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Quantum computing can provide speedups in solving many problems as theevolution of a quantum system is described by a unitary operator in anexponentially large Hilbert space. Such unitary operators change the phase oftheir eigenstates and make quantum algorithms fundamentally different fromtheir classical counterparts. Based on this unique principle of quantumcomputing, we develop a new algorithmic toolbox "quantum phase processing" thatcan directly apply arbitrary trigonometric transformations to eigenphases of aunitary operator. The quantum phase processing circuit is constructed simply,consisting of single-qubit rotations and controlled-unitaries, typically usingonly one ancilla qubit. Besides the capability of phase transformation, quantumphase processing in particular can extract the eigen-information of quantumsystems by simply measuring the ancilla qubit, making it naturally compatiblewith indirect measurement. Quantum phase processing complements anotherpowerful framework known as quantum singular value transformation and leads tomore intuitive and efficient quantum algorithms for solving problems that areparticularly phase-related. As a notable application, we propose a new quantumphase estimation algorithm without quantum Fourier transform, which requiresthe fewest ancilla qubits and matches the best performance so far. We furtherexploit the power of our method by investigating a plethora of applications inHamiltonian simulation, entanglement spectroscopy and quantum entropiesestimation, demonstrating improvements or optimality for almost all cases.

Quantum Phase Processing and its Applications in Estimating Phase and Entropies