Fault-tolerant quantum computing promises reliable access to quantum computations beyond the reach of classical methods. Achieving this goal requires both efficient quantum error correction and algorithms that make effective use of limited logical-qubit and gate resources. This talk explores these complementary challenges, focusing on quantum low-density parity-check (qLDPC) codes and the simulation of quantum many-body dynamics.

I begin by introducing quantum error correction through the toric code, discussing syndrome extraction, decoding, and the distinction between practical decoder performance and fundamental error thresholds. Statistical-mechanical mappings provide a framework for investigating optimal thresholds, while circuit-level noise and decoding latency impose additional constraints on hardware implementations.

I then discuss qLDPC codes, which offer improved encoding efficiency but often require interactions beyond nearest neighbors. I present tile-code architectures that implement syndrome extraction on two-dimensional planar lattices using strictly local gates and optimized routing. These constructions illustrate the tradeoffs between connectivity, error thresholds, and physical-qubit overhead. I also discuss ongoing investigations of the three-dimensional toric code, whose decoding problem under depolarizing noise maps to a coupled Ising and random-plaquette gauge model.

Finally, I review quantum many-body dynamics as an application of fault-tolerant quantum computers. Hamiltonian simulation enables the calculation of dynamical correlations and supports quantum phase estimation, but conventional product-formula implementations can require substantial resources for gate synthesis. Architecture-aware approaches, including the STAR architecture, suggest opportunities to improve implementation efficiency beyond standard Clifford+T decompositions. The central theme is that progress toward useful fault-tolerant quantum computation depends on coordinating code design, hardware architecture, and algorithm development.