arXiv subjects

Pak Kau Lim

Publications and source records attributed to Pak Kau Lim.

6 recordsLinked to original sources

arXiv2025

Abelian multi-cycle codes for single-shot error correction

We construct a family of quantum low-density parity-check codes locally equivalent to higher-dimensional quantum hypergraph-product (QHP) codes. Similarly to QHP codes, the proposed codes have highly redundant sets of low-weight stabilizer generators, which improves decoding accuracy in a fault-tolerant regime and gives them single-shot properties. The advantage of the new construction is that it gives shorter codes. We derive simple expressions for the dimension of the proposed codes in two important special cases, give bounds on the distances, and explicitly construct some relatively short codes. Circuit simulations for codes locally equivalent to 4-dimensional toric codes show a (pseudo)threshold close to 1.1%, better than for toric or surface codes with a similar noise model.

quant-ph↗

arXiv2025

Single-shot and two-shot decoding with generalized bicycle codes

Generalized-bicycle (GB) and more general two-block group-algebra (2BGA) quantum error-correcting codes have naturally redundant minimum-weight stabilizer generators. To use this redundancy, we constructed a large number of ``planar'' 2BGA codes over abelian groups with one and two generators, with each block row of weight 3, relatively large dimensions, distances, and maximum syndrome distance $d_{\rm S}=3$. We simulated the performance of three such codes under phenomenological noise and standard circuit noise, using sliding window sequential decoding protocol covering $T\ge 1$ measurement rounds at a time, based on an in-house binary BP+OSD decoder. While true single-shot decoding ($T=1$) suffers from a significant loss of accuracy, already two-shot ($T=2$) decoding gives nearly the same logical error rates as multi-shot with much larger $T$. Comparison with the same codes but additional stabilizer generators dropped shows that redundancy significantly improves decoding accuracy for all $T\ge 1$.

quant-ph↗

arXiv2024

Topological orders with classical Lie group symmetries from coupling electron wires

We study the topological order that arises from chiral states with ${\rm SU}(N)$ or ${\rm SO}(N)$ edge-state symmetry. This extends our previous study of topological orders that descend from the bosonic $E_8$ quantum Hall state. We use exactly solvable models of coupled electron wires to construct states with ${\rm SU}(m)_n$, ${\rm SO}(m)_n$, or ${\rm Sp}(m)_n$ topological order for various levels $n$. We use our constructions to write down string operators for various non-Abelian anyons. We thereby provide a systematic, model derivation of quantum Hall states, topological superconductors, and spin liquids with emergent non-Abelian quasiparticle excitations, including those of Ising, metaplectic, and Fibonacci type.

cond-mat.str-el↗

arXiv2023

Partial fillings of the bosonic $E_8$ quantum Hall state

We study bosonic topological phases constructed from electrons. In addition to a bulk excitation energy gap, these bosonic phases also have a fermion energy gap, below which all local excitations in the bulk and on the edge are even combinations of electrons. We focus on chiral phases, in which all low-energy edge excitations move in the same direction, that arise from the short-range entangled $E_8$ quantum Hall state, the bosonic analog of the filled lowest Landau level of electrons. The $E_8$ edge-state theory features an $E_8$ Kac-Moody symmetry that can be decomposed into ${\cal G}_A \times {\cal G}_B$ subalgebras, such as $SU(3) \times E_6$, $SO(M) \times SO(16-M)$, and $G_2 \times F_4$. (Here, $\{SO(M) \}$, $\{SU(N)\}$, and $\{E_8, G_2, F_4 \}$ denote orthogonal, unitary, and exceptional Lie algebras.) Using these symmetry decompositions, we construct exactly solvable coupled-wire model Hamiltonians for families of long-range entangled ${\cal G}_A$ or ${\cal G}_B$ bosonic fractional quantum Hall states that ``partially fill" the $E_8$ state and are pairwise related by a generalized particle-hole symmetry. These long-range entangled states feature either Abelian or non-Abelian topological order. Some support the emergence of non-local Dirac and Majorana fermions, Ising anyons, metaplectic anyons, Fibonacci anyons, as well as deconfined $\mathbb{Z}_2$ gauge fluxes and charges.

cond-mat.str-el↗

arXiv2021

Disentangling (2+1)d Topological States of Matter with the Entanglement Negativity

We use the entanglement negativity, a bipartite measure of entanglement in mixed quantum states, to study how multipartite entanglement constrains the real-space structure of the ground state wavefunctions of $(2+1)$-dimensional topological phases. We focus on the (Abelian) Laughlin and (non-Abelian) Moore-Read states at filling fraction $ν=1/m$. We show that a combination of entanglement negativities, calculated with respect to specific cylinder and torus geometries, determines a necessary condition for when a topological state can be disentangled, i.e., factorized into a tensor product of states defined on cylinder subregions. This condition, which requires the ground state to lie in a definite topological sector, is sufficient for the Laughlin state. On the other hand, we find that a general Moore-Read ground state cannot be disentangled even when the disentangling condition holds.

cond-mat.str-el↗

arXiv2019

Topological defects in general quantum LDPC codes

We consider the structure of defects carrying quantum information in general quantum low-density parity-check (LDPC) codes. These generalize the corresponding constructions for topological quantum codes, without the need for locality. Relation of such defects to (generalized) topological entanglement entropy is also discussed.

quant-ph↗