\([[4,2,2]]\) Four-qubit code

\([[4,2,2]]\) Four-qubit code[1,2]

Alternative names: \(C_4\) code, Little Shor code.

Description

Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error.

The code admits stabilizer generators \(\{XXXX, ZZZZ\} \), and its stabilizer generator matrix blocks, \(H_{X}=H_{Z}=(1,1,1,1)\), are both the parity-check matrix of the \([4,3,2]\) SPC code.

A basis of codewords is \begin{align} \begin{split} |\overline{00}\rangle = (|0000\rangle + |1111\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{01}\rangle = (|0011\rangle + |1100\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{10}\rangle = (|0101\rangle + |1010\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{11}\rangle = (|0110\rangle + |1001\rangle)/\sqrt{2}~. \end{split} \tag*{(1)}\end{align} This code is the smallest instance of the toric code, and its various single-qubit subcodes are small planar surface codes. It is the unique code for its parameters, up to equivalence [3; Thm. 8].

The subcode \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) is a \([[4,1,2]]\) code known as the Leung-Nielsen-Chuang-Yamamoto code [4], whose \(\pm\)-basis codewords can be written as \begin{align} |\overline{\pm}\rangle = \frac{1}{2}(|00\rangle \pm |11\rangle)^{\otimes 2}~.\tag*{(2)}\\ \end{align} This code can be thought of as a concatenation of a two-qubit bit-flip with a two-qubit phase-flip code.

Protection

Detects a single-qubit error [1] or single erasure [2]. Not able to correct arbitrary single-qubit errors because \( \lfloor \frac{d-1}{2} \rfloor =0 \).

The \([[4,1,2]]\) subcodes \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [4] and \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [5] approximately correct a single AD error, with the latter being a constant excitation code.

An equivalent version of this code can suppress errors in adiabatic quantum computation by being used as an excited-state space of a particular Hamiltonian [6].

Magic

Various magic-state distillation protocols exist for the \([[4,2,2]]\) qubit code and the \(C_6\) code in what are known as Meier-Eastin-Knill (MEK) protocols [7]. For example, the magic-state yield parameter is \(\gamma = \log_2 5 \approx 2.322\) for a protocol using the \([[10,2,2]]\) code [8; Box 2]; see also [9; Table IV].

Transversal Gates

A tensor product of Hadamard gates applies a Hadamard gate to both logical qubits, and a tensor product of \(S=\sqrt{Z}\) gates applies a \(CZ\) gate followed by a logical \(Z\) on both qubits [10] (see also [11]).This code is the only four-qubit subspace to house a transversal representation of the Clifford group but not of the single-qubit unitary group [12; Eq. (104)]. Its \(n\)-block version is a \([[4n,2n,2]]\) code, which houses a signed permutation representation of the \(n\)-qubit Clifford group [12; Sec. 3.1 and Appx. B].A transversal \(CZ\) gate is realized by the rotation \(\sqrt{Z}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}\).Adding \(XYZI\) to the stabilizer group produces a \([[4,1,2]]\) subcode that admits weight-two transversal logical Pauli operations [13].

Gates

Some inter-block gates can be weight-two (two-body) with the help of perturbative gadgets, making it possible to suppress errors in adiabatic quantum computation [6].Logical Clifford circuits for various qubit connectivities [14].

Decoding

Erasure decoder [15].

Fault Tolerance

Preparation of certain states, both magic and non-magic, along with transversal gates can be performed fault-tolerantly, but requires post-selection because the code cannot correct errors [10]. Magic states can be injected into surface and color codes since the code is a small instance of both [16].Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [17] admitting a post-selected threshold [18,19] (see also Ref. [20]).Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [21,22]. This scheme can be augmented to fit on a 1D qubit lattice with nearest-neighbor connectivity [23].Fault-tolerant implementation of the Deutsch-Josza algorithm [24].

Realizations

See also [25; Tab. I] for more details on each experimental realization.\([[4,1,2]]\) subcodes implemented in linear optical networks [26,27].Trapped-ion device by IonQ [28].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [29–32].The subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [33] and \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [34], treated as a planar surface code, has been realized in superconducting-circuit devices.Logical gates between two copies of the subcode \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\), interpreted as lattice surgery between planar surface codes, realized in superconducting circuits [35].Logical gates for the \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) subcode, treated as a planar code, realized in superconducting circuits [36].The CZ magic state has been realized on an IBM heavy-hex superconducting circuit device [16].Logical Clifford gates for a twist-defect surface code that is single-qubit Clifford equivalent to a \([[4,1,2]]\) realized in a trapped ion device by Quantinuum [37].CPC gadgets for the \([[4,2,2]]\) code have been implemented on the IBM 5Q superconducting device [38].An FPGA implementation of the collision clustering decoder [39] realized on a Rigetti superconducting device [40].Neutral atom arrays: error detection, erasure correction, and post-selected fault-tolerant circuits demonstrated on 24 logical qubits on a 256-qubit device by Atom Computing, with each qubit encoded in the \([[4,2,2]]\) code [41]. The device also ran the Bernstein-Vazirani algorithm on up to 28 logical qubits encoded in a \([[4,1,2]]\) subcode [41]. Post-selected fault-tolerant realization of a benchmarking protocol [10], preparation of the ground state of the single-impurity Anderson impurity model, and post-selected fault-tolerant logical Bell-state preparation demonstrated on one copy of the \([[4,2,2]]\) code on a device by Infleqtion [25]. Error correction with mid-circuit erasure measurements and logical teleportation demonstrated by the Thompson group [42]. The \([[4,2,2]]\) code has been implemented on a star topology using superconducting devices and microwave cavities [43].

Cousins

Twist-defect surface code— A small 6.6.6 color code is a \([[4,1,2]]\) subcode with three weight-three stabilizer generators [44; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [37] via a single-qubit Clifford circuit.
Quantum parity code (QPC)— The \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) \([[4,1,2]]\) subcode is the smallest QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip repetition code. An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [5,45,46].
\([[5,1,3]]\) Five-qubit perfect code— The \([[4,2,2]]\) code can be derived from the five-qubit code using a protocol that converts an \([[n,k,d]]\) code into an \([[n-1, k+1, d-1]]\) code [47; Sec. 3.5][48; Fig. 3].
Amplitude-damping (AD) code— The \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto subcode \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) (approximately) corrects a single AD error [4].
Binomial code— \([[4,1,2]]\) subcode consisting of \(|\overline{00}\rangle\) and any other codeword reduces to the \(0,2,4\) binomial code when the basis labels in each codeword are written as in base-ten. Such a mapping can be generalized [49].
Heavy-hexagon code— Magic states prepared using the \([[4,1,2]]\) subcode can be injected into the heavy-hex code [16,32]. The \(d=2\) heavy-hex code is also closely related to the \([[4,1,2]]\) subcode.
Kitaev surface code— Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [50].
Concatenated qubit code— The \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) \([[4,1,2]]\) subcode is the smallest QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip repetition code. Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [17] admitting a post-selected threshold [18,19] (see also Ref. [20]). Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [21,22]. This scheme can be augmented to fit on a 1D qubit lattice with nearest-neighbor connectivity [23]. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [50]. The \([[4,2,2]]\) code can be concatenated with two copies of the surface code to yield the 4.6.12 color code [50]. An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [5,45,46]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [51]. Recursively concatenating a \([[4,1,2]]\) subcode attains a threshold [52,53].
Codeword stabilized (CWS) code— A \([[4,1,2]]\) subcode can be thought of as a CWS code [54].
\([[7,1,3]]\) Steane code— The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [55].
\([[6,4,2]]\) error-detecting code— The \([[6,4,2]]\) error-detecting code can be constructed out of two \([[4,2,2]]\) codes in the quantum Lego code framework [55].
Coherent-parity-check (CPC) code— CPC gadgets for the \([[4,2,2]]\) code have been implemented on the IBM 5Q superconducting device [38].
\([n,n-1,2]\) Single parity-check (SPC) code— The \([[4,2,2]]\) code is constructed from the \([4,3,2]\) SPC code via the CSS construction.
Dual-rail quantum code— An \([[8,1,2]]\) QPC correcting a single AD error is equivalent to a concatenation of the \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) (constant-excitation) subcode of the \([[4,2,2]]\) code with the dual-rail code [5,45,46]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [51].
Concatenated GKP code— Recursively concatenating the \(C_6\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [56].
Tensor-network code— The Steane and \([[6,4,2]]\) error-detecting codes can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [55].
Ladder Floquet code— The smallest example of the ladder Floquet code is a dynamical version of the \([[4,2,2]]\) code [57,58]. The \([[4,2,2]]\) can be Floquetified in various ways [59,60].
\([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code— The \([[8,3,4]]_{f}\) Hamming Majorana code is a Majorana stabilizer code obtained by combining two four-qubit codes [61].
Jump code— The subcode \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [5] is a \(((4,2,1))_2\) jump code correcting a single AD error. This code can be extended to a \(((4,3,1))_2\) jump code \(\{|\overline{01}\rangle,|\overline{10}\rangle,|\overline{11}\rangle\}\) [62].
Hybrid stabilizer code— The \([[4,2,2]]\) codewords can be modified by signs to yield a \([[4,1:1,2]]\) hybrid stabilizer code [63].
Four-qubit single-deletion code— A basis of codewords for the four-qubit single-deletion code consists of the \(|\overline{00}\rangle\) and \(|\overline{01}\rangle+|\overline{10}\rangle+|\overline{11}\rangle\)states of the four-qubit code.
\([[12,2,4]]\) carbon code— The carbon code is a concatenation of the \([[4,2,2]]\) code and the \(C_6\) code.
Numerically optimized four-qubit AD code— The numerically optimized four-qubit AD code that can correct a single AD error with higher entanglement fidelity than the \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto code subcode of the \([[4,2,2]]\) code.
\([[6,2,2]]\) \(C_6\) code— Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [17] admitting a post-selected threshold [18,19] (see also Ref. [20]) and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [7]. Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [21,22]. This scheme can be augmented to fit on a 1D qubit lattice with nearest-neighbor connectivity [23].
Hypergraph product (HGP) code— There is a fault-tolerant universal computation scheme for hypergraph-product codes concatenated with the \([[4,2,2]]\) code [64].
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code— Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [21,22]. This scheme can be augmented to fit on a 1D qubit lattice with nearest-neighbor connectivity [23].
\([[16,6,4]]\) Tesseract color code— The \([[16,4,2,4]]\) tesseract subsystem color code with particular gauge fixing can be obtained from four copies of the \([[4,2,2]]\) code [65].
\([[4,1,1,2]]\) Four-qubit subsystem code— The \([[4,1,1,2]]\) code can be obtained by picking one of the logical qubits of the \([[4,2,2]]\) four-qubit code to be a gauge qubit; e.g., see Ref. [58]. One particular gauge configuration has gauge operators \(\{XXII,IIXX,ZIZI,IZIZ\}\).

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

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 Abelian topological Topological Hamiltonian-based Qubit QECC Quantum

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

Parents

Rotated surface code

The \([[4,2,2]]\) code is the smallest rotated toric code. The subcodes \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\) [35], \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [33], \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [34], and \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) [36] are small planar rotated surface codes.

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

The \([[4,2,2]]\) code can be interpreted as a 2D color code on a square of the 4.8.8 or 4.6.12 tilings, or on a trapezoidal patch that makes up two-thirds of a hexagon of the 6.6.6 tiling [16,66]. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [50]. Concatenating the \([[4,2,2]]\) code with two copies of the surface code yields the 4.6.12 color code [50]. A small 6.6.6 color code is a \([[4,1,2]]\) subcode with three weight-three stabilizer generators [44; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [37] via a single-qubit Clifford circuit.

\([[2^D,D,2]]\) hypercube quantum codeSelf-complementary qubit Amplitude-damping (AD) AQECC Color Quantum RM code Generalized homological-product QLDPC CSS Stabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

The \([[4,2,2]]\) code is a hypercube code for \(D=2\).

\([[2m,2m-2,2]]\) error-detecting codeSelf-complementary qubit Amplitude-damping (AD) AQECC Color Qubit Generalized homological-product QLDPC CSS Stabilizer Hamiltonian-based Quantum MDS Concatenated quantum Small-distance block quantum QECC Quantum

The \([[2m,2m-2,2]]\) error-detecting code for \(m=2\) reduces to the \([[4,2,2]]\) code.

Brickwork \(XS\) stabilizer codeQubit Abelian topological Topological Hamiltonian-based QECC Quantum

The \([[4,2,2]]\) code can be interpreted as a brickwork code on a square of the overlapping rectangular tilings [67].

\([[4,2,2]]_{G}\) four group-qudit codeTopological Hamiltonian-based Small-distance block quantum QECC Quantum

The four group-qudit code reduces to the four-rotor code for \(G= \mathbb{Z}_2\).

Quantum polar codeEAQECC Quantum

\([[4,2,2]]\) code is a small quantum polar code [68].