Kitaev surface code[1][2][3]

Description

A family of abelian topological CSS stabilizer codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a cellulation of a two-dimensional surface (with a qubit located at each edge of the cellulation). Toric code often either refers to the construction on the two-dimensional torus or is an alternative name for the general construction. The construction on surfaces with boundaries is often called the planar code [4].

The original construction can be naturally extended to arbitrary \(D\)-dimensional manifolds [5][6]. Given a cellulation, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers are associated with \((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \(i+1\)-faces. Such extensions are often called the \(D\)-dimensional surface or \(D\)-dimensional toric codes.

The stabilizers of the surface code on the 2-dimensional torus are generated by star operators \(A_v\) and plaquette operators \(B_p\). Each star operator is a product of four Pauli-\(X\) operators on the edges adjacent to a vertex \(v\) of the lattice; each plaquette operator is a product of four Pauli-\(Z\) operators applied to the edges adjacent to a face, or plaquette, \(p\) of the lattice (Figure I).

(float)
Figure I: Stabilizer generators and logical operators of the 2D surface code on a torus. The star operators \(A_v\) and the plaquette operators \(B_p\) generate the stabilizer group of the toric code. The logical operators are strings that wrap around the torus.

The two-dimensional toric code encodes two logical qubits. We denote by \(\overline{X}_i,\overline{Z}_i\) the logical Pauli-\(X\) and Pauli-\(Z\) operator of the \(i\)-th logical qubit. They can are represented by strings of Pauli-\(X\) operators or Pauli-\(Z\) operators that wrap around the torus as shown in Figure I.

Protection

Toric code on an \(L\times L\) torus is a \([[2L^2,2,L]]\) CSS code, and there exists a planar code with \([[L^2,1,L]]\) [7]. More generally, the code distance is related to the homology of the cellulation [8].

Rate

Rate depends on the underlying cellulation and manifold. For general 2D manifolds, \(kd^2\leq c(\log k)^2 n\) for some constant \(c\) [9], meaning that (1) 2D surface codes with bounded geometry have distance scaling at most as \(O(\sqrt{n})\) [10][11], and (2) surface codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in \(n\). Higher-dimensional hyperbolic manifolds (see code children below) yield distances scaling more favorably. Loewner's theorem provides an upper bound for any bounded-geometry surface code [5].

Encoding

Unitary [12][13][14].Dissipative [15].Stabilizer measurement-based [16].

Transversal Gates

Transversal Pauli gates exist and are based on non-trivial loops on surface. Transversal Clifford gates can be done on folded surface codes [17].

Gates

Clifford gates can be implemented via lattice surgery [7][18][19][20], twist-based lattice surgery [21], or braiding defects [22][23][24][25]. Non-Clifford gates require magic state distillation [26], Dehn twists [27], or just-in-time decoding [28].

Decoding

Minimum weight perfect-matching (MWPM) [8][29] and pipeline MWPM [30][31], a modification accounting for correlations between events.Union-find [32].Renormalization group [33][34].Tensor network [35].Markov-chain Monte Carlo [36].Cellular automaton [37].Machine learning [38][39][40].

Code Capacity Threshold

For correlated Pauli noise, bounds on code capacity thresholds can be obtained by mapping the effect of noise on the code to a statistical mechanical model. The first such threshold, based on the planar code, is \(0.017\%\) [8].\(10.9\%\) (\(10.31\%\)) with pure \(Z\)-dephasing noise for square tiling using tensor-network [41][35] (minimum-weight perfect matching [42]) decoder. \(18.9\%\) with depolarizing noise for square tiling [43].\(50\%\) with erasure errors for square tiling [44].\(3.3\%\) with phenomenological noise for square tiling [45].

Threshold

\(0.5-1.1\%\) for various error models [46][47].

Realizations

Distance-two surface codes have been implemented by Andersen et al. [48], Erhard et al. [49], Marques et al. [50], Google Quantum AI [51], and both planar and toric versions by Bluvstein et al. [52]. Distance-three surface code implemented at ETH Zurich [53] and on the Zuchongzhi 2.1 superconducting quantum processor [54]. Signatures of corresponding topological phase of matter detected in superconducting circuits [55] and two-dimensional arrays of Rydberg atoms [56].

Notes

Surfmap framework provides a way to stitch the surface code to various superconducting-circuit geometries by assigning each superconducting qubit to be either a physical or ancilla qubit, designing stabilizer measurement circuits, and scheduling stabilizer measurements [57].2D and 3D surface code visualization tool. Tutorials from error-correction perspective by J. Haah and condensed-matter perspective by M. Levin and C. Nayak.

Parents

Children

Cousins

  • Hypergraph product code — Planar (toric) code obtained from hypergraph product of two repetition (cyclic) codes.
  • Color code — Color code is equivalent to surface code in several ways [59][60]. For example, the color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D-1\)-dimensional surface code.
  • Double-semion code — There is a logical basis for the toric and double-semion codes where each codeword is a superposition of states corresponding to all noncontractible loops of a particular homotopy type. The superposition is equal for the toric code, whereas some loops appear with a \(-1\) coefficient for the double semion.
  • Haah cubic code — The energy of any partial implementation of code 1 is proportional to the boundary length similar to the 4D toric code, which can potentially surpress the effects of thermal errors, but it is currently an open problem.
  • Heavy-hexagon code — Surface code stabilizers are used to measure the Z-type stabilizers of the code.
  • Honeycomb code — Measurement of each check operator involves two qubits and projects the state of the two qubits to a two-dimensional subspace, which we regard as an effective qubit. These effective qubits form a surface code on a hexagonal superlattice. Electric and magnetic operators on the embedded surface code correspond to outer logical operators of the Floquet code. In fact, outer logical operators transition back and forth from magnetic to electric surface code operators under the measurement dynamics.
  • Lifted-product (LP) code — A lifted product code for the ring \(R=\mathbb{F}_2[x,y]/(x^L-1,y^L-1)\) is the toric code.
  • Modular-qudit surface code — The qudit surface code with \(q=2\) is the surface code.
  • Raussendorf-Bravyi-Harrington (RBH) code — Without symmetry protection, one of 2D boundaries of the cubic RBH code is effectively a 2D toric code.
  • Translationally-invariant stabilizer code — Translation-invariant 2D qubit topological stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth Clifford circuit [61][62][63].

Zoo code information

Internal code ID: surface

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Zoo Code ID: surface

Cite as:
“Kitaev surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/surface
BibTeX:
@incollection{eczoo_surface, title={Kitaev surface code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/surface} }
Permanent link:
https://errorcorrectionzoo.org/c/surface

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Cite as:

“Kitaev surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/surface

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/surface/surface/surface.yml.