An entangling Hubbard quasiparticle collider on a digital quantum processor

AI summary

Overview

Research area: Quantum simulation of strongly correlated fermionic matter — specifically quasiparticle scattering in the one-dimensional Fermi–Hubbard model, implemented on superconducting quantum hardware.

Technical level: Advanced. The paper assumes familiarity with the Hubbard model, Bethe ansatz scattering matrices, Jordan–Wigner mapping, Trotterization, and entanglement entropy decompositions. The conceptual narrative, however, is accessible to anyone who knows basic quantum mechanics.

Scope: The paper reports a hardware experiment in which a doublon and a holon are prepared as directed wave packets on a 21-site Hubbard chain encoded in 42 qubits of the ibm_boston processor, collided, and analyzed for interaction-dependent transmission and collision-generated number entanglement.

What This Paper Is About

Scattering experiments are how physicists probe interactions, but in strongly correlated materials the collisions happen between quasiparticles — collective excitations such as the doubly occupied "doublon" and the empty "holon" of a half-filled Mott insulator — and these collisions are normally only accessible indirectly through spectroscopy. This paper asks whether a digital quantum processor can act as a controllable collider for such quasiparticles: create a doublon and a holon on demand, launch them at each other with chosen momenta, and read out what happens site by site and moment by moment.

The goal is two-fold: to reproduce the exactly known Bethe-ansatz scattering matrix of the one-dimensional Hubbard model on quantum hardware without free parameters, and to detect the entanglement that the collision generates between the two halves of the chain — something a conventional cross-section measurement cannot capture.

Key Contributions

1. First measurement of the Hubbard quasiparticle transmission probability on quantum hardware. The authors collide a doublon and a holon on a 21-site chain (42 qubits) and extract the transmission probability as a function of interaction strength, finding agreement with the Bethe-ansatz result with no free parameters.

2. A tunable quasiparticle collider protocol. They introduce a two-phase time-evolution scheme: in phase 1 (0 < T ≤ 1) hopping toward the chain edges is switched off so each excitation becomes a directed wave packet moving only toward the center; in phase 2 (1 < T ≤ 4) all hopping is restored and the packets collide.

3. Tomography-free detection of collision-generated number entanglement. The superposition of transmitted and reflected channels produces a particle-number fluctuation in the left half of the chain, from which the number entropy is inferred using only Z-basis snapshots — no state tomography.

4. A controlled comparison with identical particles. Replacing the holon with a second doublon removes the exchange channel by Pauli exclusion, and the doublon–doublon collision shows neither the interaction dependence nor the number fluctuation, isolating the effect as specific to the particle–antiparticle pair.

Main Findings

• Interaction-tuned crossover from transmission to reflection. At U = 1 the doublon and holon packets pass through each other; at U = 7 they are reflected back; at U = 3.5 the post-collision charge density nearly vanishes. The collision is elastic at all measured U — outgoing packet speeds match incoming ones — consistent with integrability of the 1D Hubbard model.

• Quantitative agreement with the Bethe-ansatz S-matrix. The measured transmission probability follows 𝒯(θ) = θ²/(1 + θ²) with θ = 2J(sin k₁ − sin k₂)/U. Substituting the maximum relative group velocity c = 4J gives a parameter-free curve that matches the data; fitting c as a free parameter yields c ≈ 3.6J, attributed to the spread of momenta around k = π/2 in the prepared packet.

• The vanishing density at U = 3.5 is superposition, not annihilation. The total doublon number increases monotonically with U rather than dipping, ruling out doublon–holon recombination. Instead, the doublon and holon each spread over both sides of the chain, consistent with |ψ_out⟩ ~ t|d_R h_L⟩ + r|d_L h_R⟩.

• Collision generates number entanglement peaking at 𝒯 = 1/2. The transmitted and reflected components differ in left-half particle number by two, giving number entropy S_num = −𝒯 ln 𝒯 − (1−𝒯) ln(1−𝒯), which reaches ln 2 at 𝒯 = 1/2, and particle-number variance Var(N_L) = 4𝒯(1−𝒯). Figure 1c places the entropy maximum at U/J = 4.

• Doublon–doublon collisions show no such signatures. For two identical doublons the dynamics are nearly identical across all U (apart from a small U-dependent decay), and the left-half number variance stays close to zero at T = 3.2 for all U, in contrast to the large variance in the doublon–holon case.

• Coherence is inferred, not directly verified. The authors state explicitly that the measurement establishes the presence and relative weights of the two components but does not directly confirm coherence between them; the coherent-superposition interpretation rests on unitarity and the match to the predicted U dependence.

Methodology in Plain English

Encoding the chain. Each Hubbard site uses two qubits, one for spin-up and one for spin-down, so 21 sites require 42 qubits. Fermions are mapped to qubits with a Jordan–Wigner transformation using a "zigzag" ordering in which same-spin orbitals of neighboring sites sit on physically adjacent qubits; this makes every hopping term a nearest-neighbor two-qubit gate.

Preparing the collision. The starting state is a Néel state (|↑↓↑⋯↓↑⟩). A doublon is created on one site and a holon on another, equidistant from the chain center. Simply letting the state evolve would spread each excitation in both directions, so the circuit first runs a phase with hopping toward the edges disabled, producing directed wave packets described by Bessel functions of the first kind. Comparing to that exact semi-infinite-chain solution, the authors find more than 93% of the wave-function weight sits in the leading packet for T ≤ 4, so trailing packets are neglected.

Time evolution. Evolution uses a custom second-order Trotter decomposition in which each step is a hopping layer, an interaction-plus-fermionic-swap layer, and another hopping layer. The number of Trotter steps is fixed at N = 20, with time steps adaptively refined around the collision. Because adjacent steps' hopping layers merge, N steps cost a two-qubit circuit depth of 5N + 2. An even number of steps is always used so the spin-up/spin-down layout returns to its original configuration.

Error handling. The qubit layout is chosen by excluding qubits and couplers whose readout and CZ error rates exceed thresholds, then running Bell-state fidelity tests on remaining couplers to pick the best 42-qubit chain. Each data point uses 70,000 shots split across 70 Pauli twirling instances, combined with dynamical decoupling and readout twirling. Post-processing exploits conserved total particle number (to estimate and correct hardware error rates) and the combined inversion–particle-hole symmetry ℐ𝒞, which the initial doublon–holon configuration respects via pseudospin (η-pairing) symmetry. For symmetrization purposes the excitations are defined on the first 20 sites, so the "center" lies between sites 10 and 11 and ℐ: i → 21 − i.

Background subtraction. Because the Néel state is not the exact ground state, every measurement is paired with a reference run of the same circuit without the excitations; the difference is denoted ΔA = A − A_bg.

Tomography-free readout. Only Z-basis snapshots are taken, from which site-resolved charge, doublon (⟨n_{i↑}n_{i↓}⟩), and holon (⟨(1−n_{i↑})(1−n_{i↓})⟩) densities are reconstructed.

Parameter scan and cross-checks. Full time dynamics are measured at U = 1, 3.5, and 7; the interaction dependence of the scattering outcome is probed at fixed T = 3.2 for U = 1 through 8 in integer steps plus U = 3.5. Results are compared against matrix-product-state TEBD simulations using TeNPy, with bond dimension χ = 1024 in the site-resolved comparisons.

Why This Matters

Impact on research. This is the first hardware measurement of a Hubbard-model quasiparticle transmission probability, and it validates the Bethe-ansatz S-matrix against experiment across weak-to-strong coupling. More broadly, it shows that digital quantum processors can do something cold-atom and solid-state experiments cannot easily do: pick the flavor of each colliding quasiparticle, program its momentum, and tune the interaction continuously from weak to strong coupling. The demonstration that entanglement generated by scattering can be read out from simple occupancy statistics — without full state tomography — is a practical template for entanglement metrology on near-term devices.

Real-world applications (speculative, as the paper is fundamental physics):

• Benchmarking and validating quantum hardware and error-mitigation stacks against exactly solvable strongly correlated models.
• Informing transport modeling in correlated electron materials, where quasiparticle collisions set conductivity and relaxation rates.
• Providing a testbed for entanglement-generation protocols relevant to quantum information tasks on superconducting processors.
• Guiding the design of future quantum simulators for lattice gauge theories and other scattering problems already explored on trapped-ion and superconducting platforms.

Industry relevance. The work uses a commercial IBM superconducting processor and openly documents layout selection, Pauli twirling, dynamical decoupling, readout twirling, and symmetry-based post-processing. These are directly reusable engineering recipes for anyone running long-time correlated-fermion circuits on cloud quantum hardware, and the resource accounting (42 qubits, 20 Trotter steps, depth 5N + 2, 70,000 shots per point) gives a concrete benchmark for hardware roadmaps.

Future Directions

• Directly confirm coherence. The authors note the measurement shows only the presence and weights of the transmitted and reflected components. An interference experiment on the post-collision wave function, once longer simulation times are accessible, would test whether the superposition is genuinely coherent.
• Extend simulation time. The current window (T ≤ 4, N = 20 Trotter steps) is limited by circuit depth and hardware error; longer times would enable interference probes and access to later-time scattering dynamics.
• Generalize the collider to other quasiparticle flavors and models. The protocol's premise — choosing collision partners on demand — invites extension to other charge, spin, or pairing excitations, and to higher dimensions or multi-particle collisions.
• Push beyond two-body scattering. The exact Bethe-ansatz calibration here is two-body; whether the platform can resolve few-body scattering and entanglement structure where analytic results are unavailable is an open question the paper implicitly raises.

Target Audience

Quantum simulation researchers and condensed-matter theorists interested in strongly correlated fermions; experimentalists working on superconducting quantum processors who want concrete recipes for layout selection, Trotter compilation, and symmetry-based error mitigation; and quantum-information scientists interested in tomography-free entanglement detection. Graduate students with a background in many-body physics or quantum computing will get the most from the technical details, while the introduction and discussion are readable by a broader physics audience.

Authors’ abstract

Scattering of quasiparticles provides direct access to the interactions that govern strongly correlated systems. In this work, we realize the quasiparticle scattering experiment of the Fermi-Hubbard model on a digital quantum processor by colliding a doublon and a holon in the half-filled chain of 21 sites encoded on 42 qubits of the ibm_boston processor. For the first time, we measure the transmission probability of quasiparticles of the Hubbard model on quantum hardware, which agrees with the Bethe-ansatz scattering matrix without free parameters. Beyond the scattering probabilities, the collision generates entanglement by superposing the transmitted and reflected components, which we infer from the particle-number fluctuation. In sharp contrast, a control collision between two doublons shows neither the interaction dependence nor the number fluctuation, as expected for indistinguishable particles. Our work demonstrates that the ability to create quasiparticles of a chosen flavor on demand and program their motion and interactions makes a digital quantum processor a new experimental toolset for probing the entanglement structure generated by quasiparticle scattering.