Can Quantum Search take place in Nature ?
Lov K. Grover
• Searching – – some background and the algorithm
• Applications
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Problem Structure
• Most of theoretical computer science is about
making use of the structure of a problem to
devise efficient ways of solving the problem.
• Slab design problem is
reduced to bin-packing
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Problems without structure
• Searching an unsorted dictionary.
• Classically, not much scope for
improvement over exhaustive searching.
Find fastest pizza delivery time?
Is there a pizza delivery time of
14.9 minutes?
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Quantum Search
You are given a dictionary with 1,000,000 words.
Crossword Puzzle Challenge: _ _ r _ n h _
Answer - piranha
• Classical system needs to examine ~ 500,000 words.
• Quantum mechanical superposition is in multiple
states simultaneously.
• Quantum search algorithm does exhaustive search
over N items in O(N) steps (about 1,000 steps).
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Walsh-Hadamard Transformation (W)
H
1
2
1 1 
1  1


• N = 2n state quantum system built out of n 2-state
quantum systems called qubits.
• In W, each of the n qubits is independently
transformed by H.
• W-1=W
Wt0=1/√N for any t
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Selective Inversion
1
0
It  
0

0
0 0 0
1 0 0
0  1 0

0 0 1
• Diagonal matrix with all 1’s on diagonal except certain
selected (target) elements where it is -1 (denoted by It ).
• Circuit to selectively invert amplitudes in states where f(x)
is 1 (amplitude unchanged in states where f(x)=0)
f(x)
Input
x (n qubits)
x (n qubits)
0 /1
B (1 qubit)
1
2
1
1
 
CNOT
Output
B XOR f(x)
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Quantum Search Algorithm
• Encode N states with log2N qubits.
• Start with all qubits in 0 state.
• Apply the following operations:
(WI 0W )( I t ) . . (WI 0W )( I t ) (WI 0W )( I t )W | 0


 N 4 repetitions
• It and Is are selective inversions of source & target states.
• Observe the state.
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Optimality of quantum search algorithm
Given the following block -
f(x)
0/1
We are allowed to hook up O(log N) hardware.
Problem - find the single point at which f(x) ≠ 0.
•Classically we need N steps.
•Quantum mechanically, we need only √N steps.
Quantum search algorithm is best possible algorithm for
exhaustive searching.
- Chris Zalka, Phys. Rev. A, 1999
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Amplitude Amplification




| t V I sV I tV  I sV I tV s | 2  1Vts

1
1
 repetitions
- Vts can be amplified by a factor of 2 by iterating the
operations IsV-1ItV,  times.
- True for any unitary operation V.
-Choosing V as the Walsh-Hadamard transformation gives
the quantum search algorithm.
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Quantum Searching is tolerant of
certain kinds of errors




| t V I sV 1 I tV  I sV 1 I tV s | 2  1Vts

 repetitions
• V can be arbitrary.
• Is & It can be replaced by arbitrary phase shifts.
• This flexibility indicates that the algorithm will
be more widely applicable.
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Biological computation
• DNA & protein synthesis in cells requires the
same type of computation as in computers.
• In computers, binary representation is most
efficient (information theory).
• Mystery – DNA computation in cells uses an
alphabet of size 4 and protein synthesis uses an
alphabet of size 19 (19 amino acids).
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Does biological computation use
quantum searching?
• Mystery – DNA computation in cells uses an
alphabet of size 4 and protein synthesis uses
an alphabet of size 20 (20 amino acids).
• Observation – (Apoorva Patel, 2000) –
Quantum searching is most efficient when
carried out for databases of size:
4 - (requires 1 query)
10 - (requires 2 queries)
20 – ( requires 3 queries)
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Spatial Quantum Searching
• Quantum search type
algorithms can be used to
search a distributed
database using only
neighbor to neighbor
communications.with a
square-root speedup
over a classical database.
(Aaronson 2001)

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Spatial Quantum Searching
• A robot needs to go to one marked
cell in a multidimensional array of
N items.
• A classical robot will need O(N)
steps to find and reach the marked
cell.
• A quantum robot will need only
O(√N) steps

ON steps


O√N steps
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Photosynthesis
• Photosynthesis in plants is extremely efficient –
unlike other chemical & biological reactions, it is
able to transport almost 100% of the photons to
the desired locations.
• This used to be a mystery – one possible
explanation was given by Fleming et al (Nature –
April 14, 2007) where they analyzed the way the
energy flows in the cells as a spatial quantum
search problem.
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Quantum Computing?
•
Can FMO act as a dedicated quantum
computational device?
•
No strict one-to-one mapping onto
quantum search algorithms such
as Grover’s algorithm
But
– Coherent superposition states allow
the complex to sample many states
very rapidly.
– A trap ‘ decoheres’ the system
•
Is the system ‘designed’ to preserve
coherence?
“Quantum Mechanics Helps in Searching for a Needle in a Haystack.”
Grover, PRL, 79:325 1997
Does Nature use Quantum Searching?
Original algorithm - Database search & function inversion
Applications:
• Collision problem & Element Distinctness
• Communication algorithms
• Precision Measurements
• Pendulum Modes
Natural applications
• Genetic Pattern Matching (??)
• Photosynthesis (??)
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