Simultaneously Optimizing Symmetry Shifts and Tensor Factorizations for Cost-Efficient Fault-Tolerant Quantum Simulations of Electronic Hamiltonians

In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous best techniques for industrially and biologically relevant molecules, including the nitrogenase cofactor (FeMoCo) and cytochrome P450. Benchmark calculations comparing our method with the original double-factorization method show a 75% reduction in the block-encoding scaling constant. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the journey to the feasibility of practical Hamiltonian simulations.