from $\mathcal{Q}^T$ to $\mathcal{Q}^{\rm Sym}$, embedded with Steane syndrome extraction procedure. After the circuit is executed, the logical state is re-encoded in $\mathcal{Q}^{\rm Sym}$, and one round of error correction has been completed.

Theorems & Definitions (8)

• Definition 1
• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Example 1