[Paper Review] Quantum Simulation of Molecules without Fermionic Encoding of the Wave Function

David A. Mazziotti, Scott E. Smart+1 more|arXiv (Cornell University)|Jan 27, 2021

Quantum Computing Algorithms and ArchitectureComputer Science96 references16 citations

TL;DR

This paper proposes a new approach to quantum molecular simulation that bypasses fermionic encoding of the wave function by expressing the energy as a unique functional of the two-electron reduced density matrix (2-RDM) derived directly from unencoded N-qubit-particle wave functions. The method preserves entanglement complexity and enables potentially more efficient quantum computations, demonstrated by computing the ground-state energy and 2-RDM of H4 without fermionic encoding.

ABSTRACT

Molecular simulations generally require fermionic encoding in which fermion statistics are encoded into the qubit representation of the wave function. Recent calculations suggest that fermionic encoding of the wave function can be bypassed, leading to more efficient quantum computations. Here we show that the energy can be expressed as a functional of the two-electron reduced density matrix (2-RDM) where the 2-RDM is a unique functional of the unencoded $N$-qubit-particle wave function. Contrasts are made with current hardware-efficient methods. An application to computing the ground-state energy and 2-RDM of H$_{4}$ is presented.

Motivation & Objective

• To eliminate the need for fermionic encoding in quantum molecular simulations.
• To establish a unique, isomorphic mapping between N-fermion wave functions and N-qubit-particle wave functions.
• To demonstrate that the 2-RDM can be computed directly from unencoded qubit wave functions with preserved entanglement complexity.
• To enable more efficient quantum computations by avoiding the overhead of fermionic encoding.

Proposed method

• The energy is expressed as a functional of the 2-RDM using the equation E = Tr(2K 2D), where 2K is the reduced Hamiltonian and 2D is the 2-RDM.
• A bijective (isomorphic) mapping is proven between the Hilbert spaces of N-fermion and N-qubit-particle wave functions, preserving particle number and structure.
• The N-qubit-particle wave function is constructed using symmetric wedge products (∨), contrasting with the antisymmetric wedge products (∧) used in fermionic wave functions.
• The 2-RDM is computed directly from the unencoded qubit wave function coefficients, which are identical in form to fermionic coefficients.
• The method avoids fermionic encoding of both the wave function and the 2-RDM, reducing computational overhead.
• The approach leverages the fact that expansion coefficients span the same vector space, enabling direct functional mapping from qubit to fermionic 2-RDM.

Experimental results

Research questions

• RQ1 Can the 2-RDM be uniquely expressed as a functional of an unencoded N-qubit-particle wave function?
• RQ2 Does bypassing fermionic encoding preserve the entanglement complexity of the wave function?
• RQ3 Is there an isomorphic mapping between N-fermion and N-qubit-particle wave functions that preserves particle number and structure?
• RQ4 Can the ground-state energy and 2-RDM of H4 be computed without fermionic encoding of the wave function?
• RQ5 How does this method compare in efficiency to current hardware-efficient variational quantum algorithms?

Key findings

• The 2-RDM is uniquely determined as a functional of the unencoded N-qubit-particle wave function, with a proven bijective mapping between fermionic and qubit-particle wave functions.
• The mapping preserves particle number and vector space structure, ensuring that the entanglement complexity of the wave function remains unchanged.
• The method avoids fermionic encoding of both the wave function and the 2-RDM, potentially reducing quantum circuit depth and resource overhead.
• The ground-state energy and 2-RDM of H4 were successfully computed using the unencoded qubit wave function, demonstrating the feasibility of the approach.
• The approach enables direct measurement of the 2-RDM from unencoded qubit states with potentially non-exponential scaling in N.
• The framework provides a path to more efficient quantum simulations by sidestepping the need for fermionic encoding while maintaining exactness in the 2-RDM representation.