# Quantum Phase Estimation

**What is Quantum Phase Estimation **

The quantum phase estimation algorithm, as its name suggests, estimates the phase of a quantum system. QPE, as it is abbreviated, provides an approximation, however the accuracy of the approximation can be tuned by adding or removing what are called “counting qubits.” The tradeoff on large, fault-tolerant quantum computers (FTQC) is that adding counting qubits will improve the accuracy of the approximation at the expense of longer runtimes. Sacrificing accuracy by removing counting qubits, therefore, will shorten runtimes. An article published in A Bit of Qubit titled “Quantum Phase Estimation: More Qubits, More Accuracy: Determine Phase of an Eigenvector of a Unitary Operator” by Saptashwa Bhattacharyya dives deeper into the role of the counting qubits.

The QPE algorithm utilizes two quantum registers. The first is used as the aforementioned counting qubits, while the second encodes the quantum state that we want to estimate the phase of. The quantum circuit has four distinguishable sections:

- Each counting qubit is rotated to the Fourier basis.
- The counting qubits receive phase kickback from controlled-U gates on the 2nd register.
- The inverse Quantum Fourier Transform (iQFT) transforms the 1
^{st}register back to the computational basis. - The measurement results are used to classically calculate the estimated phase.

QPE is not to be confused with optimal quantum phase estimation, which refers to optical interferometry.

## What is Quantum Phase Estimation

QPE is one of the most important quantum algorithms. Unlike classical algorithms, which only need to be useful, quantum algorithms also need to be advantageous. QPE is both useful and advantageous because:

- It is applicable to real-world problems.
- It promises exponential computational speedups over classical alternatives.
- It lends its computational efficiency to other algorithms, such as Shor’s Factoring and Harrow–Hassidim–Lloyd (HHL) algorithms, by serving as a subroutine in them.

Since its introduction in 1995, QPE has evolved into a family of algorithms. Because its circuit depth exceeds the coherence times of today’s qubits, research continues into variations that might be executable on Noisy Intermediate-Scale Quantum (NISQ) computers.

## Applications of Quantum Phase Estimation

QPE doesn’t just have applications. Because of its exponential speedup, it has powerful applications. These applications include:

- Within Shor’s Algorithm, factoring numbers and threatening public cryptosystems
- In quantum chemistry, calculating the ground state energies of molecules
- Hamiltonian simulation, which is the simulation of quantum systems
- Sampling, which is the efficient generation of samples from probability distributions
- Entanglement spectroscopy, which allows the study of quantum entanglement
- Entropy estimation, which quantifies the entanglement of quantum states

And, as previously noted, QPE is one of the most powerful subroutines in quantum computing. Therefore, QPE effectively enables all of the applications of all the algorithms that rely upon it.

## Quantum Phase Estimation in Practice

QPE remains primarily theoretical. Its circuits are simply too deep to run on real hardware. Even if today’s qubits could maintain their coherence long enough to execute all the necessary operations, current error rates are still too high and the results would be sheer noise anyway. Consequently, experimentation is generally limited to quantum computer simulators, which limit the size of the problem to, at most, several dozen qubits. This is why QPE is classified as an FTQC algorithm; it will require large numbers of very-low-error qubits to be useful.

Another challenge to using QPE experimentally is that quantum error mitigation (EM) is not effective with large quantum circuits. Therefore, with neither quantum error correction (QEC) nor error mitigation as current options, only quantum error suppression techniques can be effective while either waiting for FTQC hardware or for smaller QPE variations.

For more information on the effects of noise, and why QPE can’t be implemented in the NISQ era, check out “Simulation and analysis of quantum phase estimation algorithm in the presence of incoherent quantum noise channels” by Muhammad Faizan and Muhammad Faryad from the Lahore University of Management Sciences.