  This paper reports a quantum algorithm for simulating quantum chemical
systems of N molecular orbitals(MOs) using $4\log N +2$ qubits. The number of
multi-electron configurations scales exponentially with the number of MOs and
is the primary bottleneck in calculating the energy of a many-electron system.
This paper introduces qubitized Hamiltonian downfolding(QHD) by combining the
techniques of qubitized quantum walks and Hamiltonian downfolding to reduce the
active space dimension systematically. At each stage of QHD, the number of
many-electron configurations is reduced by $1/4$ by decoupling the molecular
orbital (MO) farthest from the highest occupied MO (HOMO). The sequence of such
downfolding steps enables us to scale towards the low-energy HOMO-LUMO window.
For each stage of downfolding, we map the \emph{decoupling condition} i.e., a
many-body normal-ordered Bloch equation to a system of quadratic polynomial
equations. These downfolding equations can be solved using a non-linear least
squares (NLLS) approach within error $\epsilon$. Each step of NLLS involves a
Hessian inversion and comprises a quantum linear system problem(QLSP). We
describe quantum circuits that block-encode the Hessian using qubitization
oracles. Subsequently, we implement the Chebyshev expansion for solving the
QLSP within error $\epsilon'$, utilizing a sequence of qubitized quantum walks.
Starting from an N-orbital system the gate complexity of each downfolding
circuit scales as $O(N^{2}\log^{2}(1/\epsilon'))$ and for downfolding all the
MOs involve $O(N^3/\epsilon^{2})$ oracle queries.
