3D surface code

3D surface code [1,2]

Alternative Names: 3D toric code, 3D cubic code, Bosonic-charge bosonic-loop (BcBl) surface code, Solid 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 [6,7] places 3D surface codes on the edges of a 3D cubic lattice and welds their rough boundaries at each vertex. Welding identifies the qubits of the joined boundaries, keeps all \(X\)-type stabilizers, and takes as \(Z\)-type stabilizers all \(Z\)-type operators commuting with them [6]. This first gives a code with no logical qubits, so a welded \(Z\)-type stabilizer generator is then promoted to a logical operator to encode one qubit [6]. A \(Z\) error on a welded qubit excites an \(X\)-type stabilizer in every code meeting there, so \(X\)-type excitations split at each weld [6]. The string-like \(Z\)-type logical operator becomes a branching string whose energy barrier is lower bounded by that of the Ising model on the lattice [6]. 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 [8].

Related models [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]. Membrane-like logical operators have an energy barrier of order \(O(L)\), while string-like logical operators have a constant energy barrier [6].

The welded surface code on a cube of side length \(L\) has parameters \([[O(L^3),1,O(L^{4/3})]]\) and an energy barrier of order \(O(L^{2/3})\) [6,7]. The welded surface code is translation invariant only over a length that grows with system size, which evades the no-go theorems for translationally invariant 3D stabilizer codes [6,7].

A tensor-network representation of the 3D surface-code ground state is stable under a generating set of uniform local tensor perturbations [11]. The phase diagram of the perturbed tensor network maps to that of a 3D Ising gauge theory.

Transversal and Permutation-Based 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 [8].

Gates

There is a CZ gate for the 3D toric code on a Klein bottle \(\times S^1\) [12].Lattice surgery [8].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 [8].Fault-tolerant Hadamard gate using teleportation and error correction [8].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 [8,15–20], 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].Local decoder based on Toom’s rule for bit-flip errors and MWPM decoder for phase-flip errors in the 3D toric code, with extensions to welded surface codes [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 [8].

Code Capacity Threshold

Independent \(X,Z\) noise: \(12\%\) for the bit-flip channel with a local decoder based on Toom’s rule and \(3\%\) for the phase-flip channel with MWPM decoding for the 3D toric code [26]. RG decoding yields \(17.2\%\) for the surface-like logical operator of the 3D cubic code. The optimal threshold for its line-like logical operator is about \(3.3\%\) [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 welded surface code [26].

Threshold

Phenomenological noise model for the 3D toric code: \(2.90(2)\%\) under MWPM syndrome repair followed by BP-OSD decoding [29], \(7.1\%\) under improved BP-OSD [30], and \(2.6\%\) under flip decoder [24]. For the surface-like logical operator of the 3D cubic code, RG decoding yields \(7.3\%\) [4]. For 3D surface code: \(3.08(4)\%\) under MWPM syndrome repair followed by BP-OSD decoding [29]. Optimal thresholds of \(11\%\) under \(X\)-type and \(2\%\) under \(Z\)-type noise for the 3D toric code are 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 [8].
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— A tensor-network representation of the 3D surface-code ground state is stable under a generating set of uniform local tensor perturbations [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 welded surface code has a power-law energy barrier [6,7] and is partially self-correcting [7]. 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].
Recursive 3D self-correcting code— The local complexes that replace each square in the recursive 3D self-correcting code construction resemble pieces of thickness-one 3D surface code slabs with membrane-condensing front and back boundaries [38; Remark 3.7].
Sierpinski prism model code— The Sierpinski prism model code admits a topological defect network construction out of 3D surface codes on triangular prisms [39,40].
Haah cubic code (CC)— The Haah B-code admits a topological defect network construction out of two copies of the 3D surface code [39].
X-cube model code— The X-cube model admits a topological defect network construction out of 3D surface codes [39,40].
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 [15,41,42]. This process can be viewed as an ungauging [43–52] of certain symmetries. This mapping can also be done via code concatenation [8]. 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 [9].
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— 3D fractal surface codes are obtained by removing qubits from the 3D surface code on a cubic lattice.
Three-fermion (3F) Walker-Wang model code— The 3F Walker-Wang model code is obtained from two copies of the 3D surface code on the same cubic lattice [53,54]. The copies are twisted together by decorating the face terms with \(X\)-type operators on the edges lying over and under each face [54]. Flux of one copy then confines the point excitations of the other [54].
3D subsystem color code— The 3D subsystem color code can be ungauged [43–52] to obtain six copies of \(\mathbb{Z}_2\) gauge theory with one-form symmetries [48].
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 codeQubit Stabilizer Hamiltonian-based QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeQLDPC Qubit CSS Generalized homological-product 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 [55].