Reducing molecular electronic Hamiltonian simulation cost for Linear Combination of Unitaries approaches



Ignacio Loaiza, Alireza Marefat Khah, Nathan Wiebe, Artur F. Izmaylov


We consider different Linear Combination of Unitaries (LCU) decompositions for molecular electronic structure Hamiltonians. Using these LCU decompositions for Hamiltonian simulation on a quantum computer, the main figure of merit is the 1-norm of their coefficients, which is associated with the quantum circuit complexity. It is derived that the lowest possible LCU 1-norm for a given Hamiltonian is half of its spectral range. This lowest norm decomposition is practically unattainable for general Hamiltonians; therefore, multiple practical techniques to generate LCU decompositions are proposed and assessed. A technique using symmetries to reduce the 1-norm further is also introduced. In addition to considering LCU in the Schrödinger picture, we extend it to the interaction picture, which substantially further reduces the 1-norm.



Thanks for the presentation. Do I understand it correctly that simulate a Hamiltonian with a QC you need to represent it as a linear combination of unitaries? It just seems that mathematically Hermitian and unitary matrices belong to completely different space. 

Also, what's the definition of a molecular structure Hamiltonian - basically N-particle system with Coulomb forces or something else?


1) There are several simulation methods, a few of them (Taylor expansion and Qubitization) require representing the system hermitian Hamiltonian as a linear combination of unitaries.
2) The space for unitaries and hermitian operators is the same, it is 2^N dimensional Hilbert space of states for N qubits.
3) Molecular electronic Hamiltonian is a second-quantized form of T_e+ V_ee+V_en+V_nn after the Born-Oppenheimer separation in the total molecular Hamiltonian, here T_e is a kinetic energy of electrons, V_ee-V_nn corresponding Coulomb interactions between electrons and nuclei constituting the molecule. 


Thank you, Professor Izmaylov. In your opinion, what is the most impressive computation done to date in this space (of molecular electronic simulation) quantum and classical? Thank you again. It's probably the most useful presentation I've seen on this topic.


LCU based techniques are aimed to be used in fault-tolerant algorithms (assuming we have error-correcting hardware), since such hardware is not yet available, there have not been any impressive calculations with these techniques. The hope is that we will be able to do these calculations in the future. 

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