Lewis, Dylan; Wiersema, Roeland; Carrasquilla, Juan; Bose, Sougato; (2025) Geodesic algorithm for unitary gate design with time-independent Hamiltonians. Physical Review A , 111 (5) , Article 052618. 10.1103/PhysRevA.111.052618.

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Abstract

Larger multiqubit quantum gates allow shallower, more efficient quantum circuits, which could decrease the prohibitive effect of noise on algorithms for noisy intermediate-scale quantum (NISQ) devices and fault-tolerant error correction schemes. Such multiqubit gates can potentially be generated by time-independent Hamiltonians comprising only physical (one- and two-local) interaction terms. Here, we present an algorithm that finds the time-independent Hamiltonian for a target quantum gate on n qubits by using the geodesic on the Riemannian manifold of SU(2n). Differential programming is used to determine how the Hamiltonian should be updated to follow the geodesic to the target unitary as closely as possible. We show that our geodesic algorithm outperforms gradient descent methods for standard multiqubit gates such as Toffoli and Fredkin. The geodesic algorithm is then used to find previously unavailable multiqubit gates implementing high-fidelity parity checks, which could be used in a wide array of quantum codes and increase the clock speed of fault-tolerant quantum computers. The geodesic algorithm is demonstrated on an example relevant to current experimental hardware, illustrating a circuit speed up.

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