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Investigating Topological Quantum Matter: Machine Learning Topological Phases, Topological Quantum Codes, Interplay of Disorder and Topology via Transport Phenomena and Phase Transitions https://doi.org/10.17863/CAM.120787 Noormandipour, Mohammadreza Slager, Robert-Jan Future quantum technologies must meet three inter-locking demands: (i) faithful yet compact representations of strongly–entangled quantum matter, (ii) scalable error-mitigation protocols that tame spatially correlated noise, and (iii) near-term algorithms that coax useful optimisation and learning behaviour from noisy, intermediate-scale quantum processors. This dissertation attacks all three challenges through a single computational lens that marries variational wave-function design, clause-density-optimal Max-SAT decoding, parameterised quantum-circuit optimisation and disorder-driven band-topology. Part I, casts the spin-1/2 Kitaev honeycomb model into a neural framework by training a restricted Boltzmann machine (RBM) on stochastic-reconfiguration Monte-Carlo data. A custom PyTorch code with translation projection, split real/imaginary learning rates, and polynomial pre-training reaches ground state energies within 0.09% of exact values and 99.8% overlap on the smallest topological lattice (3 ×3), mapping out the optimal hidden-unit density α. Updating RBM weights according to spin-rotation strings lets us create, move, and fuse vortex pairs; resulting plaquette values and energy gaps match the Majorana solution. Quantum-state tomography on a 2×2 cluster recovers the ground state with 97% fidelity and confirms the vortex-braiding protocol. Thus RBMs emerge as precise, symmetry-compatible variational states - and practical anyon simulators - for non-Abelian spin liquids. Part II then turns to fault-tolerance and hybrid optimisation: Chapter 2 recasts maximum-likelihood decoding of CSS stabiliser codes as a weighted Max-3-SAT instance that can be solved in the clause-sparse, algorithmically easy regime, and Chapter 3 develops a depth-1 Born-machine circuit that minimises a kernel-maximum-mean-discrepancy loss to discover every symmetry-related rigid motion between two point clouds, extending naturally to a classically intractable quantum kernel. Part III, finally, probes the robustness of multi-gap Euler semimetals: Chapter 4 defines a real-space Euler marker that reveals disorder-induced unbraiding of quaternion-charged nodes and maps the ensuing quantum criticality to two-dimensional percolation. First, Part I develops a neural-network approach to the spin-1/2 Kitaev honeycomb model. After reviewing the model’s bond-dependent interactions and the Majorana-fermion solution that yields a three-fold degenerate ground state on the torus, we introduce a restricted Boltzmann machine (RBM) variational ansatz whose complex amplitudes Φ(Ξ;Ω) are trained by stochastic-reconfiguration Monte-Carlo. A custom PyTorch implementation featuring translation-projection, separate optimisation schedules for real and imaginary parameters, and a stabilising polynomial pre-training phase achieves ground state energies within 0.09% of exact values and >99.8% overlap on the minimal 3 ×3 lattice, while systematically exploring hidden-unit density α. By identifying spin-rotation string operators with explicit updates of RBM weights, the network can create, transport, and annihilate vortex pairs; plaquette-operator expectation values and energy splittings agree with the analytical Majorana spectrum. Quantum-state tomography is demonstrated on a 2 ×2 cluster, reconstructing the ground state wave-function from synthetic measurement data at 97% fidelity and validating the subsequent vortex-braiding protocol. Altogether, the chapter establishes RBMs as accurate, symmetry-respecting representations for non-Abelian spin liquids and provides practical tools for simulating anyon dynamics in frustrated magnets. Part II turns to fault-tolerance. Chapter 2 maps maximum-likelihood decoding of arbitrary CSS codes-including biased, spatially varying data and measurement noise-onto a weighted Max-3-SAT instance: each syndrome equation becomes a hard clause, while qubit and syndrome error probabilities are encoded as soft clause weights. All resulting formulas have clause density α≤4, which lies deep inside the computationally “easy” phase of satisfiability, ensuring polynomial scaling. We demonstrate, for triangular 6.6.6 colour codes, a depolarising threshold of (15.20 ±0.05)%, two percentage points above belief-propagation with ordered statistics decoding (BP-OSD); we obtain the optimal logical-error suppression p^(d/2)-where BP-OSD saturates at p^(0.75d/2)-and achieve lower logical error rates than BP-OSD on IBM’s bicycle quantum-LDPC codes and on toric codes, while preserving the same empirical runtime scaling O(n^3/2). Because only three-literal clauses arise, the decoder can be compiled into FPGA or ASIC hardware, opening a viable route to sub-microsecond, real-time decoding of large-distance codes. Chapter 3 reformulates rigid point-set matching as distribution learning on SO(d): a depth-1, n-qubit Born machine minimises a kernel maximum-mean-discrepancy loss that coincides with negative kernel correlation. The circuit uncovers all symmetry-related optimal rotations for highly symmetric shapes, generalises to three-dimensional data via a provably characteristic yet classically intractable quantum kernel, and outperforms annealing-based quantum alignment five-fold in accuracy. Part III is devoted to the stability of fragile topology under disorder. Chapter 4 constructs three- and four-band lattice Hamiltonians whose two-band subspaces carry non-trivial Euler class and tracks their response to quenched randomness. Average density of states, Kubo conductivities and a newly defined real-space Euler marker reveal that closing a protecting gap unbraids quaternion-charged nodes, driving a transition to a diffusive metal with critical exponents z = 0.7 ±0.1 and ν= 1.4 ±0.1, matching two-dimensional percolation. Edge modes rooted in a π-Zak phase survive throughout the unbraiding regime-unlike the sub-critical demise of Weyl-arc states-while locally time reversal-breaking disorder nucleates Chern-insulator puddles whose Chern markers satisfy ⟨|C|⟩≈χ, converting fragile Euler topology into quantised domains. Collectively the thesis delivers four broadly applicable advances: (i) a symmetry-exact neural-network representation of Kitaev spin-liquid states; (ii) a clause-density-optimal, hardware-compilable Max-SAT decoder for topological codes; and (iii) a quantum-circuit protocol for symmetric shape matching; (iv) and a real-space Euler marker that diagnoses disorder-driven topological transitions. Throughout, topology plays the binding role: from the anyonic fusion channels of Kitaev and quantum-Hall states, through the homological structure of stabiliser codes, to the Euler-class protection of semimetallic nodes, topological invariants guide both the model design and the computational strategies. By exploiting this unifying theme the dissertation advances quantum-state modelling, error-correction decoding and hybrid quantum–classical optimisation, thereby strengthening the practical foundations required for scalable, resilient quantum technologies. University of Cambridge 2025-05-07 Thesis Doctoral Doctor of Philosophy (PhD) eng https://www.repository.cam.ac.uk/handle/1810/388395 controlled.access https://www.repository.cam.ac.uk/bitstreams/2adf7062-29df-4e0d-9d52-1f0b7ecd9f50/download 50a8180a429ae4a437dd01f0019b87c3 https://www.repository.cam.ac.uk/bitstreams/26ab8674-cf29-4042-bd03-91fa8fed0cb8/download 87eda9de84448d1f82354d60eee3eb5f http://purl.org/NET/rdflicense/allrightsreserved Born machine circuits Chern marker Color Codes CSS codes Disorder-driven phase transitions Error correction / fault tolerance Euler semimetals Fragile topology Hybrid quantum–classical algorithms Kitaev honeycomb model LDPC Codes Maximum Satisfiablity Non-Abelian spin liquids Quantum Kernels Quantum Machine Learning Quantum technologies Topological invariants Topological phases of matter Variational methods