[Paper Review] Determination of Chain Strength induced by Embedding in D-Wave Quantum Annealer

Hunpyo Lee|arXiv (Cornell University)|Sep 25, 2022

Laser-Matter Interactions and Applications4 citations

TL;DR

This study proposes a systematic method to determine optimal chain strength $J_c$ in D-Wave quantum annealers by analyzing energy gaps and ground-state observation probabilities in ordered and disordered Ising models. Using exact diagonalization and simulated annealing, it finds that $J_c \approx 0.25\Delta_c/\Delta_s$ for ordered systems and $J_c \approx 2.1E_g$ for maximally disordered systems, with sensitivity increasing under higher disorder.

ABSTRACT

The D-wave quantum annealer requires embedding with ferromagnetic (FM) chains connected by several qubits, because it cannot capture exact long-range coupling between qubits, and retains the specific architecture that depends on the hardware type. Therefore, determination of the chain strength $J_c$ required to sustain FM order of qubits in the chains is crucial for the accuracy of quantum annealing. In this study, we devise combinatorial optimization problems with ordered and disordered qubits for various embeddings to predict appropriate $J_c$ values. We analyze the energy interval $Δ_s$ and $Δ_c$ between ground and first excited states in the combinatorial optimization problems without and with chains respectively, using the exact approach. We also measure the probability $p$ that the exact ground energy per site $E_g$ is observed in many simulated annealing shots. We demonstrate that the determination of $J_c$ is increasingly sensitive with growing disorder of qubits in the combinatorial optimization problems. In addition, the values of appropriate $J_c$, where the values of $p$ are at a maximum, increase with decreasing $Δ_s$. Finally, the appropriate value of $J_c$ is shown to be observed at approximately $Δ_c/Δ_s=0.25$ and $2.1 E_g$ in the ordered and disordered qubits, respectively.

Motivation & Objective

• To determine the optimal chain strength $J_c$ in D-Wave quantum annealers that prevents chain breaking and clustering during quantum annealing.
• To investigate how $J_c$ sensitivity varies with system disorder and energy gap structure in combinatorial optimization problems.
• To establish a quantitative relationship between $J_c$, the energy gap $\Delta_s$ (without chains), and $\Delta_c$ (with chains), for both ordered and disordered qubit configurations.
• To validate the optimal $J_c$ values using simulated annealing and compare them with results from actual D-Wave hardware behavior.

Proposed method

• Construct 2D $L \times L$ square Ising models with nearest-neighbor ($J_1$) and diagonal ($J_2$) interactions to simulate combinatorial optimization problems.
• Embed these problems into a 3D architecture by forming ferromagnetic (FM) chains along the $z$-direction, using $J_c$ to enforce qubit clustering.
• Compute the exact energy spectrum for $4 \times 4$ systems with nine qubits in chains to determine $J_c^*$ (chain breaking) and $J_c^{**}$ (chain clustering) thresholds.
• Use simulated annealing (SA) over 2000 shots to measure the probability $p$ of observing the exact ground energy per site $E_g$ for various $J_c$ values.
• Systematically vary $J_2/J_1$ to tune the energy gap $\Delta_s$ between ground and first excited states, and analyze its impact on $J_c$ optimization.
• Introduce a disorder parameter $x$ to control the ratio of $J_2 = 0.25$ and $1 - J_2$ in diagonal bonds, enabling study of disordered qubit configurations.

Experimental results

Research questions

• RQ1 What is the optimal $J_c$ value that maximizes the probability $p$ of observing the true ground state energy $E_g$ in quantum annealing for ordered and disordered Ising models?
• RQ2 How does the sensitivity of $J_c$ to parameter variation change with increasing system disorder and decreasing $\Delta_s$?
• RQ3 What is the quantitative relationship between $J_c$, $\Delta_c$ (energy gap with chains), and $\Delta_s$ (energy gap without chains) in both ordered and disordered systems?
• RQ4 How do the results from simulated annealing compare to actual D-Wave hardware behavior, particularly regarding chain breaking and ground state observation?

Key findings

• The optimal $J_c$ for ordered systems occurs at $\Delta_c / \Delta_s = 0.25$, where the probability $p$ of observing $E_g$ is maximized.
• For maximally disordered systems with $\Delta_s \approx 0$, the optimal $J_c$ is approximately $2.1E_g$, where $E_g$ is the ground-state energy per site.
• The sensitivity of $J_c$ to parameter variation increases significantly with system disorder, especially when $\Delta_s \approx 0$.
• The probability $p$ of observing $E_g$ decreases with increasing $J_2/J_1$, but remains highest at $J_c/J_1 = 2.0$ across all tested $J_2/J_1$ values.
• The number of successful ground-state observations $N$ in 2000 SA shots is highest at $J_c/J_1 = 2.1$ for $x = 0.2$, $2.3$ for $x = 0.4$, and $2.4$ for $x = 0.5$, indicating increasing optimal $J_c$ with disorder.
• Despite hardware-induced chain breaking in actual D-Wave systems, the overall trend of $p$ vs. $J_c$ remains qualitatively consistent with simulated annealing results.