26th Australian Institute of Physics Congress and the Australian and New Zealand Conference on Optics and Photonics

Sum of Squares Representations for the Hubbard Model

Not scheduled

20m

Belinda Hutchinson Building (The University of Sydney )

Contributed Oral AIP | Quantum Science and Technology (QST)

Description

Fast simulation algorithms for large systems of correlated electrons will revolutionise our predictive capabilities for chemical phenomena and is perhaps the most compelling promise of fault-tolerant quantum computers. Moreover, a suite of common electronic behaviour, including superconductivity and magnetism, is captured by the Hubbard model, which describes electron interactions on a lattice. Consequently, formulating efficient quantum simulations of Hubbard Hamiltonians has been a long-standing research imperative.

Simulation efficiency is strongly dependent on the chosen representation of the Hamiltonian when it is encoded on the quantum computer. In a technique called spectral amplification, we encode the square root of a positive Hamiltonian to magnify its spectrum near zero, yielding improved efficiency in low energy regimes. Of course, preparing positive representations of Hamiltonians and computing matrix square roots a priori entails large computational overheads, but since electronic structures like the Hubbard Hamiltonian are polynomials in fermionic operators, they are amenable to sum of squares representations. Specifically, for a sufficiently large constant $\beta > 0$, the shifted Hamiltonian, $\hat{H} + \beta I$, can be decomposed as a sum of squares of low degree polynomials. If the SOS is simple, then its square root can be efficiently encoded. If $\beta$ is small, then the low energy spectrum of the SOS is close to zero. When both objectives are achieved simultaneously, spectral amplification yields a net reduction in the cost of simulation at low spectra.

We present an SOS representation of the Hubbard Hamiltonian which, when paired with spectral amplification, improves simulation efficiency by a factor of 2 to 4, with the exact amount depending on the model's hopping amplitude and on-site interaction strength. While these results further the applicability of fault tolerant quantum computers for chemical simulations, an open question is whether alternative SOS representations can achieve asymptotic scaling improvements.