Algorithms

Deutsch-Jozsa: The First Quantum Speedup

The Deutsch-Jozsa algorithm separates constant from balanced Boolean functions in a single query, where classical deterministic algorithms need up to 2ⁿ⁻¹ + 1. This tutorial derives the algorithm from first principles, explains phase kickback, and walks through the full Qiskit implementation plus the Deutsch n=1 special case.

intermediate · ~22 min · prereq: Gates & Circuits track (Tutorials 4-7)

Bernstein-Vazirani and Simon: Learning Hidden Structure in One (or O(n)) Queries

Bernstein-Vazirani learns a hidden bit string in a single query. Simon's algorithm learns a hidden shift with O(n) queries where classical algorithms need exponentially many — and was the direct inspiration for Shor's factoring algorithm. This tutorial derives both from scratch with complete Qiskit implementations.

intermediate · ~23 min · prereq: Tutorial 8: Deutsch-Jozsa

Grover's Search and Amplitude Amplification

Grover's algorithm finds a marked element in an unstructured list of N items with O(√N) queries — a provable quadratic speedup. This tutorial derives the algorithm geometrically as a rotation in a 2D subspace, gives the exact optimal iteration count, and shows how amplitude amplification generalizes the trick far beyond search.

intermediate · ~24 min · prereq: Tutorial 9: Bernstein-Vazirani and Simon

Quantum Fourier Transform and Phase Estimation

The QFT is the quantum cousin of the classical discrete Fourier transform — but it runs in O(n²) instead of O(n·2ⁿ), which is where many quantum speedups ultimately come from. This tutorial derives the QFT circuit, explains Quantum Phase Estimation (the subroutine inside Shor, HHL, and VQE), and delivers complete Qiskit implementations.

intermediate · ~25 min · prereq: Tutorial 10: Grover and Amplitude Amplification

Shor's Algorithm: Factoring, Order-Finding, and the End of RSA

Shor's factoring algorithm reduces integer factorization to the problem of finding the multiplicative order of a random element mod N — and uses quantum phase estimation to solve that in polynomial time. This tutorial derives the full algorithm, runs a small instance in Qiskit, and honestly assesses the real-world resource requirements to break RSA-2048.

advanced · ~28 min · prereq: Tutorial 11: QFT and Phase Estimation

Block Encoding: How Modern Quantum Algorithms Get Arbitrary Matrices Into Quantum Circuits

Block encoding is the most important quantum-algorithm primitive of the past decade. It is the technique that lets you embed an arbitrary matrix A into a unitary U so a quantum computer can act with A on a state — making Hamiltonian simulation, linear-system solvers, and the entire QSVT framework possible. This tutorial defines block encoding precisely, gives the key constructions (LCU, qubitization, sparse-matrix), and shows why this is the abstraction that ate quantum algorithms after 2015.

advanced · ~19 min · prereq: Tutorial 11: QFT and Phase Estimation

Quantum Singular Value Transformation: The Framework That Unified Modern Quantum Algorithms

QSVT is the algorithmic technique that takes a block encoding of A and produces a block encoding of p(A) for any low-degree polynomial p — turning matrix arithmetic on quantum computers into a uniform framework. Hamiltonian simulation, linear-system solvers, eigenvalue estimation, and ground-state preparation are all polynomial choices in QSVT. This tutorial builds the construction from qubitization and shows the four-line argument that makes QSVT the most cited quantum-algorithm technique of the late 2010s.

advanced · ~19 min · prereq: Tutorial 29: Block Encoding

Hamiltonian Simulation: From Trotter to Qubitization, the Modern Picture

Hamiltonian simulation — computing the quantum dynamics e^(-iHt) for a chemistry, materials, or condensed-matter Hamiltonian — is the original quantum-computing application Feynman pitched in 1981 and arguably the application most likely to deliver real value first. This tutorial covers the four standard algorithm families (product formulas, LCU/Taylor series, qubitization, QSVT-based simulation), their cost scalings, and when to reach for each.

advanced · ~21 min · prereq: Tutorial 29: Block Encoding, Tutorial 30: Quantum Singular Value Transformation

Amplitude Estimation: The Quadratic-Speedup Primitive Behind Quantum Monte Carlo

Amplitude estimation extends Grover's algorithm from 'find a marked element' to 'estimate the probability of a marked element' with quadratic precision speedup. It is the primitive that powers quantum Monte Carlo, financial pricing, and any algorithm that needs to estimate an expectation value to high precision. This tutorial covers the canonical Brassard-Hoyer-Mosca-Tapp construction, the modern iterative and maximum-likelihood variants, and the hard truth about whether real applications hit the quadratic speedup in practice.

advanced · ~18 min · prereq: Tutorial 10: Grover and Amplitude Amplification, Tutorial 11: QFT and Phase Estimation

Quantum Walks: Discrete and Continuous Time

Quantum walks are the quantum analog of classical random walks — and they spread quadratically faster on simple graphs, a polynomial speedup that underlies Grover, Shor's discrete-log, and several recent quantum algorithms. This tutorial covers discrete-time quantum walks (Szegedy and coined walks), continuous-time quantum walks, the speedup origin in interference, and where quantum walks turn into algorithmic primitives in 2026.

advanced · ~17 min · prereq: Tutorial 10: Grover and Amplitude Amplification, Tutorial 30: Quantum Singular Value Transformation

HHL: The Quantum Algorithm for Linear Systems and What Survived Dequantization

The Harrow-Hassidim-Lloyd algorithm (HHL, 2009) was the first quantum algorithm with exponential speedup over the best classical methods for solving sparse linear systems. It also became the most-discussed algorithm in machine learning, finance, and many applied domains — until Tang's 2018 dequantization revealed the exponential speedup depended critically on QRAM-like input access. This tutorial covers the algorithm, what HHL actually computes, the dequantization story, and the remaining regimes where HHL gives genuine speedups.

advanced · ~16 min · prereq: Tutorial 11: QFT and Phase Estimation, Tutorial 41: Tang Dequantization

LCU: The Linear Combination of Unitaries Framework

Linear combination of unitaries (LCU) is a quantum-algorithm framework that lets you implement non-unitary linear operations on a quantum state, with cost proportional to the number of unitary terms. Together with block encoding (tutorial 29) and qubitization (tutorial 30), LCU is the structural backbone of modern Hamiltonian simulation, HHL-style algorithms, and many other quantum-algorithm primitives. This tutorial covers the LCU lemma, the cost structure, and where LCU shows up in production algorithms.

advanced · ~14 min · prereq: Tutorial 29: Block Encoding, Tutorial 31: Hamiltonian Simulation

Quantum Phase Estimation Precision: How Many Qubits, How Many Queries

Quantum phase estimation (QPE) is the workhorse of quantum factoring, HHL, quantum chemistry, and more. The precision-resource tradeoff is exactly: t bits of phase precision require 2^t controlled-U queries. This tutorial covers the resource analysis, the Heisenberg limit, the iterative QPE variant that uses fewer ancilla qubits, and the modern QSVT-based replacement that beats QPE for many applications.

advanced · ~14 min · prereq: Tutorial 11: QFT and Phase Estimation, Tutorial 30: Quantum Singular Value Transformation