Quantum implementation of non-unitary operations with biorthogonal representations

TL;DR

This paper introduces a biorthogonal representation-based dilation method for implementing non-unitary quantum operations, especially effective for operators with eigenvalues exceeding one, offering advantages over existing techniques in specific scenarios.

Contribution

A novel dilation approach using biorthogonal representations that improves efficiency for certain non-unitary operators in quantum computing.

Findings

Outperforms existing methods for operators with eigenvalues > 1

Optimal for small-dimensional non-unitary operators regardless of summands

Complements the Linear Combination of Unitaries (LCU) method

Abstract

Motivated by the contemporary advances in quantum implementation of non-unitary operations, we propose a new dilation method based on the biorthogonal representation of the non-unitary operator, mapping it to an isomorphic unitary matrix in the orthonormal computational basis. The proposed method excels in implementing non-unitary operators whose eigenvalues have absolute values exceeding one, when compared to other dilation and decomposition techniques. Unlike the Linear Combination of Unitaries (LCU) method, which becomes less efficient as the number of unitary summands grows, the proposed technique is optimal for small-dimensional non-unitary operators regardless of the number of unitary summands. Thus, it can complement the LCU method for implementing general non-unitary operators arising in positive only open quantum systems and pseudo-Hermitian systems.