Kitaev surface code[1–3]  

Also known as Kitaev toric code.

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 [4–6]. Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices.

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).

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. The original planar code on a square-lattice patch with different boundary conditions on the vertical and horizontal edges is a \([[L^2+(L-1)^2,1,L]]\) CSS code [4].

Coherent physical errors are expected to become incoherent logical errors under syndrome measurement; see corroborating numerical studies performed via the Majorana mapping [7] as well as analytical bounds [8].

Rate

Both the planar and toric codes saturate the BPT bound, which states that \(k d^2 = O(L^2)\) for codes on a 2D lattice of length \(O(L)\).

Encoding

A depth-\(L^2\) circuit that grows the code out of a small patch on an \(L\times L\) square lattice using CNOT gates (i.e., "local moves") [9,10].Graph-state based adaptive circuit [11,12].For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit [13,14] or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates [10,15,16] (matching lower bounds [17,18]).Lindbladian-based dissipative encoding for the toric code [19] that does not give a speedup relative to circuit-based encoders [20].Stabilizer measurement-based circuit of linear depth [9,21].Any geometrically local unitary circuit on a lattice \(\Lambda\) that prepares a state whose energy density with respect to the surface code Hamiltonian is \(\epsilon\) must have depth of order \(\Omega(\min(\sqrt{|\Lambda|},1/\epsilon^{\frac{1-\alpha}{2}}))\) for any \(\alpha>0\) [22].Single-shot state preparation [23].

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 [24].

Gates

Clifford gates can be implemented via lattice surgery [25–28], twist-based lattice surgery [29], or braiding defects [30–35]. Non-Clifford gates require magic state distillation [36], Dehn twists [37], or just-in-time decoding [38]. Non-stabilizer surface-code states can be prepared by augmenting the code with a quantum double model [39].

Decoding

Degenerate maximum-likelihood (ML) [9], which takes time of order \(O(n^2)\) under independent \(X,Z\) noise for the surface code [40].Minimum weight perfect-matching (MWPM) [9,41] (based on work by Edmonds on finding a matching in a graph [42,43]), which takes time up to polynomial in \(n\) for the surface code. For the case of the surface code, minimum-weight decoding reduces to MWPM [9,42,44].Modified MWPM decoders: pipeline MWPM (accounting for correlations between events) [45,46], parity blossom MWPM and fusion blossom MWPM [47], and a modification utilizing the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) [48].Belief perfect matching is a combination of belief-propagation and MWPM [49].Renormalization group (RG) [50–52].Markov-chain Monte Carlo [53].Tensor network [40].Cellular automaton [54,55].Neural network [56–59] and reinforcement learning [60].Union-find [61]. A subsequent modification utilizes the continuous signal obtained in the physical implementation of the stabilizer measurement (as opposed to discretizing the signal into a syndrome bit) [48]. Belief union find is a combination of belief-propagation and union-find [49].Decoders can be augmented with a pre-decoder [62,63], which can allow for some processing to be done inside the cryogenic environment of the quantum system [64].Sliding-window [65,66] and parallel-window [65] parallelizable decoders can be combined with many inner decoders, such as MWPM or union-find.Generalized belief propagation (GBP) [67] based on a classical version [68].

Fault Tolerance

Transversal (non-Clifford) CCZ gate by bringing 2D surface codes together and using just-in-time decoding [38]. Gate can be simulated by taking 2D slices out of 3D surface codes [69].Homomorphic measurement protocols for arbitrary surface codes [70].Non-geometrically local connectivity can reduce overhead cost [71].Fault-tolerant post-selection framework yields magic states with low overhead [72].Framework of fault tolerance utilizing ZX-calculus that is applicable to MBQC, FBQC, and conventional computation versions of the surface code [73].Single-shot state preparation [23] and MWPM decoding [74].Syndrome extraction circuits consisting of CNOT gates and ancillary measurements [32], two-body measurements based on the Majorana mapping [75,76]. Circuits can be optimized to specific architectures [77] using spacetime circuit codes.

Code Capacity Threshold

Independent \(X,Z\) noise: \(p_X = 10.31\%\) under MWPM decoding [78] (see also Ref. [40]), \(9.9\%\) under BP-OSD decoding [79], and \(8.9\%\) under GBP decoding [67]. The threshold under ML decoding corresponds to the value of critical point of the two-dimensional random-bond Ising model on the Nishimori line [9,80] (see also [81]), calculated to be \(10.94 \pm 0.02\%\) in Ref. [82], \(10.93(2)\%\) in Ref. [83], and estimated to be between \(10.9\%\) and \(11\%\) in Ref. [40]. Above values are for one type of noise only, and ML threshold for combined \(X\) and \(Z\) noise is \(2p_X - p_X^2 \approx 20.6\%\).Depolarizing noise: between \(17\%\) and \(18.5\%\) under tensor-network decoding [40], \(14\%\) under GBP decoding [67], \(16.5\%\) under recursive MWPM [84], and between \(15\%\) and \(16\%\) under RG [50], Markov-chain [53], or MWPM [85] decoding. The threshold under ML decoding corresponds to the value of critical point in the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) [86] (see also APS Physics viewpoint [87]).Erasure noise: \(50\%\) for square tiling [88]. There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition [89].

Threshold

\(1.8\%\) for circuit-level depolarizing noise under optimal decoder [90]. \(0.57\%\) for depolarizing noise on data and syndrome qubits as well initialization, gate, and measurement errors under MPWM decoding [32]. For this model, a logical qubit with a \(10^{-14}\) logical error rate requires between \(10^3\) to \(10^4\) physical qubits and a target gate fidelity above \(99.9\%\). Later work showed that arbitrarily large computations are possible for a physical error rate of approximately \(10^{-4}\) [91].\(0.35\%\) for circuit-level independent \(X,Z\) noise under optimal decoder [90].Phenomenological noise: \(3.3\%\) for square tiling [92].Phenomenological noise model for the 2D toric code: \(2.93(2)\%\) using several rounds of syndrome measurement [78].\(0.5-2.9\%\) for various noise models [93] (see also Refs. [78,94]).

Realizations

One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [95]. Signatures of corresponding topological phase of matter detected in superconducting circuits [96] and two-dimensional Rydberg atomic arrays [97]. Ground state of the toric code has been implemented with and without twists, and the non-Abelian braiding behavior of the twists, which realize Ising anyons, has been demonstrated [98].

Notes

Hardware requirements for implementing surface code QEC can be reduced by utilizing structure in the time slices of the QEC circuits [99]. 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 [100].Tutorials from error-correction perspective by J. Haah and condensed-matter perspective by M. Levin and C. Nayak.

Parents

Children

  • \([[4,2,2]]\) CSS code — \([[4,2,2]]\) code is the smallest toric code.
  • Rotated surface code — Rotated surface codes can be obtained using the same procedure as for the original surface codes but considering slightly different combinatorial surfaces [6,103] than those considered in the original proposal.

Cousins

  • Majorana stabilizer code — The Majorana mapping can be used to construct efficient algorithms for simulating rounds of error correction for the surface code [7].
  • Quantum-double code — A quantum-double model with \(G=\mathbb{Z}_2\) is the surface code. Non-stabilizer surface-code states can be prepared by augmenting the surface code with a quantum double model [39].
  • 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 [104–106].
  • Floquet color code — The ISG of the Floquet color code is the stabilizer group of one of three realizations of the \(\mathbb{Z}_2\) 2D surface code.
  • Honeycomb Floquet code — Measurement of each check operator of the honeycomb Floquet code 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. Inspired by this code, stabilizer measurement circuits consisting of two-body measurements have been designed for the surface code [75,76].
  • Spacetime circuit code — Stabilizer generators of a spacetime code are called detectors in Refs. [107,108].
  • Raussendorf-Bravyi-Harrington (RBH) cluster-state code — The RBH state encodes the temporal gate operations on the surface code into a third spatial dimension [109,110]. In addition, one of possible 2D boundaries of the RBH code is effectively a 2D toric code.
  • Color code — The 3D color code is equivalent to multiple decoupled copies of the 2D surface code [111–113]. Conversely, the 2D color code can condense to form the 2D surface code in nine different ways, i.e., by adding two body hopping terms along one of its three hexagonal directions to the stabilizer group and then taking the center of the resulting nonabelian group [114].
  • Heavy-hexagon code — Surface code stabilizers are used to measure the Z-type stabilizers of the code.
  • Kitaev honeycomb code — The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code. This code can be obtained from the square-lattice surface code by gauging out the anyon \(em\) [115; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [115; Fig. 12].
  • Subsystem surface code
  • Three-fermion (3F) subsystem code — One version of the 3F subsystem code can be obtained from two copies of the square-lattice surface code by gauging out the anyons \(e_1m_1e_2\) and \(e_2m_2\) [115; Sec. 7.4].
  • Double-semion stabilizer code — The double semion phase also has a realization in terms of qubits [116] that can be compared to the qubit surface code. There is a logical basis for both 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.

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Zoo Code ID: surface

Cite as:
“Kitaev surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/surface
BibTeX:
@incollection{eczoo_surface, title={Kitaev surface code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/surface} }
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“Kitaev surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/surface

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