Quantum implementation of non-unitary operations with biorthogonal representations

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.