  Quantum algorithms are still challenging to solve linear systems of equations
on real devices. This challenge arises from the need for deep circuits and
numerous ancilla qubits. We introduce the quantum conjugate gradient (QCG)
method using the quantum eigenvalue transformation (QET). The circuit depth of
this algorithm depends on the square root of the coefficient matrix's condition
number $\kappa$, representing a square root improvement compared to the
previous quantum algorithms, while the total query complexity worsens. The
number of ancilla qubits is constant, similar to other QET-based algorithms.
Additionally, to implement the QCG method efficiently, we devise a QET-based
technique that uses only the positive side of the polynomial (denoted by $P(x)$
for $x\in[0,1]$). We conduct numerical experiments by applying our algorithm to
the one-dimensional Poisson equation and successfully solve it. Based on the
numerical results, our algorithm significantly improves circuit depth,
outperforming another QET-based algorithm by three to four orders of magnitude.
