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# lecturenotes6 - CSE 599d Quantum Computing Reversible...

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CSE 599d - Quantum Computing Reversible Classical Circuits and the Deutsch-Jozsa Algorithm Dave Bacon Department of Computer Science & Engineering, University of Washington Now that we’ve defined what the quantum circuit model, we can begin to construct interesting quantum circuits and think about interesting quantum algorithms. We will approach this task in basically an historical manner. A few lectures ago we saw Deutsch’s algorithm. Recall that in this algorithm we were able to show that there was a set of black box functions for which we could only solve Deutsch’s problem by querying the function twice classically, but when we used the boxes in their quantum mode, we only needed to query the function once. Now this was certainly exciting. But one query versus two query is a lot like getting half price for a matinee. Certainly it is a great way to save money, but its no way to build a fortune (Yeah, yeah, you can go on all you want about Ben Franklin, but a penny saved is most of the time a waste of money these days.) In the coming lectures we will begin to explore generalizations of the Deutsch algorithm to more qubits. For it is in the scaling of resources with the size of the problem that our fortunes will be made (or lost.) I. REVERSIBLE AND IRREVERSIBLE CLASSICAL GATES Before we turn to the Deutsch-Jozsa algorithm, it use useful to discuss, before we begin, reversible versus irreversible classical circuits. A classical gate defines a function f from some input bits { 0 , 1 } n to some output bits { 0 , 1 } m . If this function is a bijection, then we say that this circuit is reversible. Recall that a bijection means that for every input to the function there is a single unique output and for every output to the function there is a single unique input. Thus given the output of a reversible function, we can uniquely construct its input. A gate which is not reversible is called irreversible. Note that our definition of reversible requires n = m . Why consider reversible gates? Well one of the original motivations comes from Landauer’s principle. So it is useful to make a slight deviation and discuss Landauer’s principle. Suppose that we examine a single bit in our computer. This bit is a degree of freedom with two possible states. Now the rest of the computer has lots of other degrees of freedom and can be in one of many states. Now one of the fundamental theorems of physics says that phase space cannot be compressed. Lets apply this principle to the process of erasing our bit. Suppose there is a physical process which irreversibly erases a bit: 1 goes to 0 and 0 goes to 0. Now sense phase space cannot be compressed, if we perform this irreversible operation, then the phase space must “bulge out” along the other degrees of freedom. In particular this implies that the entropy of the those other degrees of freedom must increase. If you work through how much it must increase, you will find that it must increase by k B T ln 2 for a system in thermodynamic equilibrium. This is Landauer’s erasure principle: erasing a single bit of

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