Direct Estimation of Quantum Entanglement

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Algorithms
Artur Ekert
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Our golden sequence
H
H
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Circuit complexity
n qubit circuit operation described by 2n x 2n
unitary matrix size and depth of circuits grow
with n
n QUBITS
Size 12 Depth 4
Fix your building units, a finite set of adequate
gates A, B, C
of gates (n) size of the circuit (n) of
parallel units (n) depth of the circuit (n)
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Asymptotic notation for comparisons
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Asymptotic notation
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Asymptotic notation - example
and
hence it is
is both
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Quantifying growth
quadratic or polynomial
cubic or polynomial
exponential
linear
root-n
logarithmic
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Efficient quantum algorithms
B
A
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A
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A
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Quantum Hadamard Transform
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Insidious phases
Discrete set of phase gates
Control phase gates

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Quantum Fourier Transform
H
F1
H
F2
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F3
H
H
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Aside Hadamard is Fourier
Is also known as the quantum Fourier transform on
group
group
the set with operation
(addition mod 2)
the set with operation
(addition mod 2 bit by bit)
group
example for n15
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Aside Hadamard is Fourier
Quantum Fourier transform on group
Quantum Fourier transform on group
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Quantum function evaluation
Boolean function
f
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Quantum function evaluation
can be viewed as m Boolean functions

fm-1
fm-2
f0
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Query Scenario
BLACK BOX, ORACLE
very precious, you are charged fixed amount of
money each time you use it
INPUT is a function f given as an ORACLE GOAL
is to determine some properties of f
making as few queries to f as possible
f
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Quantum interferometry revisited
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U
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Phases in a new way
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U
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Deutschs Problem
?
Given
is f constant or balanced
David Deutsch
four possible oracles
f
CONSTANT
BALANCED
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Deutschs Problem
Classical
2 queries 1 auxiliary operation
f
f
Quantum
CONSTANT
1 query 2 auxiliary operations
H
H
BALANCED
f
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Deutschs Problem The Guts
H
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f
H
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Deutschs Problem The Guts
H
H
f
H
But
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Deutschs Problem The Guts
H
H
f
H
So, it is now clear what happens if f(0) and f(1)
are the same or different.
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Deutschs Problem Generalised
CLASSICAL COMPLEXITY
INPUT
queries
either constant or balanced
PROMISE
determine whether constant or balanced
OUTPUT
H
H
00000 CONSTANT
H
H
H
H
any other output BALANCED
H
H
H
H
f
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Deutschs Problem Generalised
H
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f
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Deutschs Problem Generalised
H
H
f
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Deutschs Problem Generalised
H
H
f
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Deutschs Problem Generalised
What is the amplitude for finding the register in
the 0gt state?
If f(x) constant, this has amplitude 1 i.e. it is
guaranteed
If f(x) balanced, this has amplitude 0 i.e. it
will never happen
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Deutschs Problem Generalised
CLASSICAL COMPLEXITY
INPUT
queries
either constant or balanced
PROMISE
determine whether constant or balanced
OUTPUT
H
H
00000 CONSTANT
H
H
H
H
any other output BALANCED
H
H
H
H
f
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Fair comparison?
classical deterministic
quantum 1
FAIR COMPARISON
Query in k places, if the queries had at least
one 0 and one 1 then the function is balanced,
otherwise assume it is constant. Probability that
it is balanced when declared constant is
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Bernstein-Vazirani Problem
INPUT
is of the form
PROMISE
binary string
OUTPUT
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H
f
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Search Problem
INPUT
Classical Complexity
PROMISE
binary string
OUTPUT
Searching large and unsorted database containing
2n items

• Example of a sorted database
• a phone book if you are given a name and looking
  for a telephone number
• n lookups suffice
• Example of an unsorted database
• a phone book if you are given a number and
  looking for a name
• you need to check 2n items before you succeed
  with probability P1
• you need to check 2n-1 items before you succeed
  with probability P0.5

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Grovers algorithm
It is easy to recognize a solution, although hard
to find it.
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Grovers algorithm
INPUT
Quantum Complexity
PROMISE
binary string
OUTPUT
ITERATION 1
ITERATION 2

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f
f0
f
f0
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Grovers algorithm
ITERATION
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f
f0
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Grovers algorithm
ITERATION
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f
f0
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Grovers algorithm
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f
f0
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Grovers algorithm
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H
is the state input at the start of the iterations
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Grovers algorithm
Geometric Interpretation Reflects a state about
the axis orthogonal to
So, we need to consider the composed, repeated
actions of
and
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Grovers algorithm
sin? ltaH0gt
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Grovers algorithm
Overall action Rotation by angle 2?
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Grovers algorithm
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Grovers algorithm
ITERATION 1
ITERATION 2

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f
f0
f
f0
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Grovers algorithm
After r iterations, the state is rotated by
from the hyperplane
for large n
We iterate until
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Query complexity
classical probabilistic
quantum
Quadratic speedup compared to classical search
algorithms
Cryptanalysis Attack on classical cryptographic
schemes such as DES (the Data Encryption
Standard) essentially requires a search among
2567 1016 possible keys. If these can be
checked at a rate of, say, one million keys per
second, a classical computer would need over
a thousand years to discover the correct key
while a quantum computer using Grover's algorithm
would do it in less than four minutes.
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Applications of Grover

• Most common example is an unsorted database. Not
  a common scenario!
• Finding most efficient route between two places
  on a map.
• Brute-force code breaking (such as the DES
  example weve just seen).
• Any classical algorithm with probabilistic
  outcome can be enhanced.

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Simons Problem
INPUT
Classical Complexity
PROMISE
period
OUTPUT
Example s110 is the period (in the
group)
000 001 010 011 100 101 110 111
111 010 100 110 100 110 111 010
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Fields and vector spaces over them
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Binary vectors
binary vectors
Inner product
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Binary vectors
vectors x
vectors x
a binary vector can be perpendicular to itself
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Binary vectors
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Quantum Hadamard Transform
H
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Simons algorithm
H
H
y
n qubits
n qubits
f(x)
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Simons algorithm
H
H
n qubits
n qubits
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Simons algorithm
H
H
n qubits
n qubits
Solve the system of linear equations Probability
of failure of generating linearly independent
vectors y is less than 0.75
Needs roughly n queries. Quantum complexity
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Classical Complexity Analysis
Classical approach
Randomly choose
Evaluate
Search for collisions
Average number of collisions
Probability of at least one collision
Number of queries in a classical probabilistic
approach
CLASSICAL
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Quantum Complexity Analysis 1
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Quantum Complexity Analysis 2
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Summary
Deutsch (1985), Deutsch and Jozsa (92) The
first indication that quantum computers can
perform better
Grover Polynomial separation
H
H
H
H
f0
f0
f
f
classical
quantum
Simon Exponential separation
H
H
f
classical
quantum
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More general oracles
Quantum oracles do not have to be of this form
e.g. generalized controlled-U operation
n qubits
m qubits
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Phase estimation problem
n qubits
m qubits
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Phase estimation algorithm
STEP 1
H
n qubits
m qubits
Recall Quantum Fourier Transform
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Phase estimation algorithm
STEP 2 Apply the reverse of the Quantum Fourier
Transform
H
Fy
n qubits
m qubits
But what if p has more than n bits in its binary
representation ?
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Arbitrary phases
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Arbitrary phases
Final state
geometric series!
Probability of result
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Phase estimation algorithm
Probability
0111
0000
0001
0010
0011
0100
0101
0110
1000
1001
1010
1011
1100
1101
1110
1111
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Phase estimation - solution
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Group
This is a group under multiplication mod M
For example
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Order-finding problem
For example
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Order finding problem
Input M (an m-bit integer) and a ? 1,2,, M ?
1 such that gcd(a, M ) 1 Output ordM (a),
which is the minimum r gt 0 such that ar 1 (mod
M )
No efficient classical algorithm is known for
this problem
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Order finding problem
period 6
12
18
24
6
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Solving order-finding via phase estimation
n qubits
m qubits
n qubits
m qubits
Suppose we can build a circuit that multiplies y
by the powers of a, but what about the
eigenvector u ?
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Eigenvalues and eigenvectors of U
Sum over all possible values of k - the same
Sum, different order of terms
It seems that we need to know r - do we ?
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Two or more eigenvectors
Estimate of p1 with prob. ?2
F
Fy
Estimate of p2 with prob. ?2
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Solving order-finding via phase estimation
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Solving order-finding via phase estimation
F
Fy
2m qubits
m qubits
We need to distinguish between 1/r and
1/(r1) and r is of the order of M which is an m
bit number
Randomly chosen q Co-prime with r ? Continuous
fractions
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Solving order-finding via phase estimation
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The integer factorization problem
Input M (m-bit integer we can assume it is
composite) Output p, q (each greater than 1)
such that pq M
No efficient (polynomial-time) classical
algorithm is known for this problem. Important
for public key cryptosystems
Order finding and factoring have the same
complexity. Any efficient algorithm for one is
convertible into an efficient algorithm for the
other.
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Math behind quantum factoring
latter event occurs with probability ? ½
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Truth by example
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Quantum factoring

• Proposed quantum algorithm (repeatedly do)
• randomly choose a ? 2, 3, , M?1
• compute g gcd(a,M )
• if g gt 1 then
• output g, M/g
• else
• compute r ordM(a) quantum part !
• if r is even then
• compute x a r/2 1 mod M
• compute h gcd(x,M )
• if h gt 1 then output h, M/h

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Hidden subgroup problem
Given
constant and distinct on cosets of subgroub K
Find a generating set for K in polylog G steps
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Simons algorithm revisited
INPUT
Classical Complexity
PROMISE
period
OUTPUT
Example s110 is the period (in the
group)
000 001 010 011 100 101 110 111
111 010 100 110 100 110 111 010
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Simons algorithm revisited
H
H
n qubits
n qubits
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Simons Problem as HSP
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Simons Problem as HSP
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Simons Problem as HSP
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Simons Problem as HSP
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Examples
Simons algorithm G (Z2)n K 0,r
Shors algorithm G Z and K r Z
OPEN PROBLEMS
QC NEEDS YOU
D8
S4
A
B
C
D
dihedral group
symmetric group
graph isomorphism
shortest vector in a lattice
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Open problems
Graph isomorphism
Find the shortest vector