M. Kjaergaard, M. E. Schwartz, A. Greene, G. O. Samach, A. Bengtsson, M. O’Keeffe, C. M. McNally, J. Braumüller, D. K. Kim, P. Krantz, M. Marvian, A. Melville, B. M. Niedzielski, Y. Sung, R. Winik, J. Yoder, D. Rosenberg, K. Obenland, S. Lloyd, T. P. Orlando, I. Marvian, S. Gustavsson, W. D. Oliver

Density matrix exponentiation (DME) is a general technique for using a quantum state $\rho $ to enact the quantum operation ${e}^{-i\rho \theta}$ on a target system. It was first proposed in the context of quantum machine learning, but has since been shown to have broad applications in quantum metrology and computation. No experimental demonstration of DME has been performed thus far due to its demanding circuit depths and the need to efficiently generate multiple identical copies of $\rho $ during the finite lifetime of the target system. In this work, we describe the first demonstration of the DME algorithm, which we accomplish using a superconducting quantum processor. Our demonstration relies on a 99.7% fidelity controlled-phase gate implemented using two tunable superconducting transmon qubits. We achieve a fidelity surpassing 90% at circuit depths exceeding 70 when comparing the output of the circuit executed on our quantum processor to a simulation assuming perfect operations and measurements.