Tensor Network Assisted Distributed Variational Quantum Algorithm for Large Scale Combinatorial Optimization Problem

TL;DR

The Distributed Variational Quantum Algorithm is proposed, enabling the solution of 1,000-variable instances on constrained hardware and providing a scalable, noise-resilient framework that advances the timeline for practical quantum optimization algorithms.

Abstract

Although quantum computing holds promise for solving Combinatorial Optimization Problems (COPs), the limited qubit capacity of NISQ hardware makes large-scale instances intractable. Conventional methods attempt to bridge this gap through decomposition or compression, yet they frequently fail to capture global correlations of subsystems, leading to solutions of limited quality. We propose the Distributed Variational Quantum Algorithm (DVQA) to overcome these limitations, enabling the solution of 1,000-variable instances on constrained hardware. A key innovation of DVQA is its use of the truncated higher-order singular value decomposition to preserve inter-variable dependencies without relying on complex long-range entanglement, leading to a natural form of noise localization where errors scale with subsystem size rather than total qubit count, thus reconciling scalability with accuracy. Theoretical bounds confirm the algorithm's robustness for p-local Hamiltonians. Empirically, DVQA achieves state-of-the-art performance in simulations and has been experimentally validated on the Wu Kong quantum computer for portfolio optimization. This work provides a scalable, noise-resilient framework that advances the timeline for practical quantum optimization algorithms.

Figures (5)

• Figure 1: a) Large-scale COPs with $n_v$ decision variables often require more qubits than are available on today’s quantum processing units, i.e., $n_v > n_q$, where $n_q$ denotes the number of physical qubits. To address this limitation, the problem can be partitioned into smaller subsystems with $n_k$ qubits and transformed using higher-order singular value decomposition (HOSVD), enabling the solution of tractable subproblems on limited hardware. b) Specifically, truncated HOSVD (T-HOSVD) factorizes the global coefficient tensor into lower-rank components, each of which can be executed on a smaller QPU. The global solution is then reconstructed through variational recombination of the subproblem results.
• Figure 2: Schematic overview of the DVQA framework. The architecture comprises two primary modules: quantum subsystem evolution (left, green) and tensor-network-based classical optimization (right, pink). For a second-order interaction problem (e.g., QUBO p=2), each subsystem---denoted $k'$ and $\tilde{k}$---has two circuits representing the real and imaginary parts of the distributed objective function. Orthogonal initial states $|\alpha_{k'}\rangle, |\beta_{k'}\rangle$ and $|\alpha_{\tilde{k}}\rangle, |\beta_{\tilde{k}}\rangle$, together with variational parameters $\theta=\otimes_k \theta_k$, are used to generate the circuit outputs $\Pi_{k'}^{i}$ and $\Pi_{\tilde{k}}^{i}$. These outputs are contracted within a tensor network parameterized by trainable tensors $C$ and weights $w_i$, yielding the loss function $\ell(\theta, C) = \sum_{\alpha,\beta, i} w_i\, C_\alpha\, C_\beta^* \prod_k \Pi_k^i.$ The optimization is constrained by $C^{\dagger}C = 1$ and performed using the ADAM optimizer through the Lagrange-multiplier formulation $L = \lambda(C^{\dagger}C - 1) + \ell(\theta, C)$, iteratively updating both $\theta$, and $C$. The final optimized parameters $\theta_{k'}^*, \theta_{\tilde{k}}^*$, and $C^*$ reconstruct the global quantum state $|\phi\rangle = \sum_\alpha C_\alpha^* \left( \bigotimes_{k=1}^{K} U_k(\theta_k^*)\, |\alpha_k\rangle_k \right)$, where the optimized subsystem states $U_k(\theta_k^*)|\alpha_k\rangle_k$ are variationally combined through the learned tensor network.
• Figure 3: Performance comparison of different optimization methods on MaxCut (left, Dataset 1) and portfolio optimization (right, Dataset 2). The vertical axis shows the approximation ratio and the horizontal axis lists problem instances (P1--P5); colors denote different methods (Ours, Q_Enc, DP, QA). For each problem--method pair, we perform 20 independent runs. We use a box-and-whisker plot with outliers suppressed: the box spans the interquartile range (IQR) from the first quartile $Q_1$ (25th percentile) to the third quartile $Q_3$ (75th percentile), the center line indicates the median $Q_2$ (50th percentile), and the whiskers extend to the most extreme observations within $[Q_1-1.5\,\mathrm{IQR},\,Q_3+1.5\,\mathrm{IQR}]$, where $\mathrm{IQR}=Q_3-Q_1$. In addition, colored circular markers denote the run with the best objective value among the 20 runs for each problem--method pair.
• Figure 4: Scalability (left) and runtime (right) comparison of different methods. The left panel shows the approximation ratio versus problem size (Size), and the right panel shows the end-to-end runtime (Time, in seconds) versus Size.
• Figure 5: Verification of the DVQA noise model and real-device performance. (a) For a representative QUBO instance ($p=2$), comparison between the noise-simulation results and the theoretical prediction of Eq. (8) as a function of the number of subsystems $K$. The blue markers report $\Delta H = \left|E_\text{sim} - E_g\right|$ averaged over 10 independent runs; the error bars and the shaded band both indicate standard deviation (STD) across the 10 runs. The orange squares show the Eq. (8) prediction. (b) Same as (a) but for the two-body Hamiltonian $H=\mathrm{ZIIIIIIIIZ}$; results are averaged over 10 independent runs, with error bars and shaded bands denoting STD. (c) Real quantum device results on the Origin Quantum Wu--Kong superconducting processor for the SP20 problem, comparing DVQA ($K=4$) with the original VQE setting ($K=1$). Bars indicate the measured success probability, and the annotated values report the corresponding energy.