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Magic state distillation is a crucial yet resource-intensive process in fault-tolerant quantum computation. The protocol’s overhead, defined as the number of input magic states required per output magic state with an error rate below ϵ, typically grows as O(log^γ (1/ϵ)) as ϵ → 0. Achieving smaller overheads, i.e., smaller exponents γ, is highly desirable; however, all existing protocols require polylogarithmically growing overheads with some γ > 0, and identifying the smallest achievable exponent γ for distilling magic states of qubits has remained challenging. To address this issue, we develop magic state distillation protocols for qubits with efficient, polynomial-time decoding that achieve an O(1) overhead, meaning the optimal exponent γ = 0; this improves over the previous best of γ ≈ 0.678 due to Hastings and Haah. In our construction, we employ algebraic geometry codes to explicitly present asymptotically good quantum codes for 2^10-dimensional qudits that support transversally implementable logical gates in the third level of the Clifford hierarchy. These codes can be realized by representing each 2^10-dimensional qudit as a set of 10 qubits, using stabilizer operations on qubits. We prove that the use of asymptotically good codes with non-vanishing rate and relative distance in magic state distillation leads to the constant overhead. The 10-qubit magic states distilled with these codes can be converted to and from conventional magic states for the controlled-controlled-Z (CCZ) and T gates on qubits with only a constant overhead loss, making it possible to achieve constant-overhead distillation of such standard magic states for qubits. These results resolve the fundamental open problem in quantum information theory concerning the construction of magic state distillation protocols with the optimal exponent. The talk is based on the following paper. https://arxiv.org/abs/2408.07764

In my talk, I will argue that symmetry strongly changes the behaviour of massive gravity relative to its massless sibling.   After reviewing the formulation of dRGT theory of ghost-free massive gravity, I will examine the minimal model and next-to-minimal model in spherical symmetry. Although the latter has been argued to have a good Vainshtein mechanism in spherical symmetry, I will derive a restriction on non-relativistic matter that is at odds with a reasonable phenomenology. Moreover, the theory cannot reproduce the same behaviour as GR for a scalar field collapse in the small mass limit without encountering a singularity at some point in the evolution.   Since symmetry is the epitome of non-genericity, the resolution may well be that we should forgo symmetry and focus on more generic behaviour in massive gravity to study the screening mechanism. This talk is based on https://arxiv.org/pdf/2409.18802