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Larger multi-qubit quantum gates allow shallower, more efficient quantumcircuits, which could decrease the prohibitive effect of noise on algorithmsfor noisy intermediate-scale quantum (NISQ) devices and fault-tolerant errorcorrection schemes. Such multi-qubit gates can potentially be generated bytime-independent Hamiltonians comprising only physical (one- and two-local)interaction terms. Here, we present an algorithm that finds the strengths ofthe Hamiltonian terms by using the direction of the geodesic to the targetquantum gate on the Riemannian manifold of $\mathrm{SU}(2^n)$ for $n$ qubits.Differential programming is used to determine how the Hamiltonian terms shouldbe updated in order to follow the geodesic to the target unitary as closely aspossible. We numerically compare our geodesic algorithm to gradient descentmethods and show that it finds solutions with considerably fewer steps forstandard multi-qubit gates such as Toffoli and Fredkin. The geodesic algorithmis then used to find previously unavailable multi-qubit gates implementing highfidelity parity checks, which could be used in a wide array of quantum codesand increase the clock speed of fault-tolerant quantum computers.

Geodesic Algorithm for Unitary Gate Design with Time-Independent Hamiltonians