quantum information stephen m. barnett university of strathclyde [email protected] the wolfson...
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Quantum Information
Stephen M. BarnettUniversity of Strathclyde
The Wolfson Foundation
0 Motivation1 Digital electronics2 Quantum gates3 Principles of quantum computation4 Quantum algorthims5 Errors and decoherence6 Realizations?
1. Probability and Information
2. Elements of Quantum Theory
3. Quantum Cryptography
4. Generalized Measurements
5. Entanglement
6. Quantum Information Processing
7. Quantum Computation
8. Quantum Information Theory
CMOS Device Performance
Device performance doubles roughly every 5 years!
P - solvable problems (computing time is polynomial in input size)
ClassicalDeterministicAlgorithm
ClassicalProbabilisticAlgorithm
Quantum Computing
Factoring Discrete logarithm Quantum simulations ...
Quantum algorithms: scaling of computing time with N~2n
1. F.T. to determine periodicities
f(x+r) mod N = f(x) mod N
find r Classical: O(N) = O(2n)Quantum: O(log2N) = O(n2)
2. Shor’s factoring algorithm
N = pq find p and q given N
Naïve classical (trial): O(N1/2) = O(2n/2)Best known classical: O(2^[n1/3log2/3n])Shor’s algorithm: O(polynomial[logN]) = O(polynomial n)Exponential sp
eed up!
What happens to RSA? W
hat happens to money??
…101101001…
…000111010…Input
Output“Black Box” orComputer
What is a computation?
Generation of an output number (string of bits) based on an input number.
How does the computer achieve this?
6.1 Digital electronics
Physical bit - electrical voltage +5V = 1 0V = 0
Single bit operation
NOT gate
A AA A
0
1
1
0
Two bit operations
AND gateA B
A A A B
0011
0101
B
B
0001
OR gateA B
A A A B
0011
0101
B
B
0111
Two bit operations
NAND gateA B
A A
0011
0101
B
B
1110
A B
Not all the gates are needed
A small set of gates (e.g. NAND, NOT) is universal in that any logical operation can be made from them.
6.2 Quantum gates
Single qubit operations
H 10
10
1
0
212
1
Hadamard
S1
0
1
0
i Phase
T 1)4/exp(
0
1
0
i / 8
and many more
Two qubit operations - CNOT gate
control bit
target bitTC
TC
TC
TC
TC
TC
TC
TC
01
11
10
00
11
01
10
00
CNOT gate can make entangled states
TCTCTCC
11000102
12
1
We can break up any multi-qubit unitray transformation into a sequence of two-state transformations:
jfc
heb
gda
U †3
†2
†1
123
ˆˆˆˆ
ˆˆˆˆˆ
UUUU
IUUUU
jfc
he
gda
UUba
a
ba
b
ba
b
ba
a
U 0ˆˆ
100
0
0
ˆ1
2222
22
*
22
*
1
**
**3
12
2222
22
*
22
*
2
0
0
001ˆ
0
0
1ˆˆˆ010
0
ˆ
jh
feU
jf
he
gd
UUU
ca
a
ca
c
ca
c
ca
a
U
It follows that we can realise any multi-qubit transformation as a sequence of single-qubit and two-qubit unitary transformations. This is the analogue of the universality of NAND and NOT gates in digital electronics.
The CNOT gate, together with one qubit gates are universal
control bit 1
target bit
TCCTCCabcbacba
2121
control bit 2
Exercise: Construct the Toffoli gate using just CNOT gates and single qubit gates. Try to use as few gates as possible.
V V†V
IVVVVViIe
V xx
iˆˆˆˆˆ,ˆˆˆˆ
2ˆ ††2
4/
6.3 Principles of quantum computation
Encode input onto qubit string
100101101101101001
Quantum evolution = unitary transformation
100101101ˆ100101101 U
Measurement gives output = computed function (hopefully!)
000111010100101101ˆ tmeasuremen U
A quantum computation is a (generalised) measurement
Quantum computation?
Constraints of unitarity? Consider the two bit map
BAABA 1,11,10,01,0
0,10,10,00,0
State overlap
U
Problem. Our computation requires
10,00,001,00,00010
0000
Unitary evaluation of the function f
ˆ U f)(afb
aa
b
a = input string …101101001...
b = input string, usually set to “zero” …000000000...
Exercise: Show that the states transformation is an allowed unitary transformation.
We can show this by an explicit construction:
bbafafaaUafba
f)(,0
)(00)(ˆ
II
bbafafaaU
UU
afbaf
ff
ˆˆ
)()(00ˆ
ˆˆ
)(,0
2
†
Parallel quantum computation
ˆ U f
aca
a
b
)(afaca
a
Can input a superposition of many possible bit strings a.Output is an entangled stated with values of f (a) computed for each a.
Deutsch’s algorithm
A black box that computes one of four possible one-bit functions:
We wish to know if the function is constant or balanced. We can do this by performing two computations To give f (0) and f (1) . Can we do it in one step?
A f (A)Black Box
Constant functions:
f (0) 0
f (1) 0or
f (0) 1
f (1) 1
Balanced functions:
f (0) 0
f (1) 1or
f (0) 1
f (1) 0
)0()0(1021 ff ˆ U f
102
1
102
1
A quantum computer allows solution in a single run:
+ for constant for balanced
10&102
12
1
are orthogonal states and so canbe identified without error.
Exponential speed up
Suppose our box computes a one bit function of n bits and that this function is either constant or balanced.
Constant: 0 or 1 independent of input
Balanced: 0 or 1 for exactly half of the possible inputs
Orthogonal states for constant or balanced functions so solution in ONE computation.
Guaranteed classical solution in computations2n 1 1
Quantum?
xxf
x
n
n
)(
2
2/ 12
10102 2/)1( nn
Exponential speed up.
6.4 Quantum algorithms
1. F.T. to determine periodicities
f(x+r) mod N = f(x) mod N
find r
Classical: O(N) = O(2n)Quantum: O(log2N) = O(n2)
2. Shor’s factoring algorithm
N = pq find p and q given N
Naïve classical (trial): O(N1/2) = O(2n/2)Best known classical: O(2^[n1/3log2/3n])Shor’s algorithm: O(polynomial[logN]) = O(polynomial n)
3. Grover’s search algorithm - searching a database
Classical: O(N) Quantum: O(N1/2)
Factorisation algorithm
N: Given big integer to be factorisedm: Small integer chosen at randomn = 0,1,2, …
1. Make the series FN(n) = mn mod N
2. Find the period r : FN(n+r) = FN(n)
3. The greatest common divisor of N and mr/2±1 divides N
Example: N = 15, m = 2 => FN(0) = 1 FN(1) = 2 FN(2) = 4 FN(3) = 8 FN(4) = 1 FN(5) = 2 …
=> r = 4
=> mr/2 – 1 = 3 mr/2 + 1 = 5
Both OK
Example: N = 15, m = 11 => FN(0) = 1 FN(1) = 11 FN(2) = 1 FN(3) = 11 …
=> r = 2
=> mr/2 – 1 = 10 => GCD 5 mr/2 + 1 = 12 => GCD 3
Both OK
Shor’s algorithm to factorise N
1. Find integers q and M such that: q = 2M > N2 and prepare two registers each
containing M qubits.
2. Set the qubits in the first register in the state (|0 + |1)/21/2 and those in the second in the state |0.
010002
001001
000000
where
01
21
1
0
nq
q
n
3. Choose an integer m at random and entangle the two registers so that
21
1
0
mod1
Nmnq
nq
n
This can be achieved by a unitary transformation (on a suitably programmed quantum computer) within polynomial time.
4. Fourier transform for register 1:
rqk
r
nq
k
q
n
nq
n
qkniNmkNmn/for1
period
21
1
0
1
021
1
0
/2expmodmod
5. Measurement on register 1:
=> k = multiple of q/r is obtained with high probability
=> r = q/k
6.5 Errors and decoherence
Interaction with the environment introduces noise and causes errors
Phase error
1
0
1
0
Bit flip error
0
1
1
0
ˆ U f 10
21
)0()0(0121 ff
102
1
Deutsch’s algorithm
Phase error
10
10
10
10
In this case
Bit flip error
01
01
10
10
In this case
Scaling
Probability that a given qubit has no error in time t exp( t)
Probability that none of n qubits has an error in time t )exp( tn
Let t be the time taken to perform a gate operation. For an efficient algorithm we might need n2 operations.
)exp( 3 tn
The number of required gate operations tends to grow at least logarithmically in the n )logexp( 2 tnn
1090requires about 300 qubits. This gives )107exp( 5 t
Decoherence is a real problem. We need efficient error correction!
Quantum error-correction
An error can make any change to a state so it is not obvious that error-correction is possible.
The key idea, of course, is redundancy!
11100010
11110000
33
33
This is a simultaneous eigenstate of
zzzz IIZZIZZI
with eigenvalue +1 in both cases.
If a single spin-flip error occurs 10
110001
101010
011100
111000
IZZ
ZZI
IZZ
ZZI
IZZ
ZZI
110001
110001
101010
101010
011100
011100
110001
101010
011100
111000
II
II
II
x
x
x
111000
111000
111000
We can, in fact correct any single-qubit error using the 7-qubit Steane code:
11010010111100
101101000011111100110
01100111010101000000020 2/37
00101101000011
010010111100000011001
10011000101010111111121 2/37
All the states differ in least four qubits – they are also common eigenstates of 6 operators with eigenvalue +1.
ZIZIZIZ
IZZIIZZ
IIIZZZZ
XIXIXIX
IXXIIXX
IIIXXXX
7777 10110
Any single-qubit error is detectable from a unique pattern of changes to these.“O
K, I’m co
nvinced. W
here ca
n I buy one?”
Ion-trap implementation - Cirac & Zoller, Wineland et al, Blatt et al.
Single ion qubits coupled by their centre of mass motion
0,g
0,e
1,g
1,e
2,g
2,e
Centre of mass motion acts as a ‘bus’ We can entangle the ionic qubits using thecentre of mass motion.
CofM210ee
2CofM1CofM12
1 10 ege
0,1g
0,1e1,1e
0,2g
0,2e1,2e
1,2g
1,1g
CofM21212
1 0ggee
Blatt et al Innsbruck
( Vandersypen, Steffen, Breyta, Yannoni, Cleve, Chuang, July 2000 Physical Rev. Lett. )
Nuclear spins Nuclear spins Nuclear spins Nuclear spins
• 5-spin molecule synthesized
• Pathway to 7-9 qubits
• First demonstration of a fast 5-qubit algorithm
Quantum-dot array proposal
• Well defined extendible qubit array - stable memory
• Preparable in the “000…” state• Long decoherence time (>104 operation time)• Universal set of gate operations• Single-quantum measurements
D. P. DiVincenzo, in Mesoscopic Electron Transport, eds. Sohn, Kowenhoven, Schoen (Kluwer 1997), p. 657, cond-mat/9612126; “The Physical Implementation of Quantum Computation,” quant-ph/0002077.
DiVincenzo’s criteria for implementing a quantum computer
Summary
• Quantum information is radically different to its classical counterpart. This is because the superposition principle allows for many possible states.
• Our inability to measure every property we might like leads to information security, but generalised measurements allow more possibilities than the more familiar von Neumann measurements.
• Entanglement is the quintessential quantum property. It allows us to teleport quantum information AND it underlies the speed-up of quantum algorithms.
• Quantum information technology will radically change all information processing and much else besides!