This paper concerns the problem of checking if two shallow (i.e., constant-depth) quantum circuits perform equivalent computations. Equivalence checking is a fundamental correctness question—needed, e.g., for ensuring that transformations applied to a quantum circuit do not alter its behavior. For quantum circuits, the problem is challenging because a straightforward representation on a classical computer of each circuit’s quantum state can require time and space that are exponential in the number of qubits n . The paper presents Projection-Based Equivalence Checking (PBEC), which provides decision procedures for two variants of the equivalence-checking problem. Both can be carried out on a classical computer in time and space that, for any fixed depth, is linear in n . Our key insight is that local projections can serve as constraints that fully characterize the output state of a shallow quantum circuit. The output state is the unique quantum state that satisfies all the constraints. Beyond equivalence checking, we show how to use the constraint representation to check a class of assertions, both statically and at run time. Our assertion-checking methods are sound and complete for assertions expressed as conjunctions of local projections. Our experiments showed that computing the constraint representation of a random 100-qubit 1D circuit of depth 6 takes 129.64 seconds. Equivalence checking between two random 100-qubit 1D circuits of depth 3 requires 4.46 seconds for fixed input | 0 〉 ⊗ 100 , and no more than 31.96 seconds for arbitrary inputs. Computing the constraint description for a random 100-qubit circuit of depth 3 takes 6.99 seconds for a 2D structure, compared to 10.67 seconds for a circuit with arbitrary connectivity. At depth 2, equivalence checking takes 0.20 seconds for fixed input and 0.44 seconds for arbitrary input, with similar performance for both 2D and arbitrary-connectivity circuits.

@article{paperbot3012,
  title = {Scalable Equivalence Checking and Verification of Shallow Quantum Circuits},
  author = {Nengkun Yu and Xuan Du Trinh and Thomas Reps},
  journal = {Proceedings of the ACM on Programming Languages},
  volume = {9},
  number = {OOPSLA2},
  year = {2025},
  doi = {10.1145/3763153}
}