Kitaev surface code

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).

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].

Coherent physical errors are expected to become incoherent logical errors after MWPM decoding; see corroborating numerical studies performed via the Majorana mapping [9] as well as analytical bounds [10].

Rate

Rate depends on the underlying cellulation and manifold. For general 2D manifolds, $kd^2\leq c(\log k)^2 n$ for some constant $c$ [11], meaning that (1) 2D surface codes with bounded geometry have distance scaling at most as $O(\sqrt{n})$ [12][13], 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

For an $L\times L$ lattice, deterministic state preparation can be done with a geometrically local unitary $O(L)$-depth circuit [14][15] or an $O(\log{L})$-depth unitary circuit with non-local two-qubit gates [16][17] (matching a lower bound in Ref. [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 [8][21].

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 [22].

Gates

Clifford gates can be implemented via lattice surgery [7][23][24][25], twist-based lattice surgery [26], or braiding defects [27][28][29][30]. Non-Clifford gates require magic state distillation [31], Dehn twists [32], or just-in-time decoding [33]. Non-stabilizer surface-code states can be prepared by augmenting the code with a quantum double model [34].

Decoding

Maximum-likelihood (ML) [8], which takes time of order $O(n^2)$ for independent $X,Z$ noise [35].Minimum weight perfect-matching (MWPM) [8][36] (based on work by Edmonds on finding a matching in a graph [37][38]). Pipeline MWPM [39][40] - a modification accounting for correlations between events. 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) [41]. Correlated matching modifies MWPM to include correlations between $X$ and $Z$-type errors [39]. Belief perfect matching is a combination of belief-propagation and MWPM [42].Renormalization group (RG) [43][44][45].Markov-chain Monte Carlo [46].Tensor network [35].Cellular automaton [47][48].Neural network [49][50][51][52] and reinforcement learning [53].Union-find [54]. 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) [41]. Belief union find is a combination of belief-propagation and union-find [42].Decoders can be augmented with a pre-decoder [55][56], which can allow for some processing to be done inside the cryogenic environment of the quantum system [57].Sliding-window [58][59] and parallel-window [58] parallelizable decoders can be combined with many inner decoders, such as MWPM or union-find.Generalized belief propagation (GBP) [60] based on a classical version [61].

Fault Tolerance

Transversal (non-Clifford) CCZ gate by bringing 2D surface codes together and using just-in-time decoding [33]. Gate can be simulated by taking 2D slices out of 3D surface codes [62].Homomorphic measurement protocols for arbitrary surface codes [63].Non-geometrically local connectivity can reduce overhead cost [64].Fault-tolerant post-selection framework yields magic states with low overhead [65].

Code Capacity Threshold

Independent $X,Z$ noise: $p_X = 10.31\%$ under MWPM decoding [66] (see also Ref. [35]), $9.9\%$ under BP-OSD decoding [67], and $8.9\%$ under GBP decoding [60]. The threshold under ML decoding corresponds to the value of critical point of the two-dimensional random-bond Ising model on the Nishimori line [68][8] (see also [69]), calculated to be $10.94 \pm 0.02\%$ in Ref. [70], $10.93(2)\%$ in Ref. [71], and estimated to be between $10.9\%$ and $11\%$ in Ref. [35]. 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 [35], $14\%$ under GBP decoding [60], $16.5\%$ under recursive MWPM [72], and between $15\%$ and $16\%$ under RG [43], Markov-chain [46], or MWPM [73] decoding. The threshold under ML decoding corresponds to the value of critical point in the disordered eight-vertex Ising model, calculated to be $18.9(3)\%$ [74] (see also APS Physics viewpoint [75]).Erasure noise: $50\%$ for square tiling [76]. There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition [77].

Threshold

$1.8\%$ for circuit-level depolarizing noise under optimal decoder [78]. $0.57\%$ for depolarizing noise on data and syndrome qubits as well initialization, gate, and measurement errors under MPWM decoding [29]. 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 $10^{-4}$ [79].$0.35\%$ for circuit-level independent $X,Z$ noise under optimal decoder [78].Phenomenological noise: $3.3\%$ for square tiling [80].Phenomenological noise model for the 2D toric code: $2.93(2)\%$ using several rounds of syndrome measurement [66].$0.5-2.9\%$ for various noise models [81] (see also Refs. [66][82]).

Realizations

One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [83]. Signatures of corresponding topological phase of matter detected in superconducting circuits [84] and two-dimensional Rydberg atomic arrays [85].

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 [86].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

Calderbank-Shor-Steane (CSS) stabilizer code
Clifford-deformed surface code (CDSC) — CDSC codes are deformations of the surface code via constant-depth Clifford circuits that may not be CSS.
Abelian topological code — When treated as ground states of the code Hamiltonian, the code states realize $\mathbb{Z}_2$ topological order, a topological phase of matter that also exists in $\mathbb{Z}_2$ lattice gauge theory [87]. Codewords correspond to ground state of the code Hamiltonian, and error operators correspond to spontaneous creation and annihilation of pairs of charges or vortices.

Children

Fractal surface code
Higher-dimensional surface code
Hyperbolic surface code
Projective-plane surface code
$[[4,2,2]]$ CSS code — $[[4,2,2]]$ code is the smallest toric code.

Cousins

Hypergraph product code — Planar (toric) code can be obtained from hypergraph product of two repetition (cyclic) codes ([88], Ex. 6).
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 code with a quantum double model [34].
String-net code — String-net model reduces to the surface code when the category is the group $\mathbb{Z}_2$.
Majorana stabilizer code — The Majorana mapping can be used to construct efficient algorithms for simulating rounds of error correction for the surface code [9].
Analog surface code — The analog surface code is an oscillator-into-oscillator version of the surface code.
Color code — Color code is equivalent to surface code in several ways [89][90]. 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.
Galois-qudit topological code — Surface code has been extended to Galois qudits.
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 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 [91][92].
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.
Subsystem surface 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 [93][94][95].

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Victor V. Albert (2022-06-15)
Tony Lau (2022-04-02)
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Victor V. Albert (2022-02-15)
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Victor V. Albert (2021-11-05)
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Michael Vasmer (2021-11-02)

Cite as:

“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/topological/surface/surface/surface.yml.