\([[16,4,4]]\) symplectic-double code

\([[16,4,4]]\) symplectic-double code[1–3]

Description

Self-dual doubly-even CSS code obtained by concatenating the non-CSS \([[4,2,2]]\) code, defined by stabilizer generators \(XZZX,ZXXZ\), with the symplectic double of itself. Concretely, the symplectic double of the \([[4,2,2]]\) code yields a non-CSS-seeded \([[8,4,2]]\) code, and concatenating this \([[8,4,2]]\) code with the \([[4,2,2]]\) code \(C_4\) along the \(ZX\)-duality \(\tau\) yields the \([[16,4,4]]\) concatenated symplectic double (CSD) code. This code is isomorphic to the \(L=2\), \(C_4\)-based many-hypercubes (MHC) code [1,3].

A stabilizer tableau for the code is [3] \begin{align} \begin{smallmatrix} Z & Z & Z & Z & I & I & I & I & I & I & I & I & I & I & I & I \\ I & I & I & I & Z & Z & Z & Z & I & I & I & I & I & I & I & I \\ I & I & I & I & I & I & I & I & Z & Z & Z & Z & I & I & I & I \\ I & I & I & I & I & I & I & I & I & I & I & I & Z & Z & Z & Z \\ Z & Z & I & I & Z & I & Z & I & Z & I & Z & I & Z & Z & I & I \\ Z & I & Z & I & Z & Z & I & I & Z & Z & I & I & Z & I & Z & I \\ X & X & X & X & I & I & I & I & I & I & I & I & I & I & I & I \\ I & I & I & I & X & X & X & X & I & I & I & I & I & I & I & I \\ I & I & I & I & I & I & I & I & X & X & X & X & I & I & I & I \\ I & I & I & I & I & I & I & I & I & I & I & I & X & X & X & X \\ X & X & I & I & X & I & X & I & X & I & X & I & X & X & I & I \\ X & I & X & I & X & X & I & I & X & X & I & I & X & I & X & I \end{smallmatrix}~. \tag*{(1)}\end{align} Rows 1–4 are the \(Z\)-type inner stabilizers of each \([[4,2,2]]\) block; rows 5–6 are the \(Z\)-type outer stabilizers inherited from the double cover; rows 7–10 are the \(X\)-type inner stabilizers; rows 11–12 are the \(X\)-type outer stabilizers.

Protection

Detects errors on up to 3 qubits and corrects errors on 1 qubit.

Encoding

A fault-tolerant, bare-ancilla state-preparation procedure for the logical \(|\overline{0}\rangle\) and \(|\overline{+}\rangle\) states is given in [3].

Transversal and Permutation-Based Gates

SWAP-transversal gates lifted from automorphisms of the seed \([[4,2,2]]\) code, together with the \(ZX\)-duality gates \(\overline{H}_\tau\) and \(\overline{S}_\tau\) inherited from the symplectic double, generate \(216\) unique logical gates and, with a global logical phase gate, the group \((A_8\times A_8)\rtimes(C_2\times C_2)\subset \mathrm{Sp}_8(\mathbb{F}_2)\); a targeted logical \(S\) gate on any logical qubit instead generates the full symplectic group \(\mathrm{Sp}_8(\mathbb{F}_2)\) [3].

Gates

Any logical Clifford circuit on the four logical qubits can be compiled using the SWAP-transversal gateset together with at most a small number of logical phase-gate injections via state teleportation [3].

Fault Tolerance

The transversal phase gate on the outer \([[8,4,2]]\) symplectic double code requires performing logical global phase gates on each \([[4,2,2]]\) block to remain SWAP-transversal on the concatenated code; this can be implemented fault-tolerantly, but requires two-qubit gates [3][3; Appx. D.7].An automorphism gate that swaps logical qubits between different \([[4,2,2]]\) blocks requires additional \([[4,2,2]]\) ancilla blocks to implement fault-tolerantly [3; Appx. D.8].

Realizations

Neutral atom arrays: state initialization of the \([[16,4,4]]\) doubly concatenated code (a.k.a., the level-two many-hypercube code) on a device by Infleqtion [4].

Cousins

\([[4,2,2]]\) Four-qubit code— The \([[16,4,4]]\) CSD code is obtained by concatenating the \([[4,2,2]]\) code with the symplectic double of the \([[4,2,2]]\) code, and is isomorphic to the \(L=2\), \(C_4\)-based many-hypercubes code [3].
\([[6,4,2]]\) error-detecting code— The \([[16,4,4]]\) code is a \(C_4\)-based (\([[4,2,2]]\)-based) instance of the many-hypercubes code family, complementing Goto’s original \([[6,4,2]]\)-based many-hypercube construction [1,3].
Concatenated qubit code— The \([[16,4,4]]\) code is obtained by concatenating the \([[4,2,2]]\) code with the symplectic double of the \([[4,2,2]]\) code along a \(ZX\)-duality [3].

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 Hamiltonian-based QECC Quantum

Self-dual CSS codeStabilizer Hamiltonian-based Qubit QECC Quantum

Parents

Self-dual CSS code

Bivariate bicycle (BB) codeQLDPC Qubit Generalized homological-product Lattice stabilizer CSS Stabilizer Hamiltonian-based QECC Quantum

The \([[16,4,4]]\) code is among the weight-eight self-dual bivariate bicycle (BB) codes on twisted tori enumerated in [2].

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