Quantum Physics

Quantum codes with transversal non-Clifford gates have many applications in fault tolerant quantum computation, from magic state distillation to robust IQP circuit sampling.

In the past year, there has been spectacular progress on constructing such codes with optimal parameter scaling, i.e., constant encoding rate and constant relative distance.
These constructions rely on quantum versions of algebraic geometry codes, which generalise the well-known Reed-Solomon and Reed-Muller codes. In this talk, we will describe one such construction that yields a family of good codes with a transversal control-control-Z gate, and we will highlight some of the remaining open problems in this area.

Matchgates are a well studied class of quantum circuits tied to the time dynamics of Free Fermion Hamiltonians. It is important to note however that Matchgates specifically come from representing Free Fermions with the Jordan-Wigner encoding. When we represent our fermionic systems with other encodings besides Jordan-Wigner, we still are considering the time dynamics of Free Fermion solvable Hamiltonians, but we can introduce complexity in how we encode our fermionic information. This gives us a test ground for clarifying what physical properties make time dynamics hard to simulate, even when Hamiltonians can be exactly diagonalized. In this talk I will discuss the theory behind matchgates, fermionic encodings, and recent results in the simulability of Clifford/matchgate hybrid circuits (arxiv:2312.08447, arxiv:2410.10068). These results clarify resources for Free Fermions represented beyond the Jordan-Wigner encoding, as well as an overall perspective of what it means for a state to be Gaussian.