1Zapata Computing, Inc., 100 Federal Street, 20th Floor, Boston, Massachusetts 02110, USA
2Harvard University
3Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR), 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore
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Abstract
Parameterized quantum circuits (PQCs) are a central component of many variational quantum algorithms, yet there is a lack of understanding of how their parameterization impacts algorithm performance. We initiate this discussion by using principal bundles to geometrically characterize two-qubit PQCs. On the base manifold, we use the Mannoury-Fubini-Study metric to find a simple equation relating the Ricci scalar (geometry) and concurrence (entanglement). By calculating the Ricci scalar during a variational quantum eigensolver (VQE) optimization process, this offers us a new perspective to how and why Quantum Natural Gradient outperforms the standard gradient descent. We argue that the key to the Quantum Natural Gradient’s superior performance is its ability to find regions of high negative curvature early in the optimization process. These regions of high negative curvature appear to be important in accelerating the optimization process.
Featured image: Addition of single qubit gates allows the Quantum Natural gradient to find places of negative Ricci curvature. Note that before we could not find the ground state which is entangled and so addition of single qubit gates does not change the entanglement properties of the ansatz and yet their addition allowed us to find an entangled state we could not previously find; we argue that what changed was the geometry.
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Cited by
[1] Tobias Haug and M. S. Kim, “Natural parameterized quantum circuit”, arXiv:2107.14063.
[2] Francesco Scala, Stefano Mangini, Chiara Macchiavello, Daniele Bajoni, and Dario Gerace, “Quantum variational learning for entanglement witnessing”, arXiv:2205.10429.
[3] Roeland Wiersema and Nathan Killoran, “Optimizing quantum circuits with Riemannian gradient flow”, arXiv:2202.06976.
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