[Paper Review] Single-Layer Digitized-Counterdiabatic Quantum Optimization for $p$-spin Models

Huijie Guan, Fei Zhou+5 more|arXiv (Cornell University)|Nov 11, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a single-layer digitized-counterdiabatic quantum optimization (DCQO) framework for solving $p$-spin models up to 4-local interactions using variational parameter optimization. By employing a warm-started CD ansatz with a tailored scheduling function, the method achieves unit ground-state fidelity for 100% of 2-spin, 93% of 3-spin, and 83% of 4-spin instances, including 5-, 9-, and 12-qubit factorization problems, demonstrating low overhead and feasibility for NISQ-era quantum advantage.

ABSTRACT

Quantum computing holds the potential for quantum advantage in optimization problems, which requires advances in quantum algorithms and hardware specifications. Adiabatic quantum optimization is conceptually a valid solution that suffers from limited hardware coherence times. In this sense, counterdiabatic quantum protocols provide a shortcut to this process, steering the system along its ground state with fast-changing Hamiltonian. In this work, we take full advantage of a digitized-counterdiabatic quantum optimization (DCQO) algorithm to find an optimal solution of the $p$-spin model up to 4-local interactions. We choose a suitable scheduling function and initial Hamiltonian such that a single-layer quantum circuit suffices to produce a good ground-state overlap. By further optimizing parameters using variational methods, we solve with unit accuracy 2-spin, 3-spin, and 4-spin problems for $100\%$, $93\%$, and $83\%$ of instances, respectively. As a particular case of the latter, we also solve factorization problems involving 5, 9, and 12 qubits. Due to the low computational overhead, our compact approach may become a valuable tool towards quantum advantage in the NISQ era.

Motivation & Objective

• To address the challenge of limited hardware coherence times in adiabatic quantum optimization by accelerating the evolution using counterdiabatic (CD) shortcuts.
• To develop a compact, low-overhead quantum algorithm suitable for NISQ devices that maintains high ground-state fidelity.
• To demonstrate the feasibility of solving complex $p$-spin models, including factorization problems, using a single-layer quantum circuit with variational CD parameter optimization.
• To guide hardware-efficient qubit layout and gate scheduling by analyzing circuit duration across different topologies under coherence constraints.

Proposed method

• The method uses a digitized-counterdiabatic quantum optimization (DCQO) protocol with a time-dependent Hamiltonian $\mathcal{H}(t) = (1-\lambda(t))H_i + \lambda(t)H_f$, where $H_i$ is a transverse-field initial Hamiltonian and $H_f$ is the target problem Hamiltonian.
• A scheduling function $\lambda(t) = \sin^2\left(\frac{\pi}{2} \sin^2\left(\frac{\pi t}{2\tau}\right)\right)$ ensures smooth evolution with vanishing initial and final derivative, minimizing non-adiabatic transitions.
• The counterdiabatic (CD) term $H_{CD} = \dot{\lambda} A$ is approximated variationally using a compact ansatz of Y, YZ, and ZY-type Pauli terms, avoiding nonlocal terms.
• The CD ansatz is initialized using a warm-start strategy based on the ground state of the initial Hamiltonian $| - \rangle^{\otimes n}$, improving convergence and fidelity.
• Circuit depth and duration are optimized across different qubit grid topologies (e.g., 2×N/2, 3×N/2), with gate durations set to 30ns (single-qubit) and 123ns (CZ) for hardware-aware design.
• The approach is validated on 5-, 9-, and 12-qubit factorization problems using a superconducting quantum processor with T2* ~ 5 μs, and tested on 2-, 3-, and 4-spin models.

Figure 1: Accuracy of algorithms for solving factorization problem $H^{9q}$ with different evolution schemes $U(t)$ . From top to bottom, the algorithm considers evolution with problem Hamiltonian only(a), counterdiabatic terms only (b), and both problem Hamiltonian and counterdiabatic terms(c) resp

Experimental results

Research questions

• RQ1 Can a single-layer DCQO circuit achieve high ground-state fidelity for $p$-spin models with up to 4-local interactions on NISQ devices?
• RQ2 How effective is a warm-started variational CD ansatz in reducing circuit depth while maintaining accuracy?
• RQ3 What is the optimal qubit layout and gate scheduling strategy to minimize circuit duration under coherence time constraints?
• RQ4 To what extent can this method solve prime factorization problems using only a single layer of quantum gates?
• RQ5 How does the performance of the DCQO protocol scale across different $p$-spin models (p=2,3,4) in terms of fidelity and success rate?

Key findings

• The method achieves unit ground-state fidelity for 100% of 2-spin model instances, demonstrating high reliability for quadratic optimization problems.
• For 3-spin models, the approach achieves unit fidelity in 93% of tested instances, indicating strong performance on higher-order interactions.
• In 4-spin models, including factorization problems, the method attains unit fidelity in 83% of instances, showing feasibility for complex, non-quadratic problems.
• The 5-, 9-, and 12-qubit factorization problems were successfully solved with unit accuracy using only a single-layer quantum circuit.
• The optimal hardware layout for minimizing circuit duration is a 2×N/2 grid topology, which reduces depth compared to square or 3-column layouts.
• Circuit duration analysis shows that 12-qubit systems are accessible under T2* ~ 5 μs, with gate durations and error rates consistent with current superconducting hardware.

Figure 2: Convergent curve for CD approach(a) and QAOA(b) for $H^{9q}$ with cost function being zero at global minimum. The left plot shows a convergent curve with 5 randomly generated initial points and one warm starting point from CD approach with (Y+ ZY ${}_{\text{u}}$ )-type ansatz. The right pl