⚛️ quantum physics

Engineering $XY$ and $ZZ$ interactions between two remote superconducting qubits via a multimode waveguide

This paper proposes a method to selectively engineer effective $XY$ or $ZZ$ interactions between two remote superconducting qubits coupled via a finite multimode waveguide by tuning qubit frequencies and leveraging the distinct interference patterns of odd versus even mode configurations.

Original authors: Da-Wei Wang, Yu-xi Liu

Published 2026-10-21

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CC BY 4.0

Original authors: Da-Wei Wang, Yu-xi Liu

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 world of quantum computing, the most powerful machines are not built from a single, massive processor, but from many small, delicate units called qubits working together. To solve complex problems, these qubits must be able to talk to one another, sharing information across distances. One of the most promising ways to connect them is by placing them near a long, thin wire known as a waveguide. This wire acts like a highway for tiny packets of energy called photons. When qubits are placed along this highway, they can exchange information by sending virtual signals back and forth through the wire, creating a link that allows them to act as a single, coordinated system. However, this connection is tricky. The signals traveling through the wire can create two very different types of links between the qubits. One type, often called an exchange interaction, allows the qubits to swap their states, like two people trading places. The other type, a phase interaction, changes the internal timing of the qubits without them swapping places. For a quantum computer to work correctly, engineers need to be able to choose exactly which type of link they want, turning one on while turning the other off, but doing so has proven difficult.

Researchers at Tsinghua University have now discovered a clever way to control this choice using nothing more than the specific frequencies at which the qubits operate and the natural rhythm of the waveguide itself. In their study, they examined how two superconducting qubits interact when connected by a finite section of a transmission-line waveguide. Unlike a simple, endless wire, this section of the waveguide supports a specific set of standing waves, similar to the way a plucked guitar string vibrates at specific notes. The researchers found that the number of these vibrational modes available in the waveguide creates a fundamental difference in how the qubits can be controlled. When the waveguide supports an even number of these modes, the qubits naturally form a strong swapping link, while the phase link remains weak. Conversely, when the waveguide supports an odd number of modes, the researchers found they could arrange the qubit frequencies to cancel out the swapping link almost entirely, leaving behind a strong phase link.

The key to this control lies in a specific arrangement of frequencies relative to the waveguide's modes. The researchers identified a rule they call the intermediate-mode criterion. For the swapping link to be suppressed, there must be a waveguide mode sitting directly between the frequencies of the two qubits. In a setup with an odd number of modes, there is a central mode that can sit right between the two qubits, acting as a pivot point that allows the signals to cancel each other out perfectly. This cancellation is not a simple weakening of the connection; it is a precise interference where the signals from different pathways arrive out of step and neutralize one another. In contrast, when the waveguide has an even number of modes, there is no single central mode to sit between the qubits. Instead, the qubits are always flanked by pairs of modes that reinforce the swapping link, making it impossible to turn that specific interaction off without also affecting the phase link.

To prove this theory, the team ran detailed computer simulations of the entire system, modeling the behavior of the qubits and the waveguide without simplifying the complex physics. They tested a minimal setup with just two waveguide modes and another with three. In the two-mode case, the simulations showed that the qubits exchanged energy efficiently, performing a swap operation with high fidelity, while the unwanted phase shift remained negligible. In the three-mode case, the results were strikingly different. By tuning the qubit frequencies to satisfy the intermediate-mode criterion, the researchers observed that the swapping interaction dropped to a negligible level, while the phase interaction remained strong and stable. The simulations confirmed that this state was not a result of the qubits accidentally hitting a resonance that would cause them to leak energy into unwanted states, but rather a clean, controlled cancellation of the specific signal pathways responsible for the swap.

This work establishes a clear guideline for engineers building future quantum networks. It shows that by carefully selecting the number of modes in a waveguide and assigning the qubits to specific frequencies within that range, one can selectively engineer the type of connection needed for a particular task. If a quantum operation requires the qubits to swap information, an even-mode configuration provides a natural environment for that. If the operation requires a conditional phase change, an odd-mode configuration offers a way to isolate that interaction. The findings suggest that the physical layout of the waveguide and the frequency of the qubits are not just passive background details, but active tools that can be used to design the logic of a quantum computer. While the current results are based on simulations of ideal conditions, they provide a solid theoretical foundation for building more flexible and powerful quantum processors that rely on the natural interference of waves to perform complex calculations.

Technical Summary: Engineering XY and ZZ Interactions via Multimode Waveguides

Problem Statement
Distributed quantum information processing using superconducting circuits requires coherent interactions between spatially separated qubits. While transmission-line waveguides offer a flexible platform for long-range coupling via virtual photon exchange, a significant challenge remains in selectively engineering specific interaction types (XY vs. ZZ) without additional tunable coupling elements. Conventional approaches often rely on tunable couplers to control interaction strengths and phases, which introduce extra circuit complexity and potential decoherence channels. Furthermore, in fixed-coupling architectures, suppressing one interaction channel (e.g., XY) while retaining another (e.g., ZZ) is difficult because they originate from different orders of virtual processes. The central question addressed is whether qubit frequency assignment alone, in conjunction with the mode configuration of a finite multimode waveguide, can selectively engineer these effective interactions.

Methodology
The authors model two spatially separated transmon qubits coupled to a finite-length microwave transmission-line waveguide supporting discrete standing-wave modes. The system is described by a full Hamiltonian within the rotating wave approximation, treating the qubits as anharmonic oscillators and the waveguide modes as harmonic oscillators.

The study employs two primary analytical and numerical approaches:

Perturbative Analysis: The effective Hamiltonian is derived by perturbatively eliminating the waveguide modes. The second-order virtual single-photon exchange yields the effective XY interaction ( $J_{\text{eff}}$ ), while the fourth-order virtual processes involving qubit anharmonicities yield the ZZ interaction ( $\zeta$ ). The authors analyze the interference of mode-resolved contributions ( $J_m$ ) based on qubit-mode detunings and the relative standing-wave parity factors ( $s_m$ ).
Full-Hamiltonian Simulations: To verify the perturbative predictions, the authors perform numerical simulations of the full system Hamiltonian for minimal configurations (two-mode and three-mode waveguides). They evaluate gate dynamics, population transfer, leakage, and gate fidelity for specific frequency assignments.

Key Contributions and Findings

Single-Mode Limitation: For a single-mode waveguide, the suppression of the XY interaction occurs only when the qubits satisfy a two-photon resonant condition ( $\omega_1 + \omega_2 = 2\omega_{r,0}$ ). However, under this condition, the ZZ interaction remains finite and non-negligible. Consequently, the XY and ZZ interactions cannot be simultaneously suppressed; a residual coupling always exists.
Multimode Interference and the "Intermediate-Mode Criterion": The paper establishes that the balance between XY and ZZ interactions in multimode waveguides is governed by an "intermediate-mode criterion." This criterion depends on whether a waveguide mode exists spectrally between the two qubit frequencies.
- Even-Mode Configurations: In symmetric configurations with an even number of modes (where qubit frequencies lie between two central modes), the mode-resolved XY contributions interfere constructively. This results in a strong, non-zero XY interaction. While the ZZ interaction can be made small, the system is naturally suited for XY-dominated dynamics (e.g., SWAP gates).
- Odd-Mode Configurations: In configurations with an odd number of modes, there exists a central, unpaired mode located between the two qubit frequencies. This central mode provides an additional virtual exchange pathway that can destructively interfere with the contributions from symmetric mode pairs. By appropriately assigning qubit frequencies, the total XY interaction ( $J_{\text{eff}}$ ) can be strongly suppressed ( $J_{\text{eff}} \approx 0$ ) while the ZZ interaction ( $\zeta$ ) remains finite. This enables ZZ-dominated dynamics (e.g., Controlled-Z gates).
Parity and Frequency Assignment: The study highlights that the odd-even distinction is not a universal property of the mode count alone but arises from the specific spectral and parity structure within the central finite-mode window. The standing-wave parity factors ( $s_m$ ) and the signs of the detunings determine whether mode contributions cancel or reinforce.

Results

Gate Dynamics Verification: Full-Hamiltonian simulations confirm the theoretical predictions:
- Two-Mode (Even): Simulations show coherent population transfer between $|10\rangle$ and $|01\rangle$ with high fidelity, characteristic of an XY-dominated SWAP gate. The ratio $|J_{\text{eff}}/\zeta|$ is large.
- Three-Mode (Odd): Simulations demonstrate strong suppression of population exchange ( $J_{\text{eff}} \sim 10^{-4}$ MHz) while maintaining a finite ZZ interaction ( $\zeta \sim 0.12$ MHz). The system exhibits conditional phase accumulation reaching $\pi$ with high average gate fidelity, characteristic of a ZZ-dominated CZ gate.
Leakage Analysis: The simulations indicate that while leakage to non-computational states (e.g., $|20\rangle$ , $|02\rangle$ , and two-photon states) occurs, it remains small compared to the computational subspace population, validating the dispersive regime assumption.

Significance and Claims
The paper claims to establish a frequency assignment criterion for selectively engineering long-range qubit-qubit interactions in finite multimode waveguides. The primary significance lies in demonstrating that:

Selective Control without Tunable Couplers: It is possible to switch between XY-dominated and ZZ-dominated interaction regimes solely by assigning qubit frequencies relative to the waveguide mode spectrum, without requiring active tunable coupling elements.
Mechanism of Suppression: The suppression of the XY interaction in odd-mode configurations is achieved through destructive interference of virtual photon pathways mediated by an unpaired central mode, distinct from the two-photon resonance mechanism in single-mode systems.
Practical Implications: These results provide a design principle for quantum networks and waveguide-based couplers, offering a route to realize specific logic gates (SWAP vs. CZ) by exploiting the natural interference properties of multimode waveguides.

The authors maintain a modest scope, noting that their results are based on ideal coherent dynamics. They acknowledge that practical implementation must account for waveguide loss, qubit relaxation, and dephasing, which are beyond the current scope but do not invalidate the underlying interference mechanism.

Technical Summary: Engineering XY and ZZ Interactions via Multimode Waveguides

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