Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

Abstract

What kinds of symmetry-protected topologically ordered (SPTO) ground states can be used for universal measurement-based quantum computation in a similar fashion to the 2D cluster state? 2D SPTO states are classified not only by global on-site symmetries but also by subsystem symmetries, which are fine-grained symmetries dependent on the lattice geometry. Recently, all states within so-called SPTO cluster phases on the square and hexagonal lattices have been shown to be universal, based on the presence of subsystem symmetries and associated structures of quantum cellular automata. Motivated by this observation, we analyze the computational capability of SPTO cluster phases on all vertex-translative 2D Archimedean lattices. There are four subsystem symmetries here called ribbon, cone, fractal, and 1-form symmetries, and the former three are fundamentally in one-to-one correspondence with three classes of Clifford quantum cellular automata. We conclude that nine out of the eleven Archimedean lattices support universal cluster phases protected by one of the former three symmetries, while the remaining lattices possess 1-form symmetries and have a different capability related to error correction.

Subsystem symmetry enriched topological order in three dimensions

“…Regarding the SSET model from Sec. III, the fact that it combines the universal (measurement-based) quantum computational power of the SSPT order [26][27][28][29] together with the information storage capabilities of the topological order [84][85][86] suggests some potential applications. A relevant phenomenon that may be of use is the emergence of a symmetry-protected degeneracy on the looplike excitations.…”

“…A new paradigm in the classification of gapped phases of matter has recently begun thanks to the discovery of models with novel subdimensional physics. This includes the fracton topological phases [1][2][3][4][5][6][7][8][9][10][11][12], in which topological quasiparticles are either immobile or confined to move only within subsystem such as lines or planes, as well as models with subsystem symmetries, which are symmetries that act nontrivially only on rigid subsystems of the entire system [7,8,[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. These two types of models are dual since gauging subsystem symmetries can result in fracton topological order, analogous to how topological order can be obtained by gauging a global symmetry [7,8,15].…”

“…SSPT phases display new physics compared to conventional SPT phases such as extensive edge degeneracy and unique entanglement properties. They also serve as resources for quantum computation via the paradigm of measurement-based quantum computation [23][24][25][26][27][28][29].…”

Generating Fault-Tolerant Cluster States from Crystal Structures

“…Like the foliated repetition code, this 2D cluster has no checks on dual qubits. However, this partial protection extends to more general cluster phases defined on 2D lattices supporting one-form symmetries [55].…”

Identification of Symmetry-Protected Topological States on Noisy Quantum Computers

“…Reference [13] established that the quantum-wire protocol is a uniform property of all ground states belonging to a given SPT phase of 1D spin chains. This result was subsequently extended to include measurement-based quantum gates in 1D SPT phases [14,15] and, finally, to universal MBQC in 2D SPT phases [16][17][18][19].…”