\([[16,4,4]]\) 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 that is the 4.8.8 color code on the smallest torus, twisted in that one boundary identification is shifted by half a period [2]. Equivalently, it is the concatenated symplectic double of the \([[4,2,2]]\) code, i.e., its inner \([[8,4,2]]\) symplectic double with each \(ZX\)-dual pair of qubits encoded into an outer \([[4,2,2]]\) block [3]. Automorphisms of the \([[4,2,2]]\) seed lift through this structure to logical gates realized by qubit permutations [3].

Fattening each vertex of the \(2\times 2\) square lattice into a square yields four square faces and four octagonal faces, shown in Fig. I. The octagons are two-colored in checkerboard fashion. This gives eight faces on 16 qubits and 24 edges. Gluing the shifted pair of boundaries straight instead 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 face carries an \(X\)-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.

The seed is the non-CSS presentation \(\langle XZZX, ZXXZ\rangle\) of the \([[4,2,2]]\) code, whose symplectic double is an \([[8,4,2]]\) CSS code on two copies of its four qubits [3]. Qubit \(i\) of the first copy and qubit \(i\) of the second are exchanged by the \(ZX\) duality of the double. Each such pair is encoded as the two logical qubits of one \([[4,2,2]]\) block with logical operators \(\overline{X}_1=XXII\), \(\overline{Z}_1=ZIZI\), \(\overline{X}_2=XIXI\), and \(\overline{Z}_2=ZZII\) [3]. The code is isomorphic to the \(L=2\) many-hypercubes (MHC) code built from the \([[4,2,2]]\) code [3,4]. Equivalently, it is two copies of the \([[4,2,2]]\) code concatenated with the \([[4,2,2]]\) code [1].

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 to 4 are the \(Z\)-type outer stabilizers of each \([[4,2,2]]\) block. Rows 5 and 6 are the \(Z\)-type inner stabilizers inherited from the symplectic double. Rows 7 to 10 are the \(X\)-type outer stabilizers. Rows 11 and 12 are the \(X\)-type inner 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, give \(216\) unique logical gates [3]. Adding a global logical phase gate gives the group \((A_8\times A_8)\rtimes(C_2\times C_2)\subset Sp(8,2)\) [3]. 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 \(\Omega^{+}(8,2)\) [5]. Adjoining a targeted logical \(S\) on a single logical qubit yields all of \(Sp(8,2)\) [3,6].

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 [3]. The injections are performed via state teleportation [3].

Fault Tolerance

Single-shot distance-three fault-tolerant error correction with 12 nonadaptive stabilizer measurements in six rounds, or with eight or ten adaptively chosen measurements in four rounds [1; Thm. 7].The transversal phase gate on the inner \([[8,4,2]]\) symplectic double code requires performing logical global phase gates on each \([[4,2,2]]\) block to remain SWAP-transversal on the concatenated code [3; Appx. D.7]. This can be implemented fault-tolerantly but requires two-qubit gates [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 \([[16,4,4]]\) twisted color code on a device by Infleqtion [7].

Cousins

\([[4,2,2]]\) Four-qubit code— The \([[16,4,4]]\) twisted color code is obtained by concatenating the inner symplectic double of the \([[4,2,2]]\) code with outer \([[4,2,2]]\) blocks [3]. It is isomorphic to the \(L=2\), \([[4,2,2]]\)-based many-hypercubes code [3].
\([[6,4,2]]\) error-detecting code— The \([[16,4,4]]\) code is the \(C_4\)-based (\([[4,2,2]]\)-based) analogue of the original many-hypercubes codes, which are built from the \([[6,4,2]]\) code instead [3,4].
\([[16,6,4]]\) 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, it is obtained 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 \([[16,4,4]]\) twisted color code are the two self-dual CSS gauge fixings of the \([[16,4,2,4]]\) tesseract subsystem code [8]. They are 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 codeQubit Stabilizer Hamiltonian-based QECC Quantum

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Color codeQubit CSS Stabilizer Hamiltonian-based 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 codeQubit CSS Stabilizer Hamiltonian-based QECC Quantum

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

The \([[16,4,4]]\) code is the weight-eight self-dual bivariate bicycle (BB) code with \(A=1+x+y+y^{-1}\) and \(B=\overline{A}\) on the twisted torus with lattice vectors \((0,4)\) and \((2,2)\) [9; Table I].

Concatenated qubit codeQubit Concatenated quantum QECC Quantum

Using the concatenation convention of the Zoo, the inner code is the \([[8,4,2]]\) symplectic double of the \([[4,2,2]]\) code. The \([[16,4,4]]\) twisted color code is its block concatenation with outer \([[4,2,2]]\) blocks along a \(ZX\) duality [3].

Small-distance qubit stabilizer codeQubit Stabilizer Hamiltonian-based Small-distance block quantum QECC Quantum