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



Anirban Mukherjee


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.



This paper introduces a quantum algorithm for solving the downfolding equations in Hamiltonian Renormalization Group theory using the Levenberg-Marquadt Method. The authors provide a comprehensive exploration of the quantum method, offering a detailed mathematical exposition. In their approach, they use a non-linear least squares method for the downfolding equations and employ the Child's Quantum walk approach for matrix inversion. The paper concludes with a discussion of the quantum resources necessary for block-encoding the Hessian, estimating the number of queries required for Child's quantum walk, and detailing the gate complexity for matrix inversion.


  1. One of the major assumptions in the method seems to be the use of the Levenberg-Marquadt Method (LMM) for solving the downfolding equations. How robust is this method in handling different types of systems, and are there any limitations to using LMM in this context?

  2. The paper suggests that the computational bottleneck of the approach is the inversion of the Hessian matrix. Given this, how does the quantum algorithm proposed improve upon classical methods in terms of speed and efficiency?

  3. In the final section, the paper discusses the quantum resources necessary for block-encoding the Hessian and estimates the number of queries required for Child's quantum walk. How feasible is the implementation of this algorithm given the current state of quantum computing technology?


P.S. The above questions have been generated by the AI algorithm, but they appear pretty reasonable and could be useful to the audience (especially non-experts). The paper is written in an extremely technical way with heavy usage of technical terminology and even inventing new terminology.

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