assignment3 - Physics 681-481 CS 483 Assignment#3(please...

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February 16, 2006 Physics 681-481; CS 483: Assignment #3 (please hand in after the lecture, Thursday, March 2nd) I. Other aspects of Deutsch’s problem Suppose one tried to solve Deutsch’s problem, not by the clever trick used in Chapter 2, Section B, but by doing the standard thing: Start with input and output registers in the state | 0 i| 0 i , apply a Hadamard to the input register, and then apply U f , thereby associating with the two Qbits the state | ψ i = 1 2 | 0 i| f (0) i + 1 2 | 1 i| f (1) i . (1) Given two Qbits in this state, a direct measurement only reveals the value of f at either 0 or 1 (randomly), but gives no information about whether f (0) = f (1). But is there anything more clever one can to two Qbits in the state (1) to learn whether or not f (0) = f (1), by applying a further unitary transformation before measuring? Deutsch noticed a way to do this successfully half the time: (a) Show that if you apply a Hadamard H to each Qbit prior to the measurement, then regardless of which of the four possible states (1) you have been given (corresponding to the four possible ways a function f can take one bit to one bit), there is a 50% chance that the measurement will enable you to conclude whether or not f (0) = f (1). But the other 50% of the time you learn nothing whatever from the measurement outcome, neither about whether f (0) = f
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  • Spring '08
  • Ginsparg
  • Linear Algebra, Hilbert space, Deutsch, Schmidt decomposition theorem

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