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Given that classical physics emerges from quantum physics on a macroscopic scale, and all quantum operators are unitary, how are we able to perform non-unitary operations (such as setting a register to zero) in classical computation?

I suspect that unitarity must be preserved because information leaks to the environment as heat. However if this is the case, it begs a second question, why can quantum computers not 'simply' leak information to the environment as a hack to allow non-unitary transformations?

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Quantum computers can leak information to the environment in order to perform non-unitary transformations. The problem is that this irreversibly entangles the computer's state with the environment, i.e. it is equivalent to measuring the qubits that were leaked. This will collapse the state of the computer and prevent the interference effects that are needed if you want to do a quantum computation that's not secretly just a classical computation.

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