Energy Scale Degradation in Sparse Quantum Solvers: A Barrier to Quantum Utility

TL;DR

This work exposes energy-scale degradation as a fundamental obstacle to quantum utility when solving optimization problems on sparse quantum solvers via minor-embedding. By constructing a energy-rescaling framework, the authors show that increasing problem connectivity drives the effective energy scale down by $O(\sqrt{\Delta})$, exponentially suppressing the ground-state success probability as connectivity grows. They decompose solution probability into a chain-consistency component and an energy-resolution component, proving a non-monotonic trade-off with an optimal chain strength near $\lambda\approx 2$, and provide NP-hardness results for approximating chain consistency. Experimentally, D-Wave hardware validates the theory, highlighting the roles of chain volume and chain connectivity, and motivating hardware with higher connectivity and embedding algorithms that optimize conductance. The paper also delivers conductance-based and spectral bounds on chain strength, offering practical tools to assess and mitigate energy-scale degradation in sparse quantum solvers.

Abstract

Quantum computing offers a promising route for tackling hard optimization problems by encoding them as Ising models. However, sparse qubit connectivity requires the use of minor-embedding, mapping logical qubits onto chains of physical qubits, which necessitates stronger intra-chain coupling to maintain consistency. This elevated coupling strength forces a rescaling of the Hamiltonian due to hardware-imposed limits on the allowable ranges of coupling strengths, reducing the energy gaps between competing states, thus, degrading the solver's performance. Here, we introduce a theoretical model that quantifies this degradation. We show that as the connectivity degree increases, the effective temperature rises as a polynomial function, resulting in a success probability that decays exponentially. Our analysis further establishes worst-case bounds on the energy scale degradation based on the inverse conductance of chain subgraphs, revealing two most important drivers of chain strength, \textit{chain volume} and \textit{chain connectivity}. Our findings indicate that achieving quantum advantage is inherently challenging. Experiments on D-Wave quantum annealers validate these findings, highlighting the need for hardware with improved connectivity and optimized scale-aware embedding algorithms.

Figures (4)

• Figure 1: Energy rescaling effects on chain consistency in sparse quantum solvers. (a) Illustration of the problem Ising Hamiltonian $H_p$ and its hardware embedding $H_e^\lambda$. The values on nodes and edges represent linear biases and coupling strengths, respectively. The embedding maps logical variables to physical chains: chain 1 ($C_1$) consisting of physical spins 1 and 2, and chain 2 ($C_2$) consisting of physical spins 3, 4, and 5. (b) Non-monotonic chain break probability as a function of chain strength $\lambda$ at inverse temperature $\beta = 4$. Counterintuitively, as $\lambda$ increases from 0.1 to 0.5, the break probability for chain 1 increases while decreasing for chain 2. The inset shows how energy rescaling in the sparse Ising model (solid lines) elevates chain break probabilities compared to the standard Ising model (dashed lines) for $\lambda > 2$. (c) Experimental validation using D-Wave Advantage_system4.1 quantum annealer (20 runs, 400 samples each), showing consistent non-monotonic behavior for chain 1. (d) Theoretical predictions from the sparse Ising model closely match experimental quantum annealing results.
• Figure 2: Chain consistency and solution probability trade-off as a function of chain strength $\lambda$, comparing (a) theoretical sparse Ising model with energy rescaling and (b) experimental results from D-Wave's quantum annealer for the instance in Fig. \ref{['fig:test']}. $P_{cc}$ (red) represents the chain consistency probability, which generally increases with $\lambda$. ${P_{\text{solve}}}(\beta_{\text{eff}})$ (solid blue) shows the solution probability with effective inverse temperature $\beta_{\text{eff}} = \beta/{\mathcal{E}}_\lambda$, which decreases as $\lambda$ increases due to energy scale compression. $P_{\text{solve}}^{\text{sparse}}$ (light blue) shows the actual solution probability, while the dashed blue curve shows the theoretical prediction $P_{cc} \times {P_{\text{solve}}}(\beta_{\text{eff}})$ according to Theorem \ref{['theorem:solve_probability']}. Both theoretical and experimental results demonstrate the competing effects of increasing chain consistency versus degrading energy resolution as chain strength increases, with optimal performance achieved at moderate chain strength values.
• Figure 3: Scaling behavior of required chain strength (log-log scale) to achieve specific chain consistency thresholds (2%, 10%, 50%) on D-Wave Advantage_system4.1. (a) The dominant effect: Chain strength increases polynomially with the degree of spin interactions $\Delta$ (fixed chain length $l=2$), confirming our theoretical prediction of $\lambda \sim O(\sqrt{\Delta})$. (b) The secondary effect: Chain strength shows only minor increase with chain length $l$, challenging the common assumption that chain length is the primary driver of required chain strength. Dashed lines show linear fits in log-log scale, with slopes indicating scaling exponents. These results demonstrate that the degree of spin interactions, not chain length, is the critical factor determining required chain strength in sparse quantum solvers.
• Figure 4: Embedded Ising Hamiltonian with $l$ auxiliary spins for each logical spin.

Theorems & Definitions (15)

• Lemma 3.1: Hardness of Spin Correlation Approximation
• Theorem 3.2: Intractability of Chain Consistency Approximation
• Theorem 3.3
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3: Solution Probability for $H_{\text{star}}$
• Theorem 5.1: Chain Strength via Conductance
• Theorem 5.2: Spectral Chain Strength Sufficiency
• proof
• proof