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
    • \(k = t\)
    • \(k = t-1, t-2, \ldots, 1\)
    • After measuring all \(j_k\) (\(k = t, t-1, \ldots, 1\))
  • 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
      • Note: How to construct a controlled rotation gate
    • 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
  • Stock Price Data Acquisition
  • Data Preprocessing
    • Conversion to Daily Returns
    • Expected Returns
    • Variance/Covariance Matrix
  • Portfolio Optimization
    • Creating the matrix W
    • Creating the right-hand side vector
    • Extend the matrix so that it can be handled by a quantum system
  • Calculating the minimum variance portfolio using the HHL algorithm
  • Appendix: Back Testing
  • Reference