FINDING: Surface codes are stabilizer quantum error-correcting codes whose logical operators and error syndromes are governed by the p4m wallpaper group (square lattice with reflection symmetries), with Kitaev's toric code as the canonical example; Clifford theory extends to glider representations via chains of normal subgroups, linking to hierarchical symmetry breaking. | MATH: Toric code stabilizers: $A_v = \prod_{i \in \square(v)} X_i$ , $B_p = \prod_{i \in \square(p)} Z_i$ , with $A_v^2 = B_p^2 = I$ , $[A_v, B_p] = 0$ for all vertices $v$ , plaquettes $p$ . Logical operators: $\bar{X} = \prod_{\text{non-contractible loop}} X_i$ , $\bar{Z} = \prod_{\text{non-contractible dual loop}} Z_i$ . Distance $d = O(\sqrt{N})$ for $N$ physical qubits. p4m group: generated by translations $t_x, t_y$ , 90° rotation $r$ , reflections $m_x, m_y$ ; relations $r^4 = m_x^2 = m_y^2 = (m_x m_y)^2 = 1$ , $r m_x r^{-1} = m_y$ . Glider representations: for chain $1 \triangleleft G_1 \tr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Surface codes are stabilizer quantum error-correcting codes whose logical operators and error syndromes are governed by the p4m wallpaper group (square lattice with reflection symmetries), with Kitaev's toric code as the canonical example; Clifford theory extends to glider representations via chains of normal subgroups, linking to hierarchical symmetry breaking. | MATH: Toric code stabilizers: \(A_v = \prod_{i \in \square(v)} X_i$, $B_p = \prod_{i \in \square(p)} Z_i$, with $A_v^2 = B_p^2 = I$, $[A_v, B_p] = 0$ for all vertices $v$, plaquettes $p$. Logical operators: $\bar{X} = \prod_{\text{non-contractible loop}} X_i$, $\bar{Z} = \prod_{\text{non-contractible dual loop}} Z_i$. Distance $d = O(\sqrt{N})$ for $N$ physical qubits. p4m group: generated by translations $t_x, t_y$, 90° rotation $r$, reflections $m_x, m_y$; relations $r^4 = m_x^2 = m_y^2 = (m_x m_y)^2 = 1$, $r m_x r^{-1} = m_y$. Glider representations: for chain \(1 \triangleleft G_1 \tr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 23, 20260 citations

Quantum Groups in Lattice Models

Vincent Pasquier

NATO ASI series. Series B : PhysicsJan 1, 19921 citation

Quantum Mechanics and Lattices

WORLD SCIENTIFIC eBooksMay 1, 20000 citations

Belief Propagation and Ground State Search

Hendrik Kühne

Tensor Networks have recently received much attention in the physics community, thanks to their capabilities in modeling quantum-mechanical operations.Their contraction often poses a significant challenge due to di!culties in contraction order search and memory constraints.Belief Propagation o"ers a way around these problems: it represents a memory e!cient contraction scheme, that is completely independent of the topology of the Tensor Network.This thesis aims to investigate the applicability of Belief Propagation to one of Computational Physics' most prominent problems: the ground state problem.Given a Hamiltonian in an arbitrary (possibly high-dimensional) lattice, it is not at all obvious how one can e!ciently find its ground state.Since the Belief Propagation algorithm is geometry-independent, I combine it with the Density Matrix Renormalization Group algorithm, thereby introducing BP-DMRG.I show that BP-DMRG is able to recover ground states of the transverse-field Ising model and the Heisenberg model on random lattices.Moreover, BP-DMRG o"ers insight into the e!cacy of Belief Propagation on physical states; I show that ground states of the TFI are not amenable to accurate contraction by Belief Propagation.Tensor Networks have recently received much attention in the physics community, thanks to their capabilities in modeling quantum-mechanical operations.Their contraction often poses a significant challenge due to di!culties in contraction order search and memory constraints.Belief Propagation o"ers a way around these problems: it represents a memory e!cient contraction scheme, that is completely independent of the topology of the Tensor Network.This thesis aims to investigate the applicability of Belief Propagation to one of Computational Physics' most prominent problems: the ground state problem.Given a Hamiltonian in an arbitrary (possibly high-dimensional) lattice, it is not at all obvious how one can e!ciently find its ground state.Since the Belief Propagation algorithm is geometry-independent, I combine it with the Density Matrix Renormalization Group algorithm, thereby introducing BP-DMRG.I show that BP-DMRG is able to recover ground states of the transverse-field Ising model and the Heisenberg model on random lattices.Moreover, BP-DMRG o"ers insight into the e!cacy of Belief Propagation on physical states; I show that ground states of the TFI are not amenable to accurate contraction by Belief Propagation.

Jan 1, 20250 citations

Quantum Lattice Problems

Hans A. De Raedt, Wolfgang von der Linden

Topics in applied physicsJan 1, 19926 citations

Single-layer tensor network approach for three-dimensional quantum systems

I. V. Lukin, Andrii Sotnikov

Calculation of observables with three-dimensional projected entangled pair states is generally hard, as it requires a contraction of complex multi-layer tensor networks. We utilize the multi-layer structure of these tensor networks to largely simplify the contraction. The proposed approach involves the usage of the layer structure both to simplify the search for the boundary projected entangled pair states and the single-layer mapping of the final corner transfer matrix renormalization group contraction. We benchmark our results on the cubic lattice Heisenberg model, reaching the bond dimension D = 7 , and find a good agreement with the previous results. Published by the American Physical Society 2024Calculation of observables with three-dimensional projected entangled pair states is generally hard, as it requires a contraction of complex multi-layer tensor networks. We utilize the multi-layer structure of these tensor networks to largely simplify the contraction. The proposed approach involves the usage of the layer structure both to simplify the search for the boundary projected entangled pair states and the single-layer mapping of the final corner transfer matrix renormalization group contraction. We benchmark our results on the cubic lattice Heisenberg model, reaching the bond dimension D = 7 , and find a good agreement with the previous results. Published by the American Physical Society 2024

arXiv:2405.01489Physical review. B./Physical review. BAug 19, 20246 citations

A_v = \prod_{i \in \square(v)} X_i

B_p = \prod_{i \in \square(p)} Z_i

, with

A_v^2 = B_p^2 = I

[A_v, B_p] = 0

for all vertices

v

, plaquettes

p

. Logical operators:

\bar{X} = \prod_{\text{non-contractible loop}} X_i

\bar{Z} = \prod_{\text{non-contractible dual loop}} Z_i

. Distance

d = O(\sqrt{N})

for

N

physical qubits. p4m group: generated by translations

t_x, t_y

, 90° rotation

r

, reflections

m_x, m_y

; relations

r^4 = m_x^2 = m_y^2 = (m_x m_y)^2 = 1

r m_x r^{-1} = m_y

. Glider representations: for chain \(1 \triangleleft G_1 \tr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com