Time evolution of impurity models and their universality for quantum computation

TL;DR

This paper proves that the time evolution of generic time-independent impurity Hamiltonians on qubits is universal for quantum computation, with impurity size scaling as O(S log S) for depth S.

Contribution

It demonstrates universality of time-independent impurity Hamiltonian evolution for quantum computation and characterizes impurity size scaling.

Findings

Time evolution of impurity Hamiltonians can perform universal quantum computation.

Impurity size scales as O(S log S) for depth S computations.

Universality holds for generic time-independent impurity Hamiltonians on O(N) qubits.

Abstract

Impurity Hamiltonians are systems of $N$ fermionic modes where $O(1)$ of them interact among themselves via quartic (or higher order) fermion terms, while coupling quadratically with $O(N)$ bath modes. Without the quartic interactions, these systems are classically simulable with $O(N^3)$ resources. It was proved that the time-dependent evolution of these systems can perform universal quantum computation. The question of whether or not this remains true for time-independent evolution remains open. Here, we prove that the time evolution of generic time-independent impurity Hamiltonians on $O(N)$ qubits is universal on $N$ qubits if the input state is a product state of fermions in any single particle basis. In our proof we find that for a computation of depth $S$, the size of the impurity scales as $O(S\log S)$.