Basic Quantum Algorithms

TL;DR

This chapter–length survey establishes the circuit-centric foundations of basic quantum algorithms. By detailing Deutsch, Deutsch–Jozsa, Bernstein–Vazirani, Simon, and Shor, it exposes how quantum parallelism, entanglement, and oracle-based computation enable speedups, often with precise circuit constructions and universal gate decompositions. It also contrasts quantum and classical query complexities, illustrates economical circuit variants, and discusses the role of entanglement in each algorithm. Collectively, the work provides a rigorous, math-forward pathway from linear algebra to practical quantum circuit implementations, underscoring the significance of quantum circuits for foundational algorithms and compilation techniques.

Abstract

Quantum computing is evolving so rapidly that it forces us to revisit, rewrite, and update the foundations of the theory. Basic Quantum Algorithms revisits the earliest quantum algorithms. The journey began in 1985 with Deutsch attempting to evaluate a function at two domain points simultaneously. Then, in 1992, Deutsch and Jozsa created a quantum algorithm that determines whether a Boolean function is constant or balanced. The following year, Bernstein and Vazirani realized that the same algorithm could be used to identify a specific Boolean function within a set of linear Boolean functions. In 1994, Simon introduced a novel quantum algorithm that determined whether a function was one-to-one or two-to-one exponentially faster than any classical algorithm for the same problem. That same year, Shor developed two groundbreaking quantum algorithms for integer factoring and calculating discrete logarithms, posing a threat to the widely used cryptography methods. In 1995, Kitaev proposed an alternative version of Shor's algorithms that proved valuable in numerous other applications. The following year, Grover devised a quantum search algorithm that was quadratically faster than its classical equivalent. With an emphasis on the circuit model, this work provides a detailed description of all these remarkable algorithms.

Figures (27)

• Figure 1: Bloch sphere and the location of states ${\left\vert{0}\right\rangle}$, ${\left\vert{1}\right\rangle}$, ${\left\vert{\pm}\right\rangle}$, and ${\left\vert{\pm \textrm{i}}\right\rangle}$. An arbitrary state ${\left\vert{\psi}\right\rangle}$ is shown with spherical angles $\theta$ and $\varphi$.
• Figure 2: Histogram of the probability distribution generated by measuring a qubit whose state is ${\left\vert{+}\right\rangle}$.
• Figure 3: Example of a circuit with a Hadamard gate and a meter (Reprint Courtesy of IBM Corporation ©)
• Figure 4: Output of the circuit with a Hadamard gate and a meter (Reprint Courtesy of IBM Corporation ©)
• Figure 5: Circuit with gates $H$ and CNOT (Reprint Courtesy of IBM Corporation ©)

Theorems & Definitions (11)

• Theorem 2.1
• Proposition 3.1
• proof
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Proposition 6.1
• proof
• Proposition 7.1