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

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

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

Description

A four-qubit hyperbolic self-dual CSS stabilizer code that is the smallest two-logical-qubit stabilizer code to detect a single-qubit error. It is unique for its parameters [3; Thm. 8][4; ID 9].

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}

Protection

Detects a single-qubit error [1] or single erasure [2]. It cannot correct arbitrary single-qubit errors because \( \lfloor \frac{d-1}{2} \rfloor =0 \). 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 [5].

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 [6,7]. In the inner/outer-code formulation of Ref. [7], the \([[4,2,2]]\) code is a hyperbolic self-dual inner code for quadratic distillation. 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]). A logical \(CZ\) gate is then realized by \(\sqrt{Z}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}^{\dagger}\otimes\sqrt{Z}\). With a different logical basis, transversal Hadamard swaps the two logical qubits, enabling control-SWAP and \(H^{\otimes 2}\)-measurement routines for quadratic magic-state distillation [7; Sec. I.B.1].This code is the only four-qubit subspace to house a transversal representation of the single-qubit Clifford group but not of the single-qubit unitary group [12; Eq. (104)]. Equivalently, its projector is the only extra generator of the fourth tensor-power Clifford commutant beyond qubit permutations [12; Thm. 1]. 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].

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 [5].Logical Clifford circuits for various qubit connectivities [13].

Decoding

Erasure decoder [14].

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 [15].Knill’s \(C_4/C_6\) architecture uses the \([[4,2,2]]\) code at the first level and the \(C_6\) code at higher levels, together with error-correcting teleportation [16]. Later work refined the postselected-threshold analysis [17,18] (see also Ref. [19]).Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [20]. In the optimized protocol of Ref. [20], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively.Fault-tolerant implementation of the Deutsch-Jozsa algorithm [21].

Realizations

See also [22; Tab. I] for more details on each experimental realization.Trapped-ion device by IonQ [23].Logical state preparation and flag-qubit error correction realized in superconducting-circuit devices by IBM [24–27].The CZ magic state has been realized on an IBM heavy-hex superconducting circuit device [15].CPC gadgets for the \([[4,2,2]]\) code have been implemented on the IBM 5Q superconducting device [28].An FPGA implementation of the collision clustering decoder [29] realized on a Rigetti superconducting device [30].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 [31]. 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 [22]. Error correction with mid-circuit erasure measurements and logical teleportation demonstrated by the Thompson group [32]. Logical implementation of Shor’s algorithm on a device by Infleqtion [33]. The \([[4,2,2]]\) code has been implemented on a star topology using superconducting devices and microwave cavities [34].Trapped-ion processor by AQT: modular logical-state teleportation between two four-qubit error-detecting code blocks without mid-circuit measurements [35].

Cousins

\([[5,1,3]]\) 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 [36; Sec. 3.5][37; Fig. 3].
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 [38].
Toric code— The toric code can be constructed by arranging \([[4,2,2]]\) tensors on a square lattice and recovering the star and plaquette operators by operator pushing [39].
Concatenated qubit code— Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [16] admitting a post-selected threshold [17,18] (see also Ref. [19]). Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [20]. In the optimized protocol of Ref. [20], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [38]. The \([[4,2,2]]\) code can be concatenated with two copies of the surface code to yield the 4.6.12 color code [38].
\([[6,2,2]]\) \(C_6\) code— Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [16] admitting a post-selected threshold [17,18] (see also Ref. [19]) and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [6]. Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [20]. In the optimized protocol of Ref. [20], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively.
\([[7,1,3]]\) Steane code— The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [39]. Ref. [20] also introduces a \(C_4\)/Steane concatenated code, obtained by concatenating the \([[4,2,2]]\) code with the Steane code, as an underlying code for further concatenation with quantum Hamming codes.
\([[6,4,2]]\) 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 [39].
Coherent-parity-check (CPC) code— CPC gadgets for the \([[4,2,2]]\) code have been implemented on the IBM 5Q superconducting device [28].
Jump code— A \(((4,3,1))_2\) jump code is a subcode of the \([[4,2,2]]\) code and contains the \([[4,1,2]]\) LNCY code as a subcode [40].
\([n,n-1,2]\) 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 [41–43]. More generally, an \([[m^2,1,m]]\) QPC corrects \(m-1\) AD errors [44].
Concatenated GKP code— Recursively concatenating the \(C_6\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [45].
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 [39]. The toric code can be constructed by arranging \([[4,2,2]]\) tensors on a square lattice and recovering the star and plaquette operators by operator pushing [39].
Ladder Floquet code— The smallest example of the ladder Floquet code is a dynamical version of the \([[4,2,2]]\) code [46,47]. The \([[4,2,2]]\) code can be Floquetified in various ways [48,49].
\([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code— The \([[8,3,4]]_{f}\) Hamming Majorana code is a Majorana stabilizer code obtained by combining two four-qubit codes [50].
Hybrid stabilizer code— The \([[4,2,2]]\) codewords can be modified by signs to yield a \([[4,1:1,2]]\) hybrid stabilizer code [51].
\([[12,2,4]]\) carbon code— The carbon code is a concatenation of the \([[4,2,2]]\) code and the \(C_6\) code.
\([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code— The \([[4,1,2]]\) LNCY code is obtained as the \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) \([[4,1,2]]\) subcode of the \([[4,2,2]]\) four-qubit code [52]. A \(((4,3,1))_2\) jump code is a subcode of the \([[4,2,2]]\) code and contains the \([[4,1,2]]\) LNCY code as a subcode [40].
\(((4,2,2))\) Four-qubit single-deletion code— Projecting the four-qubit code into the PI subspace yields the 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.
\([[4,1,2]]\) twist-defect code— Adding \(XYZI\) to the stabilizer group of the \([[4,2,2]]\) four-qubit code yields the \([[4,1,2]]\) twist-defect subcode [53].
\(((5,6,2))\) qubit code— Tracing out any one qubit of the \(((5,6,2))\) code projector yields a \(((4,4,2))\) code; for this code, all five such partial traces are additive and therefore locally equivalent to the \([[4,2,2]]\) code [3; Thm. 8 and Corr. 18].
Hypergraph product (HGP) code— There is a fault-tolerant universal computation scheme for hypergraph-product codes concatenated with the \([[4,2,2]]\) code in which the full syndrome measurement on the lower hypergraph product code is performed only if an error is detected at the upper four-qubit code [54].
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code— Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [20]. In the optimized protocol of Ref. [20], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively.
\([[16,6,4]]\) 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 [55].
\([[4,1,1,2]]\) 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. [47]. 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 codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum

Homological code

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

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

Parents

Rotated surface code

The \([[4,2,2]]\) code is the smallest rotated toric code [4].

Square-octagon (4.8.8) color codeColor Twist-defect color QLDPC Qubit Generalized homological-product Lattice stabilizer CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

The \([[4,2,2]]\) code can be interpreted as a 2D color code on a square of the 4.8.8 tiling [15,56]. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [38].

Honeycomb (6.6.6) color codeColor Twist-defect color QLDPC Qubit Generalized homological-product Lattice stabilizer CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

The \([[4,2,2]]\) code can be interpreted as a 2D color code on a trapezoidal patch that makes up two-thirds of a hexagon of the 6.6.6 tiling [15,56].

Truncated trihexagonal (4.6.12) color codeColor Twist-defect color QLDPC Qubit Generalized homological-product Lattice stabilizer CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

The \([[4,2,2]]\) code can be interpreted as a 2D color code on a square of the 4.6.12 tiling [15,56]. Concatenating the \([[4,2,2]]\) code with two copies of the surface code yields the self-dual 4.6.12 color code [38].

\([[2^D,D,2]]\) hypercube quantum codeSelf-complementary qubit Amplitude-damping (AD) AQECC Color Quantum RM code 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 QLDPC Qubit 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 codeTopological Hamiltonian-based Qubit QECC Quantum

The \([[4,2,2]]\) code can be interpreted as a brickwork code on a square of the overlapping rectangular tilings [57].

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

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