Sublinear-depth Trotter steps for electrons in atomic orbitals

Molecules are usually discretized in atomic orbitals: a fixed number of functions on each nucleus, not a uniform grid. Günther and Harrow (arXiv:2610.00950, version 1, submitted 1 Oct 2026) ask what that choice costs on a quantum computer when the number of orbitals N grows in the thermodynamic limit. The note below follows Definition 1, Theorems 6, 7, and 8, and Table I. The paper is a preprint. The Hamiltonian simulation page is the account of the product-formula and qubitization bounds this result sits on.

The Hamiltonian they allow

Definition 1 is a set of atomic orbitals with three controls that stay fixed while N grows. The nuclei are separated by some minimum distance. Each orbital, and its gradient away from its nucleus, decays exponentially. The overlap matrix of the set has a smallest eigenvalue bounded below by a positive constant. They keep the number of orbitals per atom constant. That is the thermodynamic limit: more atoms, same local resolution. They say the continuum limit, more orbitals on a fixed molecule, does not give a uniform lower bound on that eigenvalue, and they do not claim the theorem for it. Slater-type and Gaussian orbitals are the examples they name as fitting the definition when those parameters hold.

The electronic Hamiltonian is the usual one-body plus two-body operator in those orbitals. A naive Trotter step would touch on the order of N to the fourth Coulomb terms. Their circuit does not. They localize the orbitals, split the Coulomb interaction with a hierarchical decomposition in the style of the fast multipole method, and evaluate the resulting arithmetic with shallow quantum Fourier circuits. Theorem 6 says that one Trotter stage, for a Hamiltonian in the class they call MESH, can be implemented for time t to precision ε in depth polylogarithmic in N, t, and ε.

The error and the cost

The error of the product formula is not left at the norm of H. They invoke Childs, Su, Tran, Wiebe, and Zhu for a nested-commutator bound, the same commutator scaling stated as Theorem 11 on the simulation page. Theorem 7 is the bound for their particular decomposition. For one step of a kth-order formula, the error is at most on the order of 8 to the k, times (k plus 1) factorial, times t to the k plus 1, times an extra factor. In one dimension that factor is N times log N to the power k plus 1. In dimension d greater than 1 it is N raised to (k plus 1) times (d minus 1) over d, plus 1. Repeating short steps to reach time t and error ε, and choosing the order as a function of N, t, and ε, is Theorem 8.

Theorem 8: for a Hamiltonian in their class with N orbitals, the evolution for time t can be implemented to precision ε with a number of gates N to the power (2d minus 1) over d, times t, times (N t over ε) to a power o(1), in depth N to the power (d minus 1) over d, times t, times the same sub-power factor, and with no ancilla qubits. In three dimensions the gate exponent is 5/3 and the depth exponent is 2/3. Table I records that three-dimensional row as N to the 5/3 plus o(1) gates and N to the 2/3 plus o(1) depth, with zero ancillas. The caption says that big-O and tilde-O mark theoretical upper bounds and that a similarity mark records empirical results. Earlier atomic-orbital rows in the same table run from about N to the 2 up to N to the 10 gates. The abstract's phrase "N to the 5/3 plus o(1)" is the three-dimensional gate count with t and ε absorbed the way Table I absorbs them, not a separate theorem.

What they compare it with

Qubitization, as Low and Chuang analyze it, has query cost on the order of λ t plus a logarithm of one over ε, where λ is the factor in the block encoding and is at least the norm of H. Günther and Harrow note that this norm is on the order of N in their setting, so the query depth is on the order of N t. That is why they say current qubitization methods do not give sublinear depth here. The comparison is about depth under that normalization. It is not a claim that the gate count of a compiled block encoding has been calculated for the same Hamiltonian.

They also write that their gate scaling matches the best scaling known to them for any basis, and they attribute a gate complexity of N to the 5/3 plus o(1), at constant atom density and orbital resolution, to Berry, Wan, Baczewski, Eklund, Tikku, and Babbush. That preprint's own abstract states a different-looking expression: t times the sum of η to the 4/3 times N to the 1/3 and η to the 1/3 times N to the 2/3, times (η N t over ε) to a power o(1), where η is the particle number and N is the number of grid points. This digest does not re-derive the specialization in which that expression becomes a single power of one size parameter. The sentence to cite for the atomic-orbital algorithm is Theorem 8, not the comparison sentence.

Where the claim stops

The discussion says they analyzed scaling and did not investigate the system size at which the improvement outweighs the constants, the logarithmic factors, and the cost of the circuit constructions. That crossover is an open question they state, not a number missing from this note. They also state as open whether a further improvement, past the dimension and the decay of the Coulomb potential, is possible. Version 1 is the text these statements come from. Cite the preprint for them. This digest is a secondary source.