arXiv subjects

Tyler D. Ellison

Publications and source records attributed to Tyler D. Ellison.

16 recordsLinked to original sources

arXiv2026

Many-body chirality of topological stabilizer states

A defining feature of chirality is the distinction between a system and its mirror image. Despite extensive experimental observations of chiral phases and theoretical advances, a quantum-information theoretic characterization of chirality based solely on the entanglement structure of many-body quantum states remains elusive. Here, we introduce the notion of many-body chirality by formulating it as an obstruction to transforming a quantum state into its complex conjugate through finite-depth local operations. We rigorously establish many-body chirality for stabilizer realizations of $\mathbb{Z}_d^{(k)}$ anyon theories, proving that complex conjugation can be implemented by local quantum channels if and only if the underlying anyon data are mirror invariant. This reveals forms of chirality that evade conventional diagnostics, including examples with vanishing modular commutator, vanishing chiral central charge, and commuting-projector realizations. We further show that this obstruction is intrinsically four-partite, while invisible to tripartite entanglement structure. Finally, we prove that $\mathbb{Z}_d^{(k)}$ states with $d>2$ possess intrinsic many-body imaginarity: their complex phase structure cannot be removed by finite-depth local unitaries. Remarkably, this includes states that are not many-body chiral.

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arXiv2026

Long-range nonstabilizerness of topologically encoded states from mutual information

We study long-range nonstabilizerness (LRN), namely the obstruction to remove nonstabilizerness with shallow-depth local quantum circuits. In one-dimensional settings, the mutual information between disconnected spatial regions has proven to be a powerful tool to diagnose LRN. In this work, we focus on encoded states of two-dimensional topologically-ordered systems, and explore the ability of the mutual information to serve as a diagnostic of LRN. Focusing on the concrete setting of lattice models defined on a torus, we show that information about LRN can be gained from the analysis of the mutual information between non-overlapping regions containing non-contractible loops, and of the change of such mutual information under modular real-space transformations. We exemplify this idea in the toric code and the non-abelian string-net model with doubled Fibonacci topological order. In the former case, we show that the mutual information provides a full classification, certifying LRN for all encoded non-stabilizer states. In the latter case, instead, our approach does not lead to a full classification, as it detects LRN for all states except from a finite subset with special transformation properties under the modular group. Finally, we discuss how our results on LRN constrain the logical gates that can be implemented fault-tolerantly on the torus.

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arXiv2026

Universal quantum computation with group surface codes

We introduce group surface codes, which are a natural generalization of the $\mathbb{Z}_2$ surface code, and equivalent to quantum double models of finite groups with specific boundary conditions. We show that group surface codes can be leveraged to perform non-Clifford gates in $\mathbb{Z}_2$ surface codes, thus enabling universal computation with well-established means of performing logical Clifford gates. Moreover, for suitably chosen groups, we demonstrate that arbitrary reversible classical gates can be implemented transversally in the group surface code. We present the logical operations in terms of a set of elementary logical operations, which include transversal logical gates, a means of transferring encoded information into and out of group surface codes, and preparation and readout. By composing these elementary operations, we implement a wide variety of logical gates and provide a unified perspective on recent constructions in the literature for sliding group surface codes and preparing magic states. We furthermore use tensor networks inspired by ZX-calculus to construct spacetime implementations of the elementary operations. This spacetime perspective also allows us to establish explicit correspondences with topological gauge theories. Our work extends recent efforts in performing universal quantum computation in topological orders without the braiding of anyons, and shows how certain group surface codes allow us to bypass the restrictions set by the Bravyi-K{\"o}nig theorem, which limits the computational power of topological Pauli stabilizer models.

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arXiv2025

Finite-temperature quantum topological order in three dimensions

We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature--it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic point-like excitations. The fermionic toric code, importantly, possesses an anomalous 2-form symmetry, associated with the space-like Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to an equilibrium phase of matter that only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous 2-form symmetries. Our work therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions.

cond-mat.str-el↗

arXiv2024

Spacetime Markov length: a diagnostic for fault tolerance via mixed-state phases

We establish a correspondence between the fault-tolerance of local stabilizer codes experiencing measurement and physical errors and the mixed-state phases of decohered resource states in one higher dimension. Drawing from recent developments in mixed-state phases of matter, this motivates a diagnostic of fault-tolerance, which we refer to as the spacetime Markov length. This is a length scale determined by the decay of the (classical) conditional mutual information of repeated syndrome measurement outcomes in spacetime. The diagnostic is independent of the decoder, and its divergence signals the intrinsic breakdown of fault tolerance. As a byproduct, we find that decoherence may be useful for exposing transitions from higher-form symmetry-protected topological phases driven by both incoherent and coherent perturbations.

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arXiv2024

Topological stabilizer models on continuous variables

We construct a family of two-dimensional topological stabilizer codes on continuous variable (CV) degrees of freedom, which generalize homological rotor codes and the toric-GKP code. Our topological codes are built using the concept of boson condensation -- we start from a parent stabilizer code based on an $\mathbb{R}$ gauge theory and condense various bosonic excitations. This produces a large class of topological CV stabilizer codes, including ones that are characterized by the anyon theories of $U(1)_{2n}\times U(1)_{-2m}$ Chern-Simons theories, for arbitrary pairs of positive integers $(n,m)$. Most notably, this includes anyon theories that are non-chiral and nevertheless do not admit a gapped boundary. It is widely believed that such anyon theories cannot be realized by any stabilizer model on finite-dimensional systems. We conjecture that these CV codes go beyond codes obtained from concatenating a topological qudit code with a local encoding into CVs, and thus, constitute the first example of topological codes that are intrinsic to CV systems. Moreover, we study the Hamiltonians associated to the topological CV stabilizer codes and show that, although they have a gapless spectrum, they can become gapped with the addition of a quadratic perturbation. We show that similar methods can be used to construct a gapped Hamiltonian whose anyon theory agrees with a $U(1)_2$ Chern-Simons theory. Our work initiates the study of scalable stabilizer codes that are intrinsic to CV systems and highlights how error-correcting codes can be used to design and analyze many-body systems of CVs that model lattice gauge theories.

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arXiv2024

Chiral spin liquid in a $\mathbb{Z}_3$ Kitaev model

We study a $\mathbb{Z}_3$ Kitaev model on the honeycomb lattice with nearest neighbor interactions. Based on matrix product state simulations and symmetry considerations, we find evidence that, with ferromagnetic isotropic couplings, the model realizes a chiral spin liquid, characterized by a possible $\mathrm{U}(1)_{12}$ chiral topological order. This is supported by simulations on both cylinder and strip geometries. On infinitely long cylinders with various widths, scaling analysis of entanglement entropy and maximal correlation length suggests that the model has a gapped 2D bulk. The topological entanglement entropy is extracted and found to be in agreement with the $\mathrm{U}(1)_{12}$ topological order. On infinitely long strips with moderate widths, we find the model is critical with a central charge consistent with the chiral edge theory of the $\mathrm{U}(1)_{12}$ topological phase. We conclude by discussing several open questions.

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arXiv2024

Engineering 3D Floquet codes by rewinding

Floquet codes are a novel class of quantum error-correcting codes with dynamically generated logical qubits arising from a periodic schedule of non-commuting measurements. We utilize the interpretation of measurements in terms of condensation of topological excitations and the rewinding of measurement sequences to engineer new examples of Floquet codes. In particular, rewinding is advantageous for obtaining a desired set of instantaneous stabilizer groups on both toric and planar layouts. Our first example is a Floquet code with instantaneous stabilizer codes that have the same topological order as 3D toric code(s). This Floquet code also exhibits a splitting of the topological order of the 3D toric code under the associated sequence of measurements, i.e., an instantaneous stabilizer group of a single copy of 3D toric code in one round transforms into an instantaneous stabilizer group of two copies of 3D toric codes up to nonlocal stabilizers in the following round. We further construct boundaries for this 3D code and argue that stacking it with two copies of 3D subsystem toric code allows for a transversal implementation of the logical non-Clifford $CCZ$ gate. We also show that the coupled-layer construction of the X-cube Floquet code can be modified by a rewinding schedule such that each of the instantaneous stabilizer codes is finite-depth-equivalent to the X-cube model up to toric codes; the X-cube Floquet code exhibits a splitting of the X-cube model into a copy of the X-cube model and toric codes under the measurement sequence. Our final 3D example is a generalization of the 2D Floquet toric code on the honeycomb lattice to 3D, which has instantaneous stabilizer codes with the same topological order as the 3D fermionic toric code.

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arXiv2023

Pauli topological subsystem codes from Abelian anyon theories

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 $\mathbb{Z}_4^{(1)}$ 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.

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arXiv2023

Floquet codes with a twist

We describe a method for creating twist defects in the honeycomb Floquet code of Hastings and Haah. In particular, we construct twist defects at the endpoints of condensation defects, which are built by condensing emergent fermions along one-dimensional paths. We argue that the twist defects can be used to store and process quantum information fault tolerantly, and demonstrate that, by preparing twist defects on a system with a boundary, we obtain a planar variant of the $\mathbb{Z}_2$ Floquet code. Importantly, our construction of twist defects maintains the connectivity of the hexagonal lattice, requires only 2-body measurements, and preserves the three-round period of the measurement schedule. We furthermore generalize the twist defects to $\mathbb{Z}_N$ Floquet codes defined on $N$-dimensional qudits. As an aside, we use the $\mathbb{Z}_N$ Floquet codes and condensation defects to define Floquet codes whose instantaneous stabilizer groups are characterized by the topological order of certain Abelian twisted quantum doubles.

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arXiv2022

Pauli stabilizer models of twisted quantum doubles

We construct a Pauli stabilizer model for every two-dimensional Abelian topological order that admits a gapped boundary. Our primary example is a Pauli stabilizer model on four-dimensional qudits that belongs to the double semion (DS) phase of matter. The DS stabilizer Hamiltonian is constructed by condensing an emergent boson in a $\mathbb{Z}_4$ toric code, where the condensation is implemented by making certain two-body measurements. We rigorously verify the topological order of the DS stabilizer model by identifying an explicit finite-depth quantum circuit (with ancillary qubits) that maps its ground state subspace to that of a DS string-net model. We show that the construction of the DS stabilizer Hamiltonian generalizes to all twisted quantum doubles (TQDs) with Abelian anyons. This yields a Pauli stabilizer code on composite-dimensional qudits for each such TQD, implying that the classification of topological Pauli stabilizer codes extends well beyond stacks of toric codes - in fact, exhausting all Abelian anyon theories that admit a gapped boundary. We also demonstrate that symmetry-protected topological phases of matter characterized by type I and type II cocycles can be modeled by Pauli stabilizer Hamiltonians by gauging certain 1-form symmetries of the TQD stabilizer models.

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arXiv2022

Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond

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 ${2^n}$-dimensional qudits that feature surface anyons described by $U(1)_{2^n}$ 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.

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arXiv2021

Symmetry-protected sign problem and magic in quantum phases of matter

We introduce the concepts of a symmetry-protected sign problem and symmetry-protected magic to study the complexity of symmetry-protected topological (SPT) phases of matter. In particular, we say a state has a symmetry-protected sign problem or symmetry-protected magic, if finite-depth quantum circuits composed of symmetric gates are unable to transform the state into a non-negative real wave function or stabilizer state, respectively. We prove that states belonging to certain SPT phases have these properties, as a result of their anomalous symmetry action at a boundary. For example, we find that one-dimensional $\mathbb{Z}_2 \times \mathbb{Z}_2$ SPT states (e.g. cluster state) have a symmetry-protected sign problem, and two-dimensional $\mathbb{Z}_2$ SPT states (e.g. Levin-Gu state) have symmetry-protected magic. Furthermore, we comment on the relation between a symmetry-protected sign problem and the computational wire property of one-dimensional SPT states. In an appendix, we also introduce explicit decorated domain wall models of SPT phases, which may be of independent interest.

cond-mat.str-el↗

arXiv2021

Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions

We build exactly solvable lattice Hamiltonians for fermionic symmetry-protected topological (SPT) phases in (3+1)D classified by group supercohomology. A central benefit of our construction is that it produces an explicit finite-depth quantum circuit (FDQC) that prepares the ground state from an unentangled symmetric state. The FDQC allows us to clearly demonstrate the characteristic properties of supercohomology phases - namely, symmetry fractionalization on fermion parity flux loops - predicted by continuum formulations. By composing the corresponding FDQCs, we also recover the stacking relations of supercohomology phases. Furthermore, we derive topologically ordered gapped boundaries for the supercohomology models by extending the protecting symmetries, analogous to the construction of topologically ordered boundaries for bosonic SPT phases. Our approach relies heavily on dualities that relate certain bosonic 2-group SPT phases with supercohomology SPT phases. We develop physical motivation for the dualities in terms of explicit lattice prescriptions for gauging a 1-form symmetry and for condensing emergent fermions. We also comment on generalizations to supercohomology phases in higher dimensions and to fermionic SPT phases outside of the supercohomology framework.

cond-mat.str-el↗

arXiv2019

A tensor network approach to 2D bosonization

We present a 2D bosonization duality using the language of tensor networks. Specifically, we construct a tensor network operator (TNO) that implements an exact 2D bosonization duality. The primary benefit of the TNO is that it allows for bosonization at the level of quantum states. Thus, we use the TNO to provide an explicit algorithm for bosonizing fermionic projected entangled pair states (fPEPs). A key step in the algorithm is to account for a choice of spin-structure, encoded in a set of bonds of the bosonized fPEPS. This enables our tensor network approach to bosonization to be applied to systems on arbitrary triangulations of orientable 2D manifolds.

cond-mat.str-el↗

arXiv2019

Disentangling interacting symmetry protected phases of fermions in two dimensions

We construct fixed point lattice models for group supercohomology symmetry protected topological (SPT) phases of fermions in 2+1D. A key feature of our approach is to construct finite depth circuits of local unitaries that explicitly build the ground states from a tensor product state. We then recover the classification of fermionic SPT phases, including the group structure under stacking, from the algebraic composition rules of these circuits. Furthermore, we show that the circuits are symmetric, implying that the group supercohomology phases can be many body localized. Our strategy involves first building an auxiliary bosonic model, and then fermionizing it using the duality of Chen, Kapustin, and Radicevic. One benefit of this approach is that it clearly disentangles the role of the algebraic group supercohomology data, which is used to build the auxiliary bosonic model, from that of the spin structure, which is combinatorially encoded in the lattice and enters only in the fermionization step. In particular this allows us to study our models on 2d spatial manifolds of any topology and to define a lattice-level procedure for ungauging fermion parity.

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