chapter 7 Quantum phase estimation algorithm and its application
Chapter 7 describes the quantum phase estimation algorithm, one of the most important quantum algorithms, and the Harrow-Hassidim-Lloyd (HHL) algorithm, which uses it as a subroutine to solve simultaneous linear equations at high speed (quantum phase estimation algorithm itself has already been briefly introduced in Section 2-4).
In addition, we introduce quantum random access memory (qRAM), which is required when applying the HHL algorithm to a real problem, and an example of applying the HHL algorithm to the financial engineering problem of portfolio optimization. The contents of this chapter are expected to be applied to a wide range of fields, such as speeding up machine learning using quantum computers and high-precision energy calculations for large-scale molecules. We would like you to broaden your interest while referring to references. (Algorithms introduced in this chapter are long-term algorithms (algorithms that will only work on quantum computers with quantum error correction).)
- 7-1. Quantum Phase Estimation (QPE) Algorithm Detailed:Hydrogen Molecule as Example
- Review of Phase Estimation
- Iterative Quantum Phase Estimation
- Example: Calculation of the ground state energy of the hydrogen molecular Hamiltonian using the quantum phase estimation algorithm
- 0. (Reduce Hamiltonian size using symmetry etc.)
- 1. Accurate approximation of Hamiltonian time evolution operator \(U=e^{-iH\tau}\)
- 2. Decomposition of the control time evolution operator into a set of gates that can be easily executed on a quantum computer and implementation.
- 3. Prepare an initial state with sufficient overlap with the ground state
- 4. Measure energy eigenvalues with IQPE
- Reference
- 7-2. Harrow-Hassidim-Lloyd (HHL) Algorithm
- Problem Setup
- Algorithm Flow
- 1. prepare the input state \(|\mathbf{b}\rangle\).
- 2. Store eigenvalues of \(A\) in auxiliary clock bits using the phase estimation algorithm with unitary operation \(e^{i A }\).
- 3. Multiply the inverse of the eigenvalues by a control rotation using auxiliary clock bits
- 4. Inverse operation of quantum phase estimation to restore the auxiliary clock bits
- 5. Measure auxiliary bit \(S\).
- About the computational complexity
- Reference
- 7-3. Portofolio Optimization by HHL Algorithm