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Qubit amplitudes are defined as complex numbers. But in all tutorials I have recently read, only real numbers are used and everything works. So, if I completely forget the official 'complex' definition of a qubit, what will I loose? Are complex numbers somehow important for practical quantum computing?

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  • $\begingroup$ You lose a quantum phase which is used for encoding information in e.g. HHL, Grover, Shor and many others algorithms. Quantum computer capabilities are very restricted when you forget about the phase. $\endgroup$ Dec 26, 2021 at 15:42
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    $\begingroup$ And yet… Toffoli and Hadamard are universal. $\endgroup$
    – Mark S
    Dec 27, 2021 at 2:36
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    $\begingroup$ @MartinVesely This is not to the point, the limitation really arises when you only allow for positive numbers - QC with positive and negative numbers works just fine. (Of course, if you insist on continuous time, this implies the existence of phases.) $\endgroup$ Dec 29, 2021 at 15:25
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    $\begingroup$ Relevant publication: nature.com/articles/s41586-021-04160-4 (There's also some online talks out there.) $\endgroup$ Dec 29, 2021 at 15:28

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I'm assuming you are asking specifically about the need for complex numbers in the context of a quantum algorithm written as a decomposition in terms of quantum gates. If you are instead asking about the need for complex numbers more in general in quantum mechanics, the answer would be a bit different, depend on what precisely you mean with "need", and has probably been discussed a bunch of times already on physics.SE.

It is true that many examples will only use gates involving only real numbers, which when applied to initial states with real amplitudes, with measurements in the computational basis, allow you to forget complex numbers entirely throughout said computation.

However, even in such cases, you are still using complex numbers under the hood, even though your discretising the dynamics in such a way that you don't see it in the intermediate steps you consider. What I mean is that a "quantum gate" (i.e. a unitary operator) is just a convenient way to describe the result of some interaction, whose overall effect is to act on input states in the prescribed way. Even though, say, the Pauli $X$ gate doesn't involve complex numbers, you will have complex numbers when you consider a more fine-grained description, and look at how $X$ comes about from the Schrodinger equation with some Hamiltonian $H$.

Furthermore, even if a quantum state has only real amplitudes, to describe the full range of possible ways to interact with it, you need to take into account for the possibility of performing measurements on it which, say, project it onto states with complex (non-real) amplitudes.

This said, I think there are interesting questions to be asked about this: for example, is there any difference in computational power between algorithms involving only real amplitudes? Note that there is nothing fundamental about this: after all, you could always change your description and write such circuits in terms of complex numbers. So the question would be better stated in different terms I think. Namely, about the computational power of circuits whose gates leave invariant certain subspaces of the underlying vector space.

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Following up on a comment in @gIS's answer, there is a particular sense in which the computational power of algorithms involving only real amplitudes is very much equivalent, up to a small overhead, as the power of algorithms involving both real and complex amplitudes.

I say this because it is known that Toffoli gates and Hadamard gates are sufficient for quantum computation, and both gates do not move between the real and the complex subspace. If the state does not have any complex amplitudes prior to application of said gates, it won't afterwards.

Indeed as suggested in the other answers, if a quantum algorithm, such as the QFT/quantum phase estimation, were to want to use a gate to move between the subspaces, a trick to "keeping it real" is to dedicate an extra qubit for the complex subspace.

This trick of using Toffoli and Hadamard gates has some theoretical applications - this is used in proofs that certain problems in graph theory are BQP-complete. Unweighted adjacency matrices all have entries that are naively real (indeed, $0$ or $1$) - and allowing actions on the adjacency matrices to be drawn from Toffoli/Hadmard gates may enable one to efficiently simulate any other quantum algorithm.

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    $\begingroup$ You only increase the number of qubits by 1 to encode the real-vs-imaginary distinction. You don't double it. Because in a multi-qubit state there's still only 1 real-vs-imaginary distinction in the state vector. The fact that the extra qubit is a global thing instead of a per-qubit local thing, e.g. it doesn't play nice when factoring into tensor products, is why you'll sometimes find proofs that "real-only quantum mechanics doesn't work" (under the assumption that states have to play nice under the tensor product). $\endgroup$ Dec 27, 2021 at 19:18
  • $\begingroup$ Thanks! It's more profound than when I first intuited it. I revised based on your comment. I guess this is a paper from Aharonov that proves universality of Toffoli and Hadamard - but apparently the trick of one extra qubit goes all the way back to Bernstein and Vazirini. $\endgroup$
    – Mark S
    Dec 27, 2021 at 20:02
  • $\begingroup$ you can compute the same things using only "real gates", that's for sure. I was thinking more along the lines of whether there are advantages in terms of efficiency. For example, are there algorithms which give a quantum speedup and necessarily involve "non-real" gates? $\endgroup$
    – glS
    Dec 27, 2021 at 21:05
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    $\begingroup$ Would it not follow that, as @CraigGidney mentions, a real-only gate set such as $\{H,\text{Toffoli}\}$ only requires a single extra qubit, that there is no such algorithm that necessarily involves a non-real gate that is all that much superior to/more efficient than the real-only set? There's not a blow-up in the gate count needed for this extra qubit, as far as I understand. It's also implicit in the adjacency matrix paper I mentioned above that the overhead can be no worse than polynomial, I believe. $\endgroup$
    – Mark S
    Dec 27, 2021 at 21:27
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There is a physical meaning to the complex amplitude. The phase of the state affects what happing to the state.

For example, consider photon qubit, which might be located in a superposition of $2$ optic fibers (fiber $A$ is $|0\rangle$, fiber $B$ is $|1\rangle$) they might be in the same phase, they might be mirrors (phase $-1 / \pi / 180^\circ$) or phase $i$ from each other ($\pi/2$ or complex $i$ or $90^\circ$).

Phase has meaningful computational information, for example in Quantum Phase Estimation algorithm which is important for molecules simulation.

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