This chapter provides an overview of how to perform quantum error correction using the surface code, which is the most well-studied quantum error correcting code for practical quantum computation. We provide formulas for the code distance—which determines the resource overhead when using the surface code—as a function of the desired logical error rate and underlying physical error rate. We discuss several decoders for the surface code and the possibility of experiencing the backlog problem if the decoder is too slow.

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 potential use of quantum algorithms for cryptanalysis, that is, the breaking and weakening of cryptosystems. We discuss Shor’s algorithm for factoring and discrete logarithm, which render widely used public-key cryptosystems vulnerable to attack, given access to a sufficiently large-scale quantum computer. We present resource estimates from the literature for running Shor’s algorithm, and we discuss the outlook for postquantum cryptography, which aims to replace existing cryptosystems while being resistant to quantum attack. We also cover quantum approaches for weakening the security of cryptosystems based on Grover’s search algorithm.

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 provides an overview of how to perform a universal set of logical gates on qubits encoded with the surface code, via a procedure called lattice surgery. This is the most well-studied approach for practical fault-tolerant quantum computation. We perform a back-of-the-envelope end-to-end resource estimation for the number of physical qubits and total runtime required to run a quantum algorithm in this paradigm. This provides a method for converting logical resource estimates for quantum algorithms into physical resource estimates.

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 applications of quantum computing in the area of continuous optimization, including both convex and nonconvex optimization. We discuss quantum algorithms for computing Nash equilibria for zero-sum games and for solving linear, second-order, and semidefinite programs. These algorithms are based on quantum implementations of the multiplicative weights update method or interior point methods. We also discuss general quantum algorithms for convex optimization which can provide a speedup in cases where the objective function is much easier to evaluate than the gradient of the objective function. Finally, we cover quantum algorithms for escaping saddle points and finding local minima in nonconvex optimization problems.

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 applications of quantum computing in the area of nuclear and particle physics. We cover algorithms for simulating quantum field theories, where end-to-end problems include computing fundamental physical quantities and scattering cross sections. We also discuss simulations of nuclear physics, which encompasses individual nuclei as well as dense nucleonic matter such as neutron stars.

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 applications of quantum computing relevant to the financial services industry. We discuss quantum algorithms for the portfolio optimization problem, where one aims to choose a portfolio that maximizes expected return while minimizing risk. This problem can be formulated in several ways, and quantum solutions leverage methods for combinatorial or continuous optimization. We also discuss quantum algorithms for estimating the fair price of options and other derivatives, which are based on a quantum acceleration of Monte Carlo methods.

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 applications of quantum computing in the area of quantum chemistry, where the goal is to predict the physical properties and behaviors of atoms, molecules, and materials. We discuss algorithms for simulating electrons in molecules and materials, including both static properties such as ground state energies and dynamic properties. We also discuss algorithms for simulating static and dynamic aspects of vibrations in molecules and materials.

This chapter covers applications of quantum computing in the area of condensed matter physics. We discuss algorithms for simulating the Fermi-Hubbard model, which is used to study high-temperature superconductivity and other physical phenomena. We also discuss algorithms for simulating spin models such as the Ising model and Heisenberg model. Finally, we cover algorithms for simulating the Sachdev-Ye-Kitaev (SYK) model of strongly interacting fermions, which is used to model quantum chaos and has connections to black holes.

This chapter covers applications of quantum computing in the area of combinatorial optimization. This area is related to operations research, and it encompasses many tasks that appear in science and industry, such as scheduling, routing, and supply chain management. We cover specific problems where a quadratic quantum speedup may be available via Grover’s quantum algorithm for unstructured search. We also cover several more recent proposals for achieving superquadratic speedups, including the quantum adiabatic algorithm, the quantum approximate optimization algorithm (QAOA), and the short-path algorithm.