Independent Trivariate Bicycle Codes and Hyperbolic Codes

Abstract

Quantum error correction is necessary for fault-tolerant quantum computation. The surface code achieves a threshold of approximately $1\%$ under circuit-level depolarizing noise and planar connectivity, but it encodes only one logical qubit per patch, with an encoding rate $k/n$ that approaches zero as code size increases. Finding codes with better encoding rates, higher distances, and practical thresholds under realistic noise models is a central problem in quantum error correction.This dissertation constructs and evaluates quantum error correcting codes using algebraic, topological, and dynamic approaches.Bivariate bicycle codes, introduced by Bravyi et al., use pairs of polynomials over two cyclic shift matrices to build quantum LDPC codes. In Chapter~\ref{ch:trivariate}, we extend this construction to three independent cyclic shift matrices, introducing independent trivariate bicycle codes. Using asymmetric polynomial pairs on three-dimensional tori, we construct four codes including a $[[140,6,14]]$ code with $kd^2/n = 8.40$. In the code-capacity setting, the $[[140,6,14]]$ code achieves a pseudothreshold of $8.0\%$, exceeding the best multivariate bicycle code of Voss et al.\ ($7.9\%$). With circuit-level depolarizing noise, pseudothresholds reach $0.59\%$ for $[[140,6,14]]$ and $0.53\%$ for $[[84,6,10]]$. On the SI1000 superconducting noise model, the $[[140,6,14]]$ code achieves a per-round per-observable rate of $5.6 \times 10^{-5}$ at $p = 0.20\%$. We additionally present self-dual codes with $kd^2/n$ up to $10.0$.Hyperbolic surface codes achieve constant encoding rate by embedding qubits on negatively curved surfaces. In Chapter~\ref{ch:css_erasure}, we construct hyperbolic CSS surface codes from $\{8,3\}$, $\{10,3\}$, and $\{12,3\}$ tessellations via the Wythoff kaleidoscopic construction with the Low-Index Normal Subgroups algorithm. We extend the circuit-level erasure noise model and Wang et al.\ quadratic expansion fitting of Chang et al.\ from planar surface codes to these hyperbolic codes. Applied to the Bolza $k = 4$ semi-hyperbolic family, this pipeline yields erasure thresholds of $3.6\%$ (general model) and $4.7\%$ (tailored model) at erasure fraction $R_e = 1.0$, with erasure-to-Pauli ratios of $5.0\times$ and $6.5\times$ matching the surface code values to within $5\%$. Per-observable crossing-point analysis at $R_e = 1$ (Models 1--3) independently yields a ratio of $5.3\times$, consistent with the surface code's $5.2\times$.Floquet codes replace stabilizer measurements with periodic sequences of weight-2 measurements that dynamically generate the code space. In Chapter~\ref{ch:monolithic_floquet}, we build hyperbolic Floquet codes from the same lattices and evaluate them under correlated EM3, SDEM3, and erasure noise models. Under SDEM3 depolarizing noise motivated by Majorana tetron architectures, all three tessellation families achieve per-observable thresholds of ${\sim}1.0$--$1.2\%$. Under erasure noise motivated by spin-optical architectures, fine-grained $\{8,3\}$ codes achieve thresholds of ${\sim}8.5$--$9\%$, exceeding the ${\sim}6.3\%$ planar honeycomb threshold while encoding $k = 4$ logical qubits.Scaling quantum computers beyond a single chip will likely require distributing qubits across networked quantum processing units. In Chapter~\ref{ch:distributed}, we distribute the Floquet codes across processing units via spectral bisection and evaluate them under depolarizing and erasure noise with asymmetric local and non-local error rates. With depolarizing noise at local fidelity $99.97\%$, fine-grained codes achieve non-local pseudothresholds up to $3.0\%$. Under distributed erasure noise at $1\%$ local loss, thresholds reach $35$--$40\%$ for $\{8,3\}$.Overall, this dissertation explores how algebraic, topological, and dynamic constructions can produce practical quantum error correcting codes with improved encoding rates, distances, and thresholds under hardware-motivated noise models.