Symmetry-protected Topological Order Research Articles

Symmetry-Enforced Many-Body Separability Transitions

We study quantum many-body mixed states with a symmetry from the perspective of , i.e., whether a mixed state can be expressed as an ensemble of short-range-entangled symmetric pure states. We provide evidence for “symmetry-enforced separability transitions” in a variety of states, where in one regime the mixed state is expressible as a convex sum of symmetric short-range-entangled pure states, while in the other regime, such a representation is not feasible. We first discuss the Gibbs state of Hamiltonians that exhibit spontaneous breaking of a discrete symmetry, and argue that the associated thermal phase transition can be thought of as a symmetry-enforced separability transition. Next we study cluster states in various dimensions subjected to local decoherence, and identify several distinct mixed-state phases and associated separability phase transitions, which also provides an alternative perspective on recently discussed “average symmetry-protected topological order.” We also study decohered p+ip superconductors, and find that if the decoherence breaks the fermion parity explicitly, then the resulting mixed state can be expressed as a convex sum of nonchiral states, while a fermion parity–preserving decoherence results in a phase transition at a nonzero threshold that corresponds to spontaneous breaking of fermion parity. Finally, we briefly discuss systems that satisfy the no low-energy trivial state property, such as the recently discovered good low-density parity-check codes, and argue that the Gibbs state of such systems exhibits a temperature-tuned separability transition. Published by the American Physical Society 2024

Edge Modes and Symmetry-Protected Topological States in Open Quantum Systems

Topological order offers possibilities for processing quantum information that can be immune to imperfections. However, the question of its stability out of equilibrium is relevant for experiments, where coupling to an environment is unavoidable. In this work, we demonstrate the robustness of certain aspects of Z2×Z2 symmetry-protected topological (SPT) order against a wide class of dissipation channels in the Lindblad and quantum trajectory formalisms of an open quantum system. This is illustrated using the one-dimensional ZXZ cluster Hamiltonian along with Pauli-string jump operators. We show that certain choices of dissipation retaining strong symmetries support a steady-state manifold consisting of two nonlocal qubits and for Hamiltonian perturbations preserving the global symmetry, states in this manifold remain metastable. In contrast, this metastability is destroyed upon breaking the above-mentioned symmetry. While the localized qubits of the cluster Hamiltonian are not conserved by the Lindbladian evolution, they do correspond to weak symmetries and thus retain a memory of their initial state at all times in the quantum trajectories. We utilize this feature to construct protocols to retrieve the quantum information either by monitoring jumps or error mitigation. Our work thus proposes a novel framework to study the dynamics of dissipative SPT phases and opens up the possibility of engineering entangled states relevant to quantum information processing. Published by the American Physical Society 2024

Symmetry-protected topological order as a requirement for measurement-based quantum gate teleportation

Symmetry-protected topological order as a requirement for measurement-based quantum gate teleportation

Dissipative symmetry-protected topological order

In this work, we investigate the interplay between dissipation and symmetry-protected topological order. We considered the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki model interacting with an environment where the dissipative dynamics are described by the Lindladian master equation. The Markovian dynamics is solved by the implementation of a tensor network algorithm for mixed states in the thermodynamic limit. We observe that, for time-reversal symmetric dissipation, the resulting steady state has topological signatures even if being a mixed state. This is seen in finite string-order parameters as well as in the degeneracy pattern of singular values in the tensor network decomposition of the reduced density matrix. We also show that such features do not appear for nonsymmetric dissipation. Our work opens the way toward a generalized and more practical definition of symmetry-protected topological order for mixed states induced by dissipation.

Model-Independent Learning of Quantum Phases of Matter with Quantum Convolutional Neural Networks.

Quantum convolutional neural networks (QCNNs) have been introduced as classifiers for gapped quantum phases of matter. Here, we propose a model-independent protocol for training QCNNs to discover order parameters that are unchanged under phase-preserving perturbations. We initiate the training sequence with the fixed-point wave functions of the quantum phase and add translation-invariant noise that respects the symmetries of the system to mask the fixed-point structure on short length scales. We illustrate this approach by training the QCNN on phases protected by time-reversal symmetry in one dimension, and test it on several time-reversal symmetric models exhibiting trivial, symmetry-breaking, and symmetry-protected topological order. The QCNN discovers a set of order parameters that identifies all three phases and accurately predicts the location of the phase boundary. The proposed protocol paves the way toward hardware-efficient training of quantum phase classifiers on a programmable quantum processor.

Construction of Non-Hermitian Parent Hamiltonian from Matrix Product States.

There are various research strategies used for non-Hermitian systems, which typically involve introducing non-Hermitian terms to preexisting Hermitian Hamiltonians. It can be challenging to directly design non-Hermitian many-body models that exhibit unique features not found in Hermitian systems. In this Letter, we propose a new method for constructing non-Hermitian many-body systems by generalizing the parent Hamiltonian method into non-Hermitian regimes. This allows us to build a local Hamiltonian using given matrix product states as its left and right ground states. We demonstrate this method by constructing a non-Hermitian spin-1 model from the asymmetric Affleck-Kennedy-Lieb-Tasaki state, which preserves both chiral order and symmetry-protected topological order. Our approach opens up a new paradigm for systematically constructing and studying non-Hermitian many-body systems, providing guiding principles for exploring new properties and phenomena in non-Hermitian physics.

Measurement-based quantum simulation of Abelian lattice gauge theories

Numerical simulation of lattice gauge theories is an indispensable tool in high energy physics, and their quantum simulation is expected to become a major application of quantum computers in the future. In this work, for an Abelian lattice gauge theory in dd spacetime dimensions, we define an entangled resource state (generalized cluster state) that reflects the spacetime structure of the gauge theory. We show that sequential single-qubit measurements with the bases adapted according to the former measurement outcomes induce a deterministic Hamiltonian quantum simulation of the gauge theory on the boundary. Our construction includes the (2+1)(2+1)-dimensional Abelian lattice gauge theory simulated on three-dimensional cluster state as an example, and generalizes to the simulation of Wegner’s lattice models M_{(d,n)}M(d,n) that involve higher-form Abelian gauge fields. We demonstrate that the generalized cluster state has a symmetry-protected topological order with respect to generalized global symmetries that are related to the symmetries of the simulated gauge theories on the boundary.Our procedure can be generalized to the simulation of Kitaev’s Majorana chain on a fermionic resource state. We also study the imaginary-time quantum simulation with two-qubit measurements and post-selections, and a classical-quantum correspondence, where the statistical partition function of the model M_{(d,n)}M(d,n) is written as the overlap between the product of two-qubit measurement bases and the wave function of the generalized cluster state.

Symmetries and field tensor network states

We study the interplay between symmetry representations of the physical and virtual space on the class of tensor network states for critical spins systems known as field tensor network states (fTNSs). These are by construction infinite-dimensional tensor networks whose virtual space is described by a conformal field theory (CFT). We can represent a symmetry on the physical index as a commutator with the corresponding CFT current on the virtual space. By then studying this virtual space representation we can learn about the critical symmetry-protected topological properties of the state, akin to the classification of symmetry-protected topological order for matrix product states. We use this to analytically derive the critical symmetry-protected topological properties of the two ground states of the Majumdar-Ghosh point with respect to the previously defined symmetries.

Learning of error statistics for the detection of quantum phases

We present a binary classifier based on neural networks to detect gapped quantum phases. By considering the errors on top of a suitable reference state describing the gapped phase, we show that a neural network trained on the errors can capture the correlation between the errors and can be used to detect the phase boundaries of the gapped quantum phase. We demonstrate the application of the method for matrix product state calculations for different quantum phases exhibiting local symmetry-breaking order, symmetry-protected topological order, and intrinsic topological order.

Emergent higher-symmetry protected topological orders in the confined phase of U(1) gauge theory

We consider compact ${U}^{\ensuremath{\kappa}}(1)$ gauge theory in $3+1$ dimensions with a general $2\ensuremath{\pi}$-quantized topological term ${\ensuremath{\sum}}_{I,J=1}^{\ensuremath{\kappa}}\frac{{K}_{IJ}}{4\ensuremath{\pi}}{\ensuremath{\int}}_{{M}^{4}}{F}^{I}\ensuremath{\wedge}{F}^{J}$, where $K$ is an integer symmetric matrix with even diagonal elements and ${F}^{I}=d{A}^{I}$. At energies below the gauge charges' gaps but above the monopoles' gaps, this field theory has an emergent ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry, where ${k}_{i}$ are the diagonal elements of the Smith normal form of $K$ and ${\mathbb{Z}}_{0}^{(1)}$ is regarded as a $U(1)$ 1-symmetry. In the ${U}^{\ensuremath{\kappa}}(1)$ confined phase, the boundary can have a phase whose infrared (IR) properties are described by Chern-Simons field theory. Such a phase has a ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry that can be anomalous. To show these results, we develop a bosonic lattice model whose IR properties are described by this continuum field theory, thus acting as its ultraviolet completion. The lattice model in the aforementioned limit has an exact ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry. We find that the short-range entangled gapped phase of the lattice model, corresponding to the confined phase of the ${U}^{\ensuremath{\kappa}}(1)$ gauge theory, is a symmetry protected topological (SPT) phase for the ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry, whose SPT invariant is $\phantom{\rule{1.0pt}{0ex}}{e}^{\phantom{\rule{1.0pt}{0ex}}i\phantom{\rule{1.0pt}{0ex}}\ensuremath{\pi}{\ensuremath{\sum}}_{I,J}{K}_{IJ}{\ensuremath{\int}}_{{\mathcal{M}}^{4}}{B}_{I}\ensuremath{\smile}{B}_{J}+{B}_{I}\underset{1}{\ensuremath{\smile}}\phantom{\rule{1.0pt}{0ex}}d{B}_{J}}\phantom{\rule{1.0pt}{0ex}}{e}^{\phantom{\rule{1.0pt}{0ex}}i\phantom{\rule{1.0pt}{0ex}}\ensuremath{\pi}{\ensuremath{\sum}}_{I<J}{K}_{IJ}{\ensuremath{\int}}_{{\mathcal{M}}^{4}}\phantom{\rule{1.0pt}{0ex}}d{B}_{I}\underset{2}{\ensuremath{\smile}}\phantom{\rule{1.0pt}{0ex}}d{B}_{J}}$. Here, the background $\mathbb{R}/\mathbb{Z}$-valued 2-cochains ${B}_{I}$ satisfy $\phantom{\rule{1.0pt}{0ex}}d{B}_{I}={\ensuremath{\sum}}_{I}{B}_{I}{K}_{IJ}=0\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.28em}{0ex}}1$ and describe the symmetry twist of the ${\mathbb{Z}}_{{k}_{1}}^{(1)}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{k}_{2}}^{(1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdots}$ 1-symmetry. We apply this general result to a few examples with simple $K$ matrices. We find the nontrivial SPT order in the confined phases of these models and discuss its classifications using the fourth cohomology group of the corresponding 2-group.

Symmetry Protected Topological Order in Open Quantum Systems

We systematically investigate the robustness of symmetry protected topological (SPT) order in open quantum systems by studying the evolution of string order parameters and other probes under noisy channels. We find that one-dimensional SPT order is robust against noisy couplings to the environment that satisfy a strong symmetry condition, while it is destabilized by noise that satisfies only a weak symmetry condition, which generalizes the notion of symmetry for closed systems. We also discuss "transmutation" of SPT phases into other SPT phases of equal or lesser complexity, under noisy channels that satisfy twisted versions of the strong symmetry condition.

Topological order and entanglement dynamics in the measurement-only XZZX quantum code

We examine the dynamics of a $(1+1)$-dimensional measurement-only circuit defined by the stabilizers of the [[5,1,3]] quantum error correcting code interrupted by single-qubit Pauli measurements. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled state with a ${D}_{2}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a twofold bulk degeneracy. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase. Allowing more than one type of Pauli measurement increases the measurement-induced frustration and the SPT and SB order can be broken either simultaneously or separately at nonzero measurement rate. This yields a rich phase diagram and unanticipated critical behavior at the phase transitions. Although the correlation length exponent $\ensuremath{\nu}=\frac{4}{3}$ and the dynamical critical exponent $z=1$ are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take noninteger multiples of the percolation value. Remarkably, we identify a robust transient scaling regime for the purification dynamics of $L$ qubits. It reveals a modified dynamical critical exponent ${z}^{*}\ensuremath{\ne}z$, which is observable up to times $t\ensuremath{\sim}{L}^{{z}^{*}}$ and is reminiscent of the relaxation of critical systems into a prethermal state.

Topological edge modes without symmetry in quasiperiodically driven spin chains

We construct an example of a 1d quasiperiodically driven spin chain whose edge states can coherently store quantum information, protected by a combination of localization, dynamics, and topology. In a sharp departure from topological phases in static and periodically driven (Floquet) spin chains, this model does not rely upon microscopic symmetry protection: Instead, the edge states are protected purely by emergent dynamical symmetries. We explore the dynamical signatures of this emergent dynamical symmetry-protected topological (EDSPT) order through exact numerics, time evolving block decimation, and analytic high-frequency expansion, finding evidence that the EDSPT is a stable dynamical phase protected by bulk many-body localization up to (at least) stretched-exponentially long timescales, and possibly beyond. We argue that EDSPTs are special to the quasiperiodically driven setting, and cannot arise in Floquet systems. Moreover, we find evidence of a type of boundary critical with no known static or Floquet analogue, in which the edge spin dynamics transition from quasiperiodic to chaotic, leading to bulk thermalization.

Observation of fractional edge excitations in nanographene spin chains.

Fractionalization is a phenomenon in which strong interactions in a quantum system drive the emergence of excitations with quantum numbers that are absent in the building blocks. Outstanding examples are excitations with charge e/3 in the fractional quantum Hall effect1,2, solitons in one-dimensional conducting polymers3,4 and Majorana states in topological superconductors5. Fractionalization is also predicted to manifest itself in low-dimensional quantum magnets, such as one-dimensional antiferromagnetic S = 1 chains. The fundamental features of this system are gapped excitations in the bulk6 and, remarkably, S = 1/2 edge states at the chain termini7-9, leading to a four-fold degenerate ground state that reflects the underlying symmetry-protected topological order10,11. Here, we use on-surface synthesis12 to fabricate one-dimensional spin chains that contain the S = 1 polycyclic aromatic hydrocarbon triangulene as the building block. Using scanning tunnelling microscopy and spectroscopy at 4.5 K, we probe length-dependent magnetic excitations at the atomic scale in both open-ended and cyclic spin chains, and directly observe gapped spin excitations and fractional edge states therein. Exact diagonalization calculations provide conclusive evidence that the spin chains are described by the S = 1 bilinear-biquadratic Hamiltonian in the Haldane symmetry-protected topological phase. Our results open a bottom-up approach to study strongly correlated phases in purely organic materials, with the potential for the realization of measurement-based quantum computation13.

Symmetry-protected dissipative preparation of matrix product states

We propose and analyze a method for efficient dissipative preparation of matrix product states that exploits their symmetry properties. Specifically, we construct an explicit protocol that makes use of driven-dissipative dynamics to prepare a many-body quantum state that features symmetry-protected topological order and nontrivial edge excitations. The preparation protocol is protected from errors that respect the symmetry, allowing for robust experimental implementation without fine-tuned control. Numerical simulations show that the preparation time scales polynomially in system size $n$. Furthermore, we demonstrate that this scaling can be improved to $O({log}^{2}n)$ by using parallel preparation of individual segments and fusing them via quantum feedback. A concrete scheme using excitation of trapped neutral atoms into the Rydberg state via electromagnetically induced transparency is proposed and generalizations to a broader class of matrix product states are discussed.

Quantum scars and bulk coherence in a symmetry-protected topological phase

Formation of quantum scars in many-body systems provides a novel mechanism for enhancing coherence of weakly entangled states. At the same time, coherence of edge modes in certain symmetry protected topological (SPT) phases can persist away from the ground state. In this work we show the existence of many-body scars and their implications on bulk coherence in such an SPT phase. To this end, we study the eigenstate properties and the dynamics of an interacting spin-$1/2$ chain with three-site "cluster" terms hosting a $\mathbb{Z}_2 \times \mathbb{Z}_2$ SPT phase. Focusing on the weakly interacting regime, we find that eigenstates with volume-law entanglement coexist with area-law entangled eigenstates throughout the spectrum. We show that a subset of the latter can be constructed by virtue of repeated cluster excitations on the even or odd sublattice of the chain, resulting in an equidistant "tower" of states, analogous to the phenomenology of quantum many-body scars. We further demonstrate that these scarred eigenstates support nonthermal expectation values of local cluster operators in the bulk and exhibit signatures of topological order even at finite energy densities. Studying the dynamics for out-of-equilibrium states drawn from the noninteracting "cluster basis", we unveil that nonthermalizing bulk dynamics can be observed on long time scales if clusters on odd and even sites are energetically detuned. In this case, cluster excitations remain essentially confined to one of the two sublattices such that inhomogeneous cluster configurations cannot equilibrate and thermalization of the full system is impeded. Our work sheds light on the role of quantum many-body scars in preserving SPT order at finite temperature and the possibility of coherent bulk dynamics in models with SPT order beyond the existence of long-lived edge modes.

Gauging the Kitaev chain

We gauge the fermion parity symmetry of the Kitaev chain. While the bulk of the model becomes an Ising chain of gauge-invariant spins in a tilted field, near the boundaries the global fermion parity symmetry survives gauging, leading to local gauge-invariant Majorana operators. In the absence of vortices, the Higgs phase exhibits fermionic symmetry-protected topological (SPT) order distinct from the Kitaev chain. Moreover, the deconfined phase can be stable even in the presence of vortices. We also undertake a comprehensive study of a gently gauged model which interpolates between the ordinary and gauged Kitaev chains. This showcases rich quantum criticality and illuminates the topological nature of the Higgs phase. Even in the absence of superconducting terms, gauging leads to an SPT phase which is intrinsically gapless due to an emergent anomaly.

Subsystem Ginzburg-Landau and symmetry-protected topological order coexisting on a graph

We analyze a model demonstrating the co-existence of subsystem symmetry breaking (SSB) and symmetry-protected topological (SPT) order, or subsystem LSPT order for short. Its mathematical origin is the existence of both a subsystem and a local operator, both of which commute with the Hamiltonian but anti-commute between themselves. The reason for the exponential growth of the ground state degeneracy is attributed to the existence of subsystem symmetries, which allows one to define both the Landau order parameter and the SPT-like order for each independent loop.

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.

Quantum Computational Advantage with String Order Parameters of One-Dimensional Symmetry-Protected Topological Order.

Nonlocal games with advantageous quantum strategies give arguably the most fundamental demonstration of the power of quantum resources over their classical counterparts. Recently, certain multiplayer generalizations of nonlocal games have been used to prove unconditional separations between limited computational complexity classes of shallow-depth circuits. Here, we show advantageous strategies for these nonlocal games for generic ground states of one-dimensional symmetry-protected topological orders (SPTOs), when a discrete invariant of a SPTO known as a twist phase is nontrivial and -1. Our construction demonstrates that sufficiently large string order parameters of such SPTOs are indicative of globally constrained correlations useful for the unconditional computational separation.