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Under suitable assumptions, the algorithms in [Lin, Tong, Quantum 2020] canestimate the ground state energy and prepare the ground state of a quantumHamiltonian with near-optimal query complexities. However, this is based on ablock encoding input model of the Hamiltonian, whose implementation is known torequire a large resource overhead. We develop a tool called quantum eigenvaluetransformation of unitary matrices with real polynomials (QET-U), which uses acontrolled Hamiltonian evolution as the input model, a single ancilla qubit andno multi-qubit control operations, and is thus suitable for earlyfault-tolerant quantum devices. This leads to a simple quantum algorithm thatoutperforms all previous algorithms with a comparable circuit structure forestimating the ground state energy. For a class of quantum spin Hamiltonians,we propose a new method that exploits certain anti-commutation relations andfurther removes the need of implementing the controlled Hamiltonian evolution.Coupled with Trotter based approximation of the Hamiltonian evolution, theresulting algorithm can be very suitable for early fault-tolerant quantumdevices. We demonstrate the performance of the algorithm using IBM Qiskit forthe transverse field Ising model. If we are further allowed to use multi-qubitToffoli gates, we can then implement amplitude amplification and a new binaryamplitude estimation algorithm, which increases the circuit depth but decreasesthe total query complexity. The resulting algorithm saturates the near-optimalcomplexity for ground state preparation and energy estimating using a constantnumber of ancilla qubits (no more than 3).

Ground state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices