\([[16,4,4]]\) twisted color code

\([[16,4,4]]\) twisted color code[1,2]

Alternative Names: \([[16,4,4]]\) symplectic-double code.

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 [3,4].

A stabilizer tableau for the code is [4] \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.

Geometrically the code is the 4.8.8 color code on the smallest torus, shown in Fig. I. Fattening each vertex of the \(2\times 2\) square lattice into a square yields four square faces and four octagonal faces, the latter two-colored in checkerboard fashion, for eight faces on 16 qubits and 24 edges. The torus is twisted, in that one pair of opposite boundaries is glued only after a shift by half a period; gluing them straight yields a different code.

Figure I: The eight faces of the 4.8.8 color code on the smallest torus, four squares and four octagons, each carrying a \(X\)-type and a \(Z\)-type generator on its qubits. Shaded circles are the 16 physical qubits; a white circle is a periodic image of the shaded qubit carrying the same letter. The two dashed arrows are the boundary identifications, the second of which glues top to bottom only after a shift by half a period.

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 [4].

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, give \(216\) unique logical gates and, with a global logical phase gate, the group \((A_8\times A_8)\rtimes(C_2\times C_2)\subset Sp(8,\mathbb{F}_2)\) [4]. Only one of the \(20\) \(ZX\)-dualities admits a phase-type gate, and the permissible qubit permutations \(C_2\times S_4\) act non-faithfully with image \(D_6\). Together with all two-qubit entangling gates, they generate the logical orthogonal subgroup \(\Omega^{+}(8,\mathbb{F}_2)\) [5]. Adjoining a targeted logical \(S\) on a single logical qubit yields all of \(Sp(8,\mathbb{F}_2)\) [4,6].’

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 [4].

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 [4][4; 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 [4; 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 [7].

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 [4].
\([[6,4,2]]\) 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 [3,4].
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 [4].
\([[16,6,4]]\) Tesseract color code— The \([[16,4,4]]\) code is obtained from the tesseract color code by promoting one \(X\)-type and one \(Z\)-type weight-four logical operator to stabilizers [1,2], equivalently by fixing the weight-four gauge operators of the \([[16,4,2,4]]\) tesseract subsystem code [8].
\([[16,4,4]]\) biplane code— The \([[16,4,4]]\) biplane code and the \([[16,4,4]]\) symplectic-double code are the two self-dual CSS gauge fixings of the \([[16,4,2,4]]\) tesseract subsystem code [8], obtained by promoting the weight-six and the weight-four gauge operators, respectively.

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

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Color codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

2D color codeTwist-defect color QLDPC Qubit Generalized homological-product Lattice stabilizer CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

Square-octagon (4.8.8) color code

Parents

Square-octagon (4.8.8) color code

The \([[16,4,4]]\) code is the 4.8.8 color code on the smallest torus, whose two pairs of opposite boundaries are glued straight and with a half-period shift, respectively.

Self-dual CSS codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

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 [9].

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