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Canonical quantization is not well suited to quantize gravity, while affine quantization is. For those unfamiliar with affine quantization the talk will include a primer. This procedure is then applied to deal with three nonrenormalizable, field theoretical, problems of increasing difficulty, the last one being general relativity itself.

The need for a time-shift invariant formulation of quantum theory arises from fundamental symmetry principles as well as heuristic cosmological considerations. Such a description then leaves open the question of how to reconcile global invariance with the perception of change, locally. By introducing relative time observables, we are able to make rigorous the Page-Wootters conditional probability formalism to show how local Heisenberg evolution is compatible with global invariance.

We study approximate quantum low-density parity-check (QLDPC) codes, which are approximate quantum error-correcting codes specified as the ground space of a frustration-free local Hamiltonian, whose terms do not necessarily commute. Such codes generalize stabilizer QLDPC codes, which are exact quantum error-correcting codes with sparse, low-weight stabilizer generators (i.e. each stabilizer generator acts on a few qubits, and each qubit participates in a few stabilizer generators). Our investigation is motivated by an important question in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes with constant rate, linear distance, and constant-weight stabilizers exist? We show that obtaining such optimal scaling of parameters (modulo polylogarithmic corrections) is possible if we go beyond stabilizer codes: we prove the existence of a family of [[N,k,d,ε]] approximate QLDPC codes that encode k = Ω(N/polylog N) into N physical qubits with distance d = Ω(N/polylog N) and approximation infidelity ε = 1/polylog N. We prove the existence of an efficient encoding map, and we show that arbitrary Pauli errors can be locally detected by circuits of polylogarithmic depth. Finally, we show that the spectral gap of the code Hamiltonian is Ω(N^(-3.09)) (up to polylog(N) factors) by analyzing a spacetime circuit-to-Hamiltonian construction for a bitonic sorting network architecture that is spatially local in polylog(N) spatial dimensions. (Joint work with Elizabeth Crosson, Chinmay Nirkhe, and Henry Yuen, arXiv:1811.00277)