Connecting geometry and performance of two-qubit parameterized quantum circuits

Amara Katabarwa1, Sukin Sim1,2, Dax Enshan Koh3, and Pierre-Luc Dallaire-Demers1

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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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.

The Quantum Natural Gradient (QNG) is a version of gradient based optimization that was invented to speed up the optimization of parametrized quantum circuits. The update rule used in this scheme is $\theta_{t+1} \longmapsto \theta_t – \eta g^{+} \nabla \mathcal{L}(\theta_t)$, where $\mathcal{L}(\theta_t)$ is the cost function used, like for example the expectation value of some an operator at some iteration step $t$, and $g^{+}$ is the pseudo-inverse of the quantum natural gradient. This was shown to speed up finding optimal parameters of quantum circuits used to approximate ground states. Strangely though, $g$ involves derivates of the trial wave function and nothing about the cost function landscape; so how does it use the geometry of the Hilbert space to speed up the optimization? We study the case of two qubits where we can calculate the geometry fully and see what is happening. We find that the QNG is finding places of negative Ricci curvature that are correlated with acceleration of the optimization procedure. We present numerical evidence that this correlation is actually causal.

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