Quantum Global Minimum Finder based on Variational Quantum Search

TL;DR

The paper addresses the challenge of finding global minima for non-convex functions, where classical methods may stall at local optima. It introduces the Quantum Global Minimum Finder (QGMF), which combines binary search to shift the objective with Variational Quantum Search (VQS) to locate negative-value samples within a shifted subspace, using an O(n)-depth quantum circuit suitable for Noisy Intermediate-Scale Quantum devices. Key contributions include a detailed architecture (oracle O_f, QFT-based adder, VQS), complexity analysis showing logarithmic outer search and linear-depth inner search, and a case study applying QGMF to determine the chromatic number χ(G). The work demonstrates a scalable quantum optimization framework with potential advantages over brute-force methods and offers a pathway toward practical quantum optimization in engineering, finance, and AI on near-term hardware.

Abstract

The search for global minima is a critical challenge across multiple fields including engineering, finance, and artificial intelligence, particularly with non-convex functions that feature multiple local optima, complicating optimization efforts. We introduce the Quantum Global Minimum Finder (QGMF), an innovative quantum computing approach that efficiently identifies global minima. QGMF combines binary search techniques to shift the objective function to a suitable position and then employs Variational Quantum Search to precisely locate the global minimum within this targeted subspace. Designed with a low-depth circuit architecture, QGMF is optimized for Noisy Intermediate-Scale Quantum (NISQ) devices, utilizing the logarithmic benefits of binary search to enhance scalability and efficiency. This work demonstrates the impact of QGMF in advancing the capabilities of quantum computing to overcome complex non-convex optimization challenges effectively.

Figures (8)

• Figure 1: High-level overview of the QGMF algorithm for finding the global minimum of $f(x)$. In this specific example, since $f(x)$ is initially positive for all inputs, it needs to be shifted downward along the y-axis to generate negative values. This downward shift is accomplished by shifting the function by $s_1$. After the initial downward shift by $s_1$, the function produces a significant number of negative values (exceeding our specified threshold $T$), indicating the need for a smaller shift by $s_2$. Following this adjustment, the count of negative values of $f(x)+s_2$ becomes smaller than $T$, enabling us to easily identify the minimum among the remaining negative numbers denoted as $M_r$. Finally, we utilize Eq. (\ref{['eq:gmin']}) to calculate the value of $G_m$.
• Figure 2: Two quantum circuits used for the QGMF. (a) and (b) depict quantum circuits employed to measure $\langle Z_1\rangle$ and $\langle Z_2\rangle$, which are utilized in Eq. (\ref{['eq:obj']}). Note that the VQS part does not have an oracle.
• Figure 3: Efficiency of the QGMF in identifying the global minimum of a randomly generated oracle, indicated by the number of binary search steps required (left vertical axis) across various qubit counts. The figure contrasts this with the total number of elements a classical brute-force search must navigate to theoretically ensure pinpointing the global minimum (right vertical axis). The function $\log_2{(N)}$, where $N = 2^n$, is also depicted to illustrate the exponential growth of the search space. Note that while Appendix A discusses creating sparse quantum states with a small number of non-zero elements (calculated via the approach mentioned in Algorithm. \ref{['alg:random_oracle_builder']}), the dashed red curve represents the worst-case scenario where all possible computational basis states are non-zero. This provides a clear comparison with classical search methods by illustrating the algorithm's efficiency even when dealing with the maximum possible number of non-zero elements.
• Figure 4: An example graph with four vertices.
• Figure 5: Controlled Quantum Adder circuit $\textit{controlled-A}(1)$.