Grover's Algorithm
Searches an unstructured database of N items in O(sqrt(N)) steps, providing a provable quadratic speedup over any classical algorithm.
Reference Guide
Quantum Algorithm Guide
Every major quantum algorithm explained: what it does, how fast it is, and what to learn next
30 algorithms across search, factoring, simulation, optimisation, linear algebra, machine learning, graph theory, and cryptography. Each entry covers the problem type, speedup class, key reference, and a direct link to a tutorial or glossary entry.
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Search Quadratic -
Search Quadratic Amplitude Amplification
Generalises Grover's algorithm to amplify the probability of measuring a 'good' outcome for any quantum subroutine, not just oracle-based search.
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Search Quadratic Quantum Counting
Estimates the number of solutions to a search problem by combining Grover's algorithm with Quantum Phase Estimation.
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Search Heuristic Bernstein-Vazirani Algorithm
Finds a hidden n-bit string using a single quantum query to a linear oracle, versus n classical queries. Demonstrates phase kickback and Hadamard-based interference in their simplest form.
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Search Exponential Simon's Algorithm
Finds the period of a function with a hidden XOR structure using O(n) quantum queries versus exponentially many classical queries; the inspiration for Shor's algorithm.
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Factoring Exponential Shor's Algorithm
Factors an N-digit integer in polynomial time using the quantum Fourier transform, breaking RSA-2048 on a sufficiently large fault-tolerant quantum computer.
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Factoring Exponential Quantum Period Finding
Finds the period of a modular exponentiation function exponentially faster than any classical method; this is the core subroutine inside Shor's algorithm.
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Factoring Exponential Discrete Log Algorithm
Solves the discrete logarithm problem exponentially faster than classical methods, breaking Diffie-Hellman and elliptic curve cryptography.
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Simulation Heuristic Variational Quantum Eigensolver (VQE)
Estimates the ground-state energy of a molecule using a hybrid classical-quantum loop, making it one of the most promising near-term quantum chemistry algorithms.
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Simulation Exponential Quantum Phase Estimation (QPE)
Estimates the eigenvalue (phase) of a unitary operator with exponential precision; it is a core building block for Shor's algorithm, HHL, and quantum chemistry.
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Simulation Polynomial Trotter-Suzuki Simulation
Simulates the time evolution of a quantum Hamiltonian by decomposing it into a product of short-time evolution operators, each easy to implement as a quantum circuit.
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Simulation Polynomial Qubitization
An alternative to Trotterization for Hamiltonian simulation that achieves better asymptotic gate counts by encoding the Hamiltonian as a quantum walk.
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Optimisation Heuristic QAOA
The Quantum Approximate Optimisation Algorithm uses alternating layers of problem and mixing unitaries to find approximate solutions to combinatorial optimisation problems.
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Optimisation Heuristic Quantum Annealing
Uses quantum tunneling to escape local minima in an energy landscape, finding low-energy solutions to QUBO problems on D-Wave hardware.
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Optimisation Quadratic Quantum Branch and Bound
Applies a quantum speedup to the classical branch-and-bound optimisation framework using Grover-style search to prune the solution tree faster.
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Linear Algebra Exponential HHL Algorithm
Solves linear systems Ax=b in time O(log N) under certain conditions, offering an exponential speedup over classical solvers, with important caveats about output access.
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Linear Algebra Exponential Quantum PCA
Performs principal component analysis on a quantum-encoded density matrix, potentially achieving exponential speedup over classical PCA with QRAM-loaded data.
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Linear Algebra Heuristic Quantum Recommendation Systems
A quantum algorithm for sampling from low-rank approximations of matrices, now a famous cautionary case after Tang (2019) produced a matching classical dequantisation.
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Linear Algebra Polynomial QSVT
Quantum Singular Value Transformation is a unifying framework that subsumes most known quantum speedups, including HHL, Grover, and QPE, via polynomial transformations of matrix singular values.
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Machine Learning Heuristic Quantum SVM
Uses a quantum circuit to estimate a kernel function for classification, potentially accessing feature spaces that are hard to compute classically.
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Machine Learning Heuristic Quantum Neural Networks
Parameterised quantum circuits trained via gradient descent act as quantum analogues of neural networks for classification, regression, and generative tasks.
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Machine Learning Heuristic Quantum GAN
A quantum generative adversarial network pairs a quantum generator circuit with a discriminator to learn probability distributions over quantum or classical data.
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Machine Learning Heuristic Quantum Boltzmann Machines
Uses quantum sampling from a Boltzmann distribution to train a generative model, with quantum tunneling potentially escaping local minima in the energy landscape.
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Machine Learning Exponential QRAM-based ML Algorithms
Classical ML algorithms (k-means, k-nearest neighbours, linear regression) reformulated with quantum random access memory for potentially exponential speedup in data loading.
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Graph Algorithms Quadratic Quantum Walk Search
Uses quantum walks on graphs to achieve quadratic speedup for problems such as element distinctness and triangle finding in large graphs.
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Graph Algorithms Heuristic QAOA for Max-Cut
Applies the Quantum Approximate Optimisation Algorithm specifically to the Max-Cut graph partitioning problem, one of the canonical QAOA benchmarks.
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Graph Algorithms Quadratic Quantum Minimum Finding
Finds the minimum value in an unstructured list of N items in O(sqrt(N)) evaluations using repeated Grover iterations.
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Cryptography Qualitative BB84
The first quantum key distribution protocol, using the no-cloning theorem to distribute secret keys with information-theoretic security provable from quantum mechanics.
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Cryptography Qualitative E91
An entanglement-based QKD protocol that uses Bell inequality violations to certify that no eavesdropper has intercepted the shared key.
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Cryptography Qualitative Quantum Digital Signatures
A quantum analogue of classical digital signatures that provides unconditional security against forgery, based on quantum one-way functions rather than computational hardness.
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Cryptography Qualitative Quantum Secret Sharing
Splits a secret quantum state among multiple parties using entanglement so that only an authorised subset can reconstruct it.
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Error Correction Qualitative Quantum Error Correction
Encodes one logical qubit into multiple physical qubits (Steane code: 7 qubits; surface code: ~1000 qubits) so that errors can be detected and corrected without measuring the logical state.
All Algorithms at a Glance
| Algorithm | Category | Speedup | Problem type | Learn more |
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| Grover's Algorithm | Search | Quadratic | Unstructured Search | Tutorial: Grover's Algorithm |
| Amplitude Amplification | Search | Quadratic | Probability Amplification | Explore tutorials |
| Quantum Counting | Search | Quadratic | Solution Counting | Glossary entry |
| Bernstein-Vazirani Algorithm | Search | Linear | Hidden String Recovery | Tutorial: Bernstein-Vazirani in Qiskit |
| Simon's Algorithm | Search | Exponential | Hidden XOR Period Finding | Tutorial: Simon's Algorithm |
| Shor's Algorithm | Factoring | Exponential | Integer Factorisation | Tutorial: Shor's Algorithm |
| Quantum Period Finding | Factoring | Exponential | Period Finding | Tutorial: Shor's Algorithm |
| Discrete Log Algorithm | Factoring | Exponential | Discrete Logarithm | Glossary entry |
| Variational Quantum Eigensolver (VQE) | Simulation | Heuristic | Quantum Chemistry | Tutorial: VQE |
| Quantum Phase Estimation (QPE) | Simulation | Exponential | Eigenvalue Estimation | Tutorial: QPE in Qiskit |
| Trotter-Suzuki Simulation | Simulation | Polynomial | Hamiltonian Simulation | Tutorial: Hamiltonian Simulation in Qiskit |
| Qubitization | Simulation | Polynomial | Hamiltonian Simulation | Glossary entry |
| QAOA | Optimisation | Heuristic | Combinatorial Optimisation | Tutorial: QAOA with PennyLane |
| Quantum Annealing | Optimisation | Heuristic | QUBO Optimisation | Tutorial: D-Wave Ocean |
| Quantum Branch and Bound | Optimisation | Quadratic | Exact Combinatorial Optimisation | Glossary entry |
| HHL Algorithm | Linear Algebra | Exponential | Linear Systems | Glossary entry |
| Quantum PCA | Linear Algebra | Exponential | Dimensionality Reduction | Glossary entry |
| Quantum Recommendation Systems | Linear Algebra | Heuristic | Low-Rank Matrix Sampling | Glossary entry |
| QSVT | Linear Algebra | Polynomial | Quantum Linear Algebra | Glossary entry |
| Quantum SVM | Machine Learning | Heuristic | Binary Classification | Tutorial: Quantum SVM in PennyLane |
| Quantum Neural Networks | Machine Learning | Heuristic | Parameterised Circuit Learning | Explore tutorials |
| Quantum GAN | Machine Learning | Heuristic | Generative Modelling | Tutorial: Quantum GAN with PennyLane |
| Quantum Boltzmann Machines | Machine Learning | Heuristic | Generative Modelling | Glossary entry |
| QRAM-based ML Algorithms | Machine Learning | Exponential | Classical ML with Quantum Data Access | Glossary entry |
| Quantum Walk Search | Graph Algorithms | Quadratic | Graph Search | Glossary entry |
| QAOA for Max-Cut | Graph Algorithms | Heuristic | Graph Partitioning | Tutorial: QAOA Max-Cut in Qiskit |
| Quantum Minimum Finding | Graph Algorithms | Quadratic | Unstructured Minimum Search | Glossary entry |
| BB84 | Cryptography | Qualitative | Key Distribution | Tutorial: QKD in Python |
| E91 | Cryptography | Qualitative | Key Distribution | Tutorial: QKD in Python |
| Quantum Digital Signatures | Cryptography | Qualitative | Message Authentication | Glossary entry |
| Quantum Secret Sharing | Cryptography | Qualitative | Distributed Secret Storage | Glossary entry |
| Quantum Error Correction | Error Correction | Qualitative | Fault-Tolerant Computation | Glossary entry |
Continue Learning
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Step-by-step Tutorials
Implement Grover, Shor, VQE, QAOA, and more in Qiskit and PennyLane.
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Hardware Guide
Match each algorithm class to the right quantum hardware platform.
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Quantum Courses
Structured courses covering quantum algorithms from foundations to research level.
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Glossary
Plain-language definitions for every algorithm, gate, and concept on this page.