Can we process infinite matrices with a quantum computer?
If then, how can we do that?
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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).