An explicit family of unitaries with exponentially minimal length Pauli geodesics

Recently, Nielsen et al [1, 2, 3, 4] have proposed a geometric approach to quantum computation. They’ve shown that the size of the minimum quantum circuits implementing a unitary U, up to polynomial factors, equals to the length of minimal geodesic from identity I through U. They’ve investigated a large class of solutions to the geodesic equation, called Pauli geodesics. They’ve raised a natural question whether we can explicitly construct a family of unitaries U that have exponentially long minimal length Pauli geodesics? We give a positive answer to this question. 1 Preliminary 1.1 Pauli basis and Pauli metrics We define general Pauli matrix σ as tensor product of identity matrix or Pauli matrices X, Y or Z. We define pauli weight of a general Pauli matrix σ as the total number of X, Y and Z in σ, noted by pw(σ). Given a control Hamiltonian H, we can write H in terms of the Pauli operator expansionH = ∑