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This paper reports a quantum algorithm for simulating quantum chemicalsystems of N molecular orbitals(MOs) using $4\log N +2$ qubits. The number ofmulti-electron configurations scales exponentially with the number of MOs andis the primary bottleneck in calculating the energy of a many-electron system.This paper introduces qubitized Hamiltonian downfolding(QHD) by combining thetechniques of qubitized quantum walks and Hamiltonian downfolding to reduce theactive space dimension systematically. At each stage of QHD, the number ofmany-electron configurations is reduced by $1/4$ by decoupling the molecularorbital (MO) farthest from the highest occupied MO (HOMO). The sequence of suchdownfolding 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., amany-body normal-ordered Bloch equation to a system of quadratic polynomialequations. These downfolding equations can be solved using a non-linear leastsquares (NLLS) approach within error $\epsilon$. Each step of NLLS involves aHessian inversion and comprises a quantum linear system problem(QLSP). Wedescribe quantum circuits that block-encode the Hessian using qubitizationoracles. Subsequently, we implement the Chebyshev expansion for solving theQLSP within error $\epsilon'$, utilizing a sequence of qubitized quantum walks.Starting from an N-orbital system the gate complexity of each downfoldingcircuit scales as $O(N^{2}\log^{2}(1/\epsilon'))$ and for downfolding all theMOs involve $O(N^3/\epsilon^{2})$ oracle queries.

Hamiltonian Simulation Via Qubitized Downfolding Using $4\log N+2$ Qubits