\((2,2)\) Loop toric code

\((2,2)\) Loop toric code[1]

Alternative names: Kitaev tesseract code, 4D surface code, All-loop toric code, \((2,2)\) 4D toric code.

Description

A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((2,2)\) toric code because it admits 2D membrane \(Z\)-type and \(X\)-type logical operators. Both types of operators create 1D (i.e., loop) excitations at their edges. The code serves as a self-correcting quantum memory [1,2].

Qubits are placed on plaquettes, each \(Z\)-type stabilizer generator is supported on six plaquettes surrounding an edge, and \(X\)-type stabilizers are placed on the six plaquettes of every cube [3].

Loop toric code often either refers to the construction on the four-dimensional torus or is an alternative name for the general construction.

The construction has been extended to modular qudits [4].

Protection

Code parameters for an open hypercubic lattice of side-length \(L\) are \([[6L^4 − 12L^3 + 10L^2 - 4L + 1, 1, L^2]]\) [5].

Encoding

Lindbladian-based dissipative encoding, for which codespace is steady-state space of a Lindbladian [6].

Transversal Gates

Only logical Clifford gates can be implemented transversally when defined on a hypercubic lattice [3].

Gates

Logical \(S\) gate using physical \(CS\) gates via the Pontryagin square [4].Logical \(CZ\), \(S\), and Hadamard gates when defined on a hypercubic lattice [7].

Decoding

Local automaton decoder [8] based on Toom’s rule for the classical 2D repetition code [9–12].Local automaton decoder obtained from reinforcement learning [13].

Code Capacity Threshold

Independent \(X,Z\) noise: \(2.117\%\) with Hastings decoder [8] and \(7.3\%\) with RG decoder for 4D surface code [14]. It is conjectured via a statistical-mechanical mapping that the optimal ML decoder yields a threshold of \(11.003\%\) [15].

Threshold

Phenomenological noise model for the 4D loop toric code: \(4.35\%\) with RG decoder [14], and \(4.3\%\) under improved BP-OSD decoder [16].Gate-based depolarizing noise: \(0.31\%\) with RG decoder for 4D loop toric code [14].\(1.59\%\) for independent \(X,Z\) noise and faulty syndrome measurements using the Hastings decoder [8].

Realizations

Trapped ions: single-shot QEC realized using a \([[33,1,4]]\) rotated version of the loop toric code on the Quantinuum H2 device [5].

Cousins

Higher-dimensional homological product code— The 4D loop planar (toric) code on a hypercubic lattice can be obtained from a particular choice of chain complex from a hypergraph product of four repetition codes [17].
Repetition code— The 4D loop planar (toric) code on a hypercubic lattice can be obtained from a particular choice of chain complex from a hypergraph product of four repetition codes [17].
Campbell double homological product code— The 4D loop planar (toric) code on a hypercubic lattice can be obtained from a particular choice of chain complex from a hypergraph product of four repetition codes [17]. As such, it is a particular Campbell double homological product code [18; table I].
Self-correcting quantum code— For similar reasons as the classical 2D Ising model is a self-correcting classical memory, the 4D loop toric code is a self-correcting quantum memory due to an order \(O(n)\) energy cost of creating a logical error [1,2].
Haah cubic code (CC)— The energy of any partial implementation of CC1 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.

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

The 4D loop toric code realizes 4D \(\mathbb{Z}_2\) gauge theory with only loop excitations [4].

Lattice stabilizer codeQLDPC Stabilizer Hamiltonian-based QECC Quantum

Single-shot codeQECC Quantum

Single-shot QEC has been realized using the \([[33,1,4]]\) loop toric code on the Quantinuum H2 device [5].

Abelian topological codeTopological Hamiltonian-based QECC Quantum