Exploring topology via momentum-selective tomography

AI summary

Overview

Research area: Quantum simulation of topological phases — specifically, direct measurement of band-topology invariants (winding numbers) on a superconducting quantum processor.

Technical level: Intermediate. The core idea is explainable without heavy formalism, but the paper assumes familiarity with Bloch states, the Su-Schrieffer-Heeger (SSH) model, and chiral symmetry.

Scope (one sentence): The paper proposes and experimentally demonstrates a "star-like" lattice scheme in which an auxiliary probe qubit couples only to selected momentum states of a one-dimensional topological chain, allowing the winding phase — and hence the winding number — to be read out momentum-by-momentum on a 9-qubit superconducting chip.

What This Paper Is About

Topological invariants such as the winding number are defined as integrals over the entire Brillouin zone in momentum space, but the momentum eigenstates (Bloch states) they are built from are spread out over the whole lattice, making them very hard to prepare or resolve in a real-space experiment. Existing methods — Bloch oscillations, quantum walks, charge pumping, quench dynamics — either scan the whole Brillouin zone or fill an entire band, and none directly reports the momentum-dependent geometric information of an individual Bloch state. The authors' goal is to measure that geometric information one momentum at a time: coupling a single auxiliary probe site to a chain with spatially modulated couplings so that it addresses only the Bloch state of a chosen momentum, then extracting the winding phase from the probe's zero-energy state.

Key Contributions

1. A momentum-selective "star-like" model. An auxiliary central site couples to every site of a one-dimensional ring chain with a spatially modulated coupling η_σ(x) = η̃_{σ,q} e^(−iqx). After Fourier transformation the coupling becomes η_σ(k) ∝ δ_{q,k}, so the probe couples exclusively to the Bloch wave with momentum k = q — a direct bridge between real-space control and momentum-space topology.

2. A winding-number tomography protocol. The probe is engineered to couple only to the B sublattice so that chiral symmetry is preserved, producing a zero-energy in-gap state whose real-space profile encodes the winding phase φ_q. Because the chain is a ring with no boundary, this state is explicitly not an edge state but a probe state; the phase is extracted by a gauge-fixed Fourier transform, and the winding number follows from summing phase differences around the Brillouin zone.

3. Experimental demonstration on a highly connected superconducting processor. Using the L9 flip-chip device (nine frequency-tunable transmon qubits, 24 tunable floating couplers, central-qubit connectivity d₀ = 8, non-local links bridged by long tantalum airbridges), the authors realize an extended SSH model with L = 4 unit cells and long-range cross-cell tunneling, and apply the protocol via a Y_(π/2) pulse on the central qubit, coherent evolution, per-qubit state tomography, and nested temporal-then-spatial Fourier analysis.

4. A full topological phase diagram, including a higher winding number. Sweeping t₂/t₁ and γ₂/t₁ maps the phase diagram and resolves quantized winding numbers ν = {0, ±1, 2}, characterized by phase windings {0, ±2π, 4π}, with the ν = +2 phase additionally confirmed by edge-state dynamics under open boundary conditions.

Main Findings

• Single-momentum tomography works at the standard SSH point. At t₂/t₁ = 2 with γ₁,₂ = 0 (with t₁ = 2.0 MHz and t₂ = 4.0 MHz), the extracted φ_q versus q follows a single-turn helix whose projection onto the (cos φ_q, sin φ_q) plane closes into a unit circle encircling the origin exactly once, giving ν = +1.

• Sharp quantized phase transitions. Along a vertical cut at γ₂/t₁ = 0, varying t₂/t₁ drives a conventional transition from ν = 0 to ν = +1 across the standard-SSH boundary t₂ = t₁. Along a horizontal cut at t₂/t₁ = 0.5, increasing the long-range coupling γ₂/t₁ drives the system from ν = 0 into the higher-winding phase ν = +2, with integer steps and quantized plateaus.

• Direct resolution of a double-turn winding. In the ν = +2 phase the measured φ_q shows a continuous 4π progression across the Brillouin zone; in the trivial ν = 0 phase φ_q shows non-winding oscillations; the ν = −1 phase (reached by redefining the unit cell and therefore omitted from the plotted phase diagram) shows a 2π winding in the opposite direction.

• Edge states confirm the ν = +2 phase. Under open boundary conditions, site-resolved population dynamics following Fock-state initialization at the left edge, in the bulk, and at the right edge show qualitatively distinct evolution between edge and bulk excitation.

• Robustness against coupling disorder. Adding disorder of strength W (random variables drawn uniformly from [−0.5, 0.5], averaged over ∼15 realizations), the measured ν stays pinned at +1 up to W/t₁ ∼ 1.375, matching simulation; beyond that threshold both experiment and simulation show strong fluctuations, marking disorder-induced topological breakdown.

• A four-cell device reproduces the long-chain prediction. Data from the compact L = 4 lattice agree both with finite-size exact diagonalization and with the theoretical prediction derived in the long-chain limit L = 100.

• Device performance. Average isolated energy-relaxation time T₁ = 59.3 μs, average spin-echo dephasing time T₂^echo = 29.3 μs, average single-qubit gate fidelity 99.93%, and average two-qubit CZ-gate fidelity 99.5%.

Methodology in Plain English

The authors build a ring-shaped chain of lattice sites with a single extra "probe" site sitting in the middle, coupled to every site of the ring. Normally such a probe would talk to all momentum states at once. The trick is to make the coupling strength vary from site to site as a cosine wave with a chosen spatial frequency q. Just as a radio tuned to one frequency picks out one station, this spatial modulation means the probe only couples to the chain's Bloch state with momentum q.

Because the probe is deliberately connected only to one sublattice, the combined system has a special zero-energy state whose shape in real space carries the momentum-dependent phase φ_q of that Bloch state. The experimental sequence is: put the central qubit in a superposition with a single excitation, switch on all the engineered couplings simultaneously so the system evolves, then read out each A-sublattice qubit in the x and y bases. A Fourier transform in time filters out the zero-energy component from the mixture of energies, and a second Fourier transform in space over the lattice sites extracts φ_q. Repeating for each q traces out φ_q across the Brillouin zone, and summing the phase steps gives the winding number. The four-cell device is programmed by tuning 16 active couplers, with unused couplers parked at ZZ-free idle positions, and the spatial modulation profiles are shown for target momenta n = 1, 2, 4 (n = 3 coincides with n = 1).

Why This Matters

Impact on research. The protocol gives state-by-state access to momentum-resolved geometric information without filling an entire band or imaging full momentum space, which the authors position as an advantage over Bloch-oscillation, quantum-walk, charge-pumping, and quench-based approaches. It also shows that a compact, resource-limited lattice (L = 4) can reproduce long-chain (L = 100) topological predictions, which matters for near-term processors with few qubits.

Real-world applications (as framed by the paper):

• Robust quantum computing architectures that exploit topological protection to shield quantum information.
• Topological quantum information processing built on selectively addressable momentum states.
• A characterization and benchmarking tool for verifying that a quantum processor implements a target Hamiltonian and its band structure correctly.
• A template that the authors argue can be generalized to other symmetry classes and higher-dimensional systems such as Chern insulators, and extended to multi-particle systems.

Industry relevance. The work comes from Tencent Quantum Laboratory together with Shenzhen University and uses a foundry-style flip-chip superconducting process (200 nm α-phase tantalum film, stiffener-reinforced tantalum airbridges 200 μm long and below 5 μm in height, 10 μm indium bumps, base temperature ∼15 mK). The long-airbridge crossover approach to non-local connectivity is directly relevant to hardware teams designing high-connectivity superconducting architectures, and the tomography protocol is a candidate validation suite for such devices.

Future Directions

• Scaling to larger lattices and higher winding phases, which the authors state is possible with this architecture and illustrate in the Supplementary Material (Fig. S5).
• Generalizing beyond 1D chiral-symmetric models to other symmetry classes and to higher-dimensional topological systems such as the Chern insulator.
• Extending the star-like geometry to multi-particle systems, which the authors identify as a route to probing interacting topological matter.
• Turning topology into a resource: connecting the demonstrated robustness of measured invariants (the plateau persisting to W/t₁ ∼ 1.375) to actual schemes for topologically protected quantum computation, and understanding the breakdown beyond that disorder threshold.

Target Audience

Quantum-simulation experimentalists working with superconducting circuits; condensed-matter and cold-atom researchers interested in measuring topological invariants without full band tomography; quantum-hardware engineers concerned with high-connectivity coupler architectures and device benchmarking; and graduate students or advanced undergraduates with some background in tight-binding models and Bloch theory who want a concrete, experimentally realized example of momentum-resolved topology measurement.

Authors’ abstract

Identifying and measuring topological invariants remains a fundamental challenge due to the non-local nature of momentum eigenstates in real space. Established approaches based on Bloch oscillations, quantum walks, charge pumping and quench dynamics, typically fail to provide direct momentum-dependent geometric information of each Bloch state. Here, we propose a novel framework for implementing direct band-topology tomography via momentum-selective measurements, and experimentally demonstrate it on a superconducting processor featuring a highly connected coupling architecture. Utilizing a ``star-like" geometry in which an auxiliary probe couples to a one-dimensional topological chain with spatially modulated coupling strengths, we selectively address the desired momentum states of the chain. This approach enables the direct and robust extraction of the winding phase that contributes to the associated topological invariant. Leveraging non-local couplers bridged by long tantalum airbridges, we realize an extended Su-Schrieffer-Heeger (SSH) model and map its full topological phase diagram. Through a nested Fourier analysis over time and space, we directly extract the winding phase across Brillouin zone and resolve quantized winding numbers of $ν=\{0,\pm1,2\}$ characterized by phase winding $\{0, \pm 2π,4π\}$. Our work establishes a versatile tool for probing topological characteristics and opens a new avenue for robust quantum computing architectures based on topological protection.