An adaptive phase shift method of preparing a target eigenstate of a Hamiltonian characterizing a quantum system, the quantum system comprising a plurality of physically-realized qubits, the method comprising: preparing the quantum system in an initial state, wherein the initial state has non-zero overlap with the target eigenstate; preparing a physically-realized ancilla qubit in a zero computational basis state; and iteratively adjusting the quantum system and ancilla qubit until the state of the quantum system approximates the target eigenstate to within a desired precision, wherein adjusting the quantum system and ancilla qubit comprises, at each batch of n iterations: for each iteration in the batch: preparing the ancilla qubit in the zero computational basis state; applying a first Hadamard gate to the ancilla qubit; applying a phase gate to the ancilla qubit, the phase gate applying a phase shift between the zero computational basis state and a one computational basis state of the ancilla qubit, wherein the phase shift is the same for each iteration in the batch; applying a controlled operation to the quantum system and ancilla qubit, the controlled operation controlling evolution of the quantum system under the Hamiltonian for a time t, wherein the ancilla qubit acts as a control; applying a second Hadamard gate to the ancilla qubit; and measuring the ancilla qubit to determine an ancilla qubit state; determining an estimated current average energy of the quantum system based on (i) an average of the ancilla qubit state for the batch, (ii) the time t used in the batch and (iii) the phase shift applied by the phase gate for the batch; and updating the phase shift applied by the phase gate to the ancilla qubit for the next batch of iterations such that the updated phase shift is equal to the estimated current average energy of the quantum system multiplied by the time t. preparing the quantum system in an initial state, wherein the initial state has non-zero overlap with the target eigenstate; preparing a physically-realized ancilla qubit in a zero computational basis state; and iteratively adjusting the quantum system and ancilla qubit until the state of the quantum system approximates the target eigenstate to within a desired precision, wherein adjusting the quantum system and ancilla qubit comprises, at each batch of n iterations: for each iteration in the batch: preparing the ancilla qubit in the zero computational basis state; applying a first Hadamard gate to the ancilla qubit; applying a phase gate to the ancilla qubit, the phase gate applying a phase shift between the zero computational basis state and a one computational basis state of the ancilla qubit, wherein the phase shift is the same for each iteration in the batch; applying a controlled operation to the quantum system and ancilla qubit, the controlled operation controlling evolution of the quantum system under the Hamiltonian for a time t, wherein the ancilla qubit acts as a control; applying a second Hadamard gate to the ancilla qubit; and measuring the ancilla qubit to determine an ancilla qubit state; determining an estimated current average energy of the quantum system based on (i) an average of the ancilla qubit state for the batch, (ii) the time t used in the batch and (iii) the phase shift applied by the phase gate for the batch; and updating the phase shift applied by the phase gate to the ancilla qubit for the next batch of iterations such that the updated phase shift is equal to the estimated current average energy of the quantum system multiplied by the time t. for each iteration in the batch: preparing the ancilla qubit in the zero computational basis state; applying a first Hadamard gate to the ancilla qubit; applying a phase gate to the ancilla qubit, the phase gate applying a phase shift between the zero computational basis state and a one computational basis state of the ancilla qubit, wherein the phase shift is the same for each iteration in the batch; applying a controlled operation to the quantum system and ancilla qubit, the controlled operation controlling evolution of the quantum system under the Hamiltonian for a time t, wherein the ancilla qubit acts as a control; applying a second Hadamard gate to the ancilla qubit; and measuring the ancilla qubit to determine an ancilla qubit state; preparing the ancilla qubit in the zero computational basis state; applying a first Hadamard gate to the ancilla qubit; applying a phase gate to the ancilla qubit, the phase gate applying a phase shift between the zero computational basis state and a one computational basis state of the ancilla qubit, wherein the phase shift is the same for each iteration in the batch; applying a controlled operation to the quantum system and ancilla qubit, the controlled operation controlling evolution of the quantum system under the Hamiltonian for a time t, wherein the ancilla qubit acts as a control; applying a second Hadamard gate to the ancilla qubit; and measuring the ancilla qubit to determine an ancilla qubit state; determining an estimated current average energy of the quantum system based on (i) an average of the ancilla qubit state for the batch, (ii) the time t used in the batch and (iii) the phase shift applied by the phase gate for the batch; and updating the phase shift applied by the phase gate to the ancilla qubit for the next batch of iterations such that the updated phase shift is equal to the estimated current average energy of the quantum system multiplied by the time t.