arXiv subjects

Dominic J. Williamson

Publications and source records attributed to Dominic J. Williamson.

At least 55 records · Page 3Linked to original sources

arXiv2023

No Strings Attached: Boundaries and Defects in the Cubic Code

Haah's cubic code is the prototypical type-II fracton topological order. It instantiates the no string-like operator property that underlies the favorable scaling of its code distance and logical energy barrier. Previously, the cubic code was only explored in translation-invariant systems on infinite and periodic lattices. In these settings, the code distance scales superlinearly with the linear system size, while the number of logical qubits within the degenerate ground space exhibits a complicated functional dependence that undergoes large fluctuations within a linear envelope. Here, we extend the cubic code to systems with open boundary conditions and crystal lattice defects. We characterize the condensation of topological excitations in the vicinity of these boundaries and defects, finding that their inclusion can introduce local string-like operators and enhance the mobility of otherwise fractonic excitations. Despite this, we use these boundaries and defects to define new encodings where the number of logical qubits scales linearly without fluctuations, and the code distance scales superlinearly, with the linear system size. These include a subsystem encoding with open boundary conditions and a subspace encoding using lattice defects.

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arXiv2023

Higher-Form Subsystem Symmetry Breaking: Subdimensional Criticality and Fracton Phase Transitions

Subsystem symmetry has emerged as a powerful organizing principle for unconventional quantum phases of matter, most prominently fracton topological orders. Here, we focus on a special subclass of such symmetries, known as higher-form subsystem symmetries, which allow us to adapt tools from the study of conventional topological phases to the fracton setting. We demonstrate that certain transitions out of familiar fracton phases, including the X-cube model, can be understood in terms of the spontaneous breaking of higher-form subsystem symmetries. We find simple pictures for these seemingly complicated fracton topological phase transitions by relating them in an exact manner, via gauging, to spontaneous higher-form subsystem symmetry breaking phase transitions of decoupled stacks of lower-dimensional models. We harness this perspective to construct a sequence of unconventional subdimensional critical points in two and three spatial dimensions based on the stacking and gauging of canonical models with higher-form symmetry. Through numerous examples, we illustrate the ubiquity of coupled layer constructions in theories with higher-form subsystem symmetries.

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arXiv2023

A cellular automaton decoder for a noise-bias tailored color code

Self-correcting quantum memories demonstrate robust properties that can be exploited to improve active quantum error-correction protocols. Here we propose a cellular automaton decoder for a variation of the color code where the bases of the physical qubits are locally rotated, which we call the XYZ color code. The local transformation means our decoder demonstrates key properties of a two-dimensional fractal code if the noise acting on the system is infinitely biased towards dephasing, namely, no string-like logical operators. As such, in the high-bias limit, our local decoder reproduces the behavior of a partially self-correcting memory. At low error rates, our simulations show that the memory time diverges polynomially with system size without intervention from a global decoder, up to some critical system size that grows as the error rate is lowered. Furthermore, although we find that we cannot reproduce partially self-correcting behavior at finite bias, our numerics demonstrate improved memory times at realistic noise biases. Our results therefore motivate the design of tailored cellular automaton decoders that help to reduce the bandwidth demands of global decoding for realistic noise models.

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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

Symmetry-Protected Infinite-Temperature Quantum Memory from Subsystem Codes

We study a mechanism whereby quantum information present in the initial state of a quantum many-body system can be protected for arbitrary times due to a combination of symmetry and spatial locality. Remarkably, the mechanism is sufficiently generic that the dynamics can be fully ergodic upon resolving the protecting symmetry and fixing the encoded quantum state, resulting in an infinite-temperature quantum memory. After exemplifying the mechanism in a strongly nonintegrable two-dimensional (2D) spin model inspired by the surface code, we find it has a natural interpretation in the language of noiseless subsystems and stabilizer subsystem codes. This interpretation yields a number of further examples, including a nonintegrable Hamiltonian with quantum memory based on the Bacon-Shor code. The lifetime of the encoded quantum information in these models is infinite provided the dynamics respect the stabilizer symmetry of the underlying subsystem code. In the presence of symmetry-violating perturbations, we make contact with previous work leveraging the concept of prethermalization to show that the encoded quantum information retains a parametrically long lifetime under dynamics with an enlarged continuous symmetry group. We identify conditions on the underlying subsystem code that enable such a prethermal enhancement of the memory lifetime.

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arXiv2022

Fractionalization of subsystem symmetries in two dimensions

The fractionalization of global symmetry charges is a striking hallmark of topological quantum order. Here, we discuss the fractionalization of subsystem symmetries in two-dimensional topological phases. In line with previous no-go arguments, we show that subsystem symmetry fractionalization is not possible in many cases due to the additional rigid geometric structure of the symmetries. However, we identify a new mechanism that allows fractionalization, involving global relations between macroscopically many symmetry generators. We find that anyons can fractionalize such relations, meaning that the total charge carried under all generators involved in the global relation is non-trivial, despite the fact that these generators multiply to the identity. We first discuss the general algebraic framework needed to characterize this new type of fractionalization, and then explore this framework using a number of exactly solvable models with $\mathbb{Z}_2$ topological order, including models having line and fractal symmetries. These models all showcase another necessary property of subsystem symmetry fractionalization: fractionalized anyons must have restricted mobility when the symmetry is enforced, such that they are confined to a single line or point in the case of line and fractal symmetries, respectively. Looking forward, we expect that our identification of the importance of global relations in fractionalization will hold significance for the classification of phases with subsystem symmetries in all dimensions.

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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

On the stability of topological order in tensor network states

We construct a tensor network representation of the 3d toric code ground state that is stable to a generating set of uniform local tensor perturbations, including those that do not map to local operators on the physical Hilbert space. The stability is established by mapping the phase diagram of the perturbed tensor network to that of the 3d Ising gauge theory, which has a non-zero finite temperature transition. More generally, we find that the stability of a topological tensor network state is determined by the form of its virtual symmetries and the topological excitations created by virtual operators that break those symmetries. In particular, a dual representation of the 3d toric code ground state, as well as representations of the X-cube and cubic code ground states, for which point-like excitations are created by such operators, are found to be unstable.

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arXiv2021

Topological defect network representations of fracton stabilizer codes

A topological defect network (TDN) is formed by a network of topological defects embedded within a topological quantum field theory (TQFT). TDNs were introduced recently for the purpose of describing fracton topological phases of matter using the framework of defect TQFT. Their effectiveness has been demonstrated through numerous examples, yet a systematic construction was lacking. Here we solve this problem by formulating a method to construct TDNs for a wide range of lattice Hamiltonians. Our method takes a lattice Hamiltonian as input, applies an ungauging procedure, then creates a refined lattice within each unit cell, followed by regauging the system to produce a TDN as output. For topological Calderbank-Shor-Steane (CSS) Pauli stabilizer models, this procedure is guaranteed to produce a phase equivalent TDN. This provides TDN representations of canonical fracton models for which no such construction was previously known, including Haah's cubic code and Yoshida's infinite family of fractal spin liquid models. We demonstrate the applicability of our method beyond CSS stabilizer models by constructing TDNs for non-CSS models including Chamon's model and the semionic X-cube model.

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arXiv2021

Gauging the bulk: generalized gauging maps and holographic codes

Gauging is a general procedure for mapping a quantum many-body system with a global symmetry to one with a local gauge symmetry. We consider a generalized gauging map that does not enforce gauge symmetry at all lattice sites, and show that it is an isometry on the full input space including all charged sectors. We apply this generalized gauging map to convert global-symmetric bulk systems of holographic codes to gauge-symmetric bulk systems, and vice versa, while preserving duality with a global-symmetric boundary. We separately construct holographic codes with gauge-symmetric bulk systems by directly imposing gauge-invariance constraints onto existing holographic codes, and show that the resulting bulk gauge symmetries are dual to boundary global symmetries. Combining these ideas produces a toy model that captures several interesting features of holography - it exhibits a rudimentary sort of dynamical duality, can be modified to demonstrate the relationship between metric fluctuations and approximate error-correction, and serves as an illustration for certain no-go theorems concerning symmetries in holography. Finally, we apply the generalized gauging map to construct codes with arbitrary transversal gate sets - for any compact Lie group, we use a symmetry-preserving truncation scheme to construct covariant finite-dimensional approximate holographic codes.

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arXiv2021

Planar p-String Condensation: Chiral Fracton Phases from Fractional Quantum Hall Layers and Beyond

We present a coupled-wire construction of a model with chiral fracton topological order. The model combines the known construction of $ν=1/m$ Laughlin fractional quantum Hall states with a planar p-string condensation mechanism. The bulk of the model supports gapped immobile fracton excitations that generate a hierarchy of mobile composite excitations. Open boundaries of the model are chiral and gapless, and can be used to demonstrate a fractional quantized Hall conductance where fracton composites act as charge carriers in the bulk. The planar p-string mechanism used to construct and analyze the model generalizes to a wide class of models including those based on layers supporting non-Abelian topological order. We describe this generalization and additionally provide concrete lattice-model realizations of the mechanism.

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arXiv2021

Type-II fractons from coupled spin chains and layers

We describe a construction of topological orders from coupled lower dimensional symmetry-protected topological orders, which is closely related to gauging a subsystem symmetry. Our construction yields both conventional topological orders and exotic fracton topological orders of type-I and type-II. In particular, we find a coupled spin chain construction of Haah's cubic code, and a coupled layer construction of Yoshida's fractal spin liquids.

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arXiv2021

Fractalizing quantum codes

We introduce "fractalization", a procedure by which spin models are extended to higher-dimensional "fractal" spin models. This allows us to interpret type-II fracton phases, fractal symmetry-protected topological phases, and more, in terms of well understood lower-dimensional spin models. Fractalization is also useful for deriving new spin models and quantum codes from known ones. We construct higher dimensional generalizations of fracton models that host extended fractal excitations. Finally, by applying fractalization to a 2D subsystem code, we produce a family of locally generated 3D subsystem codes that are conjectured to saturate a quantum information storage tradeoff bound.

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arXiv2021

Fault-tolerant logical gates in holographic stabilizer codes are severely restricted

We evaluate the usefulness of holographic stabilizer codes for practical purposes by studying their allowed sets of fault-tolerantly implementable gates. We treat them as subsystem codes and show that the set of transversally implementable logical operations is contained in the Clifford group for sufficiently localized logical subsystems. As well as proving this concretely for several specific codes, we argue that this restriction naturally arises in any stabilizer subsystem code that comes close to capturing certain properties of holography. We extend these results to approximate encodings, locality-preserving gates, certain codes whose logical algebras have non-trivial centers, and discuss cases where restrictions can be made to other levels of the Clifford hierarchy. A few auxiliary results may also be of interest, including a general definition of entanglement wedge map for any subsystem code, and a thorough classification of different correctability properties for regions in a subsystem code.

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arXiv2020

Relative Anomaly in (1+1)d Rational Conformal Field Theory

We study 't Hooft anomalies of symmetry-enriched rational conformal field theories (RCFT) in (1+1)d. Such anomalies determine whether a theory can be realized in a truly (1+1)d system with on-site symmetry, or on the edge of a (2+1)d symmetry-protected topological phase. RCFTs with the identical symmetry actions on their chiral algebras may have different 't Hooft anomalies due to additional symmetry charges on local primary operators. To compute the relative anomaly, we establish a precise correspondence between (1+1)d non-chiral RCFTs and (2+1)d doubled symmetry-enriched topological (SET) phases with a choice of symmetric gapped boundary. Based on these results we derive a general formula for the relative 't Hooft anomaly in terms of algebraic data that characterizes the SET phase and its boundary.

cond-mat.str-el↗

arXiv2020

Subsystem symmetry enriched topological order in three dimensions

We introduce a model of three-dimensional (3D) topological order enriched by planar subsystem symmetries. The model is constructed starting from the 3D toric code, whose ground state can be viewed as an equal-weight superposition of two-dimensional (2D) membrane coverings. We then decorate those membranes with 2D cluster states possessing symmetry-protected topological order under line-like subsystem symmetries. This endows the decorated model with planar subsystem symmetries under which the loop-like excitations of the toric code fractionalize, resulting in an extensive degeneracy per unit length of the excitation. We also show that the value of the topological entanglement entropy is larger than that of the toric code for certain bipartitions due to the subsystem symmetry enrichment. Our model can be obtained by gauging the global symmetry of a short-range entangled model which has symmetry-protected topological order coming from an interplay of global and subsystem symmetries. We study the non-trivial action of the symmetries on boundary of this model, uncovering a mixed boundary anomaly between global and subsystem symmetries. To further study this interplay, we consider gauging several different subgroups of the total symmetry. The resulting network of models, which includes models with fracton topological order, showcases more of the possible types of subsystem symmetry enrichment that can occur in 3D.

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arXiv2020

Designer non-Abelian fractons from topological layers

We formulate a construction of type-I fracton models based on gauging planar subsystem symmetries of topologically ordered two dimensional layers that have been stacked in three ambient spatial dimensions. Via our construction, any defect of an Abelian symmetry group in a two dimensional symmetry-enriched topological order can be promoted into a fracton. This allows us to construct fracton models supporting chiral boundaries and fractons of noninteger quantum dimension. We also find a lineon model supporting non-Abelian surface fractons on its boundary.

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arXiv2020

Spurious topological entanglement entropy from subsystem symmetries

We demonstrate that linear combinations of subregion entropies with canceling boundary terms, commonly used to calculate the topological entanglement entropy, may suffer from spurious nontopological contributions even in models with zero correlation length. These spurious contributions are due to a specific kind of long-range string order, and persist throughout certain subsystem symmetry-protected phases. We introduce an entropic quantity that measures the presence of such order, and hence should serve as an order parameter for the aforementioned phases.

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