This is the true key question: what are the most important algorithms that would be accelerated by quantum computing?
Some candidates:
- Shor's algorithm: this one will actually make humanity worse off, as we will be forced into post-quantum cryptography that will likely be less efficient than existing classical cryptography to implement
- quantum algorithm for linear systems of equations, and related application of systems of linear equations
- Grover's algorithm: speedup not exponential. Still useful for anything?
- Quantum Fourier transform: TODO is the speedup exponential or not?
- Deutsch: solves an useless problem
- NISQ algorithms
Maybe there is some room for doubt because some applications might be way better in some implementations, but we should at least have a good general idea.
However, clear information on this really hard to come by, not sure why.
Whenever asked e.g. at: physics.stackexchange.com/questions/3390/can-anybody-provide-a-simple-example-of-a-quantum-computer-algorithm/3407 on Physics Stack Exchange people say the infinite mantra:
Lists:
- Quantum Algorithm Zoo: the leading list as of 2020
- quantum computing computational chemistry algorithms is the area that Ciro and many people are te most excited about is
- cstheory.stackexchange.com/questions/3888/np-intermediate-problems-with-efficient-quantum-solutions
- mathoverflow.net/questions/33597/are-there-any-known-quantum-algorithms-that-clearly-fall-outside-a-few-narrow-cla
Quantum computers are not expected to solve NP-complete problems by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16Only NP-intermediate, which includes notably integer factorization:
- quantumcomputing.stackexchange.com/questions/16506/can-quantum-computer-solve-np-complete-problems
- www.cs.virginia.edu/~robins/The_Limits_of_Quantum_Computers.pdf by Scott Aaronson
- cs.stackexchange.com/questions/130470/can-quantum-computing-help-solve-np-complete-problems
- www.quora.com/How-can-quantum-computing-help-to-solve-NP-hard-problems
The most comprehensive list is the amazing curated and commented list of quantum algorithms as of 2020.
There is no fundamental difference between them, a quantum algorithm is a quantum circuit, which can be seen as a super complicated quantum gate.
Sample implementations:
Presumably the point of it is to allow simulation in classical computers?
Era of quantum computing before we reach physical errors small enough to do perfect quantum error correction as demonstrated by the quantum threshold theorem.
A quantum algorithm that is thought to be more likely to be useful in the NISQ era of quantum computing.
TODO clear example of the computational problem that it solves.
This is a term "invented" by Ciro Santilli to refer to quantum compilers that are able to convert non-specifically-quantum (functional, since there is no state in quantum software) programs into quantum circuit.
The term is made by adding "quantum" to the more "classical" concept of "high-level synthesis", which refers to software that converts an imperative program into register transfer level hardware, typicially for FPGA applications.
One of their learning sites: www.qutube.nl/
The educational/outreach branch of QuTech.
Really weird and obscure company, good coverage: thequantuminsider.com/2020/02/06/quantum-computing-incorporated-the-first-publicly-traded-quantum-computing-stock/
Publicly traded in 2007, but only pivoted to quantum computing much later.
Funding:
Pinned article: Introduction to the OurBigBook Project
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