* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Powerpoint format
Atomic orbital wikipedia , lookup
Renormalization group wikipedia , lookup
Ensemble interpretation wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Matter wave wikipedia , lookup
Wave function wikipedia , lookup
Quantum dot cellular automaton wikipedia , lookup
Identical particles wikipedia , lookup
Renormalization wikipedia , lookup
Basil Hiley wikipedia , lookup
Scalar field theory wikipedia , lookup
Wave–particle duality wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Bra–ket notation wikipedia , lookup
Particle in a box wikipedia , lookup
Bell test experiments wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Quantum field theory wikipedia , lookup
Path integral formulation wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Quantum dot wikipedia , lookup
Coherent states wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum decoherence wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Density matrix wikipedia , lookup
Double-slit experiment wikipedia , lookup
Quantum fiction wikipedia , lookup
Bell's theorem wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum entanglement wikipedia , lookup
History of quantum field theory wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum computing wikipedia , lookup
EPR paradox wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Probability amplitude wikipedia , lookup
Quantum machine learning wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Canonical quantization wikipedia , lookup
Quantum group wikipedia , lookup
Hidden variable theory wikipedia , lookup
Quantum key distribution wikipedia , lookup
Quantum cognition wikipedia , lookup
Lecture 10: Quantum Computing Basic Quantum Physics Motivation: Waves versus particles, interference experiments Quantum Notation, Jargon, and Definitions Quantum computing Quantum logic gates Quantum software Quantum hardware R. Rao: Lecture 10 – Quantum Computing 1 Classical double-slit experiment #1 Gun shoots identical particles Large particles like tennis balls All identical With random direction At a slow firing rate R. Rao: Lecture 10 – Quantum Computing Probability of detecting particles for two slits is sum of individual slit probabilities: P12 = P1 + P2 2 Classical double-slit experiment #2 Source generates water waves At any intensity value No reflection from absorber Detector measures wave intensity (wave height)2 Waves interfere at absorber R. Rao: Lecture 10 – Quantum Computing Wave intensity I12 = |h1 + h2|2 = |h1|2 + |h2|2 + 2|h1||h2|cos h1 and h2 are complex numbers 3 Double-slit with electrons Gun shoots electrons Individual particles Indestructible All identical With random direction At a slow firing rate R. Rao: Lecture 10 – Quantum Computing Detector sees individual electrons P12 P1 + P2 P12 = |1 + 2|2 (interference!) 1 and 2 are complex numbers Electrons exhibit wave behavior? 4 Measurement: Watching the electrons Electrons scatter light Put a light at the back side of the slit wall Watch where the electron goes R. Rao: Lecture 10 – Quantum Computing Light on: no interference! P12 = P1 + P2 Light off: interference! P12 = |1 + 2|2 5 First principles of quantum mechanics 1. The probability of an event is given by the square of the absolute value of the probability amplitude for that event: P probability probability amplitude (complex) P | |2 2. When an event can occur several ways, the probability amplitude for the event is the sum of the individual probability amplitudes: 1 + 2 (no measurement interference) P |1 + 2 |2 (probability of event e.g. electron at backstop) 3. If you measure and determine which of the possible alternatives an experiment takes, then the probability of the event is the sum of the probabilities for each alternative: P12 = P1 + P2 = |1|2 + |2|2 (measurement at slits no interference) R. Rao: Lecture 10 – Quantum Computing 6 Other Examples of Quantum Phenomena Spin of an electron: spin up or spin down Can be set to a continuum of values but collapses to up or down when measured with a magnetic field Polarization of a photon: horizontal or vertical Measured using a calcite crystal; can be set to a continuum of values but collapses to horizontal or vertical upon measurement Energy levels of an ion: excited or ground state Can be put in a continuum of states in between these two but collapses to excited or ground state when measured Just as the state of a transistor can represent a bit (0 or 1), the state of a quantum system (e.g. spin) represents a qubit R. Rao: Lecture 10 – Quantum Computing 7 Quantum Notation Classical notation for a bit: x = 0 or x = 1 (only 2 values) (0,1) x=1 (1,0) x=0 Dirac notation for a quantum bit x (Qubit |x>) e.g. spin of an electron: up = |1>, down = |0>, continuum of possible values |x> |1> 1 0 0 0 1 1 x c1 0 c2 1 Qubit |x> |0> Example: c1 and c2 are two real numbers R. Rao: Lecture 10 – Quantum Computing 8 Quantum Jargon A qubit in a quantum system can exist in a linear superposition of basis states (“eigenstates”) |0> and |1>: x c1 0 c2 1 c1 c2 c1 c2 2 2 1 c1and c2 are complex numbers: c1= a1 + b1i ; c2= a2 + b2i i is the square root of –1: R. Rao: Lecture 10 – Quantum Computing i 1 (i.e. i 2 1) 9 Cliff’s notes on complex numbers Consider a complex number c = a + bi Complex conjugate of c is c* = a - bi Amplitude of c = c 2 a 2 b 2 = amplitude of c* Squared amplitude of c = a2 – (-b2) = a2 – i2b2 = c*c c can also be written as c = Aei = A (cos + i sin ) where: A a 2 b2 is the amplitude, is the phase of c, cos a a b 2 2 and sin R. Rao: Lecture 10 – Quantum Computing b a 2 b2 10 The Effect of Measurement If you measure the quantum system, the qubit (superposition of states) collapses to one of the basis states |0> and |1> x c1 0 c2 1 measure either 0 or 1 c1and c2 are called probability amplitudes because probability of getting |0> or |1> upon measurement depends on their squared amplitudes: Prob 0 c1 a b 2 2 1 2 1 Prob 1 c2 Since Prob(|0>) + Prob(|1>) = 1, R. Rao: Lecture 10 – Quantum Computing c1 c2 2 2 2 a22 b22 1 11 Unitary matrices Matrix of complex numbers: A cij c11 c12 E.g., A c c 21 22 Conjugate transpose of a matrix: A *T c*ji * * c c T 11 21 E.g., A * * * c c 12 22 T A matrix U is unitary if U* U = I I is the identity matrix R. Rao: Lecture 10 – Quantum Computing 12 Why are unitary matrices important? c1 x c2 Suppose |x> is a qubit: |x> has length 1 c 1 2 c2 2 1 Suppose |y> = U|x> (i.e. transform |x> using U) Length of |y> is: * c length(U x ) U c * 1 * 2 T c1 U c1* c2 c1 c U U 1 c2 * 2 *T Unitary matrices preserve length! Unitary transformations conserve probability R. Rao: Lecture 10 – Quantum Computing 13 Quantum Operations and Gates Quantum systems are described by Schrödinger’s wave equation Integrating this differential equation, we can show that the state |x> of a quantum system evolves as: |x>new = U|x>old i t where U exp dt ' H 0 U is a unitary matrix derived from Hamiltonian H H is a unitary matrix that represents total energy of the system Every quantum state |x> is a vector of unit length Quantum gates map unit vectors |x> to unit vectors |y> For quantum computing, we design H so that U acts like a logic gate Quantum computers are deterministic, linear, reversible, and unitary until a measurement is made R. Rao: Lecture 10 – Quantum Computing 14 1-bit quantum gates Example: NOT gate Check: UNOT is unitary U NOT F0 1I G J H1 0K U NOT U NOT 1I F 0I F G G J H0K H1J K 0I F 1I F G H1J KG H0J K 1-bit gates: S phase shift gate R Rotation gate Walsh-Hadamard gate R. Rao: Lecture 10 – Quantum Computing 1 0I F U G J H0 1K 1 F1 1I G J 1 1K 2H S UR U WH 1 1 1 2 1 1 15 1-bit quantum gates (cont.) There are infinitely many 1- bit gates, corresponding to the rotations about a sphere: e.g. R gate Another example: Square root of NOT gate U U R. Rao: Lecture 10 – Quantum Computing NOT 1 i 2 2 1 i 2 2 U NOT NOT 1 i 2 2 1 i 2 2 0 1 U NOT 1 0 16 More Notation A qubit is given by a column vector: In Dirac notation, define a row vector: c1 x c2 x c1* c2* x and x are called " bra" and " ket" of x respective ly The “bra”-“ket” of x is the inner product: x x c * 1 c1 * * c c1 c1 c2c2 1 c2 * 2 R. Rao: Lecture 10 – Quantum Computing 17 Orthonormal basis states The vectors |x and |y are orthonormal iff they are orthogonal: and normalized: Example: x y 0 x x 1 and 1 0 and 0 y y 1 0 1 1 1 0 0 1 0 1 and 0 0 1 1 0 1 1 1 0 0 1 1 0 0 and 1 1 1 0 0 1 0 0 R. Rao: Lecture 10 – Quantum Computing 18 What is a Hilbert Space? If S is a set of basis states, then the Hilbert space is the space of functions from S to complex numbers: each function produces a vector of complex numbers in this space Example: |0> and |1> define an orthonormal Hilbert space whose vectors are of the form: c1 x c2 The states of a quantum system are vectors in a particular Hilbert space Measurement of the system produces one of the orthogonal axes (a basis state or “eigenstate”) of the Hilbert space R. Rao: Lecture 10 – Quantum Computing 19 Vector Products in Dirac Notation Vector Products of basis states: 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 1 1 0 0 1 This is the matrix for the NOT gate! 0 1 1 0 1 0 R. Rao: Lecture 10 – Quantum Computing 20 Quantum Logic Gates: 1-bit gates NOT gate: U NOT 0 1I F G J H1 0K U NOT U NOT 1I F 0I F G G J H0K H1J K 0I F 1I F G H1J KG H0J K U NOT 0 1 U NOT 1 0 NOT gate in Dirac Notation: U NOT 0 1 1 0 U NOT 0 0 1 0 1 0 0 1 U NOT 1 0 1 1 1 0 1 0 R. Rao: Lecture 10 – Quantum Computing 21 2-bit Quantum Logic Gates Orthonormal Basis States: 00 , 01 , 10 and 11 Controlled NOT gate: A 0 0 1 1 B 0 1 0 1 A’ 0 0 1 1 1 0 00 0 0 0 1 01 0 0 0 0 10 1 0 0 0 11 0 1 B’ 0 1 1 0 CN gate in Dirac Notation: UCN 00 00 01 01 10 11 11 10 E.g. UCN 10 00 0 01 0 10 0 11 1 11 R. Rao: Lecture 10 – Quantum Computing 22 Two-bit Quantum Operations U 00 00 01 01 10 11 11 10 CN Operation of CN gate on arbitrary qubits x: If x c0 00 c1 01 c2 10 c3 11 , U CN x c0U CN 00 c1U CN 01 c2U CN 10 c3U CN 11 c0 00 c1 01 c2 11 c3 10 c0 c0 c1 c1 U CN c c2 3 c c 3 2 UCN switches the amplitudes of |10> and |11> R. Rao: Lecture 10 – Quantum Computing 23 Entanglement Two bits in a quantum system are entangled if measurement of one is always correlated with measurement of the other E.g. x c0 00 c3 11 (c1 and c2 are zero) If you measure the second bit of x, you know the first bit also: E.g. if second bit was 1, we know first bit must be 1 E.g. y c0 01 c3 11 (c1 and c2 are zero) The qubits in y are not entangled: cannot determine value of first bit based on value of second bit Entangled qubits cannot be factored into their components: c0 00 c3 11 a1 0 b1 1 a2 0 b2 1 R. Rao: Lecture 10 – Quantum Computing 24 No Cloning Theorem There is no unitary transform that allows us to copy a qubit Proof: Suppose U is a copying matrix: U|c0> = |cc> for all states |c> Then, if c 1 1 ( 0 1 ), U c ( 00 11 ) 2 2 but U c cc 1 1 ( 00 01 10 11 ) ( 00 11 ) 2 2 Illustration: Copy a bit using CNOT….yields an entangled state! a 0 b1 0 R. Rao: Lecture 10 – Quantum Computing 25 Three-bit Quantum Gates Controlled Controlled NOT (CCN) gate: A CCN gate A A' B B' C C' A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 A’ 0 0 0 0 1 1 1 1 B’ C’ 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 0 UCCN is an 8 x 8 matrix: 000 000 110 111 111 110 UCCN is universal: we can form any Boolean function using only CCN gates: e.g. AND if C = 0 R. Rao: Lecture 10 – Quantum Computing 26 Premise of quantum computing Simulating an N-bit quantum system on a classical computer requires an amount of computation exponential in N Need to track 2N complex amplitudes simultaneously Nature updates real systems in constant time Updates all the amplitudes simultaneously Uses quantum superposition, or quantum parallelism Quantum computation: Use nature to compute Use N qubits to represent 2N complex amplitudes Perform unitary operations on qubits Measure to get the output Harness quantum superposition to get exponential speedup R. Rao: Lecture 10 – Quantum Computing 27 Comparson of Classical versus Quantum Computing Classical Computer N particles 2N unique states Computations are sequential Select 1 state Operate on it Put it back into memory Example: N=3 3 bits 8 states Work on one number at a time x = 101 Quantum Computer N particles 2N unique states Computations occur in parallel States interact They are entangled Operate on all states at once Example: N=3 3 qbits 8 complex amplitudes Operator manipulates 8 at once x 1 a 000 b 001 c 010 ...h 111 Exponential Speedup R. Rao: Lecture 10 – Quantum Computing 28 Shor’s Quantum Factoring Algorithm Suppose you want to factor a number N Shor’s algorithm: 1. Pick random x < N. 2. Compute f = gcd(x,N); if f l, return f // f is a factor 3. Find the least r > 0 such that xr 1 (mod N). 4. Compute f1 = gcd(xr/2 – 1,N) ); if f1 l, return f1 // f1 is a factor 5. Compute f2 = gcd(xr/2 + 1,N); if f2 l, return f2 // f2 is a factor 6. Go to 1 and repeat Number of repetitions for finding a factor with prob > 0.5 is polynomial in length of N Hard part: Step 3. Find the least r such that xr 1 (mod N). r is the period of repetition of x1, x2,… (mod N). R. Rao: Lecture 10 – Quantum Computing 29 Quantum parallelism for finding period Finding the period r of repetition of x1, x2,… (mod N). 1. Prepare an equal superposition of all values of r < q = N2 q 1 r,0 q r 1 2. Chose random x and compute (xr mod N) for all r simultaneously: q r , x r mod N q r 1 3. Apply quantum Fourier Transform UQFT to superposition of states: q r exp( 2 ikr / q ) k , x mod N q r , k 0 4. Measure contents of register containing k to compute period r See Shor’s paper and tutorials on class website for more details R. Rao: Lecture 10 – Quantum Computing 30 Grover’s Database Search Algorithm Problem: Search a random list of N items for a target item xT such that the function P(xT) is true e.g. searching for a key in DES Grover’s algorithm: Amplify amplitude of target item 1. 2. 3. 4. 5. Prepare an equal superposition of all x Invert the amplitude of xj if P(xj) = 1 Subtract all amplitudes from average amplitude Repeat (2) and (3) N times 4 Measure the result Quadratic speedup over classical search (O(N) steps). R. Rao: Lecture 10 – Quantum Computing 31 Quantum Hardware Suggested Possibilities: Ion Traps (Quantum dots), Cavity QED (quantum electrodynamics), NMR: nuclear magnetic resonance QC Features Ion Trap Cavity QED NMR Qubit Energy levels in an ion within an electric field Polarization of a photon in a cavity Spin states of a nucleus in a molecule Preparation Ion cooling Prepare linearly and circularly polarized photons Set average state of spins in sample Evolution Apply laser pulses at specific frequencies Photon-photon interactions Apply radio frequency pulses Conditional logic Coupled vibrations of State of Cesium ion Nuclear spin-spin trapped ions in cavity & photon interactions in a polarization molecule R. Rao: Lecture 10 – Quantum Computing 32 Quantum Computing: Summary Basic Mechanism: Parallel computation along all possible computational paths, with selective manipulation of probability amplitudes Main Features: Problem instances encoded as states of a quantum system (e.g. spins of n electrons, polarization values of n photons etc.) 1. The system is put into a superposition of all possible states, each weighted by its probability amplitude (= a complex number ci) E.g. Qubits for 2 electrons = c1 |00> + c2 |01> + c3 |10> + c4 |11> 2. The system evolves according to quantum principles: 1. Unitary matrix operation: describes how superposition of states evolves over time when no measurement is made 2. Measurement operation: maps current superposition of states to one state based on probability = square of amplitude ci E.g. probability of seeing output bits (00) is | c1|2 R. Rao: Lecture 10 – Quantum Computing 33 Quantum Computing: Problems and Future Directions Problems: Decoherence: Environmental noise may inadvertently “measure” the system, thereby disturbing the computation (current decoherence time ~ 1 ms) Software solution: Error correcting codes may help ([Shor et al.]) Scaling: All physical implementations so far (NMR, Cavity QED, etc.) have failed to scale beyond a few qubits. Future Directions: Hardware Implementations: New physical substrates are needed that allow manipulations of large numbers of qubits (superpositions of states) with little or no decoherence New Algorithms: New ways of exploiting quantum parallelism are needed that allow solutions to NP-complete problems R. Rao: Lecture 10 – Quantum Computing 34 5-minute break… (Please fill out course evaluations) Next: Student presentations! R. Rao: Lecture 10 – Quantum Computing 35