Quantum Physics

Quantum generators are crucial objects for understanding quantum systems. They encode all the information required to predict the evolution of closed (Hamiltonians) and open memoryless dynamics (Lindbladians). Additionally, they offer structural insight into the noise affecting experimental implementations of quantum processes.

In this talk, we will discuss the definition of continuous-time random unitary evolutions characterized by stochastic Hamiltonians, showing mixing properties and efficient convergence to the uniform measure over the unitary group. We will then consider the problem of embedding a quantum map into a Markovian evolution, presenting a scheme to extrapolate the full description of the Lindbladian that is the best fit for the tomographic measurements of any (noisy) quantum channel.

Finally, we will introduce a new bound on quantum operators generated by arbitrary time-dependent Hamiltonians by leveraging a correspondence with binary trees structures.

References:

[1] E. Onorati et al., ‘Mixing properties of stochastic quantum Hamiltonians’, https://arxiv.org/abs/1606.01914

[2] E. Onorati et al., ‘Fitting quantum noise models to tomographic data’, https://arxiv.org/abs/2103.17243

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.