Session A38: Topological Quantum Information: Topological Phases, Error-Correcting Codes, and Entanglement

Hybrid Fracton Phases: Parent Orders for Liquid and Non-Liquid Quantum Phases

We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders host both (i) mobile topological quasiparticles and loop excitations, as well as (ii) point-like topological excitations with restricted mobility, with non-trivial fusion rules and mutual braiding statistics between the two sets of excitations. We study the detailed properties of hybrid fracton phases through exactly solvable models in which the resulting orders hybridize a three-dimensional Z2 topological order with (i) the X-Cube fracton order, or (ii) Haah's code. The hybrid orders presented here can also be understood as the deconfined phase of a gauge theory whose gauge group is given by an Abelian global symmetry G and subsystem symmetries of a normal subgroup N along lower-dimensional sub-regions. We further generalize the hybrid fracton orders to non-Abelian cases where G is a discrete non-Abelian finite group. The non-Abelian hybrid fracton orders host immobile, point-like non-Abelian excitations.    

Subsystem criticality & bifurcating entanglement renormalization

Gapped fracton topological phases satisfy the definition of topological quantum order and yet exhibit unconventional properties including bifurcation under entanglement renormalization transformations. A direction of current interest concerns gapless phase transition points that are related to fracton topological phases. Motivated by this, I will describe a bifurcating entanglement renomalization group flow that is based on the critical (1+1)D Ising model. I will go on to show that this defines a tensor network state with some unusual correlation function behavior.    

Classification of Non-Unitary MTC and Topological Phases

Unitary modular tensor categories (MTCs) can characterize 2+1D topological phases with their modular data. Due to the rank-finiteness for MTCs, there have been many efforts to classify MTCs by the rank. Recent studies classified MTCs (both unitary and non-unitary) completely up to rank 5, and partially for higher ranks. Meanwhile, modular data of non-unitary MTCs characterize non-unitary conformal field theories, which are relevant for the Yang-Lee singularity or Gaffnian quantum Hall states. We classify non-unitary MTCs with rank greater than 5 computationally by checking consistency conditions of category theory and present the table of non-unitary MTCs for low ranks. Importantly, the consistency conditions for non-unitary MTCs can be obtained from that of unitary MTCs by allowing negative quantum dimensions, or equivalently allowing negative first row entries of the S matrices. Our classification table contains the fusion rules, the central charges, the quantum dimensions, and the modular data of each non-unitary MTC. We believe that our table would be a useful reference for researchers who study category theory or topological phases.   

Irreducible multi-partite correlations as an order parameter for k-local nontrivial states

Geometrically nontrivial quantum states can be defined as states that cannot be prepared by a constant depth geometrically local unitary circuit starting from a product state. However, for topological phases, as well as a large class of quantum error correcting codes without an underlying geometric structure, the required circuit depth remains infinite even if we replace the condition of geometric locality with the weaker condition of k-locality. Motivated by this observation, we look for a non-geometric quantity that can capture k-local non-triviality of a given state, for example, we ask if it is possible to distinguish the ground state of the toric code from a trivial state without having access to the position label of the qubits. We observe that a fundamental property of k-local nontrivial states is the presence of irreducible many-partite correlations shared between an infinitely large number of randomly chosen parties, i.e. correlations that cannot be inferred by accessing only a finite number of parties. We introduce an order parameter designed to capture such correlations. We demonstrate the utility of our order parameter by applying it to a wide variety of examples: The toric code on a square lattice, random stabilizer states, quantum expander codes, and a particular holographic stabilizer state. We discuss general relations between this order parameter and the erasure thresholds of quantum error correcting codes as well as the classical bond percolation problem.

Observation of Symmetry-Protected Selection Rules in Periodically Driven Quantum Systems

Periodically driven (Floquet) quantum systems have recently been a focus of nonequilibrium physics by virtue of their rich dynamics. Time-periodic systems not only exhibit symmetries that resemble those in spatially periodic systems, but also display novel behavior that arises from symmetry breaking. Characterization of such dynamical symmetries is crucial, but often challenging due to limited driving strength and lack of an experimentally accessible characterization technique. Here, we show how to reveal dynamical symmetries, namely, parity, rotation, and particle-hole symmetries, by observing symmetry-induced Floquet selection rules. Notably, we exploit modulated driving to reach the strong light-matter coupling regime, and we introduce a protocol to experimentally extract the transition matrix elements between Floquet states from the system coherent evolution. By using nitrogen-vacancy centers in diamond as an experimental test bed, we execute our protocol to observe symmetry-protected dark states and dark bands, and coherent destruction of tunneling. Our work shows how one can exploit the quantum control toolkit to study dynamical symmetries that arise in the topological phases of strongly driven Floquet systems.   

A holographic view of topological stabilizer codes

We study boundaries of topological stabilizer codes and the constraints imposed on them by the emergent conservation laws that govern the bulk topological order. We show ---at the level of the boundary operator algebra without referring to a particular boundary Hamiltonian--- that these constraints forbid the boundary from being realized via a local tensor product Hilbert space. Furthermore, we demonstrate that the different ways in which the boundary Hilbert space fails to be a tensor product directly encode topological properties of the bulk. In particular, we find quantities of the boundary operator algebra that are directly related to the self and mutual statistics of bulk excitations. We demonstrate this explicitly in a variety of topological stabilizer codes, including Type I and II fracton codes.   

Pauli topological stabilizer codes from twisted quantum doubles

We construct a Pauli stabilizer code for every two-dimensional Abelian topological order that admits a gapped boundary. Our primary example is a Pauli stabilizer code, defined on four-dimensional qudits, belonging to the double semion (DS) phase of matter. We find an explicit finite-depth quantum circuit (with ancillary qubits) that maps the ground state subspace of the DS stabilizer code to that of the DS string-net model. The DS stabilizer code is constructed by condensing an emergent boson in a Z₄ toric code, which can be implemented by making certain two-body measurements. We show that the construction of the DS stabilizer code generalizes to all twisted quantum doubles with Abelian anyons, yielding models defined on composite-dimensional qudits. Our work thus extends the classification of Pauli topological stabilizer codes beyond stacks of toric codes. We also demonstrate that certain symmetry-protected topological phases can be modeled by Pauli stabilizer codes by gauging 1-form symmetries of the twisted quantum double stabilizer codes.   

Entanglement cost in topological stabilizer models at finite temperature

The notion of entanglement has been useful for charactering quantum many-body systems. From the perspective of quantum information theory, it is tempting to ask whether their entanglement structures possess any operational meanings, e.g., measuring the cost of preparing an entangled system using only local operations and classical communication (LOCC). While the answer is affirmitive for pure states in that entanglement entropy coincides with entanglement cost, the case for mixed-states such as systems described by thermal Gibbs states is less understood. To this end, we address this question by studying a recently proposed quantity dubbed κ-entanglement, which not only measures entanglement for mixed states but also quantifies the entanglement cost under positive-partial-transpose (PPT) preserving operations. In particular, we focus on Gibbs states of d-dimensional toric code models for d = 2, 3, 4, and show that their κ-entanglement coincides with entanglement negativity, which has been known to diagnose topological order at finite temperature. Our finding therefore provides an operational meaning for their long-range entanglement structure.   

Chiral central charge from a single bulk wave function

A (2+1)-dimensional gapped quantum many-body system can have a topologically protected energy current at its edge. The magnitude of this current is determined entirely by the temperature and the chiral central charge, a quantity associated with the effective field theory of the edge. We derive a formula for the chiral central charge that, akin to the topological entanglement entropy, is completely determined by the many-body ground state wave function in the bulk. According to our formula, a nonzero chiral central charge gives rise to a topological obstruction that prevents the ground state wave function from being real-valued in any local product basis.   

Multipartitioning topological phases by vertex states and quantum entanglement

We discuss multipartitions of the gapped ground states of (2+1)-dimensional topological liquids into three spatial regions that are adjacent to each other and meet at points. By considering the reduced density matrix obtained by tracing over a subset of the regions, we compute various correlation measures, such as entanglement negativity, reflected entropy, and associated spectra. We utilize the bulk-boundary correspondence to show that such multipartitions can be achieved by using what we call vertex states in (1+1)-dimensional conformal field theory -- these are a type of state used to define an interaction vertex in string field theory and can be thought of as a proper generalization of conformal boundary states. This approach allows an explicit construction of the reduced density matrix near the entangling boundaries. We find the fingerprints of topological liquid in these quantities, such as (universal pieces in) the scaling of the entanglement negativity, and a non-trivial distribution of the spectrum of the partially transposed density matrix. For reflected entropy, we test the recent claim that states the difference between reflected entropy and mutual information is given, once short range correlations are properly removed, by (c/3)\ln 2 where c is the central charge of the topological liquid that measure ungappable edge degrees of freedom. As specific examples, we consider topological chiral p-wave superconductors and Chern insulators.    

Topological graph states and quantum error correction codes

Identifying and characterizing topological order is of widespread importance to both condensed matter physics and quantum information theory. In this work, we focus on the graph-state representation of stabilizer topological quantum error correction codes (QECCs). We derive a set of necessary and sufficient conditions for a family of graph states to be in TQO-1, a class of QECC states whose code distance (i.e. the number of protected single-qubit errors) scales macroscopically with the number of physical qubits. Using these criteria, we consider a number of specific graph families and discuss which are topologically ordered and how to construct the codewords. This formalism is then employed to construct several QECCs with macroscopic distance, including a new three-dimensional topological code generated by local stabilizers that also has a macroscopic number of encoded logical qubits. The results indicate that graph states provide a fruitful approach to the construction and characterization of topological stabilizer QECCs.   

Understanding entanglement negativity in topological order at finite temperature

It has been proposed that the stability of topologically ordered states of matter at finite temperature can be diagnosed by their long-range entanglement structure using entanglement negativity, a mixed-state entanglement measure. In this work, we provide a novel connection between entanglement negativity in a topological order and an emergent symmetry-protected topological (SPT) order localized on the entanglement bipartition. This connection leads to a precise understanding of the phase transition in entanglement negativity as the temperature is increased, and as thermal fluctuations eventually destroy the long-range entanglement in the topological phase. Within this correspondence, anyons in topological order correspond to symmetry charges in SPT order, and the stability of topological order at a non-zero temperature relates to the stability of SPT order against a symmetry-breaking field. For the 4d toric code and 3d toric code with point-like charges forbidden, in which topological order exists at finite temperature, the corresponding SPT order is protected by a higher-form symmetry that is robust under a weak symmetry-breaking field. Finally, a universal scaling form of long-range entanglement negativity is derived across the finite temperature transition of topological order.    

Optimal thresholds for fracton codes and random spin models with subsystem symmetry

We study optimal error thresholds for quantum error correcting codes based on fracton models. By mapping the error-correction process for bit-flip and phase-flip noises into novel statistical models with Ising variables and random multi-body couplings, we obtain models that exhibit an unconventional subsystem symmetry instead of a more usual global symmetry. We use parallel tempering Monte Carlo simulations to obtain disorder-temperature phase diagrams, which are then used to predict optimal error thresholds for the corresponding fracton code. Remarkably, we found that the X-cube fracton code displays a minimum error threshold that is much higher than 3D topological codes such as the toric code and the color code. This result, together with the predicted absence of glass order at the Nishimori line, shows great potential for fracton phases to be used as quantum memory platforms.   

Topology of hole spins in semiconductor quantum dots

In this talk, we discuss a method to flip the heavy hole spin of semiconductor quantum dots completely by an adiabatic transport of dots in the plane of two-dimensional electron gas. We estimate the geometric spin flip time for heavy holes which turned out to be much shorter time than the experimentally reported decoherence time that may provide an alternative route to flip the topologically protected heavy hole spin before reaching decoherence. By utilizing Feynman disentangling operator scheme, we find evolution matrix of heavy hole and then discuss the influence of pure Rashba and Dresshaus spin-orbit coupling effects on the geometric spin-flip probabilities due to the adiabatic motion of dots in the 2D plane. Our study may be useful for building solid-state realization of quantum computing devices