Can we process infinite matrices with a quantum computer?
If then, how can we do that?
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$\begingroup$ why do you think it should be possible? also, define "represent" here. $\endgroup$– glS ♦Aug 17, 2018 at 15:51
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$\begingroup$ @glS Firstly, what I mean by represent is to store to make it be ready for processing(Classical computingly, loading the matrix on memory). $\endgroup$– KYHSGeekCodeAug 17, 2018 at 16:08
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$\begingroup$ @glS Also the reason why I thought it would be possible: When Schrodinger proved that his wave-function equation can derive Heigenberg's matrix mechanics, he substituted functions to infinite vectors, and operators to infinite matrices, right? Then I thought reversely. Does it sounds silly..? $\endgroup$– KYHSGeekCodeAug 17, 2018 at 16:10
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4$\begingroup$ you can always "represent" infinite matrices with finite memory, but you don't need anything quantum for that. Indeed, I can do it in this chat: let "2" represent an "infinite-dimensional" diagonal matrix with all entries equal to 2. You can probably see why this way of "representing" isn't particularly useful nor revealing. If on the other hand what you are asking is whether quantum computers can be used to store an infinite amount of information (in a useful way), then the answer is no, they cannot. $\endgroup$– glS ♦Aug 17, 2018 at 16:28
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$\begingroup$ KYHSGeekCode, I don't think it's a silly question. But to follow up on gIS, please also see en.wikipedia.org/wiki/Holevo%27s_theorem - especially the part that states "given n qubits, although they can 'carry' a larger amount of (classical) information (thanks to quantum superposition), the amount of classical information that can be retrieved, i.e. accessed, can be only up to n classical (non-quantum encoded) bits." $\endgroup$– Mark SpinelliAug 17, 2018 at 17:30
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If instead of manipulating the quantum information in qubits, your quantum computer were to do operations on qu$d$its with $d$ being infinity, then you'd essentially be processing infinite matrices on a quantum computer.
However most quantum computing hardware we have today, and even most of the experiments being done in academic labs, do operations on qubits (such as spin-1/2 nuclei), rather than on qu$d$its with an infinite value for $d$ (such as a quantum harmonic oscillator).
It is theoretically possible to do quantum gates on qu$d$its with infinite $d$, which would be processing infinite matrices, but it is not done in practice. It is hard enough to make quantum computers with ~100 qubits (which would still be processing only finite-dimensional matrices).
answered Aug 18, 2018 at 13:53
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$\begingroup$ Thanks. Then do you think quantum computers manipulating something like quantum harmonic oscilator to operate with qudits with infinite value d will be built if needed in the near future? $\endgroup$ Aug 18, 2018 at 15:30
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4$\begingroup$ These guys have confirm the realization of a quantum system with at least one hundred dimensions, formed by two entangled qudits with D = 10. Here is the paper. $\endgroup$– mandoAug 20, 2018 at 14:18
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$\begingroup$ @epr_guy That seems innovative. Can D be infinity practically? Or D is just a large number(finite) anyway even with technical advance? (I mean can we use real infinity? Or just theoretically?) $\endgroup$ Aug 23, 2018 at 23:43
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$\begingroup$ @KYHSGeekCode What do you mean by real infinity? $\omega$? $\endgroup$ Aug 24, 2018 at 0:39
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$\begingroup$ @KYHSGeekCode To excite the higher energy eigenvalues of a harmonic oscillator requires higher energy. So you have to keep in mind the energy required to operate such a system. $\endgroup$– biryaniAug 24, 2018 at 11:49