3D surface code

3D surface code[1,2]

Alternative names: 3D toric code, 3D cubic code, Bosonic-charge bosonic-loop (BcBl) surface code.

Description

A generalization of the Kitaev surface code defined on a 3D cubic lattice. Qubits are placed on edges, \(Z\)-type stabilizer generators are placed on square plaquettes oriented in all three directions, and \(X\)-type stabilizers are placed on the six edges neighboring every vertex [3].

3D toric code often either refers to the construction on the three-dimensional torus or is an alternative name for the general construction. The construction on surfaces with boundaries is often called the 3D planar code. In the open-boundary hypercubic family of [4], setting one linear dimension of the tesseract construction to \(1\) yields a single-qubit cubic code that interpolates between the planar surface code and the 4D tesseract code. There exists a rotated version of the 3D surface code, the 3D rotated surface code, akin to the (2D) rotated surface code [5]. The welded surface code [6] consists of several 3D surface codes stitched together in a way that the distance scales faster than the linear size of the system. In the rectified picture, three 3D surface codes can be supported on the same rectified cubic lattice, and the corresponding cubic-lattice realization is a gauge choice of the 3D Bacon-Shor code [7].

Related models [8,9] on lattices with certain colorability are equivalent to several copies of the 3D surface code [10].

Protection

The planar 3D surface code family on a cubic lattice of length \(L\) has parameters \([[2L(L-1)^2+L^3,1,d_X=L^2,d_Z=L]]\), while the 3D toric code has parameters \([[3L^3,3,d_X=L^2,d_Z=L]]\). Rectangular open-boundary 3D versions furnish single-qubit \([[71,1,6]]\), \([[177,1,9]]\), \([[331,1,12]]\), and \([[616,1,16]]\) codes, illustrating an \(n\propto 3d^2\) tradeoff between the 2D surface and 4D tesseract families [4].

Stability against Hamiltonian perturbations was determined using a tensor-network representation [11]. The phase diagram of the perturbed tensor network maps to that of a 3D Ising gauge theory.

Transversal Gates

For a stack of three 3D surface codes on the same rectified cubic lattice, pairwise logical \(CZ\) and triple logical \(CCZ\) gates are transversal [7].

Gates

There is a CZ gate for the 3D toric code on a Klein bottle \(\times S^1\) [12].Lattice surgery [7].Single-shot lattice surgery for the 3D toric/surface code can be formulated using the fault-complex formalism [13].3D and Hybrid 2D-3D surface code computation using lattice surgery and without magic-state distillation [7].Fault-tolerant Hadamard gate using teleportation and error correction [7].Three distance-\(d\) 3D surface/toric codes with open boundaries and cyclically permuted lattice axes admit a logical \(CCZ\) gate via transversal physical \(CCZ\) gates; concatenating each supporting qubit triple with an \([[8,3,2]]\) block yields a \([[8n,3,2d]]\) 3D toric/color family whose smallest member has parameters \([[72,3,4]]\) [14].Various inter-code \(CZ\) and \(CCZ\) gates implemented via constant-depth circuits on stacks or coupled collections of 3D surface/toric codes [7,15–20], with \(CZ\) gates formulated in terms of the slant product [21,22] or cup product [23] structures.

Decoding

Flip decoder and its modification p-flip [24].Tensor-network decoder [25].Efficient MWPM decoder for 3D toric and 3D welded surface codes handling string-like syndromes only [26].Generalization of linear-time ML erasure decoder [27] to 3D surface codes [26].Equivariant machine learning decoder [28].

Fault Tolerance

Fault-tolerant Hadamard gate using teleportation and error correction [7].

Code Capacity Threshold

Independent \(X,Z\) noise: \(12\%\) for bit-flip and \(3\%\) for phase-flip channels with MWPM decoder for 3D toric code [26], and \(17.2\%\) for the surface-like logical operator together with \(3.3\%\) for the line-like logical operator of the 3D cubic code under RG decoding [4].Erasure noise: \(24.8\%\) with generalization of linear-time ML erasure decoder [27] to 3D surface codes [26]. No threshold was observed for the 3D welded surface code [26].

Threshold

Phenomenological noise model for the 3D toric code: \(2.90(2)\%\) under BP-OSD decoder [29], \(7.1\%\) under improved BP-OSD [30], and \(2.6\%\) under flip decoder [24]. For the line-like logical operator of the 3D cubic code, RG decoding yields \(7.3\%\) [4]. For 3D surface code: \(3.08(4)\%\) under flip decoder [29]. Optimal thresholds of \(11\%\) under \(X\)-type and \(2\%\) under \(Z\)-type noise derived in Ref. [31].

Cousins

Bacon-Shor code— In the rectified-cubic construction, the resulting cubic-lattice 3D surface codes are particular gauge choices of the 3D Bacon-Shor code [7].
Chamon model code— The 3D planar and toric code on a cubic lattice can be obtained from a hypergraph product of three repetition codes [33][32; Exam. A.1]. The Chamon code is an XYZ product of three repetition codes [34; Sec. 3.4].
Rotated surface code— There exists a rotated version of the 3D surface code, akin to the (2D) rotated surface code [5].
Hamiltonian-based code— Stability of the 3D surface code against Hamiltonian perturbations was determined using a tensor-network representation [11]. The phase diagram of the perturbed tensor network maps to that of a 3D Ising gauge theory.
Higher-dimensional homological product code— The 3D planar and toric code on a cubic lattice can be obtained from a hypergraph product of three repetition codes [33][32; Exam. A.1].
Repetition code— The 3D planar and toric code on a cubic lattice can be obtained from a hypergraph product of three repetition codes [33][32; Exam. A.1].
Self-correcting quantum code— The 3D welded surface code is partially self-correcting with a power-law energy barrier [6]. The 3D toric code is a classical self-correcting memory, whose protected bit admits a membrane-like logical operator [35], but it is not a quantum self-correcting memory because the star terms thermalize [36].
Floquet 3D surface code— The G-round ISG is FDLQC-equivalent to the 3D surface code, while the other rounds are FDLQC-equivalent to two copies of the 3D surface code up to non-local stabilizers [37].
X-cube Floquet code— The B- and first R-round ISGs of the rewinding schedule are FDLQC-equivalent to the product of the X-cube model, a 3D surface code, and a 3-foliated stack of 2D surface codes up to non-local stabilizers [37].
\([[8,3,2]]\) Smallest interesting color code— Three cyclically rotated copies of the 3D surface/toric code admit a logical \(CCZ\) gate via transversal physical \(CCZ\) gates, and concatenating each such qubit triple with an \([[8,3,2]]\) block yields a 3D toric/color family with parameters \([[8n,3,2d]]\); its smallest member has parameters \([[72,3,4]]\) [14].
Sierpinski prism model code— The Sierpinski prism model code admits a topological defect network construction out of 3D surface codes on triangular prisms [38,39].
Haah cubic code (CC)— The Haah B-code admits a topological defect network construction out of two copies of the 3D surface code [38].
X-cube model code— The X-cube model admits a topological defect network construction out of 3D surface codes [38].
3D color code— On closed 3-manifolds, the 3D color code is equivalent to multiple decoupled copies of the 3D surface code via a local constant-depth Clifford circuit [15,40,41]. This process can be viewed as an ungauging [42–51] of certain symmetries. This mapping can also be done via code concatenation [7]. In contrast to the 3D surface/toric code, the original colex Hamiltonian can be viewed as both a string-net condensate and a membrane-net condensate [8].
Tetrahedral color code— A tetrahedral 3D color code with four differently colored boundaries is equivalent, via a local Clifford circuit, to three 3D surface codes attached along one boundary, with condensation of a composite electric charge on that attached boundary [15].
Klein-bottle surface code— There is a CZ gate for the 3D toric code on a Klein bottle \(\times S^1\) [12].
3D fermionic surface code— The 3D (fermionic) surface code is a CSS (non-CSS) code which realizes a \(\mathbb{Z}_2\) gauge theory in 3D (with an emergent fermion). Two copies of the 3D fermionic surface code are equivalent to a copy of the 3D surface code and a copy of the 3D fermionic surface code via anyon relabeling: the two incoming fermions, \(f_1\) and \(f_2\), can be re-organized into a boson \(f_1 f_2\) and fermion \(f_2\).
\((2,2)\) Loop toric code— Setting one linear size of the open-boundary tesseract construction to \(1\) yields the cubic/3D surface code [4].
Fractal surface code— Fractal surface codes are obtained by removing qubits from the 3D surface code on a cubic lattice.
3D subsystem color code— The 3D subsystem color code can be ungauged [42–51] to obtain six copies of \(\mathbb{Z}_2\) gauge theory with one-form symmetries [47].
3D subsystem surface code— The 3D subsystem surface code is a subsystem version of the 3D surface code.

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

Parents

Homological code

Modular-qudit 3D surface codeCSS Lattice stabilizer Generalized homological-product Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

The qudit 3D surface code reduces to the 3D surface code for \(q=2\). The 3D surface code realizes 3D \(\mathbb{Z}_2\) gauge theory with bosonic charge and loop excitations (BcBl). The welded surface code does not satisfy homogeneous topological order [52].