New Iterative Method to Enhance Precision of Quantum Phase Estimation Algorithm

The Quantum Phase Estimation (QPE) algorithm, a critical subroutine in many quantum algorithms, can be improved using an iterative method proposed by Junxu Li from Northeastern University in China. The method uses propagators over longer time spans to enhance the precision of QPE, even with limited quantum resources. This is particularly useful in the noisy intermediate-scale quantum (NISQ) era and for chemists estimating the ground energy of atoms and small molecules. The research offers a promising approach to overcoming challenges posed by limited resources and noise in quantum computing.

What is the Quantum Phase Estimation Algorithm and How Can It Be Improved?

Quantum computing, a field of study that has been gaining significant interest since its proposal by Feynman in 1982, has the potential to solve classically intractable problems. One of the most powerful quantum algorithms is the Quantum Phase Estimation (QPE) algorithm, first introduced by Kitaev in 1995. This algorithm is used to estimate the phase corresponding to an eigenvalue of a given unitary operator and is a critical subroutine in a variety of quantum algorithms, including Shor’s algorithm and the Harrow-Hassidim-Lloyd algorithm.

Despite its power, the QPE algorithm requires deep circuits with ancilla qubits, which are difficult to execute in the noisy intermediate-scale quantum (NISQ) era due to limited quantum resources. Therefore, there is a great demand for a feasible approach to improve the precision of QPE. In a recent paper, Junxu Li from the Department of Physics at Northeastern University in China proposed an iterative method to improve the precision of QPE with propagators over a variety of time spans.

How Does the Iterative Method Work?

Li’s proposed method involves using propagators over longer time spans to pinpoint the eigenenergy in a branch of comb-like ranges. By implementing various propagators over appropriate time spans, the iterative QPE can pinpoint the eigenenergy more precisely. In the standard QPE algorithm, precision is gained by adding more qubits. However, in the proposed iterative method, precision is gained in each iteration, even with only a few ancilla qubits.

This method provides a feasible and promising means toward precise estimations of eigenvalue on NISQ devices. It is particularly useful for chemists solving electronic structure problems, where QPE is implemented to estimate the ground energy of atoms and small molecules with a given Hamiltonian.

What is the Role of Propagators in Quantum Computing?

In quantum computing, a propagator describes the evolution from one time to another for a given Hamiltonian. The propagator can be written as a complex exponential due to the periodicity of the complex phase. The quantum circuit to simulate the propagator is constructed by preparing the Hamiltonian as a sum over products of Pauli spin operators, which can then be compiled into fundamental gates using Trotter-Suzuki formulas.

The non-relativistic propagator can also be obtained in path integral formulation, where the action and the integration over all paths are considered. If the initial wavefunction is known, the new wavefunction can be derived with the propagator. The propagator can also be written as a sum over the eigenstates corresponding to eigenenergies, which allows the corresponding operator to be decomposed into a diagonalized unitary operator and a unitary transformation.

How is the Quantum Phase Estimation Algorithm Implemented with Propagators?

The QPE with propagators is similar to the original QPE, where the given unitary operator is replaced by an operator that approximates the propagator. The goal is to estimate the phase corresponding to an eigenvalue of the given propagator.

In the QPE circuit with propagator, there are two registers: the first N qubits are initialized at the ground state, and the others are prepared at a certain eigenstate corresponding to an eigenenergy. The overall quantum state at the beginning is a combination of these two registers.

What are the Implications of This Research?

This research provides a promising method to improve the precision of the QPE algorithm, which is a critical subroutine in many quantum algorithms. The proposed iterative method can be implemented even with limited quantum resources, making it particularly relevant in the NISQ era.

Moreover, this method can be used to estimate the ground energy of atoms and small molecules with a given Hamiltonian, providing a powerful tool for chemists solving electronic structure problems. As quantum computing continues to develop, methods like this will be crucial in overcoming the challenges posed by limited resources and noise.