Achieving an accurate description of fermionic systems typically requires considerably many more orbitals than fermions. Previous resource analyses of quantum chemistry simulation often failed to exploit this low fermionic number information in the i
mplementation of Trotter-based approaches and overestimated the quantum-computer runtime as a result. They also depended on numerical procedures that are computationally too expensive to scale up to large systems of practical interest. Here we propose techniques that solve both problems by using various factorized decompositions of the electronic structure Hamiltonian. We showcase our techniques for the uniform electron gas, finding substantial (over 100x) improvements in Trotter error for low-filling fraction and pushing to much higher numbers of orbitals than is possible with existing methods. Finally, we calculate the T-count to perform phase-estimation on Jellium. In the low-filling regime, we observe improvements in gate complexity of over 10x compared to the best Trotter-based approach reported to date. We also report gate counts competitive with qubitization-based approaches for Wigner-Seitz values of physical interest.
We present a constructive method to create quantum circuits that implement oracles $|xrangle|yrangle|0rangle^k mapsto |xrangle|y oplus f(x)rangle|0rangle^k$ for $n$-variable Boolean functions $f$ with low $T$-count. In our method $f$ is given as a 2-
regular Boolean logic network over the gate basis ${land, oplus, 1}$. Our construction leads to circuits with a $T$-count that is at most four times the number of AND nodes in the network. In addition, we propose a SAT-based method that allows us to trade qubits for $T$ gates, and explore the space/complexity trade-off of quantum circuits. Our constructive method suggests a new upper bound for the number of $T$ gates and ancilla qubits based on the multiplicative complexity $c_land(f)$ of the oracle function $f$, which is the minimum number of AND gates that is required to realize $f$ over the gate basis ${land, oplus, 1}$. There exists a quantum circuit computing $f$ with at most $4 c_land(f)$ $T$ gates using $k = c_land(f)$ ancillae. Results known for the multiplicative complexity of Boolean functions can be transferred. We verify our method by comparing it to different state-of-the-art compilers. Finally, we present our synthesis results for Boolean functions used in quantum cryptoanalysis.
