Faster ground state preparation and high-precision ground energy estimation with fewer qubits

Examples of such algorithms include Refs. 3–6. Notice that algorithms based on Trotter product formulas such as Ref. 2 do not meet this requirement.

In fact, the Hamiltonian simulation algorithms^4–6 only require few ancilla qubits and leave the O(·) expressions in Tables I and II unchanged. Reference 3 requires $Olog(d) log(βΔ−1log(χ−1)/ϵ)log log(βΔ−1log(χ−1)/ϵ)$ additional qubits, where $H̃=∑j=1dβjUj$ with unitaries U_j costing O(Λ) elementary gates and $β=∑j|βj|$ (see Table 1 of Ref. 6 for an overview).

Note that unlike Ref. 26, where $λ$ denotes the overlap, here we write the overlap as λ.

Notice that this includes many-body Hamiltonians, as Hamiltonians consisting of n terms acting on at most k qubits are sparse with d = 2^kn.

In fact, it is sufficient to assume that the spectrum of $H̃$ is contained in [0, 1 − τ], where τ is defined in (D9). This ensures that H, as defined below, has entries with modulus at most 1.