Quantum Algorithms Explained for Beginners
What makes an algorithm quantum
A quantum algorithm is a sequence of operations on qubits. It starts by putting qubits into superposition, links them with entanglement, and then uses interference so that wrong answers cancel out and right answers become more likely when you measure.
The famous ones
- Shor's algorithm: finds the prime factors of large numbers and solves related problems. This is the one that threatens common public key cryptography.
- Grover's algorithm: speeds up searching through unsorted possibilities, but only by a square root, not by an exponential amount.
- Quantum simulation methods: model how molecules and materials behave, which is considered the most natural use of a quantum computer.
- Variational algorithms: mix a quantum circuit with a normal computer that adjusts its settings. Researchers are exploring them for chemistry and optimization on today's small machines.
Why there are so few
Designing an algorithm that beats a normal computer is hard. Most problems get no meaningful speedup. The real gains come from problems with a hidden structure that interference can exploit. This is why the field talks about a narrow set of use cases rather than a general speedup.
Algorithms need good hardware
Running Shor's algorithm on numbers used in real encryption is expected to require large, error corrected machines, far beyond what exists today. See error correction and the timeline.
The crypto connection
Shor's and Grover's algorithms are the reason the crypto industry follows quantum progress. That theme is the backdrop for QUANTUM (QNT), a memecoin and not a quantum company.
How a speedup is measured
Quantum speedups are described by how the work grows as the problem gets bigger. Grover's gives a quadratic speedup: N possibilities take about the square root of N steps. Shor's gives a far bigger gap for factoring, because the best known classical methods slow down much faster than Shor's does as numbers grow. Simulating quantum systems is the third big case. The table compares them.
| Algorithm | Problem | Speedup | Needs |
|---|---|---|---|
| Shor's (1994) | Factoring, discrete logarithms | Very large | Large error corrected machine |
| Grover's (1996) | Unstructured search | Quadratic | Long, clean circuits |
| Quantum simulation | Molecules, materials | Expected large | Mid to large error corrected machine |
| Variational | Chemistry, optimization | Unproven | Small noisy machines |
Common mistakes
- Assuming a speedup for every task. Sorting, spreadsheets, video and web browsing gain nothing.
- Ignoring data loading. Many proposals quietly assume classical data can be put into qubits cheaply, which is often the hard part.
- Skipping the classical baseline. A claimed advantage only counts against the best classical method, and those keep improving. A well known example is covered in quantum machine learning.
- Confusing a demo with a threat. Factoring 15 or 21 on a small device says nothing about breaking real keys.
What is changing in 2026
The lines between algorithms and engineering are blurring. Google researcher Craig Gidney published an estimate in May 2025 that 2048-bit RSA could be factored with fewer than a million noisy qubits in under a week, down from about 20 million qubits in his 2019 estimate with Martin Ekera. In March 2026 a Google Quantum AI team and collaborators posted estimates for the elliptic curve used by Bitcoin, reporting circuits needing roughly 1,200 to 1,450 logical qubits and under half a million physical qubits on assumed hardware. These are paper estimates, not machines. No device close to that size exists. See the elliptic curve estimates and how resource estimates work.
How to check this yourself
When you see a headline about an algorithm, ask four questions. What exact problem was solved? How big was it? What was the best classical result on the same problem? Was the claim peer reviewed or only a press release? Our guides on Shor's and Grover's go deeper, and the pioneers page covers who invented them.
Sources and further reading
- Gidney 2025: factoring RSA-2048 with fewer than a million noisy qubits (arXiv)
- Google Quantum AI and collaborators 2026: elliptic curve cryptocurrency resource estimates (IACR ePrint 2026/625)
- NIST: first three finalized post-quantum standards, 13 August 2024
Checked 2026-10-09. Research and standards change often, so check the primary documents. Nothing here is financial advice. The QNT memecoin is independent of Quantinuum Ltd, the real company, and of every lab, company and standards body named on this page.
Frequently asked questions
What is the most famous quantum algorithm?
Shor's algorithm is the most famous. It can factor large numbers efficiently on a large enough quantum computer, which is why it matters for cryptography.
Does Grover's algorithm break encryption?
Not by itself. It gives a square root speedup for searching, which weakens symmetric keys and hashes somewhat. Longer keys offset it.
Can I run a quantum algorithm today?
Yes, small ones. Several providers offer cloud access to real and simulated quantum machines, though results are limited by noise.
Are quantum algorithms faster for everything?
No. Only certain problems have known quantum speedups. Most everyday tasks gain nothing.
Who invented Shor's and Grover's algorithms?
Peter Shor published his factoring algorithm in 1994, and Lov Grover published his search algorithm in 1996.
What is the difference between a quadratic and an exponential speedup?
A quadratic speedup turns N steps into roughly the square root of N. An exponential-style speedup shrinks the work far more as problems grow, which is what Shor's gives against the best known classical factoring methods.
Can I write a quantum algorithm myself?
Yes. Open source toolkits such as Qiskit and Cirq let you build and simulate circuits in Python. See how to try one online.
Do new quantum algorithms still appear?
Yes, but slowly. Much current work lowers the cost of known algorithms, such as the 2025 and 2026 resource estimates for factoring and elliptic curves.
Keep reading
- What Is Quantum Computing? A Plain English Guide
Quantum computers use qubits instead of bits. Learn what quantum computing is, what it is good at, and why the crypto world pays attention. - Quantum Gates and Circuits Explained
What are quantum gates and circuits? A plain English guide to Hadamard, CNOT and how gates turn qubits into a working quantum program. - What Is RSA and Why Can Quantum Computers Break It?
RSA encryption relies on the difficulty of factoring big numbers. Learn how RSA works in plain English and why a large quantum computer could break it. - Hash Functions and Quantum Computers: Are They Safe?
Are hash functions like SHA-256 safe from quantum computers? Learn how Grover's algorithm affects hashing, mining and blockchains in plain English.
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