now blank and the output in the output register. It does the same thing as a classical irreversible machine but with a bunch of extra stuff in the middle.
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Even classically if you just do this in a naive fashion where every classically irreversible logical gate you perform is mapped to a reversible gate but with some extra bits, then you have this funny feature, where the amount of memory space you use is proportional to the numb
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of operations you perform. This is kind of interesting because it seems to eliminate the distinction between temporal complexity and spatial complexity. Normally in ordinary classical measures of complexity PSPACE contains PTIME because you can erase stuff all along the way in
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the classical machine, so you can do an irreversible computation classically it doesn’t take much memory space but takes a lot of time. Then in the reversible context, it’s not so clear what is going on. The amount of mem space has to be proportional to the number of operations
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you’re performing. That kind of says poly time is poly space in a reversible context.
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Relating to the thermodynamic stuff, if you’re performing your logical operations and there’s some error rate, a non-zero error rate for each operation, you have to perform error correction. Say it’s 1%, so you have to do error correction along the way and you gen all these
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error bits which you have to erase. So the erasure is a kind of a refrigeration process. The notice is heat and you’re pumping the heat out. In this physical picture you’re generating entropy. Entropy requires memory space, it’s information. If you’re performing computations,
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even on a reversible computer, but using error correction, then the amount of memory space if you like, if you include entropy is proportional to the number of operations you perform.
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Because each op generates a 1/100 or ⅓ or 1/1000th of a bit. The point is the physical space is proportional to the number of ops. Temporal and spatial complexity in reversible compute is illusory. The amount of space you need is at least proportional to the amount of time
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you’re using. It can’t be bigger either, the max amount of new memory you need, you need at least one new logic op to bring in new memory. Amount of SPACE is no greater than the amount of time TIME classically, (algos taking exp time but no greater than polynomial space).
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time crystals; if you only use reversible operations and close the system it will be periodic but you can repeat the ops for free for as long as the system remains intact
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