Description
Simulating the quantum properties of molecules and materials is a leading application for quantum computing, with the primary task being the estimation of ground-state energies. Recent advances leverage spectral amplification, a technique that encodes an operator with amplified eigenvalues relative to the system Hamiltonian. By expressing the Hamiltonian as a sum of squares and block-encoding it using oblivious amplitude amplification, we obtain an operator with eigenphases $\phi_k = \arccos(2(E_k+\beta)/\lambda - 1)$, parameterised by an offset $\beta$ and scaling factor $\lambda$. The nonlinearity of the arccosine function near zero translates to highly accurate estimation of the Hamiltonian's eigenvalues, $E_k$.
We previously applied this technique to second-quantised molecular simulations, where qubits encode the occupation of fixed orbitals, reducing the gate complexity for the benchmark FeMoco cluster by over an order of magnitude. However, simulating periodic solid-state materials requires very different methods. It is most effectively treated in first quantisation using a plane-wave basis, where quantum registers instead encode the momenta of individual electrons. In this work, we extend spectral amplification to the quantum simulation of materials.
We present a highly efficient method for expressing the first-quantised plane-wave Hamiltonian as a sum of squares. This formulation yields an asymptotic gate complexity of $\mathcal{O}\left(\eta^2\Delta^{-1.5}+\eta^{2.5}\Delta^{-1}\right)$ for estimating the ground-state energy, demonstrating a significant improvement over the prior state-of-the-art complexity of $\mathcal{O}\left(\eta^2\Delta^{-2}+\eta^3\Delta^{-1}\right)$, where $\eta$ is the number of electrons and $\Delta$ is the simulation grid spacing. Furthermore, we compile explicit gate counts for a range of benchmark material systems. We demonstrate substantial resource reductions across the board, achieving up to a 44-fold speedup for the largest system considered. These algorithmic improvements significantly lower the barrier to achieving practical quantum advantage in solid-state physics.
| I am the presenting author | Yes |
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