DA-CASE: reusable measurements for adaptive quantum subspaces

Quantum subspace methods are often compared using metrics like basis dimension or energy error, which fail to capture the dominant experimental costs associated with state preparation, measurement settings, and shot allocation. This paper introduces the Dyadic Adaptive Clifford-Algebra Subspace Eigensolver (DA-CASE) to address the question: what are the actual resource costs and savings of a fixed-reference operator bank when all methods are expressed in measurement-compatible units?

DA-CASE represents basis states as virtual directions generated from a single reference state. By reconstructing overlap, Hamiltonian, and observable matrices from a cached set of Pauli expectations on that reference, the method eliminates the need for multiple separately prepared basis states. The authors implement a dyadic commuting hierarchy to group Pauli words into fully commuting settings, reducing the number of measurement settings at the cost of increased logical circuit depth. They also evaluate covariance-aware shot allocation and mode-wise overlap regularization to manage finite-shot noise and catastrophic energy estimates.

• Resource Trade-offs: The method demonstrates that generator resolution significantly impacts the retained Pauli bank size; for an eight-qubit H4 Hamiltonian, the retained bank was reduced from 7371 to 2240 Pauli words while maintaining the same subspace and energy.
• Symmetry and Regularization: A reference-conditioned symmetry test allows for the certification of the retained span without requiring individual Pauli words to conserve the sector as global operators. Additionally, mode-wise overlap regularization effectively mitigates catastrophic energy estimates, though it introduces a bias compared to fixed-cutoff methods.
• Measurement Architecture: The dyadic commuting hierarchy successfully reduces the number of required measurement settings (e.g., from 913 to 64 for the H4 case), exposing a clear trade-off between setting counts and the logical-CX depth required for stabilizer circuits.