Simulator finds proof of higher-order quantum topology

The work moves beyond materials characterized by local order parameters. By reconstructing complex propagators from real-space measurements, the team obtained both boundary and bulk characterizations, culminating in the extraction of a three-dimensional octupole moment through third nested Wilson loops. These results establish a route to realizing, characterizing, and manipulating topologically protected states in programmable three-dimensional quantum matter.

Programmable Quantum Simulator Measures Higher-Order Topology

The experimental validation of higher-order bulk-boundary correspondence occurred using up to 728 qubits within a superconducting analogue-digital quantum simulator constructed by D-Wave Quantum in Burnaby, British Columbia, Canada. This simulator enabled the realization and probing of a hierarchy encompassing one-, two- and three-dimensional topological insulators, all governed by effective tight-binding Hamiltonians. The work extends beyond simply detecting localized boundary modes spectroscopically, a technique reliant on local density of states. It directly addresses the longstanding challenge of experimentally determining underlying bulk topology.

These models were implemented using fixed-Hamiltonian dynamics across hundreds of qubits, demonstrating the simulator’s capacity to handle complex quantum systems. The team’s approach combines local spectroscopy with bulk band projector tomography, a technique used to characterize the topological properties of materials by examining their electronic band structure. This combination allows for a comprehensive understanding of the relationship between a material’s bulk properties and its surface states.

The ability to experimentally validate higher-order bulk-boundary correspondence signifies a step forward in the field of topological materials. This work, involving researchers from institutions including the University of British Columbia and Simon Fraser University, establishes a new route for investigating quantum materials and their potential applications.

Su-Schrieffer-Heeger Chains as Foundational Topological Models

Su-Schrieffer-Heeger chains serve as foundational models for exploring higher-order topological insulators, extending the concept of robust boundary states beyond conventional materials. The one-dimensional SSH chain, originally describing polyacetylene, demonstrates quantized polarization and localized end modes carrying fractional boundary charge, establishing a minimal example of topological behavior. Researchers used this model, alongside the Benalcazar-Bernevig-Hughes (BBH) models, to create a hierarchy of topological insulators spanning dimensions and boundary codimensions from one to three.

This progression culminates in three-dimensional BBH octupole insulators exhibiting surfaces, hinges and ultimately, corner modes where charge concentrates. This approach enabled the extraction of band projectors, which are important for determining topological invariants and ultimately, the three-dimensional octupole moment. The technique relies on comparing the local density of states (LDOS) for topological and trivial models to infer fractionalization at the boundaries, a key indicator of topological protection.

BBH Models Define Hierarchy of Higher-Order Insulators

These BBH models necessitate unit cells of four orbitals in two dimensions and eight in three, alongside specific hopping terms that generate a π magnetic flux through each elementary plaquette. This involved constructing four- and eight-orbital unit cells, a necessary step for realizing the BBH models and their quantized multipole moments. This approach allowed for the experimental confirmation of the relationship between bulk topological invariants and the emergence of boundary states, a cornerstone of topological physics.

Realizing Bulk Topology with Superconducting Qubit Arrays

This approach allows for direct access to coherent dynamics generated by a static programmable Hamiltonian, enabling the reconstruction of complex propagators from real-space measurements. Complementary to realizing higher-order topological insulators is the challenge of characterizing them, often done through dynamical signatures or spectroscopic identification of boundary modes. This allows for a more general approach to extracting band topology and geometry.

This simulator, built by D-Wave Quantum, enabled the creation of quantum Su-Schrieffer-Heeger (SSH) and Benalcazar-Bernevig-Hughes (BBH) models in one, two and three dimensions. The platform’s programmable two-qubit coupling terms allow for the hosting of linear, square and cubic lattices, natively implementing a transverse-field Ising Hamiltonian.

Analogue-Digital Protocols Enable Coherent Quantum Dynamics

Using a 728-qubit superconducting analogue-digital quantum simulator, the team reconstructed complex propagators, revealing both boundary and bulk characteristics of topological insulators. Analogue-digital protocols were key to this achievement, enabling precise control over the quantum walk that simulates the tight-binding Hamiltonian governing the system’s behavior. This methodology provides a pathway to probe the bulk-boundary correspondence, linking the system’s internal topology to its surface properties.

Reconstructing Complex Propagators from Real-Space Measurements

Real-space measurements yielded complementary characterizations of both boundaries and bulk properties within the quantum simulator. Localized boundary modes were revealed through local densities of states, while bulk topological invariants were obtained via experimentally reconstructed band projectors. The analogue-digital experimental protocol involved initializing a single qubit with a rotation, evolving the qubits under a static many-body Hamiltonian for varying durations, and then measuring the qubits in either the ${\sigma }^x$ or ${\sigma }^z$ basis.

Time-delayed ${\sigma }^x$ measurements effectively approximated ${\sigma }^y$ measurements, expanding the range of accessible dynamics. Site-resolved observations provided real-space data, which was then converted to frequency space to construct a band structure and perform band projector tomography from complex-valued wavefunction measurements.

Initial tests focused on the Su-Schrieffer-Heeger (SSH) chain, the simplest topological insulator, to validate the methodology. Repeating the protocol across multiple initialization sites and evolution times allowed for sampling of the full real-space propagator, an important step in reconstructing the system’s topological properties. The researchers also employed $R_y(\pi )$ rotations to resolve quantum walks, providing further insight into the system’s dynamics and confirming the ability to probe the bulk-boundary correspondence.

Band Projectors Yield Bulk Topological Invariants

Quantum walks using $R_y(\pi )$ rotations of corner excitations confirmed the system’s behavior, while the integration of nested Wilson loop topological invariants directly revealed the bulk quadrupole topology. The team extracted these invariants by computing band projectors from energy levels obtained via the reconstructed band projectors, a process enabled by the simulator’s ability to model complex wavefunctions. To achieve these measurements, the researchers employed a grid of quasi-momentum values, extracting k-space quantities from models realized on the device.

This approach allowed for the extraction of not only band structure and band projectors, but also the topological invariants themselves, demonstrating the scalability of the technique. The data analysis began with identifying a time point corresponding to t=0, and incorporated calibration refinement to mitigate device non-idealities and accurately map measurements to the underlying tight-binding model dynamics, ensuring the fidelity of the extracted topological invariants.

Nested Wilson Loops Extract Octupole Moments in 3D

Successive evaluation of nested Wilson loops along the y and x axes allowed for the determination of the octupole moment, with the second nesting yielding the spectra along a one-dimensional path through the Brillouin zone. Analysis of the zero-flux three-dimensional SSH model revealed four bulk band levels alongside a three-band in-gap surface mode, demonstrating the system’s topological characteristics.

The researchers then computed second and third nested Wilson loops corresponding to a two-dimensional quadrupole insulator BBH model and the three-dimensional octupole insulator model, respectively. These findings demonstrate a technique for characterizing higher-order bulk topology, going beyond the standard understanding of materials defined by local order parameters and spontaneous symmetry breaking, and offer a new means of exploring quantum materials.

Bulk-Boundary Correspondence Verified via Spectroscopy

The ability to experimentally extract bulk topological invariants, quantized electric multipole insulators, required access to occupied-band Bloch states, a feat achieved through Bloch-state tomography and spectroscopy. The researchers used a superconducting QA processor to implement a transverse-field Ising model (TFIM) Hamiltonian, individually programming the couplings between qubits to define the system’s behavior. Together with spectroscopically observed boundary modes and fractionalized spectral weight, the team validated higher-order bulk-boundary correspondence in a programmable quantum simulator using up to 728 qubits, demonstrating a scalable technique for characterizing these complex systems.

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