[Paper Review] Quantum Approximate Optimization Algorithm with Cat Qubits

Pontus Vikstål, Laura García-Álvarez+2 more|arXiv (Cornell University)|May 9, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes implementing the Quantum Approximate Optimization Algorithm (QAOA) using cat qubits encoded in Kerr nonlinear resonators, leveraging their Z-biased noise properties to enhance performance. Numerical simulations show that QAOA with cat qubits achieves a higher approximation ratio for 8-qubit MaxCut problems compared to standard two-level qubits, despite equal average gate fidelities, due to favorable noise resilience in the computational subspace.

ABSTRACT

The Quantum Approximate Optimization Algorithm (QAOA) -- one of the leading algorithms for applications on intermediate-scale quantum processors -- is designed to provide approximate solutions to combinatorial optimization problems with shallow quantum circuits. Here, we study QAOA implementations with cat qubits, using coherent states with opposite amplitudes. The dominant noise mechanism, i.e., photon losses, results in $Z$-biased noise with this encoding. We consider in particular an implementation with Kerr resonators. We numerically simulate solving MaxCut problems using QAOA with cat qubits by simulating the required gates sequence acting on the Kerr non-linear resonators, and compare to the case of standard qubits, encoded in ideal two-level systems, in the presence of single-photon loss. Our results show that running QAOA with cat qubits increases the approximation ratio for random instances of MaxCut with respect to qubits encoded into two-level systems.

Motivation & Objective

• To investigate the feasibility and performance of QAOA using cat qubits encoded in Kerr nonlinear resonators for combinatorial optimization.
• To compare the algorithmic performance of QAOA with cat qubits against standard two-level qubit implementations under identical average gate fidelities and single-photon loss noise.
• To evaluate whether the Z-biased noise inherent in cat qubit encoding can lead to improved approximation ratios in QAOA for MaxCut problems.
• To explore the challenges in initializing cat states via bosonic QAOA and assess the non-convexity of the variational landscape for state preparation.

Proposed method

• Implementing QAOA on cat qubits using a driven Kerr nonlinear resonator (KNR) platform, where logical qubits are encoded in coherent states |±α⟩.
• Simulating the full master equation for a two-qubit Exact Cover problem to validate the approach as a proof of principle.
• Using the Pauli-transfer matrix formalism to model single-photon loss noise on the computational subspace for larger 8-qubit MaxCut simulations.
• Employing variational optimization of QAOA parameters (β and γ) to minimize the infidelity between the variational state and the target state.
• Comparing performance metrics—specifically the approximation ratio—between cat qubits and standard two-level qubits under identical gate fidelities and noise conditions.
• Analyzing the expectation value landscape for both bosonic QAOA and standard QAOA to assess optimization difficulty, particularly for initial state preparation.

Figure 1: The computational states that lie along the $x,y,z$ -axis implemented with cat qubits and visualized on the Bloch sphere along with their Wigner function.

Experimental results

Research questions

• RQ1 Can QAOA with cat qubits outperform standard QAOA with two-level qubits in terms of approximation ratio for MaxCut problems under equivalent gate fidelities and photon loss?
• RQ2 How does the Z-biased noise inherent in cat qubits affect the performance of QAOA in the presence of single-photon loss?
• RQ3 What are the challenges in preparing a cat state using bosonic QAOA, and how does the non-convexity of the variational landscape impact state fidelity?
• RQ4 How does the initial state preparation differ between standard QAOA (starting in |+⟩) and bosonic QAOA (starting in vacuum), and what impact does this have on algorithmic performance?

Key findings

• For 8-qubit MaxCut problems, QAOA with cat qubits achieves a higher approximation ratio than standard qubits when both systems have the same average gate fidelity and are subject to single-photon loss.
• The performance gain is attributed to the Z-biased noise profile of cat qubits, which provides favorable error-correcting properties and enhances algorithmic robustness.
• Numerical simulations of a two-qubit Exact Cover problem confirm the feasibility of QAOA with cat qubits using the full master equation.
• The variational landscape for preparing a cat state via bosonic QAOA is highly non-convex, leading to a low fidelity of 0.57 for the best-optimized parameters at p=1.
• The expectation value landscape for bosonic QAOA is highly oscillatory and less smooth than for standard QAOA, indicating greater difficulty in optimization, especially in early stages.
• Despite challenges in initialization, the overall results demonstrate that cat qubit implementations can yield superior performance in QAOA due to noise resilience, even when starting from the vacuum state.

Figure 2: The circuit diagram of QAOA with depth $p=1$ for solving a two-qubit instance of the Exact Cover problem, using the universal gate set introduced in Eq. ( 5 )-( 8 ). Here, the circuit is shown using the $\hat{X}$ -mixer.