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What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We sbow that a lowerbound on the minimal size is provided by the length of the minimal geodesic between Uand the identity, I, where length is defined by a suitable Finsler metric on the manifoldSU(2n). The geodesic curves on these manifolds have the striking property that oncean initial position and velocity are set, the remMnder of the geodesic is completelydeternfined by a second order differential equation known as the geodesic equation. Thisis in contrast with the usual case in circuit design, either classical or quantum, wherebeing given part of an optimal circuit does not obviously assist in the design of therest of the circuit. Geodesic analysis thus offers a potentially powerful approacb to theproblem of proving quantum circuit lower bounds. In this paper we construct severalFinsler metrics whose minimal length geodesics provide lower bounds on quantum circuitsize. For eacb Finsler metric we give a procedure to compute the corresponding geodesicequation. We also construct a large class of solutions to the geodesic equation, whichwe call Pauli geodesics, since they arise from isometries generated by the Pauli group.For any unitary U diagonal in the computational basis, we sbow that: (a) proposed theminimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length ofthe minimal Pauli geodesic passing from I to U is equivalent to solving an exponentialsize instance of the closest vector in a lattice problem (CVP); and (c) all but a doublyexponentially small fraction of sucb unit aries have nfinimal Pauli geodesics of exponentiallength.