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Topological Order, Quantum Codes, and Quantum Computation on Fractal Geometries

Zhu, Guanyu and Jochym-O'Connor, Tomas and Dua, Arpit (2022) Topological Order, Quantum Codes, and Quantum Computation on Fractal Geometries. PRX Quantum, 3 (3). ISSN 2691-3399. doi:10.1103/prxquantum.3.030338.

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We investigate topological order on fractal geometries embedded in n dimensions. We consider the n-dimensional lattice with holes at all length scales the corresponding fractal (Hausdorff) dimension of which is D_H = n-δ. In particular, we diagnose the existence of the topological order through the lens of quantum information and geometry, i.e., via its equivalence to a quantum error-correcting code with a macroscopic code distance or the presence of macroscopic systoles in systolic geometry. We first prove a no-go theorem that Z_N topological order cannot survive on any fractal embedded in two spatial dimensions and with D_H = 2-δ. For fractal-lattice models embedded in three dimensions (3D) or higher spatial dimensions, Z_N topological order survives if the boundaries on the holes condense only loop or, more generally, k-dimensional membrane excitations (k ≥ 2), thus predicting the existence of fractal topological quantum memories (at zero temperature) or topological codes that are embeddable in 3D. Moreover, for a class of models that contain only loop or membrane excitations and are hence self-correcting on an n-dimensional manifold, we prove that Z_N topological order survives on a large class of fractal geometries independent of the type of hole boundary and is hence extremely robust. We further construct fault-tolerant logical gates in the Z_2 version of these fractal models, which we term fractal surface codes, using their connection to global and higher-form topological symmetries equivalent to sweeping the corresponding gapped domain walls. In particular, we discover a logical controlled-controlled-Z (ccz) gate corresponding to a global symmetry in a class of fractal codes embedded in 3D with Hausdorff dimension asymptotically approaching D_H = 2+ϵ for arbitrarily small ϵ, which hence only requires a space overhead Ω(d^(2+ϵ)), where d is the code distance. This in turn leads to the surprising discovery of certain exotic gapped boundaries that only condense the combination of loop excitations and certain gapped domain walls. We further obtain logical C^(p)Z gates with p ≤ n−1 on fractal codes embedded in n dimensions. In particular, for the logical C^(n−1)Z in the nth level of the Clifford hierarchy, we can reduce the space overhead to Ω(d^(n−1+ϵ)). On the mathematical side, our findings in this paper also lead to the discovery of macroscopic relative systoles in a class of fractal geometries.

Item Type:Article
Related URLs:
URLURL TypeDescription
Zhu, Guanyu0000-0002-3375-4445
Dua, Arpit0000-0001-6348-4135
Additional Information:G.Z. thanks Maissam Barkeshli for pointing out the absence of topological order on a 2D Sierpiński carpet, which partially motivated this work. A.D. thanks Dominic J. Williamson for a discussion on 2D fractal-lattice models. We thank Nikolas Breuckmann, Nicolas Delfosse, Sam Roberts, Sergey Bravyi, Jay Gambetta, Andrew Cross, Jared Hertzberg, and Su-Kuan Chu for helpful discussions and comments. G.Z. and T.J.O. are supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704. A.D. was supported by IBM during the summer internship and later supported by the Simons Foundation through the collaboration on Ultra-Quantum Matter (651438, XC) and by the Institute for Quantum Information and Matter, a National Science Foundation (NSF) Physics Frontiers Center (Grant No. PHY-1733907).
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
U.S. Department of EnergyUNSPECIFIED
Office of ScienceUNSPECIFIED
National Science FoundationPHY-1733907
Issue or Number:3
Record Number:CaltechAUTHORS:20220926-576446200.4
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:117137
Deposited By: Melissa Ray
Deposited On:28 Sep 2022 21:40
Last Modified:28 Sep 2022 21:40

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