In quantum computing, the quantum phase estimation algorithm is a quantum algorithm to estimate the phase corresponding to an eigenvalue of a given unitary operator. Because the eigenvalues of a unitary operator always have unit modulus, they are characterized by their phase, and therefore the algorithm can be equivalently described as retrieving either the phase or the eigenvalue itself. The algorithm was initially introduced by Alexei Kitaev in 1995. Phase estimation is frequently used as a subroutine in other quantum algorithms, such as Shor's algorithm, the quantum algorithm for linear systems of equations, and the quantum counting algorithm.

Overview of the algorithm

The algorithm operates on two sets of qubits, referred to in this context as registers. The two registers contain n and m qubits, respectively. Let U be a unitary operator acting on the m-qubit register. The eigenvalues of a unitary operator have unit modulus, and are therefore characterized by their phase. Thus if is an eigenvector of U, then for some. Due to the periodicity of the complex exponential, we can always assume. The goal is producing a good approximation for \theta with a small number of gates and a high probability of success. The quantum phase estimation algorithm achieves this assuming oracular access to U, and having available as a quantum state. This means that when discussing the efficiency of the algorithm we only worry about the number of times U needs to be used, but not about the cost of implementing U itself. More precisely, the algorithm returns with high probability an approximation for \theta, within additive error \varepsilon, using qubits in the first register, and controlled-U operations. Furthermore, we can improve the success probability to 1-\Delta for any \Delta>0 by using a total of uses of controlled-U, and this is optimal.

Detailed description of the algorithm

State preparation

The initial state of the system is: where is the m-qubit state that evolves through U. We first apply the n-qubit Hadamard gate operation on the first register, which produces the state:Note that here we are switching between binary and n-ary representation for the n-qubit register: the ket |j\rangle on the right-hand side is shorthand for the n-qubit state, where is the binary decomposition of j.

Controlled-U operations

This state is then evolved through the controlled-unitary evolution U_C whose action can be written as for all. This evolution can also be written concisely as which highlights its controlled nature: it applies U^k to the second register conditionally to the first register being |k\rangle. Remembering the eigenvalue condition holding for, applying U_C to thus gives where we used. To show that U_C can also be implemented efficiently, observe that we can write, where denotes the operation of applying U^{2^\ell} to the second register conditionally to the \ell-th qubit of the first register being |1\rangle. Formally, these gates can be characterized by their action asThis equation can be interpreted as saying that the state is left unchanged when j_\ell=0, that is, when the \ell-th qubit is |0\rangle, while the gate U^k is applied to the second register when the \ell-th qubit is |1\rangle. The composition of these controlled-gates thus gives with the last step directly following from the binary decomposition. From this point onwards, the second register is left untouched, and thus it is convenient to write, with the state of the n-qubit register, which is the only one we need to consider for the rest of the algorithm.

Apply inverse quantum Fourier transform

The final part of the circuit involves applying the inverse quantum Fourier transform (QFT) on the first register of :The QFT and its inverse are characterized by their action on basis states as It follows that Decomposing the state in the computational basis as the coefficients thus equalwhere we wrote with a is the nearest integer to 2^n \theta. The difference 2^n\delta must by definition satisfy. This amounts to approximating the value of by rounding 2^n \theta to the nearest integer.

Measurement

The final step involves performing a measurement in the computational basis on the first register. This yields the outcome |y\rangle with probability It follows that if \delta=0, that is, when \theta can be written as, one always finds the outcome y=a. On the other hand, if \delta\neq0, the probability reads From this expression we can see that when \delta\neq0. To see this, we observe that from the definition of \delta we have the inequality, and thus: We conclude that the algorithm provides the best n-bit estimate (i.e., one that is within 1/2^n of the correct answer) of \theta with probability at least 4/\pi^2. By adding a number of extra qubits on the order of and truncating the extra qubits the probability can increase to.

Toy examples

Consider the simplest possible instance of the algorithm, where only n=1 qubit, on top of the qubits required to encode, is involved. Suppose the eigenvalue of reads,. The first part of the algorithm generates the one-qubit state. Applying the inverse QFT amounts in this case to applying a Hadamard gate. The final outcome probabilities are thus where, or more explicitly, Suppose \lambda=1, meaning. Then p_+=1, p_-=0, and we recover deterministically the precise value of \lambda from the measurement outcomes. The same applies if \lambda=-1. If on the other hand, then , that is, p_+=1/4 and p_-=3/4. In this case the result is not deterministic, but we still find the outcome |-\rangle as more likely, compatibly with the fact that 2/3 is closer to 1 than to 0. More generally, if, then p_+\ge 1/2 if and only if. This is consistent with the results above because in the cases, corresponding to , the phase is retrieved deterministically, and the other phases are retrieved with higher accuracy the closer they are to these two.