A milestone in the field of quantum computing will be solving problems in quantum chemistry and materials faster than state of the art classical methods. The current understanding is that achieving quantum advantage in this area will require some degree of error correction. While hardware is improving towards this milestone, optimizing quantum algorithms also brings it closer to the present. Existing methods for ground state energy estimation require circuit depths that scale as O(1/ϵ⋅polylog(1/ϵ)) to reach accuracy ϵ. In this work, we develop and analyze ground state energy estimation algorithms that use just one auxilliary qubit and for which the circuit depths scale as O(1/Δ⋅polylog(Δ/ϵ)), where Δ≥ϵ is a lower bound on the energy gap of the Hamiltonian. With this O˜(Δ/ϵ) reduction in circuit depth, relative to recent resource estimates of ground state energy estimation for the industrially-relevant molecules of ethylene-carbonate and PF⁻₆, the estimated gate count and circuit depth is reduced by a factor of 43 and 78, respectively. Furthermore, the algorithm can take advantage of larger available circuit depths to reduce the total runtime. By setting α∈[0,1] and using depth proportional to ϵ^(−α)Δ^(−1+α)_true, the resulting total runtime is O˜(ϵ^(−2+α)Δ^(1−α)_true), where Δ_true is the true energy gap of the Hamiltonian. These features make our algorithm a promising candidate for realizing quantum advantage in the era of early fault-tolerant quantum computing.