Quantum circuits for toric code and X-cube fracton model

Penghua Chen1, Bowen Yan1, and Shawn X. Cui1,2

1Department of Physics and Astronomy, Purdue University, West Lafayette
2Department of Mathematics, Purdue University, West Lafayette

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We propose a systematic and efficient quantum circuit composed solely of Clifford gates for simulating the ground state of the surface code model. This approach yields the ground state of the toric code in $\lceil 2L+2+log_{2}(d)+\frac{L}{2d} \rceil$ time steps, where $L$ refers to the system size and $d$ represents the maximum distance to constrain the application of the CNOT gates. Our algorithm reformulates the problem into a purely geometric one, facilitating its extension to attain the ground state of certain 3D topological phases, such as the 3D toric model in $3L+8$ steps and the X-cube fracton model in $12L+11$ steps. Furthermore, we introduce a gluing method involving measurements, enabling our technique to attain the ground state of the 2D toric code on an arbitrary planar lattice and paving the way to more intricate 3D topological phases.

In this paper, we introduce a systematic and efficient quantum circuit, composed solely of Clifford gates, for simulating the ground state of a general surface code with linear depth. Our algorithm reformulates the problem into a purely geometric framework, which facilitates its extension to achieve the ground state of specific 3D topological phases, such as the 3D toric model and the X-cube fracton model, while maintaining linear depth. Additionally, we introduce a gluing method that balances the simulation capabilities with the use of measurement, paving the way for more intricate simulations of 3D topological phases and even the ground state of more general Pauli Hamiltonians.

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Cited by

[1] Xie Chen, Arpit Dua, Michael Hermele, David T. Stephen, Nathanan Tantivasadakarn, Robijn Vanhove, and Jing-Yu Zhao, "Sequential quantum circuits as maps between gapped phases", Physical Review B 109 7, 075116 (2024).

[2] Nathanan Tantivasadakarn and Xie Chen, "String operators for Cheshire strings in topological phases", arXiv:2307.03180, (2023).

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