Introduction to Quantum Computing: Zero to Shor's Algorithm

Explore imaginary and complex numbers, starting with imaginary unit i for square roots of negatives, and define complex numbers as a plus ib, with addition, subtraction, multiplication, and complex conjugation.

Think of a complex number as a vector on the real and imaginary axes. Compute magnitude by Pythagoras; use polar and exponential forms in unit circle rotations.

Explore matrices as 2d arrays of numbers, define their dimensions (m by n), and learn core operations—addition, subtraction, and scalar multiplication—before linking to quantum computing applications.

Multiply matrices by dot products of rows and columns to form a new matrix, transforming a column vector, with identity and inverse matrices preserving or reverting states.

Study unitary and hermitian matrices in quantum computing, covering complex conjugates, transposes, and daggers; unitary matrices preserve vector length and have dagger inverses, while hermitian matrices equal their dagger.

Explore how certain vectors stay oriented under a transformation, becoming eigenvectors with corresponding eigenvalues as the vector is stretched; illustrated by a matrix acting on [0,3] producing [0,6].

Explore how qubits differ from classical bits, representing zero and one in a two-state column vector and enabling superposition. Learn how measurement collapses a qubit state and determines outcome probabilities.

Convert arbitrary qubit state alpha beta into Dirac notation by turning the matrix into a sum and factoring alpha and beta, yielding a linear combination of |0> and |1> kets.

Place a qubit on the Bloch sphere: zero at north pole, one at south pole, plus, minus, i, and negative i states on the equator; height predicts measurement likelihood.

Explore how single qubit gates X, Y, and Z rotate qubit states by pi radians around their axes, flipping 0 and 1, and acting as their own inverses.

Explore phase on the Bloch sphere, rotating around the Z axis with complex numbers in exponential form, and distinguish global from relative phase and their impact on qubit measurement.

Examine the Hadamard gate on the Bloch sphere, its effect on zero, one, plus, minus, i, and -i states, and how phase differences make the gate powerful for quantum algorithms.

Explore how the S gate adds a relative phase of pi on two radians and the T gate adds pi on four radians, and how their daggers invert these gates.

Represent multiple qubits with the tensor product, forming |00> and the four two-qubit states, then compute measurement probabilities as magnitudes squared from superposition and extend the state by tensoring qubits.

Illustrate a quantum circuit to apply gates to specific qubits, such as an X on the second qubit and a later Hadamard, followed by measurements.

Explore multi-qubit gates, including the C-not and Toffoli gates, and see how C-not gates create controlled versions of single-qubit gates such as Y, Z, S, T, and Hadamard.

Measure a qubit in a multi-qubit state by summing probabilities where the target qubit has the desired value, then collapse and normalize with a to keep probabilities summing to one.

Explore how a two-qubit circuit produces entanglement, including Bell states, via a CNOT gate, and distinguish maximally entangled from partially entangled states through measurements.

Explore phase kickback, where a controlled U gate on a plus state induces a relative phase e^{i theta} on the control qubit when the target is an eigenstate of U.

Explore superdense coding, a quantum protocol that sends two classical bits with one qubit via entanglement between Alice and Bob. See how gates encode and Bob decodes information through measurement.

Explore classical bit operations: not, and, or, and exclusive or. Learn how adding a control bit makes gates reversible, aligning with unitary operations in quantum computing.

Demonstrates making quantum functions reversible with unitary gates and phase oracle, using x and f(x) in a minus state. Illustrates no cloning theorem and hints at Deutsch's algorithm advantage.

Explore Deutsch's algorithm, a quantum method to decide if a one-bit function is constant or balanced with single query. Learn how the Hadamard, unitary oracle, and measurement reveal the result.

Explore how the Deutsch-Jozsa algorithm generalizes to n-bit inputs, distinguishing constant from balanced functions with a single quantum query using Hadamard gates and the oracle.

Learn the Bernstein-Vazirani algorithm to discover the secret string s from a dot-product modulo two function, using one quantum query, a phase oracle, and Hadamard gates.

Explore the quantum Fourier transform (QFT), its phase-encoded states, the QFT circuit with Hadamard, controlled and swap gates, and its role in quantum phase estimation and Shor's algorithm.

Explore the quantum phase estimation algorithm to determine eigenvalues of a unitary matrix, using the two-register circuit, phase kickback, and the inverse quantum Fourier transform, with applications to Shor's algorithm.

Explore Shor's algorithm for factoring large numbers, the backbone of rsa encryption, and how modular exponentiation's period discovery enables gcd-based factorization using quantum phase estimation.