Quantum Computing for Engineers

Table of Contents

1. Algorithmic Primitives, Subroutines, and Frameworks

   1. Frontmatter

   2. Chapter 18. Phase Kickback

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter illustrates how quantum circuits can encode classical information in the phase of a quantum state using phase kickback. As the phase kickback technique is a key mechanism used to develop fundamental quantum algorithms like quantum phase estimation, quantum amplitude amplification, and quantum signal processing, this chapter sets the stage for the subsequent chapters. The phase kickback phenomenon is demonstrated with example code and results, which can be executed on a quantum computer simulator.

   3. Chapter 19. Quantum Fourier Transform

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter presents the quantum analog of the discrete Fourier transform and explains its role in many quantum algorithms, most notably Shor’s algorithm and quantum phase estimation. The quantum Fourier transform is shown to be a quantum analog of the discrete Fourier transform. Example code demonstrates the application of the quantum Fourier transform on a quantum state, which can be executed using a quantum computer simulator.

   4. Chapter 20. Quantum Phase Estimation

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter builds on the previous two chapters to show how quantum algorithms can estimate eigenvalues of unitary operators. The phase estimation procedure is derived step-by-step, and the corresponding circuit is presented. Example code demonstrates the performance of the quantum phase estimation procedure for various eigenvalues and desired precisions.

   5. Chapter 21. Trotterization

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter covers product formulas that approximate the time evolution of a quantum system—a fundamental technique in quantum simulation. The most notable product formulas, Lie–Trotter and the family of Suzuki–Trotter formulas are presented with their approximation error bounds. The procedure for exponentiating Pauli strings is provided with the corresponding circuits. Two example codes outline the error scaling of product formulas for various parameters including the number of repetitions and order of the product formula.

   6. Chapter 22. Linear Combination of Unitaries

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter presents the linear combination of unitaries technique, which extends the class of linear operations realizable on a quantum computer beyond unitary transformations at the expense of non-zero probability of failure. The connection to block-encodings is established, and example code demonstrates the application of the linear combination of unitaries technique to form operators expressed as a sum of Pauli strings (unitaries) with its corresponding circuit.

   7. Chapter 23. Qubitization and Quantum Signal Processing

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter introduces a powerful framework for quantum algorithms: qubitization and quantum signal processing. The qubitization method is derived to demonstrate the application of Chebyshev polynomials of block-encoded matrices and their sums to quantum states. The quantum signal processing theorem is then stated as a more efficient method to apply polynomials of block-encoded matrices. Finally, the quantum eigenvalue transform and quantum singular value transform are presented, which underpin modern simulation and linear algebra algorithms.

   8. Chapter 24. Amplitude Amplification and Estimation

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter introduces amplitude estimation, a subroutine which can boost the success probability of a quantum algorithm for a quadratic improvement in overall complexity. Each step of the procedure is derived and presented visually, and circuit descriptions are given where applicable. A brief review of advanced variants of amplitude amplification is provided for special cases. Finally, the amplitude estimation procedure is derived, and its circuit structure is presented.

   9. Chapter 25. Quantum Monte Carlo

      Osama M. Raisuddin, Suvranu De

      Abstract

      This chapter presents the quantum Monte Carlo algorithm for statistical sampling and expectation value estimation. A derivation of the procedure is provided, and the final result is connected to quantum amplitude estimation to demonstrate a quadratic speedup in sample complexity over any classical Monte Carlo method.

   10. Chapter 26. Matrix-Vector Multiplications and Affine Linear Operations

       Osama M. Raisuddin, Suvranu De

       Abstract

       This chapter discusses techniques for implementing linear algebraic operations in quantum circuits, including methods for sequences of matrix-vector operations. The usage of block encodings is demonstrated with its success probability. Subsequently, block-encodings are applied to compute a sequence of matrix-vector products and techniques to minimize resource requirements, and failure probabilities are presented. Finally, two techniques for affine linear operations are presented, each with its advantages and disadvantages.