Non-variational and Phase Estimation algorithms

Some InQuanto algorithms solve, for example, subspace or projective problems non-variationally. Two examples are the Quantum Subspace Expansion, and Quantum Self Consistent Equation of Motion methods. However, it should be noted that these methods often require accurate ground states, which may be obtained through variational approaches. Here we group these algorithms with the quantum phase estimation algorithms (although they are significantly different), which can also be considered a projection based approach.

Quantum phase estimation (QPE) [11, 12, 13, 14] is a quantum algorithm used to estimate the phase \(\phi \in [0, 1)\) of a given unitary operator \(U\) and eigenstate \(|\phi\rangle\) satisfying

(15)\[U|\phi\rangle = e^{i2\pi\phi} |\phi\rangle .\]

There are two main approaches to QPE algorithms, i.e., QPE based on the quantum Fourier transform (QFT) and QPE based on classical post-processing. The former is referred to as canonical QPE. It requires as many ancilla qubits as is necessary for representing the phase to the desired precision. The latter is referred to in several ways, including iterative QPE, stochastic QPE, and statistical QPE, depending on the method of the classical post-processing. Generally, the phase value is inferred by analyzing the samples obtained with the basic measurement operation with one ancilla qubit. In InQuanto, we refer to such a classical-post-processing-based QPE as iterative QPE for convenience, although some algorithms do not necessarily involve iterative feedback loops. InQuanto supports both canonical and iterative QPE algorithms, as explained in the following sections.