Sujay Kazi

The problem of coherence distillation asks: given multiple copies of a coherent input state, produce a coherent output state with much less noise (that is, much closer to a pure coherent state), in such a way that the time evolution of the output state matches the time evolution of the input state. We thoroughly address this problem for the case of qubit states. Principally, we compute the distillation channel that is asymptotically optimal in the limit of a large number of input qubit states and compute the resulting infidelity (one minus fidelity) between the output qubit state and the desired pure state. We show that this infidelity is closely related to a resource known as purity of coherence, a quantity obtained from the Right Logarithmic Derivative (RLD) Fisher information metric. We thus demonstrate an operational meaning for purity of coherence as the quantity whose monotonicity on time-translation invariant channels sets the strongest asymptotic bound on the capability of coherence distillation for qubit states. For a special choice of input and output state, we additionally investigate a number of extensions of this primary result. In particular, we present numerical schemes to produce the exact optimal distillation protocol for a fixed value of $N$, extend the computation of the optimal protocol and the minimum infidelity to the $1/N^2$ and $1/N^3$ orders, estimate the amount of purity of coherence that is dissipated by the optimal distillation protocol, and compute the mathematical behavior of the protocol more precisely in the low-noise and high-noise limits.

Barren plateaus have emerged as a pivotal challenge for variational quantum computing. Our understanding of this phenomenon underwent a transformative shift with the recent introduction of a Lie algebraic theory capable of explaining most sources of barren plateaus. However, this theory requires either initial states or observables that lie in the circuit's Lie algebra. Focusing on parametrized matchgate circuits, in this work we are able to go beyond this assumption and provide an exact formula for the loss function variance that is valid for arbitrary input states and measurements. Our results reveal that new phenomena emerge when the Lie algebra constraint is relaxed. For instance, we find that the variance does not necessarily vanish inversely with the Lie algebra's dimension. Instead, this measure of expressiveness is replaced by a generalized expressiveness quantity: the dimension of the Lie group modules. By characterizing the operators in these modules as products of Majorana operators, we can introduce a precise notion of generalized globality and show that measuring generalized-global operators leads to barren plateaus. Our work also provides operational meaning to the generalized entanglement as we connect it with known fermionic entanglement measures, and show that it satisfies a monogamy relation. Finally, while parameterized matchgate circuits are not efficiently simulable in general, our results suggest that the structure allowing for trainability may also lead to classical simulability.