Table 3.

Comparison of the total Toffoli cost for time-evolution using QSP or product formulas for 10 uniformly spaced times starting from $t=1$ and going to $t=10$ with infidelity $ϵ=0.01$ including $50$ samples to measure the kinetic energy of the projectile

Projectile + Target	$\eta$	QSP Toffoli	Product Formula Toffoli	QSP Qubits	Product Formula Qubits
Alpha + Hydrogen (50%)	28	$2.796\times {10}^{14}$	$1.124\times {10}^{13}$	1,749	2,666
Alpha + Hydrogen (75%)	92	$1.017\times {10}^{16}$	$3.069\times {10}^{14}$	3,309	3,902
Alpha + Hydrogen	218	$9.960\times {10}^{16}$	$1.399\times {10}^{15}$	5,650	6,170
Proton + Carbon	391	$1.113\times {10}^{18}$	$1.074\times {10}^{16}$	8,841	9,284
Proton + Deuterium	1729	$1.061\times {10}^{20}$	$2.079\times {10}^{17}$	33,038	33,368

The smallest Alpha + Hydrogen system used $n_p=5$ while all other systems used $n_p=6$. For all systems, the projectile kinetic energy register used $n_n=8$ bits. The number of qubits for the product formula is estimated based on the system size plus an upper bound to the number of ancilla needed for performing the polynomial interpolation multiplications and Newton–Raphson step floating point arithmetic.