  Parameterised quantum circuits (PQCs) hold great promise for demonstrating
quantum advantages in practical applications of quantum computation. Examples
of successful applications include the variational quantum eigensolver, the
quantum approximate optimisation algorithm, and quantum machine learning.
However, before executing PQCs on real quantum devices, they undergo
compilation and optimisation procedures. Given the inherent error-proneness of
these processes, it becomes crucial to verify the equivalence between the
original PQC and its compiled or optimised version. Unfortunately, most
existing quantum circuit verifiers cannot directly handle parameterised quantum
circuits; instead, they require parameter substitution to perform verification.
  In this paper, we address the critical challenge of equivalence checking for
PQCs. We propose a novel compact representation for PQCs based on tensor
decision diagrams. Leveraging this representation, we present an algorithm for
verifying PQC equivalence without the need for instantiation. Our approach
ensures both effectiveness and efficiency, as confirmed by experimental
evaluations. The decision-diagram representations offer a powerful tool for
analysing and verifying parameterised quantum circuits, bridging the gap
between theoretical models and practical implementations.
