Note that the vector product does not have to be neither associative nor commutative.
Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples
- complex numbers, i.e. with complex number multiplication
- with the cross product
- quaternions, i.e. with the quaternion multiplication
Examples under python.
Ciro Santilli's wife was studying a bit of basic Python for some job interviews, when she noticed:Damn right, girl, damn right.
Wow,inis so powerful! You can dofor x in list,for x in dictandif x in dictall with that single word!
Ciro remembers hearing about Python online briefly. It seemed like a distant thing from the Java/C dominated (and outdated) university courses. Then some teaching assistant mentioned during some course when Ciro was at École Polytechnique that Python was a great integration tool. That sounded cool.
Then finally, when the École Polytechnique mathematics department didn't let Ciro Santilli do his internship of choice due to grades and Ciro was at an useless last moment backup internship, he learned more Python instead of doing his internship job, and was hooked.
Ciro Santilli often wonders to himself, how much of the natural sciences can one learn in a lifetime? Certainly, a very strong basis, with concrete experimental and physics, chemistry and biology should be attainable to all? How much Ciro manages to learning and teach in those areas is a kind of success metric of Ciro's life.
Discovered by Marie Curie, published July 1999.
Using Punch Cards by Bubbles Whiting (2016)
Source. Interview at the The Centre for Computing History.In this example we will initialize a quantum circuit with a single CNOT gate and see the output values.
By default, Qiskit initializes every qubit to 0 as shown in the qiskit/hello.py. But we can also initialize to arbitrary values as would be done when computing the output for various different inputs.
Output:which we should all be able to understand intuitively given our understanding of the CNOT gate and quantum state vectors.
┌──────────────────────┐
q_0: ┤0 ├──■──
│ Initialize(1,0,0,0) │┌─┴─┐
q_1: ┤1 ├┤ X ├
└──────────────────────┘└───┘
c: 2/═════════════════════════════
init: [1, 0, 0, 0]
probs: [1. 0. 0. 0.]
init: [0, 1, 0, 0]
probs: [0. 0. 0. 1.]
init: [0, 0, 1, 0]
probs: [0. 0. 1. 0.]
init: [0, 0, 0, 1]
probs: [0. 1. 0. 0.]
┌──────────────────────────────────┐
q_0: ┤0 ├──■──
│ Initialize(0.70711,0,0,0.70711) │┌─┴─┐
q_1: ┤1 ├┤ X ├
└──────────────────────────────────┘└───┘
c: 2/═════════════════════════════════════════
init: [0.7071067811865475, 0, 0, 0.7071067811865475]
probs: [0.5 0.5 0. 0. ]quantumcomputing.stackexchange.com/questions/13202/qiskit-initializing-n-qubits-with-binary-values-0s-and-1s describes how to initialize circuits qubits only with binary 0 or 1 to avoid dealing with the exponential number of elements of the quantum state vector.
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 as experiments that are hard to predict outcomes Updated 2025-06-17 +Created 1970-01-01
One possibly interesting and possibly obvious point of view, is that a quantum computer is an experimental device that executes a quantum probabilistic experiment for which the probabilities cannot be calculated theoretically efficiently by a nuclear weapon.
This is how quantum computing was originally theorized by the likes of Richard Feynman: they noticed that "Hey, here's a well formulated quantum mechanics problem, which I know the algorithm to solve (calculate the probability of outcomes), but it would take exponential time on the problem size".
The converse is then of course that if you were able to encode useful problems in such an experiment, then you have a computer that allows for exponential speedups.
This can be seen very directly by studying one specific quantum computer implementation. E.g. if you take the simplest to understand one, photonic quantum computer, you can make systems for which you need exponential time to calculate the probabilities that photons will exit through certain holes and not others.
The obvious aspect of this idea is by coming from quantum logic gates are needed because you can't compute the matrix explicitly as it grows exponentially: knowing the full explicit matrix is impossible in practice, and knowing the matrix is equivalent to knowing the probabilities of every outcome.
There are unlisted articles, also show them or only show them.