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arXiv:2311.13680 · Physical review. D/Physical review. D.

Qudit stabilizer codes, CFTs, and topological surfaces

Matthew Buican, Rajath Radhakrishnan

Published Oct 31, 2024• 4 citations
arXivHTMLPDF

Abstract

We study general maps from the space of rational conformal field theories (CFTs) with a fixed chiral algebra and associated Chern-Simons (CS) theories to the space of qudit stabilizer codes with a fixed generalized Pauli group. We consider certain natural constraints on such a map and show that the map can be described as a graph homomorphism from an orbifold graph, which captures the orbifold structure of CFTs, to a code graph, which captures the structure of self-dual stabilizer codes. By studying explicit examples, we show that this graph homomorphism cannot always be a graph embedding. However, we construct a physically motivated map from universal orbifold subgraphs of CFTs to operators in a generalized Pauli group. We show that this map results in a self-dual stabilizer code if and only if the surface operators in the bulk CS theories corresponding to the CFTs in question are self-dual. For CFTs admitting a stabilizer code description, we show that the full Abelianized generalized Pauli group can be obtained from twisted sectors of certain 0-form symmetries of the CFT. Finally, we connect our construction with SymTFTs, and we argue that many equivalences between codes that arise in our setup correspond to equivalence classes of bulk topological surfaces under fusion with invertible surfaces. Published by the American Physical Society 2024

DOI
10.1103/physrevd.110.085021

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We study general maps from the space of rational conformal field theories (CFTs) with a fixed chiral algebra and associated Chern-Simons (CS) theories to the space of qudit stabilizer codes with a fixed generalized Pauli group. We consider certain natural constraints on such a map and show that the map can be described as a graph homomorphism from an orbifold graph, which captures the orbifold structure of CFTs, to a code graph, which captures the structure of self-dual stabilizer codes. By studying explicit examples, we show that this graph homomorphism cannot always be a graph embedding. However, we construct a physically motivated map from universal orbifold subgraphs of CFTs to operators in a generalized Pauli group. We show that this map results in a self-dual stabilizer code if and only if the surface operators in the bulk CS theories corresponding to the CFTs in question are self-dual. For CFTs admitting a stabilizer code description, we show that the full Abelianized generalized Pauli group can be obtained from twisted sectors of certain 0-form symmetries of the CFT. Finally, we connect our construction with SymTFTs, and we argue that many equivalences between codes that arise in our setup correspond to equivalence classes of bulk topological surfaces under fusion with invertible surfaces. Published by the American Physical Society 2024
Abstract We give a general construction relating Narain rational conformal field theories (RCFTs) and associated 3d Chern-Simons (CS) theories to quantum stabilizer codes. Starting from an abelian CS theory with a fusion group consisting of n even-order factors, we map a boundary RCFT to an n-qubit quantum code. When the relevant ’t Hooft anomalies vanish, we can orbifold our RCFTs and describe this gauging at the level of the code. Along the way, we give CFT interpretations of the code subspace and the Hilbert space of qubits while mapping error operations to CFT defect fields.
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In this paper, we introduce an algorithm for extracting topological data from translation invariant generalized Pauli stabilizer codes in two-dimensional systems, focusing on the analysis of anyon excitations and string operators. The algorithm applies to Z d qudits, including instances where d is a nonprime number. This capability allows the identification of topological orders that differ from the Z d toric codes. It extends our understanding beyond the established theorem that Pauli stabilizer codes for Z p qudits (with p being a prime) are equivalent to finite copies of Z p toric codes and trivial stabilizers. The algorithm is designed to determine all anyons and their string operators, enabling the computation of their fusion rules, topological spins, and braiding statistics. The method converts the identification of topological orders into computational tasks, including Gaussian elimination, the Hermite normal form, and the Smith normal form of truncated Laurent polynomials. Furthermore, the algorithm provides a systematic approach for studying quantum error-correcting codes. We apply it to various codes, such as self-dual CSS quantum codes modified from the two-dimensional honeycomb color code and non-CSS quantum codes that contain the double semion topological order or the six-semion topological order. Published by the American Physical Society 2024
Stabilizer codes provide exact lattice realizations of bosonic topological orders. In contrast, systematic stabilizer descriptions of intrinsically fermionic topological phases remain much less developed. In this work, we introduce Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models whose stabilizers are built from both generalized Pauli operators and Majorana operators. As a main example, we construct an exactly solvable stabilizer realization of the fermionic toric code: an intrinsically fermionic Z2\mathbb Z_2Z2​ topological order in (2+1)(2{+}1)(2+1) dimensions, using Z8\mathbb Z_8Z8​ Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for ZDF\mathbb Z_D^FZDF​ symmetries for DDD even. Using this, we realize a stabilizer model for a nontrivial Z8F\mathbb Z_8^FZ8F​ fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.
We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories–this includes anyon theories with degenerate braiding relations and those without a gapped boundary to the vacuum. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and establishes that the classification is at least as rich as that of Abelian anyon theories. We exemplify the construction with topological subsystem codes defined on four-dimensional qudits based on the Z4(1) anyon theory with degenerate braiding relations and the chiral semion theory–both of which cannot be captured by topological stabilizer codes. The construction proceeds by "gauging out" certain anyon types of a topological stabilizer code. This amounts to defining a gauge group generated by the stabilizer group of the topological stabilizer code and a set of anyonic string operators for the anyon types that are gauged out. The resulting topological subsystem code is characterized by an anyon theory containing a proper subset of the anyons of the topological stabilizer code. We thereby show that every Abelian anyon theory is a subtheory of a stack of toric codes and a certain family of twisted quantum doubles that generalize the double semion anyon theory. We further prove a number of general statements about the logical operators of translation invariant topological subsystem codes and define their associated anyon theories in terms of higher-form symmetries.