Characterization of universal two-qubit Hamiltonians

TL;DR

This paper provides a complete characterization of 2-qubit Hamiltonians that are not universal, identifying specific structural conditions that prevent universality and discussing implications for higher qubit systems.

Contribution

It offers the first explicit classification of non-universal 2-qubit Hamiltonians, clarifying the conditions under which they fail to generate all unitaries.

Findings

Three conditions for 2-qubit Hamiltonians to be non-universal

Almost all 2-qubit Hamiltonians are universal except under specific structural constraints

Non-universal 2-qubit Hamiltonians can still be universal for more than two qubits

Abstract

Suppose we can apply a given 2-qubit Hamiltonian H to any (ordered) pair of qubits. We say H is n-universal if it can be used to approximate any unitary operation on n qubits. While it is well known that almost any 2-qubit Hamiltonian is 2-universal (Deutsch, Barenco, Ekert 1995; Lloyd 1995), an explicit characterization of the set of non-universal 2-qubit Hamiltonians has been elusive. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. In particular, there are three ways that a 2-qubit Hamiltonian H can fail to be universal: (1) H shares an eigenvector with the gate that swaps two qubits, (2) H acts on the two qubits independently (in any of a certain family of bases), or (3) H has zero trace. A 2-non-universal 2-qubit Hamiltonian can still be n-universal for some n >= 3. We give some partial results on 3-universality. Finally, we also show how our characterization of 2-universal Hamiltonians implies the well-known result that almost any 2-qubit unitary is universal.