Quantum-classical algorithms for skewed linear systems with an optimized Hadamard test

The solving of linear systems provides a rich area to investigate the use nearer-term, noisy, intermediate-scale quantum computers. In this work, we discuss hybrid quantum-classical algorithms for skewed over-determined and under-determined cases. Our input model is such that columns or rows matrix defining system are given via circuits poly-logarithmic depth number much smaller than their Hilbert space dimension. have dependence on dimension polynomial in other natural quantities. addition, present an algorithm special case factorized with run time respective dimensions. At core these Hadamard test second part paper consider optimization circuit test. Given $n$-qubit $d$-depth $\mathcal{C}$, can approximate $\langle 0|\mathcal{C}|0\rangle$ using $(n + s)$ qubits $O\left(\log s d\log (n/s) d\right)$-depth circuits, where $s\leq n$. comparison, standard implementation requires $n+1$ $O(dn)$ depth. Lattice geometries underlie recent supremacy experiments superconducting devices. We also optimize $(l_1\times l_2)$ lattice $l_1 \times l_2 = n$, 0|\mathcal{C} |0\rangle$ 1)$ $O\left(d \left(l_1 l_2\right)\right)$-depth circuits. n^2\right)$ setting. Both our methods asymptotically tight one-depth $\mathcal{C}$.