⚛️ quantum physics

Analog quantum simulation of bosonic and anyonic models with flux-driven transmons

This paper proposes a general protocol for analog quantum simulation of Bose-Hubbard and anyon-Hubbard models using lattices of flux-driven transmons, enabling tunable on-site interactions, density-dependent hopping, and complex transition phases to reproduce characteristic many-body dynamics with experimentally realistic parameters.

Original authors: Isak Lyngfelt, Jorge Fernández-Pendás, Göran Johansson, Laura Garciá-Álvarez

Published 2026-09-04

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CC BY 4.0

Original authors: Isak Lyngfelt, Jorge Fernández-Pendás, Göran Johansson, Laura Garciá-Álvarez

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

Nature is full of systems where many tiny parts interact, creating behaviors that are far too complex for even the most powerful classical computers to predict. When particles like atoms or electrons crowd together, they do not just act as individuals; they form a collective whole with new properties, such as superfluidity, where matter flows without friction. To understand these mysteries, scientists often turn to quantum simulation. Instead of trying to calculate every interaction on a screen, they build a controllable quantum machine that mimics the target system. By watching how this machine evolves, they can observe the physics of the unknown directly. This approach is particularly valuable for studying exotic particles called anyons, which exist only in two dimensions and follow rules of interaction that are stranger than those of ordinary matter, and for exploring how particles move through a grid when they bump into one another.

A team of researchers at Chalmers University of Technology in Sweden has now proposed a new way to build such a simulator using superconducting circuits. Their work focuses on a specific type of quantum bit known as a transmon, which is a tiny loop of superconducting wire containing a junction that allows electrons to tunnel through an insulator. In a standard computer, these devices are usually treated as simple two-level systems, but the researchers realized that if they are allowed to access their higher energy levels, they can act as a platform for simulating complex bosonic models. The team proposed that by carefully modulating the frequency of these transmons with an oscillating magnetic field, they could force the devices to interact in a very specific way. This modulation creates a bridge between different energy states, allowing the researchers to tune how strongly the particles repel each other and how easily they hop from one site to the next on a grid.

The core of the discovery lies in how the researchers drive the system. Instead of leaving the transmons alone, they apply a rhythmic push to their frequencies, alternating the pattern across the grid. This driving force resonates with the natural energy gaps between the different states of the particles. By adjusting the speed and strength of this drive, the scientists can make the particles behave as if they are following the rules of the Bose-Hubbard model, a famous theory that describes how bosons, a class of particles that can share the same space, organize themselves. In their proposal, they showed it is possible to simulate a grid where up to three particles can occupy a single spot, a significant step forward from previous attempts that were limited to just two. They can also tune the interaction strength, effectively turning the repulsion between particles up or down, which allows them to explore different phases of matter, from a fluid-like state to a rigid, insulating one.

Perhaps the most novel aspect of this work is the proposal to simulate anyons, particles that do not fit the standard categories of bosons or fermions. When two identical particles are swapped, bosons remain unchanged, while fermions flip their sign. Anyons, however, can acquire a complex phase shift, a kind of internal rotation, when they exchange places. The researchers found that by adding a specific phase offset to the driving signal on each transmon, they could engineer these exotic exchange rules directly into the hardware. This allows the system to mimic the behavior of interacting anyons, including their tendency to form asymmetric paths when they move, a phenomenon known as an asymmetric quantum walk. The team verified their proposal through detailed numerical simulations using parameters that are realistic for current laboratory equipment. These simulations confirmed that the driven transmon array faithfully reproduces the characteristic dynamics of the target models, including the way particles localize when interactions are strong and the unique statistical patterns that emerge from anyonic behavior.

The researchers are careful to note that their findings are based on simulations and theoretical protocols, not yet on a physical experiment. They have shown that the method works in a computer model using experimentally realistic values, such as frequencies around 4.5 gigahertz and coupling strengths of 6 to 15 megahertz. The simulations included the full complexity of the hardware, accounting for the fact that real devices have extra energy levels and that the driving forces can introduce unwanted noise. Despite these challenges, the results suggest that the approach is robust enough to capture the essential physics of the target models. The team also highlighted a practical constraint: the simulation is most reliable when the number of particles in the system is kept low enough that the transmons do not get pushed into energy states that are too high to measure or control. For the anyon model, this means limiting the system to two particles per site, while the bosonic model can handle up to three.

This work opens a path toward using superconducting circuits, which are already at the forefront of quantum computing development, to study fundamental questions in many-body physics. By proving that these circuits can be tuned to simulate both standard bosonic systems and the more exotic anyonic ones, the researchers provide a versatile tool for exploring quantum matter. The ability to control the interaction strength and the statistical phase of the particles on demand means that scientists could potentially observe phase transitions and other collective phenomena that are currently out of reach. While the paper does not claim to have solved the problem of simulating these systems perfectly, it offers a concrete and flexible blueprint for doing so, suggesting that the next generation of quantum simulators could be built using the very same hardware designed for quantum computation.

Technical Summary: Analog Quantum Simulation of Bosonic and Anyonic Models with Flux-Driven Transmons

Problem and Motivation
Quantum analog simulation offers a pathway to study many-body phenomena that are classically intractable by directly realizing target system dynamics in controllable quantum hardware. While optical lattices have successfully simulated the Bose-Hubbard (BH) and anyon-Hubbard (AH) models, superconducting circuits—specifically transmon qubits—have primarily been limited to simulating hard-core bosons (single-particle per site) or spin systems due to the large anharmonicity of transmons. Previous proposals for parametrically tunable interactions in transmons were restricted to systems with only two total particles. Furthermore, while anyonic statistics have been simulated digitally in superconducting architectures, no analog protocol for the AH model on a superconducting platform had been proposed. This work addresses the challenge of extending superconducting circuit simulations to include multi-particle bosonic states and anyonic statistics without relying on gate-based digital decomposition.

Methodology
The authors propose a protocol using a lattice of capacitively coupled, flux-tunable transmon qubits. The core mechanism involves modulating the transmon frequencies ( $\omega_i$ ) in an alternating pattern across the lattice using a sinusoidal drive: $\omega_i(t) = \omega_i^0 + \delta_i \cos(\Omega_i t + \phi_i)$ .

Effective Hamiltonian Derivation: By transforming into a rotating frame and applying the Rotating Wave Approximation (RWA), the time-dependent drive creates sidebands at integer multiples of the drive frequencies. Resonant transitions between many-body Fock states occur when the sideband frequencies cancel the energy differences between states.
Resonance Conditions: The authors derive specific resonance conditions where combinations of Bessel functions (arising from the Jacobi-Anger expansion of the drive) determine the effective transition amplitudes ( $G_m$ ). By tuning the drive parameters (amplitude $\delta_i$ , frequency $\Omega_i$ , and phase $\phi_i$ ), the protocol can match the hopping amplitudes of target models.
Bose-Hubbard Simulation: For the BH model, the protocol utilizes an alternating drive pattern to create two sublattices. By selecting drive frequencies such that the sum of Bessel orders has even parity, the authors eliminate unwanted $\pi$ phase shifts between transitions. This allows for the independent tuning of on-site energy ( $\mu$ ), on-site interaction ( $U$ ), and density-dependent hopping functions $F(\hat{n}_i - \hat{n}_j)$ . The system is restricted to at most three excitations per site ( $(\hat{a}^\dagger_i)^4 = 0$ ).
Anyon-Hubbard Simulation: To simulate the AH model, the authors introduce phases ( $\phi_i$ ) to the modulation. These phases render the transition amplitudes complex-valued. By engineering a linear ramp of effective phases along the chain, the protocol reproduces the density-dependent Peierls phase characteristic of anyons. This implementation requires a restriction to at most two particles per site ( $(\hat{a}^\dagger_i)^3 = 0$ ) to satisfy the necessary constraints on drive parameters.

Key Contributions

General Protocol: A unified framework for simulating both the BH and AH models using flux-driven transmons, capable of handling up to three bosons per site for the BH model and two for the AH model.
First Analog AH Proposal: This is the first proposal for the analog simulation of the anyon-Hubbard model on a superconducting platform.
Tunability: The scheme allows for the simultaneous tuning of on-site energy, interaction strength, and density-dependent hopping amplitudes. For the AH model, it enables the tuning of the statistical phase $\theta$ .
Density-Dependent Hopping: The protocol naturally implements correlated tunneling processes where hopping amplitudes depend on the local particle density.

Results
The authors perform numerical simulations of driven transmon arrays using experimentally realistic parameters (e.g., $\omega_0/2\pi \approx 4.5$ GHz, anharmonicity $\alpha/2\pi \approx -250$ MHz).

Bose-Hubbard Dynamics: Simulations of quantum walks with multiple particles demonstrate that the driven transmon system reproduces the characteristic dynamics of the exact BH model across a range of interaction strengths ( $U$ ), including the transition from delocalized to localized behavior.
Anyon-Hubbard Dynamics: Simulations of the AH model successfully reproduce the statistics-dependent asymmetry of the anyonic quantum walk and the interaction-dependent localization. The numerical results confirm that the effective Hamiltonian derived from the driven system matches the target AH model dynamics, including the correct phase accumulation around closed loops in the Fock-state lattice.
Validation: The simulations include counter-rotating terms to verify that the RWA holds in the considered regimes. The results show high fidelity between the simulated transmon dynamics and the exact target models.

Significance and Claims
The paper claims to provide a general and flexible protocol for analog quantum simulation using superconducting circuits, moving beyond the limitations of hard-core boson approximations. The authors emphasize that their method allows for the simulation of interacting bosonic systems with up to three particles per site and, notably, the first analog simulation of anyonic statistics on this platform. The work suggests that these simulators can serve not only as tools for studying many-body physics (such as Mott-insulator-to-superfluid transitions and anyonic quantum walks) but also as a means to characterize the underlying hardware and test the validity of effective models at larger scales. The authors remain modest regarding the scope, noting that the protocol is restricted to specific excitation manifolds and that the verification of these simulators remains a challenge when direct classical comparison is impossible.

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