Quantum Circuit for Non-Unitary Linear Transformation of Basis Sets

TL;DR

This paper presents a quantum method using SVD for non-unitary linear transformations of basis sets, enabling advanced analysis of quantum states and potentially improving energy calculations in quantum chemistry.

Contribution

It introduces a novel quantum circuit that performs non-unitary basis transformations efficiently, expanding quantum computational capabilities beyond traditional unitary operations.

Findings

Achieves $O(n)$ operational depth with about $n$ ancilla qubits.

Enables transformation between different basis spaces for wavefunctions.

Facilitates calculation of overlaps and energies of multiple eigenstates.

Abstract

This paper introduces a novel approach to implementing non-unitary linear transformations of basis on quantum computational platforms, a significant leap beyond the conventional unitary methods. By integrating Singular Value Decomposition (SVD) into the process, the method achieves an operational depth of $O(n)$ with about $n$ ancilla qubits, enhancing the computational capabilities for analyzing fermionic systems. The non-unitarity of the transformation allows us to transform a wave function from one basis to another, which can span different spaces. By this trick, we can calculate the overlap of two wavefunctions that live in different (but non-distinct Hilbert subspaces) with different basis representations. This provides the opportunity to use state specific ansatzes to calculate different energy eigenstates under orbital-optimized settings and may improve the accuracy when computing the energies of multiple eigenstates simultaneously in VQE or other framework. It allows for a deeper exploration of complex quantum states and phenomena, expanding the practical applications of quantum computing in physics and chemistry.