Quantum circuit for non-unitary linear transformation of basis sets

This paper presents a novel approach for implementing non-unitary basis transformations on quantum computational platforms, extending beyond conventional unitary methods. By integrating Singular Value Decomposition (SVD), the method achieves operational depth O(n) with about n ancilla qubits, enhancing the analysis of fermionic systems. The non-unitary transformation enables mapping a wavefunction between different bases, allowing the calculation of overlaps of states residing in distinct Hilbert subspaces. This capability supports the use of state-specific ansatzes to compute multiple energy eigenstates under orbital-optimized settings, potentially improving accuracy in variational quantum eigensolver (VQE) and related frameworks. Overall, the method expands the range of accessible quantum states and phenomena, offering new opportunities for applying quantum computing in physics and chemistry.