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Quantum Approximate Optimization Algorithm with Cat Qubits

This paper demonstrates that implementing the Quantum Approximate Optimization Algorithm (QAOA) with cat qubits in Kerr resonators, which exhibit $Z$ -biased noise due to photon losses, yields higher approximation ratios for random MaxCut instances compared to standard two-level qubits under single-photon loss.

Original authors: Pontus Vikstål, Laura García-Álvarez, Shruti Puri, Giulia Ferrini

First appeared 2024-09-17 · Updated 2026-11-11

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CC BY 4.0

Original authors: Pontus Vikstål, Laura García-Álvarez, Shruti Puri, Giulia Ferrini

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 race to build useful quantum computers, scientists are currently navigating a difficult middle ground. They have machines large enough to attempt real problems, but not yet large enough to be perfectly reliable. These devices, often called noisy intermediate-scale quantum computers, are prone to errors that can scramble their calculations before they finish. To work around this, researchers rely on a specific strategy known as the Quantum Approximate Optimization Algorithm. This method is designed to solve complex puzzles—like finding the most efficient route for a delivery fleet or the best way to split a group of people into two teams—by using short, shallow circuits that are less likely to break down under the weight of noise. The central challenge for these machines is not just running the algorithm, but finding a way to store information that resists the specific kinds of errors that plague current hardware.

A team of researchers has now explored a promising new way to store this information, using a type of quantum bit called a "cat qubit." Unlike standard quantum bits that are often compared to tiny spinning tops, cat qubits are built from light trapped inside a resonator, existing in a state that resembles two opposite waves at once. The researchers focused on a specific physical setup involving a device called a Kerr nonlinear resonator, which uses a special kind of light-matter interaction to create these states. The key feature of this approach is that the errors it produces are not random; they are biased. In a standard system, an error might flip a bit from zero to one or change its phase with equal likelihood. In this cat qubit system, the dominant error—caused by the loss of a single photon—almost exclusively affects the phase, leaving the bit-flip errors so rare they are practically negligible. This bias acts like a built-in shield, making the information much more robust against the most common form of decay.

The researchers set out to see if this bias could actually improve the performance of the optimization algorithm. They simulated the process of solving a classic problem known as MaxCut, which involves dividing the nodes of a network into two groups to maximize the connections between them. They ran these simulations on a computer, modeling the behavior of the cat qubits as they passed through the necessary sequence of operations. To ensure a fair test, they compared the cat qubits against standard quantum bits encoded in simple two-level systems, carefully tuning the standard system so that both types of bits had the exact same average reliability for their individual operations. The results were clear: when the algorithm ran on the cat qubits, it consistently found better solutions than when it ran on the standard bits. Even with the same level of noise, the biased nature of the cat qubits allowed the algorithm to maintain a higher quality of approximation, effectively getting closer to the perfect answer.

To understand why this happened, the team looked closely at how errors behaved as they moved through the algorithm's steps. They found that while the cat qubits were indeed susceptible to phase errors, the specific gates used in the algorithm handled these errors differently than they handled random noise. In the standard system, errors would spread and corrupt the calculation in all directions. In the cat qubit system, the errors remained confined to a single direction, which the algorithm could tolerate much better. The researchers also tested a different idea: whether creating a completely new version of the algorithm designed specifically for these light-based systems would be even better. Their simulations suggested that sticking with the standard algorithm, but simply running it on the superior cat qubits, was the more effective path. The study concludes that for the near future, where noise is unavoidable, using these biased qubits offers a concrete advantage, providing a clearer path to solving difficult optimization problems than current standard approaches.

Technical Summary: Quantum Approximate Optimization Algorithm with Cat Qubits

Problem Statement
The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for solving combinatorial optimization problems on intermediate-scale quantum (NISQ) processors. However, its performance is currently limited by hardware noise, which degrades the algorithm's effectiveness as circuit depth increases. While variational algorithms are designed to run on shallow circuits, reaching deeper circuits requires hardware implementations that can better tolerate noise. This work investigates whether encoding qubits into bosonic systems, specifically "cat qubits" realized via Kerr nonlinear resonators (KNRs), can offer a performance advantage over standard two-level system qubits when solving discrete optimization problems like MaxCut.

Methodology
The authors propose and simulate a QAOA implementation using cat qubits, which are encoded in the coherent states of a driven Kerr parametric oscillator with opposite amplitudes ( $|\pm\alpha\rangle$ ).

Encoding and Noise Bias: The cat qubit encoding (specifically the superposition of even and odd parity cat states) results in "biased noise." Under the dominant error mechanism of single-photon loss, the noise manifests primarily as phase ( $\hat{Z}$ ) errors, while bit-flip ( $\hat{X}$ ) errors are exponentially suppressed.
Gate Implementation: The study utilizes a universal gate set for cat qubits: $R_Z(\phi)$ , $R_X(\theta)$ , $R_Y(\phi)$ , and $R_{ZZ}(\Theta)$ . These are implemented via specific drives and interactions in KNRs.
Simulation Strategy:
1. Small Scale (Proof of Principle): The authors first simulate a two-qubit Exact Cover problem using the full Lindblad master equation to validate the approach under single-photon loss.
2. Large Scale (6, 8, 10 qubits): To overcome the exponential scaling of the Hilbert space for larger systems, the authors employ the Pauli Transfer Matrix (PTM) formalism. They characterize the error channels induced by single-photon losses on the computational subspace, converting continuous time evolution into Kraus operators.
3. Fair Comparison: To ensure a fair comparison between cat qubits and standard qubits, the authors tune the relaxation rates ( $T_1$ ) of the standard qubits such that their average gate fidelities match those of the cat qubits (specifically for $\alpha = 1.36$ ). This isolates the benefit of the noise bias from simple improvements in gate fidelity.

Key Contributions

Discrete Optimization in Bosonic Systems: Unlike previous bosonic QAOA studies that focused on continuous variable optimization, this work addresses discrete combinatorial problems (MaxCut and Exact Cover) using cat qubits.
Noise-Bias Preservation Analysis: The paper demonstrates that while the $R_{ZZ}$ gate preserves the noise bias (converting single-photon loss to $\hat{Z}$ errors), the $R_X$ gate does not; it transforms $\hat{Z}$ errors into a combination of $\hat{Z}$ and $\hat{Y}$ errors, effectively making the noise channel unbiased for that specific gate.
Performance Benchmarking: The study provides a numerical comparison of QAOA performance on cat qubits versus standard qubits for random MaxCut instances, controlling for average gate fidelity.

Results

Approximation Ratio: For random MaxCut instances on 6, 8, and 10 qubits, the QAOA implemented with cat qubits achieves a higher mean approximation ratio than the implementation with standard qubits, even when both systems operate with identical average gate fidelities.
Robustness to Depth: While the approximation ratio for both systems initially increases with circuit depth ( $p$ ) and then decreases due to noise accumulation, the cat qubit implementation maintains a superior performance across all tested depth levels.
Error Mechanism: The performance advantage is attributed to the fact that in typical MaxCut circuits, the number of $R_{ZZ}$ gates (which preserve noise bias) exceeds the number of $R_X$ gates (which do not). Consequently, the dominant error mechanism in cat qubits remains biased ( $\hat{Z}$ ), whereas standard qubits suffer from unstructured errors ( $\hat{X}, \hat{Y}, \hat{Z}$ ) that affect all gate types.
Bosonic QAOA Variant: The authors also explored a "genuinely bosonic" QAOA variant (Trotterizing a bosonic annealing Hamiltonian) but found it did not yield improvements over the standard QAOA approach using cat qubits, hypothesizing that the inefficiency stems from the difficulty of mapping the resonator state back to the qubit computational basis.

Significance and Claims
The paper claims that noise-biased qubits, such as cat qubits, are preferable to standard qubits for implementing QAOA on near-term intermediate-scale quantum processors. The primary significance lies in demonstrating that the specific structure of biased noise (favoring dephasing) can enhance algorithmic performance for discrete optimization problems, even when the hardware requires non-bias-preserving gates (like $R_X$ ). The authors suggest these results are likely problem-agnostic and that other combinatorial problems would similarly benefit from this performance separation. They conclude that this approach offers a concrete estimate of achievable approximation ratios for MaxCut using realistic Kerr resonator parameters, providing a viable path toward deeper, more effective quantum circuits in the NISQ era.

Technical Summary: Quantum Approximate Optimization Algorithm with Cat Qubits

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