Z y x θ φ |ψ>|ψ> z y x θ φ |ψ>|ψ> Quantum Computing...

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Transcript of Z y x θ φ |ψ>|ψ> z y x θ φ |ψ>|ψ> Quantum Computing...

  • Slide 1
  • z y x |>|> z y x |>|> Quantum Computing presented by Scott Erholm and Bob Wall
  • Slide 2
  • z y x |>|> Motivation for Quantum Computing Moore's Law: The number of transistors capable of being placed on a chip doubles every 18 months. If the number or transistors doubles, then individual size must decrease, or we would have gigantic processors which would requireenormous amounts of energy and generate destructive heat. Therefore, size must decrease as complexity increases. Intel's latest technique uses a 90nm process. Each transistor is 50nm wide. Some gates are a scant 1.2nm wide. 1.2nm is the same width as about 5 silicon atoms
  • Slide 3
  • z y x |>|> Number of Atoms per Bit The number of atoms needed to represent one bit has been decreasing nearly linearly. Extrapolation suggests the one-atom-per-bit level would be reached in in about 2020. Before then, we will either hit a wall and fail to keep up with Moore's pace, or we will be forced to use and exploit quantum effects in computers.
  • Slide 4
  • z y x |>|> Brief History of Quantum Computing 1982 Richard Feynman proposes quantum computer 1985 David Deutsch gives a description of a universal quantum computer Pre-1994 Quantum computing remains a marginal curiosity 1994 Peter Shor introduces quantum algorithm for factoring integers and extracting discrete logarithms in polynomial time 1995 Shor establishes theory of quantum error-correcting codes 1997 Ethan Bernstein and Umesh Vazirani construct theoretical quantum Turing machine 1998 First working 2-qubit NMR demonstrated at Berkeley. 1999 First working 3-qubit NMR demonstrated at IBM 2000 First working 5-qubit NMR demonstrated at IBM 2001 First working 7-qubit NMR demonstrated at IBM 2004 Flying Qubit is demonstrated by researchers at U. Michigan
  • Slide 5
  • z y x |>|> In normal, or classical computing, the basic unit of information is the bit A bit can be in one of two states, 0 or 1 Representing quantum information In quantum computing, the basic unit of information is the qubit After measurement, a qubit will be in one of two states, |0> or |1> |0>, |1> notation called ket, invented by Dirac States of a qubit can be represented by entities which resolve two states, such as: Photon: Vertical or Horizontal polarization Electron: Spin up or Spin down Atom: Discrete energy levels |0> and |1> are called basis states Prior to measurement however, the qubit exists in some linear combination of states: |> = |0> + |1> which is called a superposition.
  • Slide 6
  • z y x |>|> Superposition |> = |0> + |1> and coefficients are complex numbers, called the amplitudes of superposition = x 0 + iy 0 and = x 1 + iy 1 These coefficients represent the probability of the qubit being in the |0> or |1> state. Upon measurement, a qubit will collapse into either the |0> or |1> state, but before measurement, there is only the probability that the qubit is in a particular state. || 2 is the probability of |0>|| 2 is the probability of |1> || 2 + || 2 = 1 since the probabilities must sum to one. Quantum Physics is probabilistic
  • Slide 7
  • z y x |>|> Classical 2-state Probabilistic System If p is the probability of state x, then state y has a probability of (1-p). The canonical example is that of flipping a fair coin: Probability of T = , so therefore the probability of H = (1-) = . A classical probabilistic system can be thought of as a probabilistic bit, or pbit. Ignoring for the moment that it is incorrect use of the ket notation, a pbit can be represented the same way as a qubit: |outcome> = a|0> + b|1> where a = p and b = (1-p) So in the coin example, the outcome is = |T> + |H>
  • Slide 8
  • z y x |>|> Comparisons Bit is 1-D point in only one of two states, 0 and 1. Pbit is a 2-D line between the two states 0 and 1 Qubit is a 3-D sphere with 0 and 1 at the poles, and an infinite number of superpositions as points on the sphere
  • Slide 9
  • z y x |>|> Sphere representations of a qubit The reason a qubit is represented as a sphere is because the coefficients are complex numbers. The probability from 0 to 1 is the similar to classical probability, but the complex coefficient introduces an additional parameter, phase angle. The spheres below represent three different qubits with the same probability proportions of 0-ness and 1-ness, but with different phases.
  • Slide 10
  • z y x |>|> Bloch Sphere Because || 2 + || 2 = 1, the superposition equation |> = |0> + |1> may be re-written in polar coordinate fashion as: |> = e i (cos/2 |0> + e i sin/2 |1>) Since the factor e i has no observable effects, it can be ignored, so we can effectively write: |> = cos/2 |0> + e i sin/2 |1> It is important to note that the Bloch sphere is a unit spherethat is, every point on the surface of the sphere can be described by a unit vectora vector of length 1. It is easy to see then, that every possible state is a unit vector. Therefore, any state of a qubit can be described by a vector, and the vectors may be manipulated mathematically.
  • Slide 11
  • z y x |>|> Why does the phase matter? Phase becomes important when adding together the effects of a computational operation on all the terms in a superposition. For example, consider a pair of qubits A and B: | A > = |0> + |1> and | B > = |0> - |1> Individually, they both have the same probability || 2 of |0>, and since || 2 = |-| 2, they both have the same probability of |1>. Now however, consider a specific quantum computation for which the output is defined to be: [(+)/2]|0> + [(-)/2]|1>
  • Slide 12
  • z y x |>|> Why does the phase matter? (cont.) Using qubit A, the output is [(+)/2]|0> + [(-)/2]|1>, but using qubit B, the output is [(-)/2]|0> + [(+)/2]|1>. The only difference in the output is the sign of in the coefficients. As a special illustrative case, set = = 1/2. In this case, the measured answer of from qubit A will always be |0>, and the measured answer from qubit B will always be |1>. Even though A and B have the same proportions of 0-ness and 1- ness, because the phase is different, they will produce different measurements.
  • Slide 13
  • z y x |>|> Qubit gates In order to perform any real computation, we need gates to act upon the qubits! In classical computation, the familiar gates are not, and, or, xor, and others which take bits as the input(s), and combine them in a logical manner to produce an output. Since the states and operations are entirely different, traditional logic gates will not work on qubits, but there do exists some analogs. NOT Since |0> + |1> is a unit vector, it can be represented as a matrix If a NOT function is applied, the vector becomes |1> + |0>, or The NOT itself may be represented as a matrix In matrix form then, =
  • Slide 14
  • z y x |>|> Matrix manipulations Can any matrix manipulation be performed on a qubit? No, not just any manipulation The normalization condition requires that || 2 + || 2 = 1 for the state |> = |0> + |1>. This must also be true of the resultant state |'> = '|0> + '|1> after the gate has acted. The condition the gate must adhere to in order to achieve the constraint is known as the unitary condition. In order for a unit function matrix U to act, U U = I, where U is the conjugate transpose of U (obtained by transposing and then complex conjugating U), and I is the 2x2 identity matrix. This is the only constraint on quantum gates. Any unitary matrix specifies a valid quantum gate!
  • Slide 15
  • z y x |>|> Single qubit quantum gates In classical computing, the NOT gate is the only non-trivial single-bit gate. In quantum computing, there are many non-trivial single-qubit gates. Already shown the NOT (X) gate: One important gate is the phase shift (Z) gate: One most useful is the Hadamard (H) gate: The z rotation (Z ) gate rotates the vector around the z axis, : In addition to the z rotation, gates exist for x and y rotations also.
  • Slide 16
  • z y x |>|> Multiple qubit quantum gates Prototypical multiple-qubit gate is the controlled-NOT, or CNOT gate. Two qubit inputsone the control and the other the target. If the control qubit is set to 0, then the target qubit is not changed. If the control qubit is set to 1, then the target qubit is flipped. |00> |00>; |01> |01>; |10> |11>; |11> |10>; CNOT =
  • Slide 17
  • z y x |>|> Multiple qubit quantum gates cont. Quantum gates cannot be used directly to simulate classical logic gates because quantum gates are reversible (they have an inverse) while classical gates do not. However, the Toffoli gate can be used to simulate any classical using reversible quantum gates, even though the Toffoli gate is itself reversible. Toffoli gate truth table
  • Slide 18
  • z y x |>|> The state of a register of n qubits has 2 n different dimensions (rather than 2n dimensions with 2 n different values). This exponential growth of the state space is what really gives quantum computing its bang for the buck. A number of these dimensions cannot be represented as simple products of the states of individual qubits; these are referred to as entangled states. When the state of two qubits is entangled, an operation that affec