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Understanding and protecting against errors happening on a quantum device constitute one of the biggest challenges in developing a fault-tolerant quantum computing architecture. Recently, the focus of the community shifted from a static perspective, where information is encoded into a (fixed) subspace, to a dynamical perspective where one allows for the logical subspace to change over time [1], with the most famous example being Floquet codes, error-correcting protocols defined by a periodic sequence of low-weight non-commuting (Pauli) measurements [2]. Understanding and analyzing such protocols in the presence of errors is an important challenge to devise and to understand new fault-tolerant protocols. We argue that tensor network methods inspired by the ZX calculus are good tools to analyze the error-correction properties of dynamical protocols as they allow for a simple, graphical, yet complete understanding of the evolution of logical information through a circuit, also in the presence of errors [3]. In this talk, I want to give an introduction on how to represent circuits composed of Clifford unitaries and Pauli measurements graphically. First, I introduce the elementary tensors from which one can build tensor networks resembling the circuits of the above form. To understand error-correcting properties of a circuit expressed as a tensor network, I introduce the notion of *projective Pauli flow* in terms of projective symmetries of the building blocks of the network [4]. I will show how all quantities needed to perform error-correction on a circuit can be understood graphically as different types of Pauli flow. If time allows, I show how local relations amongst the elementary tensors can be used to ‘deform’ a circuit into a different, but equivalent one [3,5]. Importantly, these local deformations preserve the global Pauli flows and with that the logical action of the circuit and certain error-correction properties. This allows to transform given error-correcting circuits into new ones while preserving their error-correcting nature. [1] Delfosse, Paetznick; “Spacetime codes of Clifford circuits”, arXiv:2304.05943 [2] Hastings, Haah; “Dynamically Generated Logical Qubits”, arXiv:2107.02194 [3] Bombin et al; “Unifying flavors of fault tolerance with the ZX calculus”, arXiv:2303.08829 [4] Magdalena de la Fuente, Old, et al.; in preparation [5] Townsend-Teague, Magdalena de la Fuente, Kesselring; “Floquetifying the Colour Code”, arXiv:2307.11136

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Julio C. Magdalena de la Fuente, FU Berlin
Seminar Room 0.03, ETP
Contact: Guo-Yi Zhu