Introduction to Quantum Information Processing QIC 710 / CS 768 / PH 767 / CO 681 / AM 871

Transcription

1 Introduction to Quantum Information Processing QIC 710 / CS 768 / PH 767 / CO 681 / AM 871 Lecture 1 (2017) Jon Yard QNC 3126 jyard@uwaterloo.ca TAs Nitica Sakharwade nsakharwade@perimeterinstitute.ca Chunhao Wang chunhao.wang@uwaterloo.ca 1

2 Moore s Law number of transistors year Following trend will reach atomic scale Quantum mechanical effects occur at this scale: Measuring a state (e.g. position) disturbs it Quantum systems sometimes seem to behave as if they are in several states at once Different evolutions can interfere with each other 2

3 Quantum mechanical effects Additional nuisances to overcome? or New types of behavior to make use of? [Shor, 1994]: polynomial-time algorithm for factoring integers on a quantum computer This could be used to break most of the existing public-key cryptosystems on the internet, such as RSA 3

4 Nontechnical schematic view of quantum algorithms Classical deterministic: START FINISH Classical probabilistic: p 1 p 2 START p 3 FINISH Quantum: α 1 α 2 START α 3 POSITIVE NEGATIVE FINISH 4

5 Also with quantum information: Faster algorithms for several combinatorial search problems and for evaluating game trees (polynomial speed-up) Fast algorithms for simulating quantum mechanical systems Communication savings in distributed systems Various notions of quantum proof systems Recent experimental progress rapidly growing industry may be bringing quantum computers closer to reality Quantum information theory: Framework for modeling and overcoming quantum mechanical noise Generalizes notions in classical information theory, such as error-correcting codes entropy compression correlation entanglement quantum information theory classical information theory 5

6 This course covers the basics of quantum information processing Topics include: Introduction to the quantum information framework Quantum algorithms (including Shor s factoring algorithm and Grover s search algorithm) Computational complexity theory Density matrices and quantum operations on them Distance measures between quantum states Entropy and noiseless coding Error-correcting codes and fault-tolerance Non-locality Cryptography 6

7 General course information Background: linear algebra probability theory classical algorithms and complexity Evaluation: 5 assignments (12% each) project presentation (40%) Recommended texts: An Introduction to Quantum Computation, P. Kaye, R. Laflamme, M. Mosca (Oxford University Press, 2007). Primary reference. Quantum Computation and Quantum Information, Michael A. Nielsen and Isaac L. Chuang (Cambridge University Press, 2000). Secondary reference. Quantum Computation Since Democritus, Scott Aaronson (Cambridge University Press, 2012). Optional fun background reading. 7

8 Basic framework of quantum information 8

9 Types of information is quantum information digital or analog? probabilistic digital: 1 1 analog: 1 0 p 0 1 Probabilities p, q 0, p + q = 1 Cannot explicitly extract p and q (only statistical inference) In any concrete setting, explicit state is 0 or 1 Issue of precision (imperfect ok) q 0 r 0 r [0,1] Can explicitly extract r Issue of precision for setting & reading state Precision need not be perfect to be useful 9

10 Quantum (digital) information Amplitudes α, β C, α 2 + β 2 = 1 α Explicit state is a vector β C2 Cannot explicitly extract α and β (only statistical inference) Issue of precision (imperfect ok) 10

11 Dirac bra/ket notation Ket: ψ always denotes a column vector, e.g. ψ = Convention: 0 = 1 0, 1 = 0 1 α 1 α d C d α 0 + β 1 = α β Bra: ψ always denotes a row vector that is the conjugate transpose of ψ, e.g. ψ = (α 1 α 2 α d ) Bracket: φ ψ denotes φ, ψ, the inner product of φ and ψ 11

12 Basic operations on qubits (I) (0) Initialize qubit to 0 or to 1 (1) Apply a unitary operation U (unitary means U U = I) Examples: conjugate transpose Rotation: cos θ sin θ sin θ cos θ NOT (bit flip): σ x = X = Hadamard: H = Phase flip: σ z = Z =

13 Basic operations on qubits (II) (2) Apply a standard measurement: ψ with prob α 2 1 with prob β 2 ψ 0 and the quantum state collapses to 0 or 1 accordingly () There exist other quantum operations, but they can all be simulated by the aforementioned types Example: measurement with respect to a different orthonormal basis {ψ, ψ } 13

14 Distinguishing between two states Let be in state + = or = Question 1: can we distinguish between the two cases? Distinguishing procedure: 1. Apply H = measure This works because H + = 0 and H = 1. Question 2: can we distinguish between 0 and +? Since they re not orthogonal, they cannot be perfectly distinguished 14

15 n-qubit systems Probabilistic states: Quantum states: x, p x 0 p x = 1 x p 000 p 001 p 010 p 011 p 100 p 101 p 110 p 111 x, α x C α 2 x = 1 x ψ = α 000 α 001 α 010 α 011 α 100 α 101 α 110 α 111 Dirac notation: 000, 001, 010,, 111 are basis vectors, so ψ = σ x α x x. 15

16 Operations on n-qubit states Unitary operations: (U U = I) x α x x U α x x x = α x U x x Measurements: α x x x α α α ? 000 with prob α with prob α with prob α and the quantum state collapses 16

17 (Tensor) product states Two ways of thinking about two qubits: (α 0 + β 1 )(α 0 + β 1 ) = αα 00 + αβ 01 + βα 10 + ββ 11 This is a product state (tensor/kronecker product) A B = a 11 B a 12 B a 1n B a 21 B a 22 B a 2n B a m1 B a m2 B a mn B 17

18 Entanglement What about the following state??? = This cannot be expressed as a product state! It s an example of an entangled state which can exhibit interesting nonlocal correlations 18

19 Structure among subsystems qubits: time #1 #2 #3 U V W #4 unitary operations measurements 19

20 Quantum computations Quantum circuits: Feasible if circuit-size scales polynomially 20

21 Example of a one-qubit gate applied to a two-qubit system (do nothing) U U = u 00 u 01 u 10 u 11 Maps basis states as The resulting 4x4 matrix is U U U U 1 I U = u 00 u u 10 u u 00 u u 10 u 11 21

22 Controlled-U gates U U = u 00 u 01 u 10 u 11 Maps basis states as The resulting 4x4 matrix is U U 1 Controlled-U = u 00 u u 10 u 11 22

23 Controlled-NOT (CNOT) X a b a a b Note: control qubit may change on some input states

24 Famous single-qubit gates All these generate the Clifford group, which is related to the symmetry group of a cube or octahedron. More later on this. X = H = Hadamard Pauli Y = 0 i i 0 Z = P = i Phase An important non-clifford gate is the T-gate T = e 2πi/8 24