This chapter explores what we could do with a computer whose operating system is quantum mechanics, rather than classical mechanics. One of the answers is: factorise primes really quickly. We will explain why this is interesting.

This chapter covers quantum algorithmic primitives for loading classical data into a quantum algorithm. These primitives are important in many quantum algorithms, and they are especially essential for algorithms for big-data problems in the area of machine learning. We cover quantum random access memory (QRAM), an operation that allows a quantum algorithm to query a classical database in superposition. We carefully detail caveats and nuances that appear for realizing fast large-scale QRAM and what this means for algorithms that rely upon QRAM. We also cover primitives for preparing arbitrary quantum states given a list of the amplitudes stored in a classical database, and for performing a block-encoding of a matrix, given a list of its entries stored in a classical database.

This chapter covers the multiplicative weights update method, a quantum algorithmic primitive for certain continuous optimization problems. This method is a framework for classical algorithms, but it can be made quantum by incorporating the quantum algorithmic primitive of Gibbs sampling and amplitude amplification. The framework can be applied to solve linear programs and related convex problems, or generalized to handle matrix-valued weights and used to solve semidefinite programs.

This chapter covers quantum algorithmic primitives related to linear algebra. We discuss block-encodings, a versatile and abstract access model that features in many quantum algorithms. We explain how block-encodings can be manipulated, for example by taking products or linear combinations. We discuss the techniques of quantum signal processing, qubitization, and quantum singular value transformation, which unify many quantum algorithms into a common framework.

In the Preface, we motivate the book by discussing the history of quantum computing and the development of the field of quantum algorithms over the past several decades. We argue that the present moment calls for adopting an end-to-end lens in how we study quantum algorithms, and we discuss the contents of the book and how to use it.

This chapter covers the quantum adiabatic algorithm, a quantum algorithmic primitive for preparing the ground state of a Hamiltonian. The quantum adiabatic algorithm is a prominent ingredient in quantum algorithms for end-to-end problems in combinatorial optimization and simulation of physical systems. For example, it can be used to prepare the electronic ground state of a molecule, which is used as an input to quantum phase estimation to estimate the ground state energy.

This chapter covers quantum linear system solvers, which are quantum algorithmic primitives for solving a linear system of equations. The linear system problem is encountered in many real-world situations, and quantum linear system solvers are a prominent ingredient in quantum algorithms in the areas of machine learning and continuous optimization. Quantum linear systems solvers do not themselves solve end-to-end problems because their output is a quantum state, which is one of its major caveats.

This chapter covers the quantum algorithmic primitive called quantum gradient estimation, where the goal is to output an estimate for the gradient of a multivariate function. This primitive features in other primitives, for example, quantum tomography. It also features in several quantum algorithms for end-to-end problems in continuous optimization, finance, and machine learning, among other areas. The size of the speedup it provides depends on how the algorithm can access the function, and how difficult the gradient is to estimate classically.

This chapter covers the quantum algorithmic primitive of approximate tensor network contraction. Tensor networks are a powerful classical method for representing complex classical data as a network of individual tensor objects. To evaluate the tensor network, it must be contracted, which can be computationally challenging. A quantum algorithm for approximate tensor network contraction can provide a quantum speedup for contracting tensor networks that satisfy certain conditions.

This chapter covers quantum tomography, a quantum algorithmic primitive that enables a quantum algorithm to learn a full classical description of a quantum state. Generally, the goal of a quantum tomography procedure is to obtain this description using as few copies of the state as possible. The optimal number of copies may depend on what kind of measurements are allowed and what error metric is being used, and in most cases, quantum tomography procedures have been developed with provably optimal complexity.

This chapter covers the quantum algorithmic primitive of Hamiltonian simulation, which aims to digitally simulate the evolution of a quantum state forward in time according to a Hamiltonian. There are several approaches to Hamiltonian simulation, which are best suited to different situations. We cover approaches for time-independent Hamiltonian simulation based on product formulas, the randomized compiling approach called qDRIFT, and quantum signal processing. We also discuss a method that leverages linear combination of unitaries and truncation of Taylor and Dyson series, which is well suited for time-dependent Hamiltonian simulation

This chapter covers the quantum algorithmic primitive called quantum phase estimation. Quantum phase estimation is an essential quantum algorithmic primitive that computes an estimate for the eigenvalue of a unitary operator, given as input an eigenstate of the operator. It features prominently in many end-to-end quantum algorithms, for example, computing ground state energies of physical systems in the areas of condensed matter physics and quantum chemistry. We carefully discuss nuances of quantum phase estimation that appear when it is applied to a superposition of eigenstates with different eigenvalues.

This chapter covers quantum interior point methods, which are quantum algorithmic primitives for application to convex optimization problems, particularly linear, second-order, and semidefinite programs. Interior point methods are a successful classical iterative technique that solve a linear system of equations at each iteration. Quantum interior point methods replace this step with quantum a quantum linear system solver combined with quantum tomography, potentially offering a polynomial speedup.

This chapter covers the quantum algorithmic primitive called Gibbs sampling. Gibbs sampling accomplishes the task of preparing a digital representation of the thermal state, also known as the Gibbs state, of a quantum system in thermal equilibrium. Gibbs sampling is an important ingredient in quantum algorithms to simulate physical systems. We cover multiple approaches to Gibbs sampling, including algorithms that are analogues of classical Markov chain Monte Carlo algorithms.

This chapter covers the quantum Fourier transform, which is an essential quantum algorithmic primitive that efficiently applies a discrete Fourier transform to the amplitudes of a quantum state. It features prominently in quantum phase estimation and Shor’s algorithm for factoring and computing discrete logarithms.

This chapter covers the quantum algorithmic primitives of amplitude amplification and amplitude estimation. Amplitude amplification is a generalization of Grover’s quantum algorithm for the unstructured search problem. Amplitude estimation can be understood in a similar framework, where it utilizes quantum phase estimation to estimate the value of the amplitude or probability associated with a quantum state. Both amplitude amplification and amplitude estimation provide a quadratic speedup over their classical counterparts, and feature prominently as an ingredient in many end-to-end algorithms.

This chapter covers variational quantum algorithms, which act as a primitive ingredient for larger quantum algorithms in several application areas, including quantum chemistry, combinatorial optimization, and machine learning. Variational quantum algorithms are parameterized quantum circuits where the parameters are trained to optimize a certain cost function. They are often shallow circuits, which potentially makes them suitable for near-term devices that are not error corrected.