02685nas a2200253 4500008004100000245007700041210006900118260001400187520182700201653002102028653002702049653004202076653003502118653003002153653003102183653005202214100002302266700001602289700002102305700002202326700003002348700001602378856003702394 2022 eng d00aCodimension-2 defects and higher symmetries in (3+1)D topological phases0 aCodimension2 defects and higher symmetries in 31D topological ph c8/15/20223 a

(3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of Z2 gauge theory with fermionic charges, in Z2×Z2 gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral (Dn) and alternating (A6) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an H4 cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D A6 gauge theory.

10aFOS: Mathematics10aFOS: Physical sciences10aHigh Energy Physics - Theory (hep-th)10aMathematical Physics (math-ph)10aQuantum Algebra (math.QA)10aQuantum Physics (quant-ph)10aStrongly Correlated Electrons (cond-mat.str-el)1 aBarkeshli, Maissam1 aChen, Yu-An1 aHuang, Sheng-Jie1 aKobayashi, Ryohei1 aTantivasadakarn, Nathanan1 aZhu, Guanyu uhttps://arxiv.org/abs/2208.0736701455nas a2200205 4500008004100000245006000041210005900101260001500160520078600175653002100961653002700982653003501009653003001044653003101074653005201105100001601157700002501173700001401198856003701212 2022 eng d00aError-correcting codes for fermionic quantum simulation0 aErrorcorrecting codes for fermionic quantum simulation c10/16/20223 a

We provide ways to simulate fermions by qubits on 2d lattices using Z2 gauge theories (stabilizer codes). By studying the symplectic automorphisms of the Pauli module over the Laurent polynomial ring, we develop a systematic way to increase the code distances of stabilizer codes. We identify a family of stabilizer codes that can be used to simulate fermions with code distances of d=2,3,4,5,6,7 such that any ⌊d−12⌋-qubit error can be corrected. In particular, we demonstrate three stabilizer codes with code distances of d=3, d=4, and d=5, respectively, with all stabilizers and logical operators shown explicitly. The syndromes for all Pauli errors are provided. Finally, we introduce a syndrome-matching method to compute code distances numerically.

10aFOS: Mathematics10aFOS: Physical sciences10aMathematical Physics (math-ph)10aQuantum Algebra (math.QA)10aQuantum Physics (quant-ph)10aStrongly Correlated Electrons (cond-mat.str-el)1 aChen, Yu-An1 aGorshkov, Alexey, V.1 aXu, Yijia uhttps://arxiv.org/abs/2210.0841102101nas a2200217 4500008004100000245007000041210006900111260001500180520136900195653002101564653002701585653004201612653003001654653003101684653005201715100002301767700001601790700001801806700002201824856003701846 2022 eng d00aHigher-group symmetry in finite gauge theory and stabilizer codes0 aHighergroup symmetry in finite gauge theory and stabilizer codes c11/21/20223 a

A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, we derive the d-group global symmetry and its 't Hooft anomaly for topological finite group gauge theories in (d+1) space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the [O5]∈H5(BG,U(1)) obstruction than has appeared in previous work. We also show how the d-group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We construct new logical gates in stabilizer codes using the d-group symmetry, such as the control-Z gate in (3+1)D Z2 toric code.

10aFOS: Mathematics10aFOS: Physical sciences10aHigh Energy Physics - Theory (hep-th)10aQuantum Algebra (math.QA)10aQuantum Physics (quant-ph)10aStrongly Correlated Electrons (cond-mat.str-el)1 aBarkeshli, Maissam1 aChen, Yu-An1 aHsin, Po-Shen1 aKobayashi, Ryohei uhttps://arxiv.org/abs/2211.1176402246nas a2200205 4500008004100000245006600041210006600107260001400173520157400187653002701761653003101788653005201819100002301871700001601894700001501910700002001925700003001945700002801975856003702003 2022 eng d00aPauli topological subsystem codes from Abelian anyon theories0 aPauli topological subsystem codes from Abelian anyon theories c11/7/20223 a

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 Z(1)4 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.

10aFOS: Physical sciences10aQuantum Physics (quant-ph)10aStrongly Correlated Electrons (cond-mat.str-el)1 aEllison, Tyler, D.1 aChen, Yu-An1 aDua, Arpit1 aShirley, Wilbur1 aTantivasadakarn, Nathanan1 aWilliamson, Dominic, J. uhttps://arxiv.org/abs/2211.0379801959nas a2200217 4500008004100000245010400041210006900145260001400214520119900228653002701427653003501454653003101489653005201520100002001572700001601592700001501608700002301623700003001646700002801676856003701704 2022 eng d00aThree-dimensional quantum cellular automata from chiral semion surface topological order and beyond0 aThreedimensional quantum cellular automata from chiral semion su c2/10/20223 a

We construct a novel three-dimensional quantum cellular automaton (QCA) based on a system with short-range entangled bulk and chiral semion boundary topological order. We argue that either the QCA is nontrivial, i.e. not a finite-depth circuit of local quantum gates, or there exists a two-dimensional commuting projector Hamiltonian realizing the chiral semion topological order (characterized by U(1)2 Chern-Simons theory). Our QCA is obtained by first constructing the Walker-Wang Hamiltonian of a certain premodular tensor category of order four, then condensing the deconfined bulk boson at the level of lattice operators. We show that the resulting Hamiltonian hosts chiral semion surface topological order in the presence of a boundary and can be realized as a non-Pauli stabilizer code on qubits, from which the QCA is defined. The construction is then generalized to a class of QCAs defined by non-Pauli stabilizer codes on 2n-dimensional qudits that feature surface anyons described by U(1)2n Chern-Simons theory. Our results support the conjecture that the group of nontrivial three-dimensional QCAs is isomorphic to the Witt group of non-degenerate braided fusion categories.

10aFOS: Physical sciences10aMathematical Physics (math-ph)10aQuantum Physics (quant-ph)10aStrongly Correlated Electrons (cond-mat.str-el)1 aShirley, Wilbur1 aChen, Yu-An1 aDua, Arpit1 aEllison, Tyler, D.1 aTantivasadakarn, Nathanan1 aWilliamson, Dominic, J. uhttps://arxiv.org/abs/2202.05442