Expressivity Limits in Quantum Walk-based Optimization

TL;DR

This work analyzes the expressivity of Quantum Walk Optimization Algorithms (QWOA) through the Dynamical Lie Algebra (DLA) framework. It derives a spectral-based bound: the DLA dimension satisfies $\operatorname{dim}(\mathfrak g_{QWOA}) \le m^2+1$, where $m$ is the number of distinct eigenvalues of the problem Hamiltonian, implying a polynomial DLA for $\,\mathsf{NPO}-\mathsf{PB}$ problems. By coupling this bound with Grover-type performance limits, the authors show that QWOA must be overparameterized for certain problems outside the $\mathsf{BPPO}$ and $\mathsf{BP-APX}$ classes, while for some structured instances (e.g., complete graphs in Max-Cut) the depth remains subpolynomial, suggesting potential underparameterization. The paper grounds these insights with illustrative analyses of unstructured search, Max-Cut, and $k$-Densest Subgraph, highlighting when DLA-driven expressivity is a bottleneck and guiding future design of structure-aware ansätze. Overall, the results illuminate fundamental limits of QWOA expressivity and provide a framework for assessing when increased parameterization is necessary for optimization tasks.

Abstract

Quantum algorithms have emerged as a promising tool to solve combinatorial optimization problems. The quantum walk optimization algorithm (QWOA) is one such variational approach that has recently gained attention. In the broader context of variational quantum algorithms (VQAs), understanding the expressivity of the ansatz has proven critical for evaluating their performance. A key method to study this aspect involves analyzing the dimension of the dynamic Lie algebra (DLA). In this work, we derive novel upper bounds on the DLA dimension for QWOA applied to arbitrary optimization problems. Specifically, we show that the DLA dimension scales at most quadratically with the number of distinct eigenvalues of the problem Hamiltonian. As a consequence, our bound guarantees a polynomial DLA dimension with respect to the input size for optimization problems in the class $\mathsf{NPO}\text{-}\mathsf{PB}$. This result, coupled with recently established performance bounds for QWOA, allows us to identify complexity-theoretic conditions under which QWOA must be overparameterized to obtain optimal or approximate solutions for $\mathsf{NPO}\text{-}\mathsf{PB}$ problems.

Figures (2)

• Figure 1: A diagram relating complexity classes. Dashed lines indicate the separation between deterministic and randomized algorithms (horizontal), decision and optimization problems (first vertical line), and exact and approximation algorithms (second vertical line). Observe that, starting from $\mathsf{P}$, one can reach any other class by progressively introducing, if necessary, optimization, randomized algorithms, and approximation algorithms.
• Figure 2: Illustration of the indexing procedure. (a) An illustrative indexing map $\mathrm{id}:S' \to \{0,\dots,|S'|-1\}$ for a $4$-qubit system. (b) Adjacency matrix $H_M$ of the CTQW defined over the feasible subspace, where the feasible states are irregularly distributed (depicted in bold). (c) Adjacency matrix $\tilde{H}_M$ of the CTQW defined over the indexed subspace, which rearranges the feasible states into a contiguous block $\{0,\dots,|S'|-1\}$ (also shown in bold), enabling the walk to be implemented via the Fourier transform modulo $|S'|$.

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Definition 3: Optimal QWOA depth
• Definition 4: Random sampling algorithm
• Definition 5: $c$-approximate QWOA depth
• Definition 6: Set of generators, adapted from Ref. larocca2022diagnosing, Definition 2
• Definition 7: Dynamical Lie algebra (DLA), adapted from Ref. larocca2022diagnosing, Definition 3
• Definition 8: Overparametrization, adapted from Ref. larocca2023theory
• Theorem 1: General bound of DLA dimension for QWOA
• Corollary 1