Quantum algorithm for the calculation of transition amplitudes in hybrid tensor networks

The hybrid tensor network approach allows us to perform calculations on systems larger than the scale of a quantum computer. However, when calculating transition amplitudes, there is the problem that the number of terms to be measured increases exponentially with that of the contracted operators. This problem is caused by the fact that the contracted operators are represented as non-Hermitian operators. In this study, we propose a method for hybrid tensor network calculation that contracts non-Hermitian operators without an exponential increase in the number of terms. In the proposed method, transition amplitudes are calculated by combining the singular value decomposition of the contracted non-Hermitian operators with a Hadamard test. The method significantly extends the applicability of the hybrid tensor network approach.