Improving the trainability of VQE on NISQ computers for solving portfolio optimization using convex interpolation

TL;DR

The paper addresses the limited trainability of variational quantum eigensolvers (VQAs) on NISQ devices for large-scale combinatorial problems like portfolio optimization. It introduces convex interpolation, built on the clustering of Dicke-state basis by Hamming distance, to predict the ground-state location and enable near-solution initialization, regular cost landscapes, and recursive ansatz partitioning. The authors demonstrate a 40-qubit portfolio-optimization instance using only 10 qubits and show numerical evidence that VQE–greedy hybrids improve global–local optimization synergy, with extensions to graph bisection. This work presents a practical, architecture-aware pathway to exploit quantum advantages on NISQ hardware for real-world optimization tasks and suggests broader applicability to other large-scale combinatorial problems.

Abstract

Solving combinatorial optimization problems using variational quantum algorithms (VQAs) might be a promise application in the NISQ era. However, the limited trainability of VQAs could hinder their scalability to large problem sizes. In this paper, we improve the trainability of variational quantum eigensolver (VQE) by utilizing convex interpolation to solve portfolio optimization. Based on convex interpolation, the location of the ground state can be evaluated by learning the property of a small subset of basis states in the Hilbert space. This enlightens naturally the proposals of the strategies of close-to-solution initialization, regular cost function landscape, and recursive ansatz equilibrium partition. The successfully implementation of a $40$-qubit experiment using only $10$ superconducting qubits demonstrates the effectiveness of our proposals. Furthermore, the quantum inspiration has also spurred the development of a prototype greedy algorithm. Extensive numerical simulations indicate that the hybridization of VQE and greedy algorithms achieves a mutual complementarity, combining the advantages of both global and local optimization methods. Our proposals can be extended to improve the trainability for solving other large-scale combinatorial optimization problems that are widely used in real applications, paving the way to unleash quantum advantages of NISQ computers in the near future.

Figures (20)

• Figure 1: (a) The CCC ansatz for preparing the Dicke state $\vert D^6_3\rangle$. Each $V_i$ is constructed using a "CNOT, C-RY, CNOT" ($3$C) block depicted in the red box. The parameter is the angle of C-RY gate. Column $010101$ of the unitary $U_6$ contains all $\binom{6}{3}$ basis states. (b) The direct product decomposition of the CCC ansatz in (a). The entanglement between the two fragments are fully decomposed into $4$ product sub-ansatze. The red dashed boxes indicate the increase of $X$ gates in the upper fragment. In the lower fragment of sa$_1$ and the upper fragment of sa$_2$, the Dicke state $\vert D^3_2\rangle$ is prepared based on the relationship $\vert D^n_k\rangle = X^{\otimes n}\vert D^n_{n-k}\rangle$, as detailed in Appendix \ref{['appen: completeness and reachability proves']}.
• Figure 2: (a) The number of basis states distributed in each sub-ansatz for $\vert D^{24}_{12}\rangle$. It resembles a Gaussian distribution. The sub-ansatze closing to the edge, i.e. sa$_1$ and sa$_{11}$, contain a much smaller number of basis states. (b) The illustration of the convex interpolation. The red dashed polyline connects the true minimum energies of the sub-ansatze, represented by the red × marks. The green solid line represents the interpolated convex curve using the minimum energies of the $4$ outer sub-ansatze, denoted by the blue dots, using $scipy.interpolate.KroghInterpolator()$.
• Figure 3: (a) The matrix representation of column $0101...01$ of the unitary $U_n$, with colored squares representing its elements. (b) The interpolated convex curve. The target region contains the ground state. The subspaces within the target region (between the blue dashed line) encompass the basis states corresponding to the blue squares in (a). (c) The initialized probability distribution. Its principal component (between the blue dashed lines) covers the target region.
• Figure 4: The ratio and variance curves of sub-ansatze sa$_1$ to sa$_8$ of $\vert D^{18}_9\rangle$ with respect to the $21$ parameter values sampled uniformly on the interval $[0,\pi]$, i.e. $0$, $\pi/20$, $\pi/10$, $...$, $\pi$. Each sub-ansatz (principal component) has its own optimal ratio interval, i.e. the interval producing the larger ratios, corresponding to the concave (hill) part of the curves. As the subscript of the sub-ansatz increases from $1$ to $\frac{n}{2}-1$, the optimal ratio interval migrates to the small parameter and then to the large parameter. This is consistent with the propagation mode of the "$1$" information. The variance changes in basically the same way as the ratio. The values of the variances may seem small, e.g. $10^{-5}$, yet they are compressed by BP. This is in accordance with the exponential vanishing $1/2^{18}\approx 10^{-5}$. The initial state $\bigotimes^9_{i=1}\vert 01\rangle$ is assigned to sa$_4$. This results in a ratio close to $1$ when the parameter is initialized not far from $0$. However, in this scenario, the principal component is mainly focused on basis state $\bigotimes^9_{i=1}\vert 01\rangle$.
• Figure 5: Comparison between close-to-solution initialization and random initialization. $\delta_c$ represents the optimal ratio of each sub-ansatz based on close-to-solution initialization, while $\delta_r$ (with error bars) represents the average ratio, and $\sigma_r$ (with error bars) represents the average variance of the $20$ simulations based on random initialization. As can be seen, close-to-solution initialization is overall better than random initialization.