⚛️ quantum physics

Exploring topology via momentum-selective tomography

This paper proposes and experimentally demonstrates a novel momentum-selective tomography framework on a superconducting processor, enabling the direct extraction of winding phases and the mapping of topological phase diagrams for extended Su-Schrieffer-Heeger models.

Original authors: Sainan Huai, Ling Lin, Kunliang Bu, Tianqi Cai, Zhiwen Zong, Xiaopei Yang, Yuan Li, Yulei Ye, Zhenxing Zhang, Shengyu Zhang, Chaohong Lee

Published 2026-10-08

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CC BY 4.0

Original authors: Sainan Huai, Ling Lin, Kunliang Bu, Tianqi Cai, Zhiwen Zong, Xiaopei Yang, Yuan Li, Yulei Ye, Zhenxing Zhang, Shengyu Zhang, Chaohong Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the strange world of quantum physics, particles do not just sit in one place; they exist as waves that spread out across space. When these waves move through a repeating pattern, like atoms arranged in a crystal, they form specific shapes known as energy bands. For decades, physicists have known that these bands can possess a hidden, global property called topology. Think of this property like the number of holes in a piece of dough: a donut has one hole, a pretzel has three, and a bagel has one, but you cannot turn a donut into a pretzel without tearing the dough. In the quantum world, this "hole count" is called a topological invariant. It is a number that describes the entire system at once, making it incredibly stable against small disturbances. This stability is why scientists are so interested in these properties; they could be the key to building quantum computers that do not crash when faced with tiny errors. However, measuring this global "hole count" has been a major hurdle. Because the waves are spread out, it is difficult to look at just one specific part of the pattern to see how the shape twists. Previous methods required watching the entire system change over time or filling it up completely, which is like trying to understand the shape of a knot by watching the whole rope wiggle rather than looking at the knot itself.

A team of researchers has now found a way to look directly at these twists, one piece at a time, using a specialized quantum computer. They built a device with nine superconducting qubits, which are tiny circuits that act like artificial atoms, arranged in a ring. To this ring, they added a central probe, creating a star-like shape where the center connects to every point on the ring. The researchers did not just connect the center to the ring with a single wire; they carefully adjusted the strength of each connection so that it varied in a smooth, wave-like pattern around the circle. By tuning this pattern, they discovered they could make the central probe talk to only one specific type of wave traveling around the ring, ignoring all the others. This is a crucial breakthrough because it allows them to isolate a single state of motion without disturbing the rest of the system.

Once they isolated a specific wave, they let the system evolve and then measured the state of the qubits on the ring. Because they knew exactly which wave they had selected, they could read the phase, or the specific timing, of that wave's twist. By repeating this process for many different waves around the ring, they were able to map out how the twist changed from one point to the next. When they added up all these changes, they could calculate the total number of twists, which is the topological invariant. The team tested this method on a model known as the Su-Schrieffer-Heeger model, which is a standard way to study these properties. They successfully measured the twist numbers for different configurations, finding values of zero, one, and even two. A value of two means the wave twists around twice as it goes around the ring, a more complex shape than the single twist usually seen.

The researchers also tested how well their method held up when the system was imperfect. In the real world, machines are never perfect, and small errors or "disorder" are always present. They intentionally introduced random variations into the connections between the qubits to simulate a messy environment. Even with these imperfections, the measured twist numbers remained stable and did not change, confirming that the method is robust. This resilience is exactly what is needed for future quantum technologies, where protecting information from noise is essential. The team demonstrated that they could map out the entire landscape of possible shapes, showing where the system switches from having no twists to having one or two. They verified their findings by comparing the experimental data with computer simulations, and the two matched perfectly.

This work provides a new tool for exploring the hidden geometry of quantum matter. Instead of relying on indirect clues or watching the whole system blur together, scientists can now pick out specific parts of the quantum landscape and measure their shape directly. The researchers showed that this approach works on a small, compact device with just nine qubits, suggesting it could be scaled up to larger systems. By proving that they can extract these global properties from local measurements, they have opened a new path for understanding and using topological states. This could eventually lead to quantum computers that are naturally protected against errors, using the very same "hole count" properties that make the system so stable in the first place. The experiment stands as a clear demonstration that the abstract mathematics of topology can be measured directly in a physical device, turning a theoretical concept into a tangible reality.

Technical Summary: Exploring Topology via Momentum-Selective Tomography

Problem Statement
The identification and measurement of topological invariants, such as the Chern number and winding number, remain a fundamental challenge in condensed matter physics and quantum simulation. These invariants are defined over the Brillouin zone in momentum space and characterize the global geometric features of energy bands. However, direct observation in experimental settings is hindered by the non-local nature of momentum eigenstates (Bloch states) in real space. Existing approaches, including Bloch oscillations, quantum walks, charge pumping, and quench dynamics, typically fail to provide direct, momentum-dependent geometric information for individual Bloch states. While bulk-boundary correspondence offers experimental signatures via boundary states, it does not provide a comprehensive view of the underlying bulk band topology. Furthermore, previous methods often require scanning the entire Brillouin zone or completely filling Bloch bands (e.g., via sudden quenches to flat bands), raising the question of whether one can directly measure the momentum-dependent geometric information of specific Bloch states without full tomography.

Methodology
The authors propose and experimentally demonstrate a novel framework for direct band-topology tomography using momentum-selective measurements. The core of the scheme involves a "star-like" lattice model where an auxiliary probe site couples to a one-dimensional (1D) topological chain with spatially and periodically modulated coupling strengths.

Theoretical Framework:
- Star-like Model: A central auxiliary site (probe) couples to every site of a 1D ring chain. The coupling strength $\eta(x)$ is modulated spatially as $\eta(x) = \tilde{\eta} e^{-iqx}$ .
- Momentum Selectivity: Through Fourier transformation, this spatial modulation ensures the probe couples exclusively to Bloch states with momentum $k=q$ . By tuning the modulation frequency $q$ , specific momentum states can be addressed.
- Winding Phase Extraction: The protocol targets the extended Su-Schrieffer-Heeger (SSH) model. By preserving chiral symmetry (coupling the probe only to the B-sublattice), a zero-energy eigenstate emerges in the gap. The real-space wavefunction of this zero-energy state encodes the winding phase $\phi_q$ of the target momentum.
- Nested Fourier Analysis: The experiment utilizes a two-step spectral analysis. First, a temporal Fourier transform of the dynamical evolution isolates the zero-energy component from the complex dynamics. Second, a spatial Fourier transform of this isolated component extracts the winding phase $\phi_q$ . The winding number $\nu$ is then calculated by integrating the phase accumulation across the Brillouin zone.
Experimental Platform:
- The scheme is implemented on a 9-qubit superconducting quantum processor (L9) featuring a flip-chip architecture with a highly connected coupling network.
- Architecture: The device uses 24 independently tunable floating couplers and long tantalum airbridges to realize non-local connections without line-crossing overhead. This allows a central qubit ( $Q_0$ ) to couple to up to 8 outer qubits, forming the required star-like geometry.
- Model Realization: The outer 8 qubits form an extended SSH model with 4 unit cells, including intra-cell ( $t_1$ ), inter-cell ( $t_2$ ), and cross-cell ( $\gamma_2$ ) hoppings. The central qubit couples to the B-sublattice sites with programmable spatial modulation.
- Protocol: The system is initialized in a superposition of the vacuum and a single excitation on the probe. It evolves under the engineered Hamiltonian, followed by simultaneous single-qubit quantum state tomography (measuring $\langle \sigma_x \rangle$ and $\langle \sigma_y \rangle$ ) on the A-sublattice qubits.

Key Results

Direct Phase Extraction: The experiment successfully extracted the winding phase $\phi_q$ for specific momenta across the Brillouin zone. The data showed a distinct single-turn helix for the trivial-to-topological transition ( $\nu=1$ ), with the projection onto the $(\cos \phi_q, \sin \phi_q)$ plane encircling the origin exactly once.
Topological Phase Diagram Mapping: The authors mapped the full topological phase diagram of the extended SSH model by varying coupling strengths $t_2/t_1$ and $\gamma_2/t_1$ . They experimentally resolved quantized winding numbers $\nu \in \{0, 1, 2\}$ . (Note: While the $\nu = -1$ phase is theoretically accessible by re-defining the unit cell, it was not displayed in the experimental phase diagram).
High-Winding Phases: The protocol successfully characterized a higher-winding phase with $\nu = 2$ , evidenced by a continuous $4\pi$ phase progression across the Brillouin zone, corresponding to a double-turn helical winding.
Robustness: The measured winding numbers remained quantized and stable against moderate disorder (up to $W/t_1 \sim 1.375$ ). The variance was strongly suppressed, confirming that local errors merely shift individual phases without altering the global topological invariant, provided the bulk gap remains open.
Finite-Size Scalability: Despite using a compact system ( $L=4$ unit cells), the experimental results matched theoretical predictions derived for the long-chain limit ( $L=100$ ), validating the efficiency of the protocol for resource-limited near-term processors.

Significance and Claims
The paper establishes momentum-selective excitation as a robust tool for band-topology tomography. The authors claim this approach offers distinct advantages over existing techniques:

Directness: Unlike methods based on Bloch oscillations or quantum walks, it explicitly resolves the momentum-dependent geometric information (winding phase) of individual Bloch states.
Efficiency: In contrast to full tomography via quantum quench, it bypasses the need to fill the entire energy band.
Robustness: The extraction principle, relying on nested Fourier analysis of globally protected phases, ensures resilience against system imperfections and moderate disorder.

The work demonstrates a versatile technical path for probing topological invariants in complex quantum systems. The authors suggest this capability could serve as a foundation for exploring topological quantum information processing and potentially aid in the development of robust quantum computing architectures based on topological protection. The framework is noted to be generalizable to other symmetry classes and higher-dimensional systems, such as Chern insulators, and extendable to multi-particle systems.

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