Calderbank-Shor-Steane (CSS) codes are a class of quantum error correction codes that contains the toric code and fracton models. A procedure called foliation defines a cluster state for a given CSS code. We use the CSS chain complex and its tensor product with other chain complexes to describe the topological structure in the foliated cluster state, and argue that it has a symmetry-protected topological order protected by generalized global symmetries supported on cycles in the foliated CSS chain complex. We demonstrate the so-called anomaly inflow between CSS codes and corresponding foliated cluster states by explicitly showing the equality of the gauge transformations of the bulk and boundary partition functions defined as functionals of defect world-volumes. We show that the bulk and boundary defects are related via measurement of the bulk system. Further, we provide a procedure to obtain statistical models associated with general CSS codes via the foliated cluster state, and derive a generalization of the Kramers-Wannier-Wegner duality for such statistical models with insertion of twist defects. We also study the measurement-assisted gauging method with cluster-state entanglers for CSS/fracton models based on recent proposals in the literature, and demonstrate a non-invertible fusion of duality operators. Using the cluster-state entanglers, we construct the so-called strange correlator for general CSS/fracton models. Finally, we introduce a new family of subsystem-symmetric quantum models each of which is self-dual under the generalized Kramers-Wannier-Wegner duality transformation, which becomes a non-invertible symmetry.
Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting
X- and
Z-type checks. We introduce CSS-like codes on
G-valued qudits for any finite group
G that reduce to qubit CSS codes for
G=Z2 yet generalize the Kitaev quantum double model for general groups. The
X-checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while
Z-checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when
G is non-Abelian and simple, every
G-covariant group-CSS code with suitably upper-bounded
Z-check weight and lower-bounded
Z-distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.
Previous work [] has demonstrated that quantum simulation of Abelian lattice gauge theories (Wegner models including the toric code in a limit) in general dimensions can be achieved by local adaptive measurements on symmetry-protected topological (SPT) states with higher-form generalized global symmetries. The entanglement structure of the resource SPT state reflects the geometric structure of the gauge theory. In this work we explicitly demonstrate the anomaly inflow mechanism between the deconfining phase of the simulated gauge theory on the boundary and the SPT state in the bulk by showing that the anomalous gauge variation of the boundary state obtained by bulk measurement matches that of the bulk theory. Moreover, we construct the resource state and the measurement pattern for the measurement-based quantum simulation of a lattice gauge theory with a matter field (Fradkin-Shenker model), where a simple scheme to protect gauge invariance of the simulated state against errors is proposed. We further consider taking an overlap between the wave function of the resource state for lattice gauge theories and that of a parameterized product state, and we derive precise dualities between partition functions with insertion of defects corresponding to gauging higher-form global symmetries, as well as measurement-induced phases where states induced by a partial overlap possess different (symmetry-protected) topological orders. Measurement-assisted operators to dualize quantum Hamiltonians of lattice gauge theories and their noninvertibility are also presented. Published by the American Physical Society 2024
Defects associated with non-invertible symmetries have attracted significant attention in recent years. Among them, Kramers-Wannier (KW) duality defects have been investigated in both classical statistical systems and quantum Hamiltonian models. Aasen et al. analyzed duality defects in the 2D Ising model and in statistical models built from fusion categories, while Koide et al. later constructed a duality defect in 4D lattice gauge theory. In this work, we extend these developments by providing a systematic construction of KW duality defects/KW defects for a broad class of models formulated within the chain complex framework. Our construction employs the strange correlator, an overlap between a topologically ordered state and a product state, to realize these KW defects.
Constructing new quantum codes and understanding their error resilience are central challenges in the development of robust quantum memories. Topological codes are particularly promising due to their favorable error-correcting properties and their connections to phases of matter in many-body physics. In this thesis, we explore the interplay between classical Ising models and quantum error correction through subsystem symmetries, fracton topological order, and Kramers-Wannier-type duality. We study two three-dimensional classical self-dual Ising models with subsystem symmetries, the Tetrahedral Ising model and the Fractal Ising model, investigating their thermal behavior, their relation to fracton phases through subsystem-symmetry gauging, and the properties of the resulting fracton codes. Using a statistical-mechanical mapping, we determine the optimal code-capacity threshold of the Checkerboard code to be
0.107(3), which saturates the theoretical limit for CSS codes and represents the highest optimal error threshold among known three-dimensional codes. We relate this saturation to a generalized entropy relation for classical spin models satisfying a Kramers-Wannier-type duality, and argue how this prediction extends to CSS codes with zero encoding rate whose
X- and
Z-noise models map to classically dual spin models. These findings establish fracton codes as highly resilient candidates for quantum memories and demonstrate the power of the statistical-mechanical framework, together with its duality predictions, in analyzing and constructing robust quantum error-correcting codes.
Fracton phases are new types of phases of matter characterized by subsystem global symmetry, which is a generalized global symmetry whose symmetry operator is partially topological. Their continuum low-energy effective descriptions admit two different formulations: an exotic quantum field theory (QFT) using exotic tensor gauge fields, and a foliated QFT constructed from a foliation structure and foliated gauge fields. For certain fracton QFTs, these two descriptions are equivalent, which is called the foliated-exotic duality. In this dissertation, we extend the foliated-exotic duality by combining it with the anomaly inflow mechanism for 't Hooft anomalies of subsystem symmetries. This dissertation has two main results. First, we discuss the exotic and foliated
BF theories in 2+1 dimensions, which exhibit the mixed 't Hooft anomaly of
ZN×ZN subsystem symmetry. This anomaly is captured by a subsystem symmetry-protected topological (SSPT) phase for
ZN×ZN subsystem symmetry in one dimension higher. By extending the foliated-exotic duality in the fractonic
BF theory to the SSPT phase, we establish the field correspondences in the SSPT phase and construct the foliated description of the SSPT phase. Second, we discuss the exotic
ϕ-theory in 2+1 dimensions -- a fractonic gapless scalar field theory, which has the 't Hooft anomaly of
U(1)×U(1) subsystem symmetry. The anomaly is captured by an SSPT phase for
U(1)×U(1) subsystem symmetry in 3+1 dimensions via the anomaly inflow mechanism. Extending the foliated-exotic duality to the
ϕ-theory, we establish field correspondences in the
ϕ-theory and construct the foliated
ϕ-theory that is equivalent to the exotic
ϕ-theory. This provides the first example of the foliated-exotic duality in gapless theories.
Although it is well-known that all quantum states can be produced by single-qubit measurements on the cluster states, it is not a simple task to explicitly find which measurement patterns on the cluster states can generate different quantum states. In this paper, we introduce a general scheme to find measurement patterns corresponding to Calderbank--Shor--Steane (CSS) topological states containing Kitaev's toric code states and color code states on different lattices and in different dimensions. Furthermore, we find a measurement pattern for generating non-Abelian anyons where measurement-induced defects on a toric code state play the role of Ising anyons. We also support our scheme by a graphical notation where, by following a few simple graphical transformations, one will be able to convert a CSS topological state to a cluster state. Our scheme can also be used for experimental realization of anyons on cluster states.