Transcript Document

Quantum Information
Stephen M. Barnett
University of Strathclyde
[email protected]
The Wolfson Foundation
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
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1
2
3
4
5
6
Motivation
Digital electronics
Quantum gates
Principles of quantum computation
Quantum algorthims
Errors and decoherence
Realizations?
CMOS Device Performance
Device performance doubles roughly every 5 years!
P - solvable problems (computing time is polynomial in input size)
Classical
Deterministic
Algorithm
Classical
Probabilistic
Algorithm
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)
What is a computation?
Generation of an output number (string of bits) based on an
input number.
…101101001…
“Black Box” or
Computer
Input
How does the computer achieve this?
Output
…000111010…
6.1 Digital electronics
Physical bit - electrical voltage
+5V = 1
0V = 0
Single bit operation
A
A
NOT gate
A
A
0
1
1
0
Two bit operations
A
A B
B
0
0
1
1
A B
0
1
0
1
A
A B
B
0
0
1
1
AB
0
1
0
1
A B
0
0
0
1
AND gate
AB
0
1
1
1
OR gate
Two bit operations
A
A B
B
0
0
1
1
A B
0
1
0
1
A B
1
1
1
0
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.
NAND gate
6.2 Quantum gates
Single qubit operations
H
0
1
S
T

1
2
1
2
0
0
1
1
Hadamard
0
0

1
i1
Phase
0
0

1
exp(i / 4) 1
/8
and many more
Two qubit operations - CNOT gate
control bit
0C 0T 0C 0T
0 C1T
0 C1T

1C 0T
1C1T
1C1T
1C 0T
target bit
CNOT gate can make entangled states
1
2
0


1

1
0

0
C
C
T
2
C
0
T
1C1T

We can break up any multi-qubit unitray transformation into a sequence of twostate transformations:
a

Uˆ   b
c






ˆ
U1  





g

h
j 
d
e
f
a*
Uˆ 3Uˆ 2Uˆ 1Uˆ  Iˆ
† ˆ† ˆ†
ˆ
ˆ
U U U U
1
b*
2
2
a b
a
2
2
a b
0
a b
b
a b
0
2
2
2
2
2
3

0


 a d  g  



ˆ
ˆ
0   U1U   0 e h 
 c f  j  




1







Uˆ 2  




a *
2
a   c
0
c
2
a   c
1 0

ˆ
 U 3   0 e*
 0 h*

2
0
1
2


2
2 


 1 d  g  
a c 


ˆ
ˆ
ˆ
0
  U 2U1U   0 e h 
 a

 0 f  j  


2
2 
a  c 

c*
0 

*
f  
j * 
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
Exercise:
Construct the Toffoli gate using just CNOT gates and single qubit
gates. Try to use as few gates as possible.
control bit 1
control bit 2
target bit
a
b
C1
C2
c T a
b
C1
C2
c  ab T

Vˆ
e
ˆ
V
Vˆ
†
ˆ
V
i / 4
2


ˆI  iˆ  Vˆ 2  ˆ , VˆVˆ †  Vˆ †Vˆ  Iˆ
x
x
6.3 Principles of quantum computation
Encode input onto qubit string
101101001
 1 0 1 1 0 1 0 0 1
Quantum evolution = unitary transformation
1 0 11 0 1 0 0 1
Uˆ 1 0 1 1 0 1 0 0 1
Measurement gives output = computed function (hopefully!)
Uˆ 1 0 1 1 0 1 0 0 1 measuremen
t 000111010
A quantum computation is a (generalised) measurement
Quantum computation?
Constraints of unitarity? Consider the two bit map
AB  A A  B
0,0  0,0 1,0  1,0
0,1  0,0
1,1  1,1
State overlap



ˆ

U
      



Problem. Our computation requires
0 0 0 0
0 1 0 0
0,0 0,1  0  0,0 0,0  1
Unitary evaluation of the function f
a
ˆ
Uf
b
a
b  f (a)
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:
Uˆ f 

a


a a   f (a) 0  0 f (a) 
b b


b  0, f ( a )



Uˆ †f  Uˆ f
Uˆ 2f 

a
 Iˆ  Iˆ


a a   0 0  f (a) f (a) 
b b


b  0, f ( a )



Parallel quantum computation
c
a
a
a
ˆ
Uf
c
a
a f (a)
a
b
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
f (A)
A black box that computes one of four possible one-bit functions:
Constant functions:
f (0)  0
or
f (1)  0
f (0)  1
f (1)  1
Balanced functions:
f (0)  0
or
f (1)  1
f (0)  1
f (1)  0
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 quantum computer allows solution in a single run:
1
2
1
2
0  1 
ˆ
Uf
0  1 
1
2
0  1 
&
1
2
0  1 
are orthogonal states and so can
be identified without error.
1
2
0

 1  f (0)  f (0)
+ for constant
 for balanced

Exponential speed up
Suppose our box computes a one bit function of n
either constant or balanced.
bits and that this function is
Balanced: 0 or 1 for exactly
half of the possible inputs
Constant: 0 or 1
independent of input
Guaranteed classical solution in 2 n1  1
computations
Quantum?
2
( n1) / 2
Orthogonal states for
constant or balanced
functions so solution in
ONE computation.
0  1  0  1 
n
2
n / 2

x2 n
 1
f ( x)
x
Exponential speed up.
6.4 Quantum algorithms
1. F.T. to determine periodicities
f(x+r) mod N = f(x) mod N
Classical: O(N) = O(2n)
Quantum: O(log2N) = O(n2)
find r
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
Example: N = 15, m = 2
Example: N = 15, m = 11
N: Given big integer to be factorised
=> FN(0) = 1
=> FN(0) = 1
m: Small integer chosen at random
FN(1) = 2
FN(1) = 11
n = 0,1,2, …
FN(2) = 4
FN(2) = 1
FN(3) = 8
FN(3) = 11
FN(4) = 1
…
n
1. Make the series FN(n) = m mod N
FN(5) = 2
…
2. Find the period r : FN(n+r) = FN(n)
=> r = 4
=> r = 2
3. The greatest common divisor of N and mr/2±1 divides N
=> mr/2 – 1 = 3
mr/2 + 1 = 5
Both OK
=> 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>.
 
q 1

q
1
n 0
where
0  00 000
1  00 001
2  00 010

n
1
0
2
3. Choose an integer m at random and entangle the two registers so that
 
1
q
q 1

n
n 0
1
m n mod N
2
This can be achieved by a unitary transformation (on a suitably programmed
quantum computer) within polynomial time.
4. Fourier transform for register 1:
q 1

q 1
n
n 0
n
1
m mod N
2

q 1

n 0 k 0
k
n
m
mod N exp i 2kn / q 
1




2 
1 for k  q / r
period r
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
0
0

1 1
Bit flip error
0
1

1
0
Deutsch’s algorithm
1
2
1
2
0  1 
ˆ
Uf
0  1 
Phase error
0 1 0 1

0 1 0 1
In this case
1
2
 1  0  f (0)
 f (0)
Bit flip error
0 1 1  0

0 1 1  0
In this case

Scaling
exp( t)
Probability that a given qubit has no error in time t
Probability that none of n qubits has an error in time t
Let t be the time taken to perform a gate operation.
For an efficient algorithm we might need n2 operations.
The number of required gate operations tends to
grow at least logarithmically in the n
10
90
requires about 300 qubits. This gives
exp( nt )
exp(n t )
3
exp(n log nt )
2
exp(7 105 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!
0 3  000
1 3  111
 0 3   1 3   000   111
This is a simultaneous eigenstate of
ZZI   z  z  I
with eigenvalue +1 in both cases.
IZZ  I  z  z
If a single spin-flip error occurs
0 1
 000   111 
ZZI
  100   011 
ZZI
  010   101 
 x IZZ
I  I  100   011  000
111

011

100 
I  IZZ
  000   111
x  I  010   101
  010   101 
 x ZZI
I  I  001   110
IZZ

 110
001
000
 111
  001   110 
We can, in fact correct any single-qubit error using the 7-qubit Steane code:
IIIXXXX
0 7  23 / 2  0000000  1010101 0110011
IXXIIXX
 1100110  0001111 1011010
XIXIXIX
IIIZZZZ
IZZIIZZ
ZIZIZIZ
 0111100  1101001
 07   17  1 07   17
1 7  2 3 / 2  1111111 0101010  1001100
 0011001 1110000  0100101
 1000011 0010110
All the states differ in least four qubits – they are also common eigenstates of 6
operators with eigenvalue +1.
Any single-qubit error is detectable from a unique pattern of changes to these.

Ion-trap implementation - Cirac & Zoller, Wineland et al, Blatt et al.
Single ion qubits coupled by their centre of mass motion
e,0
g ,0
e,1
g ,1
e,2
g ,2
Centre of mass motion acts as a ‘bus’
e1 ,0
g1 ,0
e1 ,1
e
g1 ,1
1
2
e2 ,0
g 2 ,0
We can entangle the
ionic qubits using the
centre of mass motion.
e
1
1
0
e
2
CofM
0
CofM

 g 1 1 CofM e
2
e2 ,1
g 2 ,1
1
2
 e 1 e 2  g 1 g 2  0 CofM
Blatt et al Innsbruck
Nuclear spins
( Vandersypen, Steffen, Breyta, Yannoni, Cleve, Chuang, July 2000 Physical Rev. Lett. )
• 5-spin molecule synthesized
• First demonstration of a fast
5-qubit algorithm
• Pathway to 7-9 qubits
Quantum-dot array proposal
DiVincenzo’s criteria for implementing a quantum computer
•
•
•
•
•
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.
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!