groverʼs quantum search algorithm and a classical analogue · 2018. 9. 27. ·...

AbstractQuantum computation is currently a popular topic inphysics, not only because it may offer an improvementon classical computation, but also because it sheds lighton very fundamental issues in quantum mechanics. Inthis paper, we will explain Grover’s quantum searchalgorithm, which allows a quantum computer to perform

a database search of N elements in O N( ) operations.It is important to observe that Grover’s algorithmexploits the principle of superposition, but does notrequire quantum entanglement. Because superposition isa classical concept as well, there exists a classicalanalogue to quantum search involving a system ofcoupled oscillators.

I. Quantum Search of an Unstructured Database

Consider a database of N = 2n elements

(strings) of length l bits and suppose we wish to search

the database for a particular element, s . Assume we

have a simple classical computer consisting of a CPU

that manipulates data and a memory that stores the

database. The CPU contains an n bit index register that

refers to the individual elements in the database. One

iteration of a search algorithm might proceed as follows:

(1) Load the database entry corresponding to

the index value into the CPU

(2) Compare this string to s

(3) If they are the same, output the value of the

index and then stop. If not, increment the

index and begin again at step (1).

Evidently, this requires at most N −1 comparisons, i.e.

it takes O(N) operations to find the solution s . For

example, the database could consist of names in

alphabetical order with corresponding telephone

numbers. In this case, the database is unstructured with

respect to the telephone numbers. We might imagine

that we have a particular phone number but need to

match a name to that number. To do so it would be

necessary to search the database as described above.

The same task could be accomplished using a

quantum computer instead. The quantum computer will

also consist of a CPU and a Nl bit memory. Now

however, the CPU contains four registers: an n qubit

“index” register initialized to 0 ; an l qubit “compare”

register that always remains in the state s ; an l qubit

“data” register initialized to 0 ; and a 1 qubit register

initialized to the state 0 − 1( ) / 2 . Returning to our

example, each l qubit item corresponds to a telephone

number, and each n qubit item corresponds to a name.

The CPU performs a LOAD operation as follows. If the

index register is in state x and the data register is in

state d , then the contents of the xth memory cell, dx ,

are added (bitwise, modulo 2) to the data register:

d → d ⊕ dx . Suppose the CPU is in the state

xindex{ s

compare}0data{

0 − 12

.

Then, applying the LOAD operation, this becomes

Groverʼs Quantum Search Algorithm and A Classical AnalogueAARON J. AMSEL ʻ03

DARTMOUTH UNDERGRADUATE JOURNAL OF SCIENCE24 SPRING 2003 25

.

Next, the 2nd and 3rd registers are compared and if they

are the same, a bit flip is applied to the 4th register. The

result is

x s dx0 − 12

→− x s dx

0 − 12

if s = dx

x s dx0 − 12

if s ↑ dx

⎧

⎨⎪

⎩⎪

.

Since dx ⊕ dx = 0 , applying the LOAD operation again

resets the data register to 0 . So the net effect thus far

has been

x s 0 0 − 12

→− x s 0 0 − 1

2if s = dx

x s 0 0 − 12

if s ↑ dx

⎧

⎨⎪

⎩⎪

Now, defining the function

f (x) =1 if dx = s

0 if dx ↑ s⎧⎨⎩

the above operations can be written succinctly as

x Ô⎯ →⎯ (−1) f (x ) x .

The operator Ô represents the action of an “oracle,” a

black box that recognizes the solution and marks it with

a phase shift.

Now that the workings of the oracle have been

described, let us move on to the actual procedure of

Grover’s quantum search. The algorithm begins by

applying the Hadamard transform to the initial state

0 ⊗n . This puts the computer in the superposition state

ψ =1N

xx= 0

N −1

∑ .

Next we repeatedly apply the Grover operator Ĝ , which

consists of these steps:

(1) Apply the oracle Ô

(2) Apply the Hadamard transform Ĥ⊗n

(3) Perform a phase shift of –1 on every basis

state except 0 . (The corresponding

operator is given by 2 0 0 − Î )

(4) Apply the Hadamard transform Ĥ⊗n

From this definition it is straightforward to show that the

Grover operator may be written as

Ĝ = 2ψ ψ − Î( )Ô .

To interpret the action of the Grover operator, it is

helpful to define

x⊥ =1N −1

xx↑xo∑ ,

where xo is the solution to the search problem, i.e.

f (xo ) =1 . Then the initial state ψ may be rewritten

as

ψ =1N

xo +N −1N

x⊥ .

With the definition

cos θ2⎛⎝⎞⎠ =

N −1N

,

we can rewrite the initial state again as

ψ = sin θ2⎛⎝

⎞⎠ xo + cos

θ2⎛⎝

⎞⎠ x⊥ .

If the Grover operator Ĝ is then applied to ψ , some

simple algebra shows that


Ĝψ = sin 32θ

⎛⎝

⎞⎠ xo + cos

32θ

⎛⎝

⎞⎠ x⊥ .

Hence, in the xo , x⊥ basis, Ĝ rotates ψ by angle θ

toward xo . Applying the Grover operator repeatedly

brings the state vector closer to xo , and so an

observation in the xo , x⊥ basis produces the solution

to the search problem with high probability.

It is important to know how many calls to the

oracle are required to complete the search. To rotate the

state vector to approximately xo , ψ must be rotated

by angle π 2 − θ 2( ) . Given that each application of

the Grover operator rotates the state vector by angle θ ,

the total number of calls to the oracle, R , is given by

R =

π2−θ2

θ.

With the reasonable assumption that N >>1, we can

approximate 1/ N = sin θ / 2( ) ≈ θ / 2 . Using this in

the expression for R above and dropping a negligible

term, we obtain

R ≈ π4

N .

Therefore, Grover’s quantum search requires O N( )

oracle consultations, which is significantly more

efficient than the classical version. More generally, it

can be shown that for a search of N elements with M

solutions,

R ≈ π4

NM

.

Furthermore, it has been proven that the algorithm

described here is optimal, that is, no quantum algorithm

can perform a search in fewer than O N( ) calls to the

oracle. For those interested in more reading on quantum

search or on quantum computation in general, we

recommend the book by Nielsen and Chuang [1].

II. A Classical Analogue of Quantum Search

Now we will investigate a classical analogue of

quantum search involving a system of coupled

oscillators. Grover studied the system of coupled

pendulums shown in Figure 1 [2]. There are N small

pendulums whose motion is coupled by being attached

to a larger pendulum. One of these, which we will refer

to as the “marked” pendulum, is distinct from the rest in

that it has a different length and mass. Grover showed

that for appropriate parameters, most of the system’s

energy will be concentrated in the marked pendulum

after O N( ) cycles. The analogy to a search algorithm

is this: suppose that initially we do not know which

pendulum is marked. Then, if the system is set in

motion and the amplitudes of the small pendulums are

observed after O N( ) cycles, it should be possible to

identify the marked pendulum.

We will now discuss in detail a system almost

identical to this one. Our system has a large mass M

attached to a spring of constantK . Attached to the large

mass are N smaller oscillators each with spring constant

k . Of these, there is one marked oscillator with massDARTMOUTH UNDERGRADUATE JOURNAL OF SCIENCE26 SPRING 2003 27

m̃ , while the remaining N −1 have mass m . The

equations of motion for this system are

M ˙̇X = −KX + k xi − X( )i=1

N

∑

m˙̇xi = −k xi − X( ) i ↑ Nm ˜̇̇x = −k x̃ − X( )

where X is the position of the large mass, xi is the

position of the ith unmarked small mass, and x̃ is the

position of the marked mass. Note that since the N −1

unmarked small masses are completely identical, all the

xi are the same. In the future, we will just use the

coordinate x to denote the position of the unmarked

small masses. Now, defining a new time coordinate

τ =km̃t ⇒ d

2

dt 2=km̃

d2

dτ 2,

the equations of motion become

Mm̃˙̇X = − K

kX + (N −1)x + x̃ − NX

mm̃˙̇x = − x − X( )

˜̇̇x = − x̃ − X( )

Next it is useful to set M̃ = M / N , K̃ = K / N , and

x̃' = x̃ / N . Hence, we obtain

M̃m̃˙̇X = − K̃

k+1

⎛

⎝⎜ ⎞

⎠⎟ + (1 − 1

N)x + x̃'

Nmm̃˙̇x = − x − X( )

˜̇̇x' = − x̃' − XN

⎛⎝

⎞⎠

For notational convenience, let M̃→ M , K̃→ K ,

x̃'→ x̃ . The equations of motion can be written more

compactly in matrix form. To this end, introduce the

column vector

vx =

x̃

x

X

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

.

Then the equations of motion are

1 0 0

0 m m̃ 0

0 0 Mm̃

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟

d2

dτ 2

x̃

x

X

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

−1 0 1 N0 −1 11

N 1 −K k +1( )

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

x̃

x

X

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

,

where we have assumed N >>1 and dropped a term of

order 1/ N . To further simplify, define a new

coordinate

x̃'

x'

X'

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

1 0 0

0 m m̃ 0

0 0 M m̃

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

x̃

x

X

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

.

This transforms the equations of motion to

d2

dτ 2

x̃'

x'

X'

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

−1 0 β N0 −α αββN αβ − γ +1( )β

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

x̃'

x'

X'

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

where α = m̃ /m , β = m̃ /M , and γ = K / k . Thus, the

equation of motion for this system is of the form

v̇̇x(τ) = Avx(τ ) ,

where A is a real matrix. Assume a solution of the form

vx(τ) = Re civuie

iω iτ

i=1

3

∑⎡⎣⎢

⎤

⎦⎥,

where ci are complex coefficients,vui are the

eigenvectors of A , and −ω i2 are the eigenvalues of A .


It is then straightforward to show that the general

solution is

vx(τ) = Re UΛ(τ )U −1 vx (0)[ ] ,

where U is the matrix whose columns are the

eigenvectors of A and

Λ(τ ) =

eiω1τ 0 0

0 eiω2τ 0

0 0 eiω 3τ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

.

This is actually the solution in the primed coordinate

system, so transforming back to the “real” coordinates

the solution is

vx(τ) = M Re UΛ(τ )U −1 vx ' (0)[ ] ,

where

M =

N 0 0

0 α 0

0 0 β

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

.

We will choose the initial condition

vx' (0) =

1/ N

α

0

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

,

which, in the actual coordinates, corresponds to

displacing all the small masses (including the marked

mass) a distance of 1 unit. Also, at this point we can

reverse the rescaling of the system parameters, so that

the “real” parameters are α = m̃ /m , β = Nm̃ /M , and

γ = K / Nk . Note that β and γ should be of order 1.

We can get an idea about how to specify the

parameters α , β , and γ by looking at the

eigenfrequencies of matrixA . To do this, we make the

large N approximation

A ≈

−1 0 0

0 −α αβ

0 αβ − γ +1( )β

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

and set

det

ω2 − 1 0 0

0 ω 2 − α αβ

0 αβ ω 2 − γ + 1( )β

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 0 .

First, it is clear that the x̃ mode has frequency ω̃ =1 ,

which makes sense given that the time coordinate is

scaled by the frequency of the marked mass. The

remaining two eigenfrequencies are given by

ω±2 =

12α + (1 +γ )β[ ] ± 12

α + (1 + γ )β[ ]2 − 4αβγ .

To ensure transfer of energy from the x , X system to

the x̃ system, we impose the resonance condition

ω+ = 1 or ω− = 1. Simple algebra will show

thatω− = 1 is obtained by setting

γ =1β+

1α − 1

.

The equations of motion were solved as

described above using Mathematica software. For

example, a plot of the three system coordinates vs. τ is

given in Figure 2. Clearly the amplitude of the marked

mass increases dramatically, while the amplitudes of the

other masses remain negligible in comparison. The total

energy of a simple harmonic oscillator is the sum of the

potential energy (kx 2 / 2) and the kinetic energy

(mẋ2 / 2 ). In our system then, the energies are given by


ε̃ = ˜̇x2 + ( x̃ − X)2 marked mass[ ]

ε = N −1( ) 1αẋ2 + (x − X)2⎛⎝

⎞⎠ N −1 unmarked masses[ ]

Ε = N 1βẊ2 + γX2

⎛

⎝⎜ ⎞

⎠large mass[ ]

where we have dropped the constant factor k / 2

common in each case. The total energy of the system is

ET = ε̃ + ε + Ε = constant . These energies have been

plotted in Figure 3. The plot demonstrates that a

significant portion (but not all) of the system’s energy is

transferred to the marked mass, as expected. Finally, it

is interesting to compare the number of periods required

for the marked mass to reach its peak amplitude with the

number of oscillators, N . We provide a log-log plot of

these quantities for three values of N in Figure 4. The

curve is a straight line with slope almost exactly 1/ 2 .

This implies that it would indeed take O N( ) cycles to

transfer most of the energy to the marked mass.

It is interesting to consider whether or not it is

possible to do better than O N( ) . The proof that the

quantum search is optimal assumes unitary (i.e. linear)

evolution. For the classical analogue though, it would

be conceivable to introduce nonlinearities, through

anharmonic springs or damping for example. Could this

be done in such a way as to speed up the energy

transfer? Could it even be possible to transfer all the

energy to the marked mass in one consultation of the

oracle, i.e. in a single oscillation of the marked mass?

These are questions for further investigation. We then

might relate insights gained from this study back to

quantum search, perhaps finding an improvement on

Grover’s algorithm through some form of nonlinear

quantum mechanics [3].

I am pleased to acknowledge Professor Miles

Blencowe, who proposed this research and guided my

work over the past two years. I greatly appreciate the

time and effort he devoted to this project.

REFERENCES

1. Nielsen, M. and Chuang, I. (2000), Quantum Computation andQuantum Information. (Cambridge, UK: Cambridge UniversityPress).

2. Grover, L. and Sengupta, A. (2001) A Classical Analog ofQuantum Search. http://lanl.arxiv.org/pdf/quant-ph/0109123

3. Abrams and Lloyd. (1998) Nonlinear Quantum MechanicsImplies Polynomial-time Solution for NP-complete and #PProblems. http://lanl.arxiv.org/abs/quant-ph/?9801041

FIGURES

Figure 1. The system of coupled pendulums studied by Grover is depicted here. There are small pendulums attached to a larger pendulum of mass M and length L. Of the small pendulums, have length and mass

, while the marked pendulum has length and mass .


Figure 2. A plot of the system coordinates vs. is given here for the parameters and . The curve with rapidly increasing amplitude is of

course , the marked mass. Curves for and are also shown, but are just barely visible due to their small amplitudes.

Figure 3. The total energies are given here for the parameters and . In this plot, is the thick black curve, is the thin black curve, is the thin

gray curve, and is the dashed line.


Figure 4. Given here is a log-log plot of , the number of periods for the marked mass to reach peak amplitude, vs. , the number of small oscillators. The slope of the line indicates that the exponent is 1/2.

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groverʼs quantum search algorithm and a classical analogue · 2018. 9. 27. ·...

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