On the Quantum Circuit Complexity Equivalence

Nielsen [3] recently asked the following question: ”What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation U , without the use of ancilla qubits?” Nielsen was able to prove that a lower bound on the minimal size circuit is provided by the length of the geodesic between the identity I and U , where the length is defined by a suitable Finsler metric on SU(2). We prove that the minimum circuit size that simulates U is in linear relation with the geodesic length and simulation parameters, for the given Finsler structure F . As a corollary we prove the highest lower bound of O( 4 p d2Fp(I, U)LFp(I, Ũ)) and the lowest upper bound of Ω(n 4d3Fp(I, U)), for the standard simulation technique. Therefore, our results show that by standard simulation one can not expect a better then n times improvement in the upper bound over the result from Nielsen, Dowling, Gu and Doherty [4]. Moreover, our equivalence result can be applied to the arbitrary path on the manifold including the one that is generated adiabatically.