INTRODUCTION TO QUANTUM COMPUTING

Transcription

1 INTRODUCTION TO QUANTUM COMPUTING Writen by: Eleanor Rieffel and Wolfgang Polak Presented by: Anthony Luaders

2 OUTLINE: Introduction Notation Experiment Quantum Bit Quantum Key Distribution Multiple Qubits Entangled Particles Recent News Suggested Reading Sources

3 INTRODUCTION 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.

4 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 space

5 QUANTUM 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 algorithm

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7 BAR-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 consistent

8 NOTATION: 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 is the transformation that maps 0 to (0, 0) T

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10 EXPERIMENT Need A strong light source such as a laser pointer Three polarization filters (can be picked up at any camera supply store) Purpose Named A, B and C Polarized horizontally at 45 degrees Demonstrates some of the principles of quantum mechanics through photons and their polarization

11 STEP 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 source

12 STEP TWO Insert filter C between the filter A and the screen closer to the screen The intensity of the output drops to zero

13 STEP 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 light

14 THE 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 place

15 MEASURING 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 = 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 = a + b results in Now changes to

16 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 = ill be projected onto with probability 0 Thus no photons get through

17 UNDERSTANDING: 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 1/2

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19 QUANTUM BIT: QUBIT A quantum bit, or qubit, is a unit vector in a twodimensional 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 1

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21 QUANTUM KEY DISTRIBUTION In 1984, Bennett and Brassard described the first quantum key distribution scheme Sequences of single qubits can be used to transmit private keys on insecure channels Consider Alice and Bob want to communicate privately They are connected by an ordinary bidirectional open channel They are also connected by a unidirectional quantum channel Both channels can be observed by Eve the ease dropper

22 ESTABLISHING A SECRET KEY Alice sends a sequence of bits to Bob For each bit Alice randomly uses on of the following two bases for encoding each bit s or

23 BOB RECEIVES EACH PHOTON Bob measures the state of the photon he receives by randomly picking either basis Now, Alice and Bob communicate on the open channel by using the same encoding and decoding With this information they can determine which bits were transmitted correctly They will use these keys and discard all others On average they will agree on 50% of all bits transmitted

24 EVE THE EASE DROPPER Suppose Eve measures the state of the photon transmitted by Alice She resends new photons with the measured state She will use the wrong basis roughly 50% of the time Bob measures the resent qubit with the correct basis There will be a 25% probability that he measures the wrong value Thus, ease dropping on the quantum channel is bound to introduce a high error rate that both Alice and Bob can detect

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26 MULTIPLE QUBITS Classical Physics states the possible states of a system of n particles form a vector space of 2n dimensions Based on individual states that can be described by a vector of two-dimensional vector space A quantum system is a different story n qubits have a state space of 2 n dimensions This is what suggest a possible an exponential speed-up of computation on quantum computers over classical computers

27 CARTESIAN VS. TENSOR PRODUCT Classically particles combine through the cartesian product Let V and W be 2 two-dimensional complex vector spaces V = {v 1,v 2 } and W = {w 1,w 2 } The cartesian product of these two is {v 1, v 2, w 1, w 2 } Quantum states combine through the tensor product The tensor product of V and W is {v1 w1,v1 w2,v2 w1,v2 w2} Thus, { 0, 1 }, has basis { 0 0, 0 1, 1 0, 1 1 } Which can be written more compactly as { 00, 01, 10, 11 }

28 A BASIS FOR A 3-QUBIT SYSTEM Has 8 basis vectors In general an n-qubit system has 2 n basis vectors

29 A STATE THAT CANNOT BE DESCRIBED The state is an example We cannot find a 1, a 2, b 1, b 2 such that (a b 1 1 ) (a b 2 1 ) = a 1 b 2 = 0 implies that either a 1 a 2 = 0 or b 1 b 2 = 0 These states have no classical counterpart These states provide the exponential growth of quantum state spaces They are known as entangled states

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31 ENTANGLED PARTICLES If the measurement of one particle has an effect on the other it is considered to be entangled Take the state The probability that the first bit is measured to be 0 is 1/2 if the second bit has not been measured If the second bit had been measured, the probability that the first bit is measured as 0 is either 1 or 0, depending on whether the second bit was measured as 0 or 1 respectively Therefore this state is entangled

32 THE EPR PARADOX Einstein, Podolsky and Rosen proposed an experiment that uses entangled particles in a manner that seemed to violate fundamental principles of relativity. Image Alice has one pair of 1/ / 2 11 and Bob takes the other

33 ALICE AND BOB They are arbitrarily far apart Alice measures her particle and observes state 0 If Bob measures his particle he will also observe 0 This change occurs instantaneously Is it possible for Alice and Bob to communicate faster than the speed of light? No, there is no way for either of them to use this mechanism to communicate Why?

34 WHY: EPR? Einstein, Podolsky and Rosen proposed that each particle has some internal state that completely determines what the result of any given measurement will be. This is an example of a local hidden variable theory An actual experiment showed that Bell s inequality is violated. Quantum mechanics cannot be explained by any local hidden variable theory

35 WHY: CAUSE AND EFFECT? Stated earlier that Alice affects a measurement performed by Bob, this is incorrect It is possible to set up the EPR scenario so that one observer sees Alice measure first, then Bob In this same scenario another observer sees Bob measure first, then Alice Thus, cause and effect cannot be compatible with both observers seeing cause from different people This resolves the EPR paradox All we can say is that Alice and Bob will observe the same random behavior

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37 RECENT NEWS Four qubits 6cm by 6cm chip that holds nine quantum devices 10 qubits could be possible by the end of the year Perfect Diamonds Scientist have developed a new way to manipulate atoms Works at room temperature Cheaper than creating atoms

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39 SUGGESTED READING An Introduction to Quantum Computing for Non- Physicist by Eleanor Rieffel and Wolfgang Polak Approaching Quantum Computing by Dan C. Marinescu and Gabriela M. Marinescu Classical and Quantum Information by Dan C. Marinescu Programming the Universe by Seth Lloyd

40 OUTLINE: Introduction Notation Experiment Quantum Bit Quantum Key Distribution Multiple Qubits Entangled Particles Recent News Suggested Reading Sources

41 SOURCES Rieffel, Eleanor, and Wolfgang Polak. "An Introduction to Quantum Computing for Non-Physicists." AMC Computing Surveys 32.3 (2000): Palmer Science, Jason. "BBC News - Quantum Computing Device Hints at Powerful Future." BBC - Homepage. 22 Mar Web. 11 Apr < Powell, Devin. "Diamond Could Store Quantum Information." Science News. 24 Mar Web. 11 Apr < generic/id/71588/title/ Diamond_could_store_quantum_information>.