We can look at this quantum circuit again. This is the quantum circuit that realizes the Deutsch algorithm. The key part is the application of UF; is a function evaluation. We can see this circuit as a computing. Let me recall some information task we have known from the beginning, which is a discrimination. Suppose there's a preparation that give you a state, psi or phi with some probability, that we prepare the two detectors and a detection event. The first one conclude state phi, detection event in the second detector conclude state psi. This defines the problem of state discrimination and how difficult is it. We have limitations in quantum mechanics and the success probability of discriminating between two content state. Success probability is actually limited and cannot be larger than one-half. For now, just so we can pull this a [inaudible] probability one-half and one-half then we have a one-quarter and psi and phi. That's the maximum possible probability in the discrimination between two quantum states. We can translate this one to the scenario of discriminating between two unitary transformations. Quantum mechanics, we have a state dynamics and measurement, and there are two possibilities in dynamics, suppose there is a box. Now we have state which is centered to this box and here are two possible unitary transformations, U or V. Then we prepare two detectors. Detection event in the first detector conclude, U has been prepared in the box and detection event in the second one here, it concludes V has been applied. This is the scenario of discriminating between two unitary transformations. After this transformation, after this box, there are two possibilities. The first possibility is U psi or the second one is V psi. So this is precisely state combinations. So we want to discriminate between two states and we hope that these two states, there are possibly [inaudible]. So we are looking for the first best and optimal input state such that these two state optimally discriminated. Overall, we apply this formula and then it is concerned with the quantity, U [inaudible] minus [inaudible] one. We want to maximize this quantity over all possible input state. This define a scenario of discriminating between two data transmission or the process here. Here the goal was to find if the function is constant or balanced, or this is equivalent to saying that we want to classify or we want to know if f is two possibility is constant or balanced, then the goal of the circuit here is to say "discriminate between the two cases." So take a closer look at how this looks like. The first case, suppose f is constant then we can write down U directly, if you see U works for these two bits as x and y plus fx. Then this Uf can be written as x and y in maps x to x but it maps y plus fx and y. Then in maps y to y plus x then suppose f is constant and how Uf looks like, then this would be zero. Tensor. Suppose f is constant, then we have the same values in here. Suppose this guy is a zero, then you have y y, y y, and then is identity. If f is 1, then is just a flip of y, which is x operation and this x operation is there. Then there are two cases; U_f is either identity times identity or identity times x operation. Which means U_f corresponds to a local unit of transformation, which cannot generate internal state. Then second, suppose f is constant or is a balanced. For instance, we assume that as an example, f(0) is zero, and f(1) is one. Then U_f can be written, plug in these numbers here then you can see that this one times, since f(0) is zero, we have identity and one, then we have x operation. This is precisely CNOT. In the other case, f(0) is one and f(1) is zero. Then this case U_f is identity times x, and U_CNOT identity times x. In this case, f is balanced, what happened? This unitaries must be entirely unitaries. Basically, these are the U_f with a balance of function f is equivalent to U_CNOT case up to local unit of transformations. Then we had this input here, and we had output here. Now we can interpret the circuit as follows. If we choose input 0, 0, then we can have [inaudible] state here, so zero and one, which is this guy, this must be either 0 or 1. Then by discriminating these two states here, by measurement, we can actually discriminate or find if this function is entangled-engaged or non-entangling local unitary transformations. We can interpret this problem as the problem of discriminating between classes of unitary transformations like this one. It's interesting to remark that we have information task here which is quantum channel discrimination. Here we have a computing and some mathematics problem, I put here math problems. This math problem corresponds to the Deutsch problem. The computing is this a circuit. Information processing is this discrimination task. This information task can be interpreted as in terms of computing; it happened that this problem can solve this Deutsch problem. This one is devised to solve this problem. This actually corresponds to information task of discriminating between unitary transformations. This math problem can be approached in alternative ways and the first one is by computing. We write down the circuit for the purpose of solving this mathematical problem. Or this guy can be formulated as a channel discrimination problem and can be approached by the information processing. There's very close relation between computing and information task. Although we are not intending to solve any problem and it happened that this information task correspond to certain strategy to solve some mathematics problem. This Deutsch algorithm is a very good example where we can see the relation of information processing and computing and some mathematics problems.
