INTRODUCTION TO QUANTUM COMPUTING Writen by: Eleanor Rieffel and Wolfgang Polak Presented by: Anthony LuadersOUTLINE: ① Introduction ② Notation ③ Experiment ④ Quantum Bit ⑤ Quantum Key Distribution ⑥ Multiple Qubits ⑦ Entangled Particles ⑧ Recent News ⑨ Suggested Reading ⑩ SourcesINTRODUCTION 1980’s, Richard Feynman observed that certain quantum mechanical effects cannot be simulated on a classical computer. 1994, Peter Shor described a polynomial time quantum algorithm for factoring integers.CLASSIC COMPUTING The time it takes to do certain computations can be decreased using parallel processors Exponential decrease in amount of time Exponential increase in the number of processors Exponential increase in the amount of physical spaceQUANTUM COMPUTING The time it takes to do certain computations can be decreased using parallel processors Exponential decrease in amount of time Linear increase in the number of processors Linear increase in the amount of physical space This is known as quantum parallelism There is a catch… While massive parallel computation can be preformed, access to the results is restricted Fix… Shor’s factorization algorithm Grover’s search algorithmOUTLINE: ① Introduction ② Notation ③ Experiment ④ Quantum Bit ⑤ Quantum Key Distribution ⑥ Multiple Qubits ⑦ Entangled Particles ⑧ Recent News ⑨ Suggested Reading ⑩ SourcesBAR-KET NOTATION The matching bra, ⟨x|, denotes the conjugate transpose of |x⟩ Example: The orthonormal basis{|0⟩, |1⟩} can be expressed as {(1, 0)T , (0, 1)T } Any complex linear combination of |0⟩ and |1⟩, (a|0⟩ + b|1⟩), can be written (a, b)T Note the order of the basis vectors is arbitrary, but it must be consistentNOTATION: INNER AND OUTER PRODUCT The inner product ⟨x|y⟩ - found by combining ⟨x| and | y⟩ as in ⟨x|| y⟩ Example |0⟩ is a unit vector. ⟨0|0⟩ = 1 Since |0⟩ and |1⟩ are orthogonal we have ⟨0|1⟩ = 0 The outer product | x⟩⟨y| - found by combining ⟨y| and | x⟩ Example |0⟩⟨1| is the transformation that maps |1⟩ to |0⟩ |0⟩⟨1| is the transformation that maps |0⟩ to (0, 0)TOUTLINE: ① Introduction ② Notation ③ Experiment ④ Quantum Bit ⑤ Quantum Key Distribution ⑥ Multiple Qubits ⑦ Entangled Particles ⑧ Recent News ⑨ Suggested Reading ⑩ SourcesEXPERIMENT Need A strong light source such as a laser pointer Three polarization filters (can be picked up at any camera supply store) Named A, B and C Polarized horizontally at 45 degrees Purpose Demonstrates some of the principles of quantum mechanics through photons and their polarizationSTEP ONE Shine the laser (light source) at a projection screen Insert filter A between the laser and the screen closer to the light source Assume the incoming light is randomly polarized The output now has half the intensity of the incoming light sourceSTEP TWO Insert filter C between the filter A and the screen closer to the screen The intensity of the output drops to zeroSTEP THREE Insert filter B between filter A and filter C There will be a small amount of light visible on the screen Exactly one eighth of the original lightTHE EXPLANATION A photon’s polarization state can be modeled by a unit vector a|↑⟩ + b|→⟩ |→⟩ (horizontal polarization) |↑⟩ (vertical polarization) a and b are complex numbers |a|2 + |b|2 = 1 The measurement postulate of quantum mechanics states that any device measuring a two-dimensional system has an associated orthonormal basis with respect to which the quantum measurement takes placeMEASURING THE STATE Measurement of a state transforms the state into one of the measuring device’s associated basis vector The probability that the state is measured as basis vector|u⟩ is |u|2 |ψ⟩ = a|↑⟩ + b|→⟩ has a probability of |↑⟩ with probability |a|2 |→⟩ with probability |b|2 Measurement of the quantum state will change the state to the result of the measurement Measuring |ψ ⟩ = a|↑⟩ + b|→⟩ results in |↑⟩ Now ψ changes to |↑⟩UNDERSTANDING: STEP ONE & TWO Filter A measures the photon polarization with respect to the basis vector |→⟩ Photons that pass through all have polarization |→⟩ Those that are reflected all have polarization |↑⟩ Assume that the light source produces photons with random polarization Filter A will measure 50% of all the photons as horizontally polarized. Filter C will measure these photons with respect to|↑⟩ The state |→⟩ = 0|↑⟩ + 1|→⟩ ill be projected onto |↑⟩ with probability 0 Thus no photons get throughUNDERSTANDING: STEP THREE Filter B measures the quantum state with respect to the basis This can be rewritten as {|↗⟩, |↖⟩} The photons measured as |↗⟩ pass through 50% of the photons passing through A will pass through B and be in state |↗⟩ Photons will be measured by filter C as |↑⟩ with probability 1/2OUTLINE: ① Introduction ② Notation ③ Experiment ④ Quantum Bit ⑤ Quantum Key Distribution ⑥ Multiple Qubits ⑦ Entangled Particles ⑧ Recent News ⑨ Suggested Reading ⑩ SourcesQUANTUM BIT: QUBIT A quantum bit, or qubit, is a unit vector in a two-dimensional complex vector space for which a particular basis, denoted by {|0⟩, |1⟩}, has been fixed A classical bit can have a state of either 0 or 1 A qubit can have a state of either 0, 1 or both Having a state of both is known as superposition No good classical explination Cannot be viewed as between 0 and 1 Cannot be viewed as a hidden unknown that represents either 0 or 1OUTLINE: ① Introduction ② Notation ③ Experiment ④ Quantum Bit ⑤ Quantum Key Distribution ⑥ Multiple Qubits ⑦ Entangled Particles ⑧ Recent News ⑨ Suggested Reading ⑩ SourcesQUANTUM KEY DISTRIBUTION In
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