Scaling of variational quantum circuit depth for condensed matter systems

Carlos Bravo-Prieto1,2, Josep Lumbreras-Zarapico1, Luca Tagliacozzo1, and José I. Latorre1,3,4

1Departament de Física Quàntica i Astrofísica and Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain.
2Barcelona Supercomputing Center, Barcelona, Spain.
3Center for Quantum Technologies, National University of Singapore, Singapore.
4Technology Innovation Institute, Abu Dhabi, UAE.

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We benchmark the accuracy of a variational quantum eigensolver based on a finite-depth quantum circuit encoding ground state of local Hamiltonians. We show that in gapped phases, the accuracy improves exponentially with the depth of the circuit. When trying to encode the ground state of conformally invariant Hamiltonians, we observe two regimes. A $\textit{finite-depth}$ regime, where the accuracy improves slowly with the number of layers, and a $\textit{finite-size}$ regime where it improves again exponentially. The cross-over between the two regimes happens at a critical number of layers whose value increases linearly with the size of the system. We discuss the implication of these observations in the context of comparing different variational ansatz and their effectiveness in describing critical ground states.

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