\([[9,1,3]]\) Surface-17 code

\([[9,1,3]]\) Surface-17 code[1]

Alternative names: \([[9,1,3]]\) rotated surface code.

Description

A \([[9,1,3]]\) rotated surface code named for the sum of its 9 data qubits and 8 syndrome qubits. It uses the smallest number of qubits to perform fault-tolerant error correction on a surface code with parallel syndrome extraction.

Protection

Independent correction of single-qubit \(X\) and \(Z\) errors. Correction for some two-qubit \(X\) and \(Z\) errors. Admits pseudo-thresholds of \(\approx 10^{-4}\) under depolarizing noise.

Encoding

Measurement-free fault-tolerant logical zero state preparation in nearest-neighbor qubit connectivity [2].Fault-tolerant logical zero and logical plus state preparation in all-to-all and 2D grid connectivity with flag qubits [3].

Transversal Gates

Pauli gates, CNOT gate, and \(H\) gate (with relabeling).

Decoding

Lookup table [1].Syndrome extraction using Toffoli gates and qubit reset [4].

Fault Tolerance

Measurement-free fault-tolerant logical zero state preparation in nearest-neighbor qubit connectivity [2].Fault-tolerant logical zero and logical plus state preparation in all-to-all and 2D grid connectivity with flag qubits [3].

Realizations

Implemented at ETH Zurich by the Wallraff group [5] and on the Zuchongzhi 2.1 superconducting quantum processor [6]. Both experimental error rates are above the pseudo-threshold for this code relative to a single qubit; see Physics viewpoint for a summary [7]. Magic state have been created on the latter processor [8]. Lattice surgery on the surface-17 code has been realized by splitting the code into two repetition codes by the Wallraff group [9]. The device noise can be used to develop a decoder without relying a theoretical noise model [10].

Notes

Subject of various numerical studies examining the system for noises and architectures specific to trapped ions [1,11,12] and superconducting circuits [13–15]

Cousins

\([[9,1,3]]\) Shor code— Both Shor's code and surface-17 are \([[9,1,3]]\) codes, but they are distinct (e.g., they have different quantum weight enumerators).
\([[30,8,3]]\) Bring code— Bring's code and the surface-17 code have been compared numerically [16].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Homological code

Kitaev surface codeCDSC Twist-defect surface Lattice stabilizer Generalized homological-product QLDPC CSS Stabilizer Qubit Abelian topological Topological Hamiltonian-based QECC Quantum

Rotated surface codeQubit CSS Generalized homological-product QLDPC Stabilizer Hamiltonian-based Concatenated quantum Single-shot QECC Quantum

Parents

Rotated surface code

Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum