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Unformatted text preview: CSE 599d - Quantum Computing When Quantum Computers Fall Apart Dave Bacon Department of Computer Science & Engineering, University of Washington In this lecture we are going to begin discussing what happens to quantum computers when we move away from the closed system view of quantum theory and encounter the reality that real quantum system are open quantum systems. Open quantum systems interact with their environment and decohere. When we try to control real quantum systems, we aren’t able to perfectly control them. When we try to perform projective measurements on a real quantum systems, we don’t perform exactly perfect projective measurements. When we try to prepare real quantum systems into a particular state, we don’t succeed in preparing this state with perfect certainty. All of these issues must be addressed if we are really going to (1) build a quantum computer, (2) accept that quantum computation is a valid model deserving of the moniker digital computer. I. QUANTUM NOISE Suppose that our happy qubit | ψ i is sitting there minding its business, when the cold hard reality that is not alone in the universe shows its head. In particular another qubit, in the state √ 1- p | i + √ p | 1 i comes along and interacts with our qubit by a controlled-NOT, controlled from the extra qubit. What will the effect of this evolution be on our qubit? Well we can calculate the Kraus operators: A = h | B C X p 1- p | i B + √ p | 1 i B = p 1- pI A 1 = h 1 | B C X p 1- p | i B + √ p | 1 i B = √ pX (1) The evolution of our initial state ρ = | ψ ih ψ | is then given by ρ → A ρA + A 1 ρA 1 = (1- p ) ρ + pXρX. (2) We can interpret this evolution as describing the procedure of doing nothing to our qubit with probability 1- p and with probability p applying the unitary operator X to our qubit. This is an example of quantum noise on our qubit. But notice that this quantum noise is very similar to what we might call classical noise in the computational basis. In the computational basis, this represents nothing more that flipping the bit. Now suppose that we replace the controlled-NOT in this operation with a controlled- U operation. Then we can similarly calculate that A = h | B C X p 1- p | i B + √ p | 1 i B = p 1- pI A 1 = h 1 | B C X p 1- p | i B + √ p | 1 i B = √ pU (3) The evolution of this superoperator can be interpreted as doing nothing with probability 1- p and applying U with probability p . Now if U is the Z gate, then, this evolution does something kind of strange to our qubit, when expressed in the computational basis. In particular the Z gate does not change the amplitude of a state in the computational basis: α | i + β | 1 i → α | i - β | 1 i . Thus if we are talking about measurements in the computational basis the effect of this “noise” does not change the probabilities of the two outcomes. But if we examine the density matrix, then something has happened to our system: | α | 2 αβ * α * β | β | 2 → (1...
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- Fall '08
- Computer Science, Hilbert space, Quantum entanglement, Qubit, Kraus operators