Grovers Algorithm

Grover’s Algorithm
Quantum Search Algorithm in O(
√
N) complexity
C. Haaland
December 30, 2017

Outline
Classical Search
Quantum Mechanics Overview
Notation
Mathematics
QM Background
Explanation
References

Classical Search
Motivation
“Imagine a phone directory containing N names arranged in
completely random order. In order to ﬁnd someone’s phone
number with a probability of 0.5, any classical algorithm
(whether deterministic or probabilistic) will need to look at a
minimum of N/2 names.”

Classical Search
Motivation
“Imagine a phone directory containing N names arranged in
completely random order. In order to ﬁnd someone’s phone
number with a probability of 0.5, any classical algorithm
(whether deterministic or probabilistic) will need to look at a
minimum of N/2 names.”
Consider the function F : {0, 1}3 → {0, 1},
F(x, y, z) =(x ∨ y ∨ z)∧
(¬x ∨ ¬y ∨ z)∧
(x ∨ ¬y ∨ ¬z)∧
(¬x ∨ y ∨ ¬z)
Question: For what values of the input does F(x, y, z) = 1?

Classical Search
Complexity
Problems are O(N) on a classical computer
Lov Grover published a quantum algorithm in 1996 with
complexity O(
√
N).
For large N, this represents a signiﬁcant improvement over the
classical case
Quantum algorithm cannot guarantee the correct answer, only
returns the solution with high probability

Notation
R is set of real numbers

Notation
C is set of complex numbers (e.g. x + iy)

Notation
Rn (Cn) indicates column vector of n real (complex) numbers
respectively

Notation
respectively
z∗ is the complex conjugate
If z = x + iy then z∗
= x − iy

Notation
respectively
= x − iy
|ψ is Dirac notation for a column vector (ket vector)

Notation
respectively
= x − iy
|ψ is Dirac notation for a column vector (ket vector)
ψ| is Dirac notation for a row vector (bra vector)

Notation (cont’d)
A ∈ Rm×n(Cm×n) means A is a matrix of real (complex)
numbers with m rows and n columns

Notation (cont’d)
When m = n we call A an operator

Notation (cont’d)
A† is Hermitian/adjoint/conjugate transpose of A

Notation (cont’d)
If a matrix A has elements Aij the matrix A†
has entries A∗
ji
An example is ψ| = |ψ †

Notation (cont’d)
If a matrix A has elements Aij the matrix A†
has entries A∗
ji
An example is ψ| = |ψ †
The quantity φ|ψ is called the inner product
The quantity is a scalar with value
n
i=1 φ∗
i ψi

Vector Norm
The norm/length/magnitude squared of |ψ ∈ Cn is deﬁned as
ψ|ψ =
n
i=1
ψ∗
i ψi =
n
i=1
|ψi |2

Vector Norm
The norm/length/magnitude squared of |ψ ∈ Cn is deﬁned as
ψ|ψ =
n
i=1
ψ∗
i ψi =
n
i=1
|ψi |2
This will always be 1 for a valid quantum mechanical state

Matrices
From the deﬁnition of the adjoint
(A|ψ )†
= (|ψ )†
A†
= ψ|A†
where A ∈ Cn×n

Matrices
(A|ψ )†
= (|ψ )†
A†
= ψ|A†
where A ∈ Cn×n
Because the norm of a quantum state is 1, if |φ = U|ψ then
φ|φ = 1

Matrices
(A|ψ )†
= (|ψ )†
A†
= ψ|A†
where A ∈ Cn×n
φ|φ = 1
From this we have ( ψ|U†
)(U|ψ ) = 1 which means UT
U = I

Matrices
(A|ψ )†
= (|ψ )†
A†
= ψ|A†
where A ∈ Cn×n
φ|φ = 1
U = I
An operator satisfying this property is called unitary

Matrices
(A|ψ )†
= (|ψ )†
A†
= ψ|A†
where A ∈ Cn×n
φ|φ = 1
U = I
An operator satisfying this property is called unitary
Unitary operators are the way in which quantum states are
altered or evolved
H =
1
√
2
1 1
1 −1
, σx =
0 1
1 0
, σy =
0 i
−i 0
, σz =
1 0
0 −1

Linearity
Linearity of a function f means
f (x + y) = f (x) + f (y)
f (αx) = αf (x)

Linearity
f (x + y) = f (x) + f (y)
f (αx) = αf (x)
Matrix multiplication is linear

Linearity
f (x + y) = f (x) + f (y)
f (αx) = αf (x)
Matrix multiplication is linear
Common modeling approximation in engineering
Example: log(1 + x) is very nearly x for small x

Linearity & QM
Linearity is a fundamental property of QM
“With classical fields, we often use linear equations,
such as the differential equations that allow us to
solve for small oscillatory motion of, say, a pendu-
lum. In such a classical case, the linear equation is
an approximation; a pendulum with twice the am-
plitude of oscillation will not oscillate at exactly the
same frequency, for example. Hence we cannot take
the solution derived at one amplitude of oscillation of
the pendulum and merely scale it up for larger ampli-
tudes of oscillation, except as a first approximation.
We should emphasize right away, however, that, in
quantum mechanics, this linearity of the equations
with respect to the quantum mechanical amplitude
is not an approximation of any kind; it is apparently
an absolute property.”-David Miller (Prof. Stanford
University)

Quantum State
Single Particle System
Single particle can be in one of two states

Quantum State
Electron, for example, can be spin up, |0 , or spin down, |1

Quantum State
Particle encodes a single quantum bit or qubit

Quantum State
A qubit is a vector in C2
represented as
|0 =
1
0
, |1 =
0
1

Quantum State
represented as
|0 =
1
0
, |1 =
0
1
Measuring, we will ﬁnd the particle in exactly one of the two
states

Quantum State
represented as
|0 =
1
0
, |1 =
0
1
states
Before measurement this is not true!

Quantum State
represented as
|0 =
1
0
, |1 =
0
1
states
Particle exists in a linear superposition of these states

Quantum State
represented as
|0 =
1
0
, |1 =
0
1
states
State will be |ψ = ψ0|0 + ψ1|1
ψ0, ψ1 ∈ C and |ψ0|2
+ |ψ1|2
= 1

Quantum State
represented as
|0 =
1
0
, |1 =
0
1
states
State will be |ψ = ψ0|0 + ψ1|1
ψ0, ψ1 ∈ C and |ψ0|2
+ |ψ1|2
= 1
Classical analogy is Schrodinger’s cat that is both alive and
dead

Quantum State
Multi-Particle System
Suppose we have n particles each of which can be in one of
two states

Quantum State
two states
The system represents n qubits of information

Quantum State
two states
There are N = 2n
possible diﬀerent states
|0 . . . 00 , |0 . . . 01 , |0 . . . 10 , . . . , |1 . . . 11

Quantum State
two states
There are N = 2n
|0 . . . 00 , |0 . . . 01 , |0 . . . 10 , . . . , |1 . . . 11
Each state is a standard basis vector in RN
(i.e. one
component equal to 1 and the rest 0)
Note: The vector components are not the same as the entries
of the Dirac notation

Quantum State
two states
There are N = 2n
|0 . . . 00 , |0 . . . 01 , |0 . . . 10 , . . . , |1 . . . 11
(i.e. one
Measuring, we will ﬁnd the system to be in one of N = 2n
possible states |ψ ∈ RN

Quantum State
two states
There are N = 2n
|0 . . . 00 , |0 . . . 01 , |0 . . . 10 , . . . , |1 . . . 11
(i.e. one
Before measurement, it will be in a linear superposition of
these states

Quantum State
two states
There are N = 2n
|0 . . . 00 , |0 . . . 01 , |0 . . . 10 , . . . , |1 . . . 11
(i.e. one
Before measurement, it will be in a linear superposition of
these states
Its state will be
|ψ = ψ0|0 . . . 00 + ψ1|0 . . . 01 + · · · + ψN−1|1 . . . 11 where
ψi ∈ C and
N−1
i=0 |ψi |2
= 1

Born Rule
What are the coeﬃcients ψi respresenting physically?

Born Rule
ψi cannot be measured, but |ψi |2
can!

Born Rule
can!
Squared magnitude is the probability of measuring the system
in state i

Born Rule
can!
in state i
This interpretation is called the Born Rule

Born Rule
can!
in state i
This is empirical and more like a postulate

Born Rule
can!
in state i
Many attempts to derive it from ﬁrst principles

Born Rule
can!
in state i
Are we sure it’s completely random?

Born Rule
can!
in state i
As far as we can tell, it is actually random (cf. Bell’s
Inequalities)

Born Rule
can!
in state i
Inequalities)
Some interpretations of this phenomenon are

Born Rule
can!
in state i
Inequalities)
Some interpretations of this phenomenon are
Copenhagen Interpretation
Many World’s Hypothesis
Quantum Decoherence

Quantum Measurement
Wave Particle Duality
A wave (of water say) impinging on a screen with two
openings will yield a diﬀraction pattern

Quantum Measurement
Each opening becomes the source of a new wave front

Quantum Measurement
Each opening becomes the source of a new wave front
The two waves interfere with each other forming alternating
peaks and troughs

Quantum Measurement (cont’d)
Perform the same experiment with an electron gun

A diﬀraction pattern is observed!
This indicates electrons are like waves

Now we have an apparatus that can shoot only a single
electron at a time

electron at a time
A diﬀraction pattern! The electron is interfering with itself

electron at a time
A measurement device is placed on one of the slits to see
which one the electron went through

electron at a time
A measurement device is placed on one of the slits to see
which one the electron went through
Diﬀraction pattern is gone.
Electron behaves like a particle that goes through one slit or
the other when measured
The act of measurement has altered the outcome

Function Inversion
Often framed in the context of unstructured database search

Function Inversion
More precisely, searches a function for a single satisfying input

Function Inversion
Given y, ﬁnd the corresponding x such that f (x) = y.
Invert the function f (·)
To search an explicit list (i.e. unstructured database), need a
function backing it

Function Inversion
Given y, ﬁnd the corresponding x such that f (x) = y.
Invert the function f (·)
To search an explicit list (i.e. unstructured database), need a
function backing it
If there is no solution or multiple solutions, algorithm does not
work out of the box

Algorithm
We search for index |ω

Algorithm
Initialize n qubits to quantum state |s ∈ RN (N = 2n) which
is the uniform superposition of states

Algorithm
|s
1
√
N
N
i=1
|x =
1
√
N
1

Algorithm
|s
1
√
N
N
i=1
|x =
1
√
N
1
Repeat r = π
4
√
N times

Algorithm
|s
1
√
N
N
i=1
|x =
1
√
N
1
Repeat r = π
4
√
N times
Deﬁne the oracle operator Uω ∈ Rn×n
as
Uω = I − 2|ω ω| = diag(1, 1, . . . , −1, , . . . , 1, 1
0,1,...,ω,...,N−2,N−1
)
Apply the operator Uω to the qubit state

Inversion About the Mean
The operator Us can be written as
Us =





2
N − 1 2
N · · · 2
N
2
N
2
N − 1 · · · 2
N
...
...
...
...
2
N
2
N · · · 2
N − 1





Applying the operator to |x gives the update
xi ← −xi +
2
N
N
j=1
xj
Adds twice the mean of the coeﬃcients to negation of each
state
State xω was already negated which boosts its value.

Grover’s Example
N=4
For the case of N = 4 we need only compute r = π
4
√
4 = 1
iteration
Uω|s = (I − 2|ω ω|)|s = |s − 2|ω ω|s = |s − 2|ω (1/
√
4)
Us |s −
2
√
4
|ω = (2|s s| − I) (|s − |ω )
= 2|s s|s − |s − |s s|ω + |ω
= 2|s − |s − |s − |ω
= |ω
Measure the system to get the answer

Grover’s Illustration (N=8, ω=3)

Grover’s Illustration (N=8, ω=3)
Graphical

Notes
If the solution does not exist, an answer is returned at random

Notes
Performing more iterations will degrade the solution
probability

Notes
Performing more iterations will degrade the solution
probability
Multiple solutions will change the optimal number of
iterations needed

The Quantum Oracle
The oracle is a (problem dependent) function such that
f (ω) = 1 and 0 otherwise

The Quantum Oracle
Each state is then multiplied with (−1)f (x)
to “mark” the
solution

The Quantum Oracle
to “mark” the
solution
Quantum oracle must be able to evaluate on the superposition
of indices

The Quantum Oracle
to “mark” the
solution
of indices
Classically, we can only evaluate one query at a time

The Quantum Oracle
to “mark” the
solution
of indices
QM has natural parallelism; if f can evaluate x or y, then we
can evaluate f 1√
2
(x + y)

The Quantum Oracle
to “mark” the
solution
of indices
can evaluate f 1√
2
(x + y)
A quantum circuit with quantum gates can be built to
evaluate the predicate

The Quantum Oracle
to “mark” the
solution
of indices
can evaluate f 1√
2
(x + y)
Gates such as Hadamard, Pauli Spin, Phase shift, CNOT etc
(all are unitary operators) used to build oracles

The Quantum Oracle
to “mark” the
solution
of indices
can evaluate f 1√
2
(x + y)
Gates such as Hadamard, Pauli Spin, Phase shift, CNOT etc
(all are unitary operators) used to build oracles
Programming a quantum computer is more like programming
an FPGA than writing software

Usefulness of Grover’s
The O(
√
N) bound assumes predicate can be evaluated on
superposition of all states

Usefulness of Grover’s
The O(
√
N) bound assumes predicate can be evaluated on
superposition of all states
f (·) must be implemented in quantum hardware using
quantum gates
May be difficult to find compact gate representation
Problems like database search must first convert to an implicit
list
Cost of evaluating f (·) may dominate

Current State of Quantum Computing
IBM claims to have a working prototype of a 50 qubit
quantum computer

quantum computer
D Wave solves optimization problems by exploiting QM eﬀects

quantum computer
D Wave solves optimization problems by exploiting QM eﬀects
Much of quantum computing is still theoretical and uses
simulation
Q#, Libquantum, IBM Q etc
Simulation requires lots of memory (e.g. 32 qubits implies
2 · 232
real numbers)

References
https://en.wikipedia.org/wiki/Grover%27s algorithm
https://quantiki.org/wiki/grovers-search-algorithm
Quantum Mechanics for Scientists and Engineers - David
Miller
https://web.eecs.umich.edu/ imarkov/pubs/jour/cise05-
grov.pdf

Grovers Algorithm

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