Global Minimization of Electronic Hamiltonian 1‑Norm via Linear Programming in the Block Invariant Symmetry Shift (BLISS) Method

The cost of encoding a system Hamiltonian in a digital
quantum
computer as a linear combination of unitaries (LCU) grows with the
1-norm of the LCU expansion. The Block Invariant Symmetry Shift (BLISS)
technique reduces this 1-norm by modifying the Hamiltonian action
on only the undesired electron-number subspaces. Previously, BLISS
required a computationally expensive nonlinear optimization that was
not guaranteed to find the global minimum. Here, we introduce various
reformulations of this optimization as a linear programming problem,
which guarantees optimality and significantly reduces the computational
cost. We apply BLISS to industrially relevant homogeneous catalysts
in active spaces of up to 76 orbitals, finding substantial reductions
in both the spectral range of the modified Hamiltonian and the 1-norms
of Pauli and fermionic LCUs. Our linear programming techniques for
obtaining the BLISS operator enable more efficient Hamiltonian simulation
and, by reducing the Hamiltonian’s spectral range, offer opportunities
for improved LCU groupings to further reduce the 1-norm.