Table 1.

Costs of exact quantum algorithms and mean-field classical algorithms for simulating fermionic dynamics

Processor	Algorithm	Observable	Space	Gate complexity
Classical	T = 0 mean-field with occ-RI-K/ACE23,24	Anything	$\tilde{\mathcal{O}}\left(N\eta \right)$	$\left(N^{4/3}{\eta }^{7/3}t+N^{5/3}{\eta }^{4/3}t\right){\left(\frac{Nt}{ϵ}\right)}^{o\left(1\right)}$
Classical	T > 0 mean-field (density matrix) with refs. 23,24	Anything	$\tilde{\mathcal{O}}\left(NM\right)$	$\left(N^{4/3}{M}^2{\eta }^{1/3}t+\frac{N^{5/3}{M}^{2}t}{{\eta }^{2/3}}\right){\left(\frac{Nt}{ϵ}\right)}^{o\left(1\right)}$
Classical	T > 0 mean-field (sampled trajectories) with refs. 23,24	Anything	$\tilde{\mathcal{O}}\left(N\eta \right)$	$\left(\frac{N^{4/3}{\eta }^{7/3}t}{{ϵ}^2}+\frac{N^{5/3}{\eta }^{4/3}t}{{ϵ}^2}\right){\left(\frac{Nt}{ϵ}\right)}^{o\left(1\right)}$
Quantum	Second-quantized Trotter grid algorithm45	Sample $∣\psi \left(t\right)⟩$	$\mathcal{O}\left(N\log N\right)$	$\left(N^{4/3}{\eta }^{1/3}t+\frac{N^{5/3}t}{{\eta }^{2/3}}\right){\left(\frac{Nt}{ϵ}\right)}^{o\left(1\right)}$
Quantum	First-quantized Trotter grid algorithm here	Sample $∣\psi \left(t\right)⟩$	$\mathcal{O}\left(\eta \log N\right)$	$\left(N^{1/3}{\eta }^{7/3}t+N^{2/3}{\eta }^{4/3}t\right){\left(\frac{Nt}{ϵ}\right)}^{o\left(1\right)}$
Quantum	Interaction picture plane wave algorithm51	Sample $∣\psi \left(t\right)⟩$	$\mathcal{O}\left(\eta \log N\right)$	$\tilde{\mathcal{O}}\left(N^{1/3}{\eta }^{8/3}t\right)$
Quantum	Grid basis algorithm from Appendix K of ref. 53	Sample $∣\psi \left(t\right)⟩$	$\mathcal{O}\left(\eta \log N\right)$	$\tilde{\mathcal{O}}\left(N^{1/3}{\eta }^{8/3}t\right)$
Quantum	New shadows procedure here	k-RDM(t)	$\mathcal{O}\left(\eta \log N\right)$	$\tilde{\mathcal{O}}\left(k^k{\eta }^{k}L{\mathcal{C}}_{\mathrm{samp}}/{ϵ}^2\right)$
Quantum	Gradient measurement61	$⟨\psi \left(t\right)∣O∣\psi \left(t\right)⟩$	$\tilde{\mathcal{O}}\left(\eta +L\right)$	$\tilde{\mathcal{O}}\left(\sqrt{L}{\mathcal{C}}_{\mathrm{samp}}\lambda /ϵ\right)$
Quantum	Gradient measurement61	$⟨\psi \left(t\right)∣H∣\psi \left(t\right)⟩$	$\tilde{\mathcal{O}}\left(\eta +L\right)$	$\tilde{\mathcal{O}}\left(\frac{\sqrt{L}{\mathcal{C}}_{\mathrm{samp}}t\left(N^{1/3}{\eta }^{5/3}+N^{2/3}{\eta }^{1/3}\right)}{ϵ}\right)$

N is the number of basis functions, η is the number of particles, ϵ is target precision, M is the number of appreciably occupied orbitals in a finite-temperature (T) simulation (M ≃ N for high T), O is any observable having norm λ that can be block encoded with cost less than time-evolution, t is the duration of evolution, L is the number of time points at which we wish to resolve quantities and ${\mathcal{C}}_{\mathrm{samp}}$ is the cost of sampling $∣\psi \left(t\right)⟩$ with a quantum algorithm. We are not accounting for the additive time-independent costs of state preparation ($\tilde{\mathcal{O}}\left(\eta N\right)$ gates using the procedure of Supplementary Note 7) or of classically reconstructing the k-RDM given measurement outcomes. Thus, this table reports gate complexities for long-time t simulations. In Supplementary Note 5 we provide a table clarifying which algorithm has optimal gate complexity as a function of N/η.