The graph isomorphism problem asks whether two graphs are equivalent up to a permutation of their vertices. Its precise computational complexity remains unresolved, and it is still unknown whether a polynomial-time algorithm exists. We investigate whether simple graph-dependent Instantaneous Quantum Polynomial-time (IQP) and Quantum Approximate Optimisation Algorithm (QAOA) circuits can distinguish non-isomorphic graphs. We first show that suitable IQP and QAOA constructions allow the underlying graph to be reconstructed using a polynomial number of samples. Building on previous work using the QAOA energy, we show that Hamming weight distributions can distinguish graph pairs at circuit depths for which the numerically computed energy gap is zero. We then apply graph invariants to the matrix of ZZ correlators on the output distribution. For the specific case of the graph spectrum, at QAOA depth $p=2$, this invariant numerically distinguishes all $10{,}486$ cospectral regular graph pairs tested on up to 18 vertices. For IQP circuits, we introduce a finer invariant based on probability multiplicities within Hamming weight sectors, which distinguishes every non-isomorphic graph pair tested numerically. Finally, we derive exact graph-theoretic characterisations of the IQP output distribution at $θ=π/2$ and $θ=π/4$. These results show that simple graph-dependent quantum circuits encode substantial structural information, while leaving open whether this information can be extracted efficiently for general graph isomorphism.
Automatic Test Pattern Generation for Robust Quantum Circuit Testing
Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Department of Computer and Information Science, University of Pennsylvania（宾夕法尼亚大学计算机与信息科学系）；Centre for Quantum Software and Information, University of Technology Sydney（悉尼科技大学量子软件与信息研究中心）
