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. There exists a rotated version of the 3D surface code, the 3D rotated surface code, akin to the (2D) rotated surface code [4]. The welded surface code [5] consists of several 3D surface codes stitched together in a way that the distance scales faster than the linear size of the system.

Related models [6,7] on lattices with certain colorability are equivalent to several copies of the 3D surface code [8].

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 \([[3L^3,3,d_X=L^2,d_Z=L]]\).

Stability against Hamiltonian perturbations was determined using a tensor-network representation [9]. The phase diagram of the perturbed tensor network maps to that of a 3D Ising gauge theory.

Transversal Gates

Transversal \(CZ\) and \(CCZ\) gates [10–16], with \(CZ\) gates formulated in terms of the slant product [17,18] or cup product [19] structures.

Gates

There is a CZ gate for the 3D toric code on a Klein bottle \(\times S^1\) [20].Lattice surgery [11].3D and Hybrid 2D-3D surface code computation using lattice surgery and without magic-state distillation [11].Fault-tolerant Hadamard gate using teleportation and error correction [11].

Decoding

Flip decoder and its modification p-flip [21].Tensor-network decoder [22].Efficient MWPM decoder for 3D toric and 3D welded surface codes handling string-like syndromes only [23].Generalization of linear-time ML erasure decoder [24] to 3D surface codes [23].Equivariant machine learning decoder [25].

Fault Tolerance

Fault-tolerant Hadamard gate using teleportation and error correction [11].

Code Capacity Threshold

Independent \(X,Z\) noise: \(12\%\) for bit-flip and \(3\%\) for phase-flip channels with MWPM decoder for 3D toric code [23], and \(17.2\%\) and \(3.3\%\) with RG decoder for 3D toric code [26].Erasure noise: \(24.8\%\) with generalization of linear-time ML erasure decoder [24] to 3D surface codes [23]. No threshold was observed for the 3D welded surface code [23].

Threshold

Phenomenological noise model for the 3D toric code: \(2.90(2)\%\) under BP-OSD decoder [27], \(7.1\%\) under improved BP-OSD [28], \(7.3\%\) under RG [26], and \(2.6\%\) under flip decoder [21]. For 3D surface code: \(3.08(4)\%\) under flip decoder [27].

Cousins

Chamon model code— The 3D planar (3D toric) code can be obtained from a hypergraph product of three repetition (cyclic) codes [30][29; Exam. A.1], while the Chamon code is an XYZ product of three repetition codes [31; Sec. 3.4].
Rotated surface code— There exists a rotated version of the 3D surface code, akin to the (2D) rotated surface code [4].
Hamiltonian-based code— Stability of the 3D surface code against Hamiltonian perturbations was determined using a tensor-network representation [9]. The phase diagram of the perturbed tensor network maps to that of a 3D Ising gauge theory.
Self-correcting quantum code— The 3D welded surface code is partially self-correcting with a power-law energy barrier [5]. The 3D toric code is a classical self-correcting memory, whose protected bit admits a membrane-like logical operator [32], but it is not a quantum self-correcting memory because the star terms thermalize [33].
Floquet 3D surface code— The ISG of the Floquet 3D surface code is that of the 3D 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. A rewinding of the original measurement sequence yields a period-six sequence whose ISGs are those of the X-cube model, some 3D surface codes, and some decoupled surface codes [34].
\([[8,3,2]]\) Smallest interesting color code— The \([[8,3,2]]\) code can be concatenated with a 3D surface code to yield a \([[O(d^3),3,2d]]\) code family that admits a transversal implementation of the logical \(CCZ\) gate [35].
Sierpinski prism model code— The Sierpinski prism model code admits a topological defect network construction out of 3D surface codes on triangular prisms [36,37].
Haah cubic code (CC)— The Haah B-code admits a topological defect network construction out of two copies of the 3D surface code [36].
X-cube model code— The X-cube model admits a topological defect network construction out of 3D surface codes [36].
3D color code— The 3D color code is equivalent to multiple decoupled copies of the 3D surface code via a local constant-depth Clifford circuit [10,38,39]. This process can be viewed as an ungauging [40–49] of certain symmetries. This mapping can also be done via code concatenation [11].
Klein-bottle surface code— There is a CZ gate for the 3D toric code on a Klein bottle \(\times S^1\) [20].
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 \(f_2\).
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 [40–49] to obtain six copies of \(\mathbb{Z}_2\) gauge theory with one-form symmetries [45].
3D subsystem 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 codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Homological code

Parents

Homological code

Higher-dimensional HGP codeGeneralized homological-product QLDPC Stabilizer Hamiltonian-based Qubit QECC Quantum

The 3D planar (3D toric) code can be obtained from a hypergraph product of three repetition (cyclic) codes [30][29; Exam. A.1].

Modular-qudit 3D surface codeLattice stabilizer Generalized homological-product QLDPC CSS 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 [50].