Description
A code based on the cluster state on a square lattice that was used in the first proposal for MBQC [2,3]. In the one-way model, the pre-entangled square-lattice cluster is a universal resource, and the computation is carried out entirely by adaptive single-qubit measurements.Protection
Random measurement outcomes induce Pauli byproduct operators that are tracked classically and propagated to later measurements or the final readout [3]. For computations longer than the available lattice extent, the original paper proposed splitting the computation into consecutive segments and stabilizing each segment using standard error-correction techniques [3].Encoding
Initialization of each qubit in the \(|+\rangle\) state followed by nearest-neighbor Ising-type entangling evolution, equivalently controlled-phase gates on the edges of the square lattice, prepares the resource state [3].Gates
Measurements in the \(Z\) basis remove qubits from the lattice to carve out the computation network. \(X\)-basis measurements propagate quantum information along a wire, adaptive equatorial-basis measurements on a five-qubit chain implement arbitrary single-qubit \(SU(2)\) rotations, and a four-qubit pattern implements CNOT between neighboring wires. Later measurement bases can depend on earlier outcomes because of the tracked Pauli byproducts [3].Universal MBQC remains possible on irregular occupied sublattices above the percolation threshold because wires and gates can be bent and stretched without changing circuit topology [3].Realizations
Encoding on 72 qubits of the Zuchongzhi 3.1 quantum processor [4].Notes
The original proposal discussed implementations using neutral atoms in optical lattices with controlled collisions and capacitively coupled quantum dots [3].Cousin
Primary Hierarchy
Parents
Square-lattice cluster-state code
References
- [1]
- H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles”, Physical Review Letters 86, 910 (2001) arXiv:quant-ph/0004051 DOI
- [2]
- R. Raussendorf, D. Browne, and H. Briegel, “The one-way quantum computer–a non-network model of quantum computation”, Journal of Modern Optics 49, 1299 (2002) arXiv:quant-ph/0108118 DOI
- [3]
- R. Raussendorf and H. J. Briegel, “A One-Way Quantum Computer”, Physical Review Letters 86, 5188 (2001) DOI
- [4]
- T. Jiang et al., “One- and two-dimensional cluster states for topological phase simulation and measurement-based quantum computation”, Nature Physics 22, 430 (2026) arXiv:2505.01978 DOI
- [5]
- D. T. Stephen, H. P. Nautrup, J. Bermejo-Vega, J. Eisert, and R. Raussendorf, “Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter”, Quantum 3, 142 (2019) arXiv:1806.08780 DOI
- [6]
- T. Devakul and D. J. Williamson, “Universal quantum computation using fractal symmetry-protected cluster phases”, Physical Review A 98, (2018) arXiv:1806.04663 DOI
- [7]
- A. K. Daniel, R. N. Alexander, and A. Miyake, “Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices”, Quantum 4, 228 (2020) arXiv:1907.13279 DOI
- [8]
- Y. You, T. Devakul, F. J. Burnell, and S. L. Sondhi, “Subsystem symmetry protected topological order”, Physical Review B 98, (2018) arXiv:1803.02369 DOI
- [9]
- R. Raussendorf, C. Okay, D.-S. Wang, D. T. Stephen, and H. P. Nautrup, “Computationally Universal Phase of Quantum Matter”, Physical Review Letters 122, (2019) arXiv:1803.00095 DOI
Page edit log
- Victor V. Albert (2024-12-06) — most recent
Cite as:
“Square-lattice cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/square_lattice_cluster