Quantitative analysis of the effectiveness of mid-anneal measurement in quantum annealing

TL;DR

This paper addresses the difficulty that coefficient tuning and hardware noise create in encoding optimal constrained solutions as ground states in quantum annealing. It introduces mid-anneal measurement and the metric $Q_\mathrm{d}$ to quantify its effectiveness, and analyzes GBP and QKP as representative problems along with Ising-model scaling to study mechanisms and scalability. The findings show mid-anneal measurement is most beneficial when desired solutions reside in low-lying excited states near the ground state, with effectiveness governed by the energy structure and state similarity, and it remains scalable to larger systems. A practical takeaway is to adopt a hybrid strategy combining standard annealing with mid-anneal measurements to mitigate encoding failures in large-scale quantum annealing hardware.

Abstract

Quantum annealing is a promising metaheuristic for solving constrained combinatorial optimization problems. However, parameter tuning difficulties and hardware noise often prevent optimal solutions from being properly encoded as the ground states of the problem Hamiltonian. This study investigates mid-anneal measurement as a mitigation approach for such situations, analyzing its effectiveness and underlying physical mechanisms. We introduce a quantitative metric to evaluate the effectiveness of mid-anneal measurement and apply it to the graph bipartitioning problem and the quadratic knapsack problem. Our findings reveal that mid-anneal measurement is most effective when the energy difference between desired solutions and ground states is small, with effectiveness strongly governed by the energy structure. Furthermore, the effectiveness increases as the Hamming distance between the ground and excited states gets small, highlighting the role of state similarity. Analysis of fully-connected Ising models demonstrates that the effectiveness of mid-anneal measurement persists with increasing system size, indicating its scalability and practical applicability to large-scale quantum annealing.

Figures (8)

• Figure 1: (Color online) Characteristics of $Q_\mathrm{d}$, which represents the effectiveness of mid-anneal measurement defined in Eq. \ref{['Eq: Q_d']}. The second factor in Eq. \ref{['Eq: Q_d']}, $\underset{t}{\mathrm{max}} P (t) - P (t=0)$, represents the improvement from the initial equal-weight superposition state. The third factor in Eq. \ref{['Eq: Q_d']}, $\underset{t}{\mathrm{max}} P (t) - P (t=\tau)$, represents the additional improvement due to mid-anneal measurement compared to standard annealing. (a), (b) Cases where mid-anneal measurement is not effective. In case (a), the second factor $\underset{t}{\mathrm{max}} P(t) - P (t=0)$ is zero, while in case (b), the third factor $\underset{t}{\mathrm{max}} P(t) - P (t=\tau)$ is zero. Therefore, $Q_\mathrm{d}$ becomes zero, indicating that mid-anneal measurement is not effective. (c) Case where mid-anneal measurement is effective. Both the first and third factors take non-zero values, resulting in a non-zero value of $Q_\mathrm{d}$, which indicates that mid-anneal measurement is effective.
• Figure 2: (Color online) Example of the relationship between the effectiveness of mid-anneal measurement and constraint coefficients. These results are from quantum annealing simulations in the adiabatic limit for QKP, $N=5$, $W=1$. (a) Dependence of $Q_\mathrm{d, f}$ on $\lambda$ and $\mu$. The red line indicates the theoretical boundary in the $\lambda - \mu$ space where the ground state of $H_\mathrm{c}$ switches from feasible to infeasible. (b) Adiabatic time evolution of the probability of obtaining feasible solutions $P_\mathrm{f}(t)$ for different values of $\mu$ at $\lambda=0.7$ (where $\mu^* = 1$).
• Figure 3: (Color online) Dependence of the effectiveness of mid-anneal measurement $Q_\mathrm{d, f}$, $Q_\mathrm{d, opt}$ on the energy difference between the minimum energies of feasible and infeasible solutions $\Delta E_\mathrm{f}$. Parameters are: $N=6$, $c=0$, $\lambda=-0.916$ for GBP; and $N=5$, $W=1$, $\lambda=0.7$ for QKP. Results are shown for quantum annealing in the adiabatic limit (static) and with annealing time $\tau$. Calculations were performed for 10 instances, with $\Delta E_\mathrm{f}$ grouped in intervals of $0.01$. The mean (solid lines) and standard deviation (error bars) were then calculated. (a) GBP, $Q_\mathrm{d, f}$; (b) GBP, $Q_\mathrm{d, opt}$; (c) QKP, $Q_\mathrm{d, f}$; (d) QKP, $Q_\mathrm{d, opt}$.
• Figure 4: (Color online) Dependence of the effectiveness of mid-anneal measurement $Q_\mathrm{d, f}$, $Q_\mathrm{d, opt}$ on the energy difference between the minimum energies of feasible and infeasible solutions $\Delta E_\mathrm{f}$. Parameters are: $N=6$ for GBP, $N=5$ for QKP. Results are from quantum annealing simulations in the adiabatic limit, comparing the effects of partitioning constraint $c$ in GBP and weight constraint $W$ in QKP. Calculations were performed for 10 instances, with $\Delta E_\mathrm{f}$ grouped in intervals of 0.01. The mean (solid lines) and standard deviation (error bars) were then calculated. $\lambda$ was set such that $\mu^*=1$. (a) GBP, $Q_\mathrm{d, f}$; (b) GBP, $Q_\mathrm{d, opt}$; (c) QKP, $Q_\mathrm{d, f}$; (d) QKP, $Q_\mathrm{d, opt}$.
• Figure 5: (Color online) Relationship between the effectiveness of mid-anneal measurement and the probability at the initial time of annealing. The vertical axis plots the maximum values of $Q_\mathrm{d, f}$ and $Q_\mathrm{d, opt}$ when $\Delta E_\mathrm{f}$ is varied for each instance, denoted as $\underset{\Delta E_\mathrm{f}}{\mathrm{max}} Q_\mathrm{d, f}$, $\underset{\Delta E_\mathrm{f}}{\mathrm{max}} Q_\mathrm{d, opt}$, which represent the potential for mid-anneal measurement to function most effectively. The results are from quantum annealing simulations in the adiabatic limit. The mean (solid lines) and standard deviation (error bars) were calculated over $10$ instances. The horizontal axis shows the initial probabilities $P_\mathrm{f}(0)$ for $Q_\mathrm{d, f}$ and $P_\mathrm{opt}(0)$ for $Q_\mathrm{d, opt}$. (a) GBP, $P_\mathrm{f}(0)$ vs $\underset{\Delta E_\mathrm{f}}{\mathrm{max}} Q_\mathrm{d, f}$; (b) GBP, $P_\mathrm{opt}(0)$ vs $\underset{\Delta E_\mathrm{f}}{\mathrm{max}} Q_\mathrm{d, opt}$; (c) QKP, $P_\mathrm{f}(0)$ vs $\underset{\Delta E_\mathrm{f}}{\mathrm{max}} Q_\mathrm{d, f}$; (d) QKP, $P_\mathrm{opt}(0)$ vs $\underset{\Delta E_\mathrm{f}}{\mathrm{max}} Q_\mathrm{d, opt}$.