Operator-centric Clifford algebra for variational eigensolvers and finite-shot adaptive selection

Research Approach

This paper presents a unified algebraic framework for n-qubit variational quantum algorithms using complex Clifford algebra. By mapping density operators, gates, and observables into a single Pauli-word algebra via the Jordan-Wigner transformation, the authors provide a consistent mathematical language for state evolution and gradient calculation. A central contribution is the derivation of a transpose-parity selection rule: for real Hamiltonians and states, any candidate Pauli word with an even number of Y factors has a zero gradient, allowing for exact pruning of the operator pool.

The authors test their framework on the open transverse-field Ising chain. They find that while a compact local operator pool is sufficient for 4 qubits, it fails to maintain accuracy at larger sizes. A more systematic 'local-three' pool, which includes all contiguous one-, two-, and three-site Pauli words with odd Y parity, achieves relative energy errors below 1.3x10^-12 for 6 qubits. Furthermore, the paper evaluates three finite-shot selection policies. While fixed-shot selection fails entirely in 100-seed tests, a 'racing' policy—which dynamically stops measuring candidates whose confidence bounds fall below the current leader—succeeds in 84% of runs while reducing the median measurement cost by 34% compared to uniform escalation.

This work provides a rigorous algebraic foundation for variational eigensolvers, clarifying the distinction between Pauli-word rotations and Spin-group rotors. The exact pruning rule and the racing policy offer practical, reproducible strategies to mitigate the measurement bottleneck in adaptive quantum algorithms. By demonstrating that shared-parameter Hamiltonian variational ansatzes require careful gradient handling at the physical-gate level, the study highlights common pitfalls in standard parameter-shift implementations.