Hybrid Quantum-Classical Hamiltonian Downfolding for Molecular Systems
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Solution Overview
Problem
The computational complexity of diagonalizing large electronic Hamiltonians in quantum computing, particularly for molecular systems, is exacerbated by the exponential scaling with the number of orbitals, making it challenging to simulate electronic Potential Energy Surfaces (PES) and construct effective Hamiltonians for applications in quantum chemistry and material design.
Innovation Solution
A hybrid quantum-classical architecture method that iteratively downfolds electronic Hamiltonians by determining projection operators, constructing qubit Hamiltonians, transforming them into polynomial equations, and solving these equations using the Levenberg-Marquadt Method, reducing the number of molecular orbitals and simplifying the similarity transformation process to linear order, thereby reducing the complexity and improving accuracy.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the full electronic Hamiltonian is diagonalized to obtain accurate eigenvalues and eigenvectors, then the computational precision is improved, but the computational complexity scales exponentially with the number of orbitals
Solution Approach 1:
The full electronic Hamiltonian is segmented into an effective Hamiltonian acting on a reduced subspace of active orbitals and a remainder. By dividing the problem into manageable segments (active space vs. full space), the method achieves accurate results for the relevant subspace without the exponential cost of diagonalizing the complete Hamiltonian.
Solution Approach 2:
The method extracts and isolates the essential physics contained in a reduced effective Hamiltonian from the full Hamiltonian. By taking out only the relevant active orbital subspace and constructing an effective Hamiltonian that captures the essential electronic structure, the approach achieves chemical accuracy without requiring full diagonalization.
2Productivity
If the number of molecular orbitals is reduced to decrease computational complexity, then the computational time is improved, but the accuracy of the PES calculation deteriorates
Solution Approach 1:
The method changes the parameter representation by transforming the full Hamiltonian into an effective Hamiltonian with optimized parameters (one-electron and two-electron integrals) in the reduced basis. This parameter transformation allows the reduced model to achieve the same accuracy as the full model by optimizing the effective parameters through the polynomial equation solving process.
3Measurement precision
If quantum phase estimation algorithm is used to estimate energy, then the accuracy is improved, but the number of physical qubits and Toffoli gates required increases extremely high
Solution Approach 1:
The method transitions from the traditional quantum phase estimation approach to a polynomial equation solving framework. By changing the dimensional approach from direct quantum phase estimation to solving polynomial equations derived from projection operators, the method achieves the same energy estimation accuracy with significantly reduced quantum resources.
4Measurement precision
If double unitary transformation or coupled cluster transformation is performed to simulate effective Hamiltonian, then the simulation accuracy is improved, but the circuit depth and complexity increase
Solution Approach 1:
The method extracts the essential transformation by directly constructing the effective Hamiltonian through projection operators and polynomial equations, rather than performing full double unitary or coupled cluster transformations. This extraction approach achieves the same simulation accuracy by focusing only on the relevant active space transformations without the overhead of complete exponential transformations.
Data Source
AI summary
Conventional Hamiltonian downfolding methods involve approximating exponential within the double unitary coupled cluster transformation which effects accuracy of the resultant Hamiltonian. Thus, the present disclosure provides a method for downfolding electronic Hamiltonians using a hybrid quantum-classical architecture wherein similarity transformation in such a way that the exponential terminates at linear order. In addition, a many body Bloch equation is defined which embodies every similarity downfolding transformation step. From the Bloch equation, a system of polynomial equations is derived for downfolding one molecular orbital. Quantum Circuits are used to facilitate solving the polynomial equations, which helps in constructing a lower dimensional Hamiltonian with one less molecular orbital at every downfolding step/iteration. The entire process gets repeated for every orbital downfolding, leading to a smaller dimensional effective Hamiltonian.