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On the D-Wave 2000Q website, it is stated that the processor environment is kept "in a high vacuum: pressure is 10 billion times lower than atmospheric pressure".

Why the pressure has to be such low? What would happen to the computer if the pressure increases?

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Pressure implies the presence of stray atoms flying around messing things up. The use of a vacuum is required to prevent this, as one of the ways of keeping the device well isolated from unwanted effects. I think that they are just intending the "10 billion times lower than atmospheric pressure" statement to demonstrate how good their vacuum is.

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The device works at cryogenic temperatures, which is so cold that all the gasses would freeze on the experiment device. Moreover, before they do so, they would conduct heat from the walls of the chamber to the experiment device and make it hard to cool the device down.

Thus, you need vacuum for being able to cool things down to a very low temperature, and once the device is cold, the presence of the cold elements even improves the vacuum, as the little rest gas you were unable to pump out before will freeze - effect known as cryopumping.

See also Why must quantum computers be kept near absolute zero?

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