Download presentation
Presentation is loading. Please wait.
1
October 1 & 3, 20071 Introduction to Quantum Computing Lecture 1 of 2 Introduction to Quantum Computing Lecture 1 of 2 http://www.cs.uwaterloo.ca/~cleve/CS497-F07 CS 497 Frontiers of Computer Science Richard Cleve David R. Cheriton School of Computer Science Institute for Quantum Computing University of Waterloo
2
2 Contents of lecture 1 1.Preliminary remarks 2.Quantum states 3.Unitary operations & measurements 4.Subsystem structure & quantum circuit diagrams 5.Introductory remarks about quantum algorithms 6.Deutsch’s parity algorithm 7.One-out-of-four search algorithm
3
3 Contents of lecture 1 1.Preliminary remarks 2.Quantum states 3.Unitary operations & measurements 4.Subsystem structure & quantum circuit diagrams 5.Introductory remarks about quantum algorithms 6.Deutsch’s parity algorithm 7.One-out-of-four search algorithm
4
4 Moore’s Law 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 Following trend … atomic scale in 15-20 years Quantum mechanical effects occur at this scale: 1975198019851990199520002005 10 4 10 5 10 6 10 7 10 8 10 9 number of transistors year
5
5 Quantum mechanical effects Additional nuisances to overcome? or New types of behavior to make use of? [Shor ’94]: polynomial-time algorithm for factoring integers on a quantum computer This could be used to break most of the existing public-key cryptosystems, including RSA, and elliptic curve crypto [Bennett, Brassard ’84]: provably secure codes with short keys
6
6 Also with quantum information: Faster algorithms for combinatorial search problems Fast algorithms for simulating quantum mechanics Communication savings in distributed systems More efficient notions of “proof systems” Quantum information theory is a generalization of the classical information theory that we all know—which is based on probability theory classical information theory quantum information theory
7
7 Contents of lecture 1 1.Preliminary remarks 2.Quantum states 3.Unitary operations & measurements 4.Subsystem structure & quantum circuit diagrams 5.Introductory remarks about quantum algorithms 6.Deutsch’s parity algorithm 7.One-out-of-four search algorithm
8
8 Classical and quantum systems Probabilistic states:Quantum states: Dirac notation: |000 , |001 , |010 , …, |111 are basis vectors, so
9
9 Dirac bra/ket notation Ket: ψ always denotes a column vector, e.g. Bracket: φ ψ denotes φ ψ , the inner product of φ and ψ Bra: ψ always denotes a row vector that is the conjugate transpose of ψ , e.g. [ * 1 * 2 * d ] Convention:
10
10 Contents of lecture 1 1.Preliminary remarks 2.Quantum states 3.Unitary operations & measurements 4.Subsystem structure & quantum circuit diagrams 5.Introductory remarks about quantum algorithms 6.Deutsch’s parity algorithm 7.One-out-of-four search algorithm
11
11 Basic operations on qubits (I) Rotation by : (0) Initialize qubit to |0 or to |1 (1) Apply a unitary operation U ( formally U † U = I ) Examples: Recall conjugate transpose NOT (bit flip): Maps |0 |1 |1 |0 Phase flip: Maps |0 |0 |1 |1
12
12 Basic operations on qubits (II) Hadamard: More examples of unitary operations: (unitary rotation) 00 11 Reflection about this line H0H0 H1H1
13
13 Basic operations on qubits (III) (3) Apply a “standard” measurement: 0 + 1 ( ) 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 0 , ψ 1 } ||2||2 ||2||2 00 11 ψ0ψ0 ψ1ψ1 … and the quantum state collapses to 0 or 1
14
14 Distinguishing between two states Question 1: can we distinguish between the two cases? Let be in state or Distinguishing procedure: 1.apply H 2.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 … but statistical difference is detectable
15
15 Operations on n -qubit states Unitary operations: … and the quantum state collapses Measurements: ( U † U = I )
16
16 Contents of lecture 1 1.Preliminary remarks 2.Quantum states 3.Unitary operations & measurements 4.Subsystem structure & quantum circuit diagrams 5.Introductory remarks about quantum algorithms 6.Deutsch’s parity algorithm 7.One-out-of-four search algorithm
17
17 Entanglement The state of the combined system their tensor product: ?? Suppose that two qubits are in states: Question: what are the states of the individual qubits for 1. ? 2. ? Answers: 1. 2.... this is an entangled state
18
18 Structure among subsystems V U W qubits: #2 #1 #4 #3 time unitary operationsmeasurements
19
19 Quantum circuits 00 11 11 00 11 00 1 0 1 0 1 1 Computation is “feasible” if circuit-size scales polynomially
20
20 Example of a one-qubit gate applied to a two-qubit system U (do nothing) The resulting 4x4 matrix is 00 0U001 0U110 1U011 1U100 0U001 0U110 1U011 1U1 Maps basis states as: Question: what happens if U is applied to the first qubit?
21
21 Controlled- U gates U 00 0001 0110 1U011 1U100 0001 0110 1U011 1U1 Maps basis states as: Resulting 4x4 matrix is controlled- U =
22
22 Controlled- NOT (CNOT) Note: “control” qubit may change on some input states! X aa bb abab aa ≡ 0 + 10 + 1 0 − 10 − 1 0 − 10 − 1 0 − 10 − 1 H H H H
23
23 Contents of lecture 1 1.Preliminary remarks 2.Quantum states 3.Unitary operations & measurements 4.Subsystem structure & quantum circuit diagrams 5.Introductory remarks about quantum algorithms 6.Deutsch’s parity algorithm 7.One-out-of-four search algorithm
24
24 Multiplication problem “Grade school” algorithm takes O(n 2 ) steps Best currently-known classical algorithm costs O(n log n loglog n) Best currently-known quantum method: same Input: two n -bit numbers (e.g. 101 and 111) Output: their product (e.g. 100011)
25
25 Factoring problem Trial division costs 2 n /2 Best currently-known classical algorithm costs O ( 2 n ⅓ log ⅔ n ) Hardness of factoring is the basis of the security of many cryptosystems (e.g. RSA) Shor’s quantum algorithm costs n 2 [ O( n 2 log n loglog n) ] Implementation would break RSA and other cryptosystems Input: an n -bit number (e.g. 100011) Output: their product (e.g. 101, 111)
26
26 How do quantum algorithms work? This is not performing “exponentially many computations at polynomial cost” But we can make some interesting tradeoffs: instead of learning about any (x, f ( x ) ) point, one can learn something about a global property of f Given a polynomial-time classical algorithm for f :{0, 1} n → T, it is straightforward to construct a quantum algorithm that creates the state: The most straightforward way of extracting information from the state yields just (x, f ( x ) ) for a random x {0, 1} n
27
27 Contents of lecture 1 1.Preliminary remarks 2.Quantum states 3.Unitary operations & measurements 4.Subsystem structure & quantum circuit diagrams 5.Introductory remarks about quantum algorithms 6.Deutsch’s parity algorithm 7.One-out-of-four search algorithm
28
28 Deutsch’s problem Let f : {0,1} → {0,1} f There are four possibilities: xf1(x)f1(x) 0 1 0 1 0 xf2(x)f2(x) 0 1 0 1 1 xf3(x)f3(x) 0 1 0 1 0 1 xf4(x)f4(x) 0 1 0 1 0 Goal: determine f ( 0 ) f ( 1 ) Any classical method requires two queries What about a quantum method?
29
29 Reversible black box for f UfUf a b a b f(a)b f(a) f alternate notation: A classical algorithm: (still requires 2 queries) ff 0 0 1 f ( 0 ) f ( 1 ) 2 queries + 1 auxiliary operation
30
30 Quantum algorithm for Deutsch H f H H 11 00 f ( 0 ) f ( 1 ) 1 query + 4 auxiliary operations How does this algorithm work? Each of the three H operations can be seen as playing a different role... 1 23
31
31 Quantum algorithm (1) H f H H 11 00 1. Creates the state 0 – 1 , which is an eigenvector of 1 23 NOT with eigenvalue –1 I with eigenvalue +1 This causes f to induce a phase shift of ( –1 ) f(x) to x f 0 – 10 – 1 xx (–1) f(x)x(–1) f(x)x 0 – 10 – 1
32
32 Quantum algorithm (2) 2. Causes f to be queried in superposition (at 0 + 1 ) f 0 – 10 – 1 00 (–1) f(0)0 + (–1) f(1)1(–1) f(0)0 + (–1) f(1)1 0 – 10 – 1 H xf1(x)f1(x) 0 1 0 1 0 xf2(x)f2(x) 0 1 0 1 1 xf3(x)f3(x) 0 1 0 1 0 1 xf4(x)f4(x) 0 1 0 1 0 (0 + 1)(0 + 1) (0 – 1)(0 – 1)
33
33 Quantum algorithm (3) 3. Distinguishes between ( 0 + 1 ) and ( 0 – 1 ) H ( 0 + 1 ) 0 ( 0 – 1 ) 1 H
34
34 Summary of Deutsch’s algorithm Hf H H 11 00 f ( 0 ) f ( 1 ) 1 23 constructs eigenvector so f -queries induce phases: x ( –1 ) f(x) x produces superpositions of inputs to f : 0 + 1 extracts phase differences from ( –1 ) f( 0 ) 0 + ( –1 ) f( 1 ) 1 Makes only one query, whereas two are needed classically
35
35 Contents of lecture 1 1.Preliminary remarks 2.Quantum states 3.Unitary operations & measurements 4.Subsystem structure & quantum circuit diagrams 5.Introductory remarks about quantum algorithms 6.Deutsch’s parity algorithm 7.One-out-of-four search algorithm
36
36 One-out-of-four search Let f : {0,1} 2 → {0,1} have the property that there is exactly one x {0,1} 2 for which f (x) = 1 Four possibilities: xf 00 (x) 00 01 10 11 1 0 Goal: find x {0,1} 2 for which f (x) = 1 xf 01 (x) 00 01 10 11 0 1 0 xf 10 (x) 00 01 10 11 0 1 0 xf 11 (x) 00 01 10 11 0 1 What is the minimum number of queries classically? ____ Quantumly? ____
37
37 Quantum algorithm (I) f x1x1 x2x2 yy x2x2 x1x1 y f ( x 1,x 2 ) ( ( –1 ) f( 00 ) 00 + ( –1 ) f( 01 ) 01 + ( –1 ) f( 10 ) 10 + ( –1 ) f( 11 ) 11 )( 0 – 1 ) Output state of query? Black box for 1-4 search: Start by creating phases in superposition of all inputs to f : Input state to query? f H H H 11 00 00 ( 00 + 01 + 10 + 11 )( 0 – 1 )
38
38 Quantum algorithm (II) Output state of the first two qubits in the four cases: f H H H 11 00 00 Case of f 00 ? ψ 01 = + 00 – 01 + 10 + 11 ψ 10 = + 00 + 01 – 10 + 11 ψ 11 = + 00 + 01 + 10 – 11 What noteworthy property do these states have? U ψ 00 = – 00 + 01 + 10 + 11 Case of f 01 ? Case of f 10 ? Case of f 11 ? Orthogonal! Apply the U that maps ψ 00 , ψ 01 , ψ 10 , ψ 11 to 00 , 01 , 10 , 11 (resp.)
39
39
Similar presentations
