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

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