Constant depth fault-tolerant Clifford circuits for multi-qubit large block codes

Abstract

Fault-tolerant quantum computation (FTQC) schemes using large block codes that encode <i>k</i> > 1 qubits in <i>n</i> physical qubits can potentially reduce the resource overhead to a great extent because of their high encoding rate. However, the fault-tolerant (FT) logical operations for the encoded qubits are difficult to find and implement, which usually takes not only a very large resource overhead but also long <i>in situ</i> computation time. In this paper, we focus on Calderbank–Shor–Steane [[<i>n, k, d</i>]] (CSS) codes and their logical FT Clifford circuits. We show that the depth of an arbitrary logical Clifford circuit can be implemented fault-tolerantly in O(1) steps <i>in situ</i> via either Knill or Steane syndrome measurement circuit, with the qualified ancilla states efficiently prepared. Particularly, for those codes satisfying <i>k/n</i> ~ Θ(1), the resource scaling for Clifford circuits implementation on the logical level can be the same as on the physical level up to a constant, which is independent of code distance d. With a suitable pipeline to produce ancilla states, our scheme requires only a modest resource cost in physical qubits, physical gates, and computation time for very large scale FTQC.