1
Introduction to Introduction to Quantum Information ProcessingQuantum Information Processing
CS 467 / CS 667CS 467 / CS 667Phys 667 / Phys 767Phys 667 / Phys 767C&O 481 / C&O 681C&O 481 / C&O 681
Richard Cleve DC [email protected]
Lecture 1 (2008)
2
Mooreās LawMooreās Law
ā¢ Measuring a state (e.g. position) disturbs itā¢ Quantum systems sometimes seem to behave
as if they are in several states at onceā¢ Different evolutions can interfere with each other
Following trend ā¦ atomic scale in 15-20 years
Quantum mechanical effects occur at this scale:
1975 1980 1985 1990 1995 2000 2005104
105
106
107
108
109
number of transistors
year
3
Quantum mechanical effectsQuantum mechanical effectsAdditional nuisances to overcome?
orNew 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
4
FINISHFINISH
FINISHFINISH
Quantum algorithmsQuantum algorithms
Classical deterministic:Classical deterministic:
Classical probabilistic:Classical probabilistic:
STARTSTART FINISHFINISH
Quantum:Quantum:
POSITIVEPOSITIVENEGATIVENEGATIVE
Ī±Ī±11
Ī±Ī±33
Ī±Ī±22
pp11
pp33
pp22
STARTSTART
STARTSTART
5
Also with quantum information:Also with quantum information:ā¢ Faster algorithms for combinatorial search [Grover ā96]
ā¢ Unbreakable codes with short keys [Bennett, Brassard ā84]
ā¢ Communication savings in distributed systems [C, Buhrman ā97]
ā¢ More efficient āproof systemsā [Watrous ā99]
ā¦ and an extensive quantum information theory arises, which generalizes classical information theory classical
informationtheory
quantum information
theory
For example: a theory of quantum error-correcting codes
6
This course covers the basics of This course covers the basics of quantum information processingquantum information processing
Topics include:ā¢ Quantum algorithms and complexity theoryā¢ Quantum information theoryā¢ Quantum error-correcting codesā¢ Physical implementations*ā¢ Quantum cryptographyā¢ Quantum nonlocality and communication complexity
7
General course informationGeneral course informationBackground:ā¢ classical algorithms and complexityclassical algorithms and complexityā¢ linear algebralinear algebraā¢ probability theoryprobability theory
Evaluation:ā¢ 5 assignments (12% each)5 assignments (12% each)ā¢ project presentation (40%)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.
8
Basic framework of quantum information
9
qp
Types of informationTypes of informationis quantum information digital or analog?is quantum information digital or analog?
digital:
0
1
00
1
1
analog:
0
1
r [0,1]
r
ā¢ Can explicitly extract rā¢ Issue of precision for setting & reading state ā¢ Precision need not be perfect to be useful
ā¢ Probabilities p, q 0, p + q = 1
ā¢ Cannot explicitly extract p and q (only statistical inference)ā¢ In any concrete setting, explicit state is 0 or 1ā¢ Issue of precision (imperfect ok)
probabilistic
10
Quantum (digital) informationQuantum (digital) information
0
1
00
1
1
ā¢ Amplitudes , C, ||2 + ||2 = 1
ā¢ Explicit state is
ā¢ Cannot explicitly extract and (only statistical inference)
ā¢ Issue of precision (imperfect ok)
Ī²
Ī±
11
Dirac bra/ket notationDirac bra/ket notation
d
2
1
Ket: Ļ always denotes a column vector, e.g.
Bracket: ĻĻ denotes ĻĻ, the inner
product of Ļ and Ļ
Bra: Ļ always denotes a row vector that is the
conjugate transpose of Ļ, e.g. [ *1 *
2 *d ]
0
10
1
01Convention:
12
Basic operations on qubits (I)Basic operations on qubits (I)
01
10Xx
11
11
2
1H
10
01Zz
cossin
sincosRotation:
Hadamard: Phase flip:
NOT (bit flip):
(0) Initialize qubit to |0 or to |1
(1) Apply a unitary operation U (Uā U = I )
Examples:
13
Basic operations on qubits (II)Basic operations on qubits (II)(3) Apply a āstandardā measurement:
0 + 1
2
2
probwith1
probwith0
() There exist other quantum operations, but they can all be āsimulatedā by the aforementioned types
Example: measurement with respect to a different orthonormal basis {Ļ, Ļā²}
||2
||2
0
1
Ļ
Ļā²
ā¦ and the quantum state collapses
14
Distinguishing between two statesDistinguishing between two states
Question 1: can we distinguish between the two cases?
Let be in state or
Distinguishing procedure:1. apply H2. measure
This works because H + = 0 and H ā = 1
Question 2: can we distinguish between 0 and +?
Since theyāre not orthogonal, they cannot be perfectly distinguished ā¦
102
1 10
2
1
15
nn-qubit systems-qubit systems
111
110
101
100
011
010
001
000
p
p
p
p
p
p
p
pProbabilistic states:
1x
xp
0 xpx,
111
110
101
100
011
010
001
000
Ī±
Ī±
Ī±
Ī±
Ī±
Ī±
Ī±
Ī±Quantum states:
12
xxĪ±
CĪ±x x ,
x
x xĪ±Ļ
Dirac notation: |000, |001, |010, ā¦, |111 are basis vectors,
so
16
Operations on Operations on nn-qubit states-qubit states
Unitary operations:
xx
xx xĪ±xĪ± U
ā¦ and the quantum state collapses
x
x xĪ±
Measurements:
2
111
2
001
2
000
probwith111
probwith001
probwith000
Ī±
Ī±
Ī±
(Uā U = I )
111
001
000
Ī±
Ī±
Ī±
?
17
EntanglementEntanglement
11'10'01'00'1'0'10 Ī²Ī²Ī²Ī±Ī±Ī²Ī±Ī±Ī²Ī±Ī²Ī±
Product state (tensor/Kronecker product):
11002
12
1 Example of an entangled state:
ā¦ can exhibit interesting ānonlocalā correlations:
18
Structure among subsystemsStructure among subsystems
V
UW
qubits:
#2
#1
#4
#3
time
unitary operations measurements
19
Quantum computationsQuantum computations
0
1
1
0
1
0
1
0
1
0
1
1
Quantum circuits:
āFeasibleā if circuit-size scales polynomially
20
Example of a one-qubit gate Example of a one-qubit gate applied to a two-qubit systemapplied to a two-qubit system
1110
0100
uu
uuU
U
(do nothing)
The resulting 4x4 matrix is
1110
0100
1110
0100
00
00
00
00
uu
uu
uu
uu
UI00 0U0 01 0U1 10 1U0 11 1U1
Maps basis states as:
21
Controlled-Controlled-UU gates gates
1110
0100
00
00
0010
0001
uu
uu
U
00 00 01 01 10 1U0 11 1U1
Maps basis states as:
Resulting 4x4 matrix is controlled-U =
1110
0100
uu
uuU
22
Controlled-NOT (CNOT)Controlled-NOT (CNOT)
X
a
b ab
a
ā”
Note: ācontrolā qubit may change on some input states
0 + 1
0 ā 10 ā 1
0 ā 1
23
Introduction to Introduction to Quantum Information ProcessingQuantum Information Processing
CS 467 / CS 667CS 467 / CS 667Phys 667 / Phys 767Phys 667 / Phys 767C&O 481 / C&O 681C&O 481 / C&O 681
Richard Cleve DC [email protected]
Lecture 2 (2008)
24
Superdense coding
25
How much classical information in How much classical information in nn qubits?qubits?
2n1 complex numbers apparently needed to describe an arbitrary n-qubit pure quantum state:
000000 + 001001 + 010010 + + 111111
Does this mean that an exponential amount of classical information is somehow stored in n qubits?
Not in an operational sense ...
For example, Holevoās Theorem (from 1973) implies: one cannot convey more than n classical bits of information in n qubits
26
Holevoās TheoremHolevoās Theorem
UĻn qubits
b1
b2
b3
bn
Easy case:
b1b2 ... bn certainly
cannot convey more than n bits!
Hard case (the general case):
Ļn qubits
b1
b2
b3
bn
U00
000
m qubits
bn+1
bn+2
bn+3
bn+4
bn+m
The difficult proof is beyond the scope of this course
27
Superdense coding (prelude)Superdense coding (prelude)
By Holevoās Theorem, this is impossible
Alice Bob
ab
Suppose that Alice wants to convey two classical bits to Bob sending just one qubit
ab
28
Superdense codingSuperdense coding
How can this help?
Alice Bob
ab
In superdense coding, Bob is allowed to send a qubit to Alice first
ab
29
How superdense coding worksHow superdense coding works1. Bob creates the state 00 + 11 and sends the first qubit
to Alice
2. Alice:
01
10X
10
01Zif a = 1 then apply X to qubit
if b = 1 then apply Z to qubit send the qubit back to Bob
ab state00 00 +
11
01 00 ā 11
10 01 + 10
11 01 ā 10
3. Bob measures the two qubits in the Bell basis
Bell basis
30
Measurement in the Bell basisMeasurement in the Bell basis
H
Specifically, Bob applies
to his two qubits ...
input output00 + 11
00
01 + 10
01
00 ā 11
10
01 ā 10
11 and then measures them, yielding ab
This concludes superdense coding
31
Introduction to Introduction to Quantum Information ProcessingQuantum Information Processing
CS 467 / CS 667CS 467 / CS 667Phys 667 / Phys 767Phys 667 / Phys 767C&O 481 / C&O 681C&O 481 / C&O 681
Richard Cleve DC [email protected]
Lecture 3 (2008)
32
Teleportation
33
RecapRecap
ā¢ n-qubit quantum state: 2n-dimensional unit vector
ā¢ Unitary op: 2n2n linear operation U such that Uā U = I (where Uā denotes the conjugate transpose of U )
U0000 = the 1st column of U
U0001 = the 2nd column of U the columns of U
: : : : : : are orthonormal
U1111 = the (2n)th column of U
34
Incomplete measurements (I)Incomplete measurements (I)Measurements up until now are with respect to orthogonal one-dimensional subspaces:
0 1
2
The orthogonal subspaces can have other dimensions:
span of 0 and 1
2
(qutrit)
35
Incomplete measurements (II)Incomplete measurements (II)
Such a measurement on 0 0 + 1 1 + 2 2
results in 00 + 11 with prob 02 + 12
2 with prob 22
(renormalized)
36
Measuring the first qubit of a Measuring the first qubit of a two-qubit systemtwo-qubit system
Result is the mixture 0000 + 0101 with prob 002 + 012
1010 + 1111 with prob 102 + 112
Defined as the incomplete measurement with respect to the two dimensional subspaces:ā¢ span of 00 & 01 (all states with first qubit 0), andā¢ span of 10 & 11 (all states with first qubit 1)
0000 + 0101 + 1010 + 1111
37
Easy exercise: show that measuring the first qubit and then measuring the second qubit gives the same result as measuring both qubits at once
38
Teleportation (prelude)Teleportation (prelude)Suppose Alice wishes to convey a qubit to Bob by sending just classical bits
0 + 1
0 + 1
If Alice knows and , she can send approximations of them ābut this still requires infinitely many bits for perfect precision
Moreover, if Alice does not know or , she can at best acquire one bit about them by a measurement
39
Teleportation scenarioTeleportation scenario
0 + 1(1/2)(00 + 11)
In teleportation, Alice and Bob also start with a Bell state
and Alice can send two classical bits to Bob
Note that the initial state of the three qubit system is:
(1/2)(0 + 1)(00 + 11) = (1/2)(000 + 011 + 100 + 111)
40
How teleportation worksHow teleportation works
(0 + 1)(00 + 11) (omitting the 1/2
factor)
= 000 + 011 + 100 + 111
= Ā½(00 + 11)(0 + 1)+ Ā½(01 + 10)(1 + 0) + Ā½(00 ā 11)(0 ā 1)+ Ā½(01 ā 10)(1 ā 0)
Initial state:
Protocol: Alice measures her two qubits in the Bell basis and sends the result to Bob (who then ācorrectsā his state)
41
What Alice does specificallyWhat Alice does specifically
Alice applies to her two qubits, yielding:
Then Alice sends her two classical bits to Bob, who then adjusts his qubit to be 0 + 1 whatever case occurs
Ā½00(0 + 1)+ Ā½01(1 + 0)+ Ā½10(0 ā 1)+ Ā½11(1 ā 0)
(00, 0 + 1) with prob.
Ā¼ (01, 1 + 0) with
prob. Ā¼ (10, 0 ā 1) with prob. Ā¼ (11, 1 ā 0) with prob. Ā¼
H
42
Bobās adjustment procedureBobās adjustment procedure
01
10X
10
01Z
if b = 1 he applies X to qubit
if a = 1 he applies Z to qubit
Bob receives two classical bits a, b from Alice, and:
00, 0 + 101, X(1 + 0) = 0 + 110, Z(0 ā 1) = 0 + 111, ZX(1 ā 0) = 0 + 1
yielding:
Note that Bob acquires the correct state in each case
43
Summary of teleportationSummary of teleportation
H
X Z
0 + 1
00 + 11
0 + 1ba
Alice
Bob
Quantum circuit exercise: try to work through the details of the analysis of this teleportation protocol
44
No-cloning theorem
45
ClassicalClassical information can be copied information can be copied
0
a a
a 0
a a
a
What about quantum information?
Ļ
0
Ļ
Ļ ?
46
works fine for Ļ = 0 and Ļ = 1
... but it fails for Ļ = (1/2)(0 + 1) ...
... where it yields output (1/2)(00 + 11)
instead of ĻĻ = (1/4)(00 + 01 + 10 + 11)
Candidate:
47
No-cloning theoremNo-cloning theoremTheorem: there is no valid quantum operation that maps an arbitrary state Ļ to ĻĻ
Proof:
Let Ļ and Ļā² be two input
states, yielding outputs ĻĻg and Ļā²Ļā²gā² respectively
Since U preserves inner products:
ĻĻā² = ĻĻā²ĻĻā²ggā² so
ĻĻā²(1ā ĻĻā²ggā²) = 0 so
ĻĻā² = 0 or 1
Ļ
0
0
Ļ
Ļ
g
U