Quantum Implementation of Non-unitary Operations with Biorthogonal Representations

We propose a new dilation method for implementing non-unitary operators on a quantum computer using the biorthogonal framework. By selecting an appropriate biorthogonal basis for the non-unitary operator, a unitary counterpart can be constructed in the biorthogonal representation, enabling its implementation in the orthonormal computational basis. When compared to other dilation and decomposition methods, the proposed method is particularly efficient for non-contraction, non-unitary operators. In contrast to the Linear Combination of Unitaries (LCU) method, the efficiency of the biorthogonal dilation technique is not constrained by the number of unitary summands but instead by the dimensionality of the non-unitary operator. The proposed method complements the LCU method for implementing general non-unitary operators that arise in positive-only open quantum systems, pseudo-Hermitian, and general non-Hermitian systems.