Kitaev surface code[1–4]  

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). Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices.

The construction on closed surfaces (surfaces with boundaries) is called the toric code (planar code [4,5]). There are two types of stabilizers one can put on edges, yielding open (a.k.a. rough or primal) and closed (a.k.a. smooth or dual) boundaries. A mixed boundary consists of an interleaving of the two stabilizer types [6].

Protection

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 [5]. Code size \(k = 2^{2g}\) for a torus of genus \(g\), and such higher genus surfaces have been investigated [7].

Topological order and gauge-theory analogy

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 [8]. This order does not persist at nonzero temperature [9,10].

Pauli noise operators can be organized into anyonic strings of the gauge theory, which cause excitations of the ground-state subspace. The inability of local errors to distinguish the codewords translates to the "topologically protected" degeneracy of the ground state, rigorously formulated by the TQO-1 condition. The joint \(+1\)-eigenspace of the \(Z\)-type Paulis corresponds to the subspace that conserves \(\mathbb{Z}_2\) flux, while the joint \(+1\)-eigenspace of \(X\)-type operators corresponds to the subspace that preserves \(\mathbb{Z}_2\) gauge symmetry (a one-form symmetry). Logical Pauli operators correspond to non-contractible Wilson loops in the case of closed boundaries, and to paths connecting different types of boundaries in the case of open boundaries.

Behavior under Hamiltonian \(X\)-type and \(Z\)-type perturbations is related to an anisotropic 3D gauge Higgs model [11,12]. In order to corrupt logical states, any local noise must bring the code state out of the topological order [10].

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") [13,14].Teleportation-based state injection into the planar code [15].Graph-state based adaptive circuit [16,17].For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit [18,19] or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates [14,20,21] (matching lower bounds [22–24]).Stabilizer measurement-based circuit of linear depth [13,25].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 \(\alpha>0\) [26].Single-shot state preparation [27].Various techniques to generate lattices useful for particular architectures [28] or removing lattice defects [29,30] exist.

Transversal Gates

Transversal Clifford gates can be done on folded surface codes [31].

Gates

Clifford gates can be implemented via lattice surgery [32–35]. Non-Clifford gates require magic state distillation [36], Dehn twists [37,38], or just-in-time decoding [39,40]. Non-stabilizer surface-code states can be prepared by augmenting the code with a quantum double model [41]. ZX calculus [42,43] can be used to reduce the complexity of surface-code lattice surgery diagrams [44] and to reduce T-gate counts in magic-state distillation protocols [45,46]. Transversal injection method to prepare non-stabilizer states [47]. Certain gates can be performed adiabatically [48,49], yielding an instance of holonomic quantum computation [50]. Fault-tolerant gates should be interpretable as monodromies under a particular notion of parallel transport [51]. A combination of fold-transversal gates, Dehn twists and single-shot logical Pauli measurements generates the logical Clifford group [52].

Decoding

Using data from multiple syndrome measurements prior to decoding allows for correcting syndrome measurement errors. The surface code requires order \(O(d)\) extraction rounds in order to gain a reliable estimate. Syndrome measurements are distance-preserving because syndrome extraction circuits can be designed to avoid hook errors [13].Syndrome extraction circuits consist of CNOT gates and ancillary measurements since this is a stabilizer code [53]. Measurement schedules can be optimized using spacetime circuit codes to yield what is known as the 3CX surface code [54]. Schedules can also be optimized via ZX calculus [42,43]. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed [55–57], with the scheme in Ref. [56] being a special case of DWR.Expanding diamonds decoder correcting errors of some maximum fractal dimension [58]. The sub-threshold failure probability scales as \((p/p_{\text{th}})^{d^\beta}\), where \(p_{\text{th}}\) is the threshold and \(\beta = \log_3 2\).Minimum weight perfect-matching (MWPM) [13,59] (based on work by Edmonds on finding a matching in a graph [60,61]), 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 [13,60,62]. MWPM solves the MPE decoding problem exactly for independent \(X\) and \(Z\) noise. MPE decoding is \(NP\)-hard in general for the surface code [63].Bravyi-Suchara-Vargo (BSV) tensor network decoder [64] approximately solves the ML decoding problem under independent \(X,Z\) noise for the surface code and takes time of order \(O(n^2)\) [64]. ML decoding [13] is \(\#P\)-hard in general for the surface code [63].Union-find decoder [65] uses the union-find data structure [66–68], solving the MPE decoding problem exactly for low-weight errors under depolarizing noise. 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) [69]. Belief union find is a combination of belief-propagation and union-find [70]. Strictly local (as opposed to partially local) union find [71] has a worst-case runtime of order \(O(d^3)\) in the distance \(d\).Modified MWPM decoders: pipeline MWPM (accounting for correlations between events) [72,73]; modification tailored to asymmetric noise [74]; parity blossom MWPM and fusion blossom MWPM [75], 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) [69]; belief perfect matching (a combination of belief-propagation and MWPM) [70]; spanning tree matching (STM) and rapid-fire (RFire) decoders [76]; ordered decoding based on MWPM [77]. Combining, or harmonizing, various decoders can improve performance [78]. One such example is the Libra decoder [79], a combination of MWPM decoders combined with matching synthesis.Renormalization group (RG) [80–82]; see Ref. [83] for the planar surface code.Linear-time ML erasure decoder [84].Markov-chain Monte Carlo [85].Cellular automaton decoders [86–88]; see also [89].Neural network [90–92], reinforcement learning [93–95], and transformer-based [96,97] decoders.Lightweight low-latency look-up table (LILLIPUT) decoder for small surface codes [98].Decoders can be augmented with a pre-decoder [99,100], which can allow for some processing to be done inside the cryogenic environment of the quantum system [101].Sliding-window [102,103], parallel-window [102], and predictive-window [104] parallelizable decoders, designed to overcome the backlog problem, can be combined with many inner decoders, such as MWPM or union-find.Modifications of BP: generalized belief propagation (GBP) [105], based on a classical version [106]; AMBP4, a quaternary version [107] of the MBP decoder [108] of complexity \(O(n\log\log n)\); blockBP, a combination of BP and tensor-network decoders [109]; machine-learning inspired modifications [110]. See Ref. [111] for a review of BP decoders. The min-sum decoder, a simple variant of BP, cannot be used to attain the benefits of codes with distance greater than 9 [112].A color-code decoder can be used for the surface code [113].Progressive-Proximity Bit-Flipping (PPBF) decoder [114].Collision clustering decoder [115].Quasi-local Lindbladian decoder based on the approximate Petz theorem [116].Exclusive decoder family incorporating post-selection on decoding instances deemed not too difficult [117].

Fault Tolerance

Transversal (non-Clifford) CCZ gate by bringing 2D surface codes together and using just-in-time decoding [39,40]. Gate can be simulated by taking 2D slices out of 3D surface codes [118].Flag fault-tolerant syndrome extraction [119].Homomorphic measurement protocols for arbitrary surface codes [120].Non-geometrically local connectivity can reduce overhead cost [121].Magic-state distillation protocols [53,122–124] leading up to magic-state cultivation [125].Framework of fault tolerance utilizing ZX calculus [42,43] that is applicable to MBQC, FBQC, and conventional computation versions of the surface code [126].Single-shot state preparation [27] and MWPM decoding [127].Syndrome extraction circuits consisting of CNOT gates and ancillary measurements [53]. Measurement schedules can be optimized using spacetime circuit codes to yield what is known as the 3CX surface code [54]. Schedules can also be optimized via ZX calculus [42,43]. Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed [55–57], with the scheme in Ref. [56] being a special case of DWR.

Threshold

Circuit-level noise: \(1.8\%\) under correlated CNOT-gate errors and single-qubit-gate depolarizing noise [128] with optimal decoder [129], and \(0.35\%\) under independent \(X,Z\) noise with optimal decoder [129]. Also, \(0.57\%\) for depolarizing noise on data and syndrome qubits as well initialization, gate, and measurement errors under MPWM decoding [53]. 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 \(10^{-4}\) [130]. Thresholds of \(0.5-2.9\%\) have been observed for various noise models [129,131–136]. A threshold of \(0.41\%\) when concatenated with the \([[4,2,2]]\) code [137].Phenomenological noise: \(3.3\%\) for square tiling [138], and \(2.93(2)\%\) using several rounds of syndrome measurement [131].Quasistatic phase damping and readout noise: \(2.85\%\) [139].Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [140].Various thresholds for various measurement schedules, including that of the 3CX surface code, have been obtained [141].

Realizations

Signatures of corresponding topological phase of matter detected in superconducting circuits [142] and two-dimensional Rydberg atomic arrays [143]. Measurement schedules associated with the 3CX surface code realized in superconducting qubits on the Willow device by Google Quantum AI [144].

Notes

Introduction to computation with the surface code [145,146].Tutorials from error-correction perspective by A. Kubica and J. Haah and condensed-matter perspective by M. Levin and C. Nayak. Review of surface code decoders [147].Hardware requirements for implementing surface code QEC can be reduced by utilizing structure in the time slices of the QEC circuits [148]. Various optimization and calibration routines exist [149].

Parents

Children

  • Rotated surface code — The lattice of the rotated surface code can be obtained by taking the medial graph of the surface code lattice (treated as a graph) and applying a similar procedure to construct the check operators [6,150][151; Fig. 8]. Applying the quantum Tanner transformation to the surface code yields the rotated surface code [152,153]. The rotated surface code presents certain savings over the original surface code [154].
  • Toric code — The toric code is the surface code on a 2D torus.
  • 2D hyperbolic surface code

Cousins

  • Layer code — Layer codes are combinations of constant-rate QLDPC codes with surface codes built using lattice surgery.
  • Long-range enhanced surface code (LRESC) — LRESCs reduce to planar surface codes when a trivial LDPC code is used in the hypergraph product.
  • La-cross code — La-cross codes at \(k=1\) yield the toric (planar surface) code and periodic (open) boundary conditions.
  • Majorana stabilizer code — The Majorana mapping can be used to construct efficient algorithms for simulating rounds of error correction for the surface code [155].
  • 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 [41].
  • Hamiltonian-based code — While codewords of the surface code form ground states of the code's stabilizer Hamiltonian, they can also be ground states of other gapless Hamiltonians [156].
  • Self-correcting quantum code — Various candidates for self-correcting quantum memories have been constructed by coupling neighboring anyons so as to prevent them from spreading [87,89,157–159]
  • \(t\)-design — Unitary \(t\)-designs can be generated via coherent errors, syndrome extraction, and correction [160].
  • Asymmetric quantum code — The surface code on a hexagonal lattice is an asymmetric CSS code [161].
  • 2D lattice stabilizer code — Translation-invariant 2D qubit lattice stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth Clifford circuit [162–164]. There exists an algorithm with which one can determine the fusion and braiding rules of a 2D translationally invariant qubit code, and decompose the given code into copies of the surface code [165].
  • Dynamical automorphism (DA) code — One of the instantaneous stabilizer codes of the 2D DA color code are stacks of surface codes
  • Floquet color code — The ISG of the Floquet color code is the stabilizer group of one of nine realizations of the \(\mathbb{Z}_2\) 2D surface code.
  • X-cube Floquet code — The ISG of the X-cube Floquet code can be that of the X-cube model code or that of several decoupled surface codes.
  • 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 the honeycomb Floquet code, various weight-two measurement schemes have been designed [55–57], with the scheme in Ref. [56] being a special case of DWR. Numerical comparisons have been performed [166].
  • Spacetime circuit code — Stabilizer generators of a spacetime code are called detectors in Refs. [167,168].
  • Majorana surface code — The Majorana surface code is a Majorana stabilizer analogue of the surface code.
  • Neural network quantum code — Reinforcement learners can be used to optimize the geometry of the surface code to be more suited to a noise channel [169].
  • Raussendorf-Bravyi-Harrington (RBH) cluster-state code — The RBH state encodes the temporal gate operations on the surface code into a third spatial dimension [132,170]. In addition, one of possible 2D boundaries of the RBH code is effectively a 2D toric code.
  • \([[144,12,12]]\) gross code — The gross code requires less physical and ancilla qubits (for syndrome extraction) than the surface code with the same number of logical qubits and distance. The gross code is equivalent to 8 copies of the surface code via a constant-depth Clifford circuit, and is an element of a larger family of 2D stabilizer codes [171]. An architecture combining the surface and gross codes was proposed in [172].
  • \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code — Quantum Hamming codes can be concatened with surface codes [173].
  • 2D color code — The 2D color code is equivalent to multiple decoupled copies of the 2D surface code via a local constant-depth Clifford circuit [174–176]. This process can be viewed as an ungauging [177–179,179] of certain symmetries. 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 triangular directions to the stabilizer group and then taking the center of the resulting nonabelian group [180]. Both the surface and 2D color codes can be constructed from two distinct types of lattices, namely, 4-valent and 3-valent 3-colorable lattices, respectively [181].
  • Generalized five-squares code — Decoding of five-squares codes leads to a mapping of these codes to two copies of the surface code [182,183].
  • Heavy-hexagon code — Surface code stabilizers are used to measure the Z-type stabilizers of the code. There are various ways to embed the surface code into the heavy-hex lattice [184].
  • 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 \(em\) [185; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [185; Fig. 12].
  • 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 \(e_2m_2\) [185; Sec. 7.4].
  • Subsystem surface code
  • Fracton stabilizer code — Foliated type-I fracton phase codes can be grown by foliation, i.e., stacking copies of the 2D surface code; see [186; Eq. (D32)].

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Zoo Code ID: surface

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“Kitaev surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/surface
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@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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