Measuring higher-order bulk topology in a programmable quantum simulator

AI summary

Overview

Research area: Quantum simulation of topological matter — specifically higher-order topological insulators (Su-Schrieffer-Heeger chains and Benalcazar-Bernevig-Hughes quadrupole and octupole models) implemented on a programmable superconducting quantum annealing processor.

Technical level: Advanced. The paper assumes familiarity with tight-binding models, Bloch states, Wilson loops, and quantum-annealing hardware.

Scope (one sentence): The authors realize one-, two-, and three-dimensional topological insulator models on up to 728 qubits of a D-Wave Advantage2 processor and extract both boundary signatures (local densities of states, fractionalized corner charge) and bulk topological invariants (Zak phase, quadrupole moment, and a three-dimensional octupole moment from third nested Wilson loops).

What This Paper Is About

Topological insulators are defined by global properties of their quantum states rather than by local order parameters, and their topology shows up as robust states at boundaries. Higher-order versions go further: a two-dimensional quadrupole insulator has corner modes even though its bulk and edges are gapped, and a three-dimensional octupole insulator has corner modes beneath gapped surfaces and hinges. Detecting those corner modes spectroscopically has been done before, but measuring the bulk invariant that predicts them — for example via nested Wilson loops — had remained an outstanding experimental challenge. This paper builds a programmable quantum simulator that does both sides of that bulk-boundary correspondence in up to three dimensions.

Key Contributions

1. Real-space, three-dimensional realization at a new scale. The authors implement SSH and BBH models in one, two, and three dimensions using up to 728 qubits of a superconducting quantum annealing processor, compared with prior direct real-space realizations reported at up to 36 qubits in up to two dimensions.
2. An analogue-digital protocol for coherent single-particle dynamics. Rather than global quenches or quasi-static measurements (the previous uses of such hardware cited by the authors), the work uses newly developed analogue-digital protocols to access coherent dynamics from a static programmable transverse-field Ising Hamiltonian, engineered so that the low-energy effective model is a noninteracting tight-binding Hamiltonian.
3. Spectral projector tomography from complex propagators. By combining R_y(π/2) excitation with phase-delayed readout, the authors reconstruct the complex propagator (equivalently the single-particle Green's function) site by site, and from its Lehmann representation extract band projectors at each momentum.
4. Bulk topological invariants measured experimentally, up to a third nested Wilson loop. The measured projectors yield a Zak phase in 1D, quantized nested polarizations p_y^{ν_x} = p_x^{ν_y} = 1/2 in 2D, and a ZYX octupole invariant of 1/2 in 3D — the central technical result.

Main Findings

• SSH chain (1D): With Δ = 2.0 GHz and γ, λ varied between 50 and 150 MHz, real-space σ^x measurements show transport in the uniform (γ = λ) and trivial (γ > λ) cases versus a persisting boundary excitation in the topological (γ < λ) case. The reconstructed local density of states shows in-gap end modes only in the topological case.
• Fractionalized boundary charge: Comparing topological and trivial open chains at the same chemical potential yields a deficit that "approaches 1/2 at each end but does not reach it," attributed to peak broadening from the finite time window, decoherence, and the finite length of end states.
• Zak phase jump: The experimentally reconstructed (d_x, d_y) path encircles the origin for γ < λ but not γ > λ, and the Zak phase jumps from π to 0 mod 2π as γ/λ passes unity.
• 2D quadrupole insulator: Bulk spectra along the finite-size path M_L–Γ_L–X_L–M_L show two twofold-degenerate manifolds whose gap at M closes as γ/λ → 1. The LDOS shows gapped bulk and edge spectra with an in-gap corner mode and a corner spectral-weight deficit approaching 1/2 per corner.
• Gapped Wannier spectrum and quantized quadrupole: Wilson loops along x and y (equated by graph automorphism) produce a gapped Wannier spectrum in the quadrupole regime and one concentrated near zero in the trivial regime; the nested polarizations are both 1/2 and jump to 0 (modulo 1) at γ = λ.
• 3D octupole insulator: Using a 3×3×3 lattice of eight-orbital unit cells (216 dimers, 432 physical qubits), the LDOS resolves gapped bulk, face, and hinge states with in-gap corner modes, with a charge deficit approaching 1/2 at each corner.
• Staggered-potential test of the octupole moment: Detuning each site by ±50 MHz in a bipartite pattern at {γ, λ} = {50, 250} MHz has almost no effect when γ > λ, but for γ < λ it forces a single multipole orientation and splits the corner modes into alternating positive and negative charges — direct evidence of an octupole moment.
• Third nested Wilson loop: The ZYX octupole invariant takes the value 1/2 and shows a phase transition as γ/λ is tuned through 1 from below, with error bars given as 95% bootstrap confidence intervals.
• Zero-flux comparison cases: The 2D SSH model (all hopping signs equal) resolves a four-band-level spectrum with two gaps hosting topologically protected in-gap edge modes; the 3D SSH model shows four bulk band levels with a three-band in-gap surface mode.
• Classical tractability acknowledged: The dynamics are restricted to the single-excitation sector, whose dimension grows only linearly with lattice size, so these particular experiments remain classically tractable.

Methodology in Plain English

The hardware natively implements a transverse-field Ising Hamiltonian with individually programmable couplings. When the transverse field Δ is much larger than every coupling |J_ij|, processes that change the number of excitations are off-resonant, so the dynamics inside the one-excitation subspace are governed by an effective noninteracting tight-binding model whose hopping amplitudes are the programmed J_ij. The lattice geometry is therefore drawn directly into the qubit connectivity graph.

To probe it, the team flips a single qubit with an R_y(θ_e) rotation, lets the system evolve for a chosen time t, and reads out all sites at once. A R_y(π) pulse with z-basis readout traces a quantum walk; a R_y(π/2) pulse with x-basis readout, supplemented by time-shifted measurements, recovers both the real and imaginary parts of the complex propagation amplitude, because in the weak-coupling limit σ^y_r(t) ≈ σ^x_r(t + 1/(4Δ)). The time step is δt = 1/(8Δ) = 62.5 ps.

Fourier transforming the complex return amplitude in time gives the local density of states; Fourier transforming in space gives band structure and, via a Lehmann-representation fit, the band projectors. From those projectors the authors compute (nested) Wilson loops and the associated topological invariants. Data-analysis details include choosing the time origin by maximizing a site-averaged integrated LDOS, symmetrizing the LDOS about a central frequency Δ̃ (within 1% of Δ = 2 GHz in all dimensionalities) to compensate for an observed underpopulation of higher frequencies, and using lattice symmetries and gauge transformations to average propagators.

Because the hardware graph cannot host large square and cubic lattices with the auxiliary source and detector qubits required, each logical qubit is encoded in a strongly coupled two-qubit dimer (Δ_1 = 3.0 GHz, J_d = 2.5 GHz), giving an effective tunnelling rate Δ = 2.0 GHz and effective hopping proportional to the inter-dimer couplings. Experiments ran on a D-Wave Advantage2 processor in Burnaby, BC, with 200 shots per embedding per QPU call, time series of 20 ns (1D), 10 ns (2D), and 6 ns (3D), and bootstrapping over 200 resamplings of symmetries. In 2D a skew L = 13 lattice of 364 dimers was used for bulk spectroscopy and L = 6 for LDOS; in 3D the z direction was made periodic to gain interior cells.

Why This Matters

The paper closes a loop that is usually left open: boundary modes are observed and the bulk invariant that predicts them is extracted from the same experimental data, in one, two, and three dimensions. The authors frame this as a step toward self-consistent realization and measurement of programmable quantum matter, and argue that the three-dimensional geometry required for the octupole invariants "remains beyond the demonstrated state of the art for modalities such as gate-model superconducting qubits and trapped ions."

Potential applications (the paper itself does not report applied use cases; the following are plausible downstream directions rather than results in the paper):

• Benchmarking and validating topological invariants computed from noisy experimental data, useful for verifying future topological qubit proposals.
• Characterizing metamaterials and classical-wave analogues of higher-order topological insulators, where the same nested-Wilson-loop diagnostics apply.
• Providing a testbed for studying how disorder, decoherence, and finite size distort quantized multipole moments.
• Establishing bulk-property measurement protocols that could transfer to other quantum simulation platforms.

Industry relevance: The work comes largely from D-Wave Quantum, and its main industrial significance is a demonstration that a programmable annealing processor can do coherent single-excitation spectroscopy and tomography — a mode of operation distinct from optimization — at a scale (hundreds of qubits, three dimensions) that competing platforms have not demonstrated for this problem.

Future Directions

• Go beyond the single-excitation sector. The authors explicitly propose realizing multiple-excitation sectors in two and three dimensions to access a Hilbert space that grows exponentially with system size, opening routes to interaction-driven topological phases, correlated states in topological flat bands, and nonequilibrium many-body phenomena.
• Scale up three-dimensional lattices. The L = 3 octupole lattice forces a periodic z dimension, a truncated 6 ns time series, and a coarse band structure; larger lattices would sharpen the invariants and remove finite-size artifacts such as the residual gap at R_L when γ = λ.
• Improve invariant extraction under decoherence. The measured fractional charges approach but do not reach 1/2, and the authors identify finite time windows, relaxation, and finite mode length as causes; better error mitigation or longer coherence would tighten the quantization.
• Push toward the interacting regime. Extending the analogue-digital protocol to interacting Hamiltonians is the natural next step implied by the outlook, but the paper does not report such an experiment.

Target Audience

Quantum-simulation researchers working on topological matter; superconducting-qubit and quantum-annealing hardware specialists interested in coherent analogue-digital protocols; condensed-matter theorists who study higher-order topology and would benefit from an experimental handle on nested Wilson loops; and graduate students or advanced readers comfortable with tight-binding models and band topology who want a concrete example of measuring a topological invariant rather than inferring it from edge spectra.

Authors’ abstract

Topological insulators lie outside the standard Landau paradigm: Instead of being characterized by local order parameters and spontaneous symmetry breaking, their defining properties are encoded in the global structure of their quantum states and manifested as robust boundary states. Crystalline higher-order topological insulators can extend this principle by hosting localized corner modes in two and higher dimensions. Although these boundary modes can be detected spectroscopically through the local density of states, determining the underlying bulk topology, for example through nested Wilson loops, remains an outstanding experimental challenge. Here we use up to 728 qubits in a superconducting analogue-digital quantum simulator to realize and probe a hierarchy of one-, two-, and three-dimensional topological insulators governed by effective tight-binding Hamiltonians. By reconstructing complex propagators from real-space measurements, we obtain complementary boundary and bulk characterizations: local densities of states reveal localized boundary modes, whereas experimentally reconstructed band projectors yield bulk topological invariants, culminating in a three-dimensional octupole moment extracted through third nested Wilson loops. These results establish a route to realizing, characterizing, and manipulating topologically protected states in programmable three-dimensional quantum matter.