Quantum Searching & Related Algorithms
Lov K. Grover, Bell Labs, Alcatel-Lucent
• Searching – quantum & classical
• Quantum Searching
• Fixed Point Searching
• The search algorithm combines the two main building
blocks for quantum algorithms---fast transforms and
amplitude amplification---and is deceptively simple.
- David Meyer (Three views of the search algorithm)
Classical Searching out of 5 items
NO
ITEM 2
ITEM 3
ITEM 4
ITEM 5
ITEM 3
ITEM 4
ITEM 5
ITEM 4
ITEM 5
NO
ITEM 1
NO
ITEM 1
ITEM 2
AHA!
ITEM 1
ITEM 2
ITEM 3
ITEM 5
Quantum Mechanical Search
NO
NO
NO
AHA!
NO
Design a scheme so that chance of being in AHA! state is high
AHA!
NO
NO
NO
Now if the system is observed, there is a high
probability of observing AHA! state.
NO
Search – Quantum & Classical
In amplitude amplification, amplitude in target state
is amplified. (after h iterations, the probability
of success is |sin(2hUts)2|) .
In classical searching probabilities in non-target states
is reduced (e.g. after h iterations, the probability of
success is 1- (1-|Uts|2)h ).
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
Observe the state.
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
However, only optimal for exhaustive search of 1 in N items.
Quantum searching amidst uncertainty
• Quantum search algorithm is optimal only if
number of solutions is known.
Puzzle - Find a solution if the number of
solutions is either 1 or 2 with equal probability.
(Only one observation allowed)
½+½(1-(½)t/4)
½(sin2(t)+sin2(2t))
Maximum
success
probability = 3/4
Fixed
point searching
converges to 1.
Fixed Point Quantum Searching
• Fixed point – point of monotonic convergence (no overshoot)
• Fixed point
•
Target state of
(standard) quantum search
• Iterative quantum procedures cannot have fixed points
(Reason – Unitary transformations have eigenvalues
of modulus unity so inherently periodic).
• Fixed points achieved by
1. Using measurements
2. Iterating with slightly different unitary operations in
different iterations.
Slightly different operations in
different iterations
• If |Vts|2 = 1-d,
denote /3 phase shift of t & s state by Rt & Rs.
•
|VRsV † RtV|ts2 = 1-d3
| V(RsV†RtV)(RsV†R†tV )(R†sV†RtV)(RsV†RtV)|ts2= 1-d9
• Non-periodic sequence and can hence have fixed-points
Error correction - idea e
|t>
• U takes us to within e of the target state.
|<t|U|s>|2 =1- e
U|s>
|s>
then URsU†RtU takes us to within e3 |t>e3
of target
URsU†RtU|s>
|<t|URsU†RtU|s>|2=1-e3
|s>
• Can cancel errors in any unitary U by URsU†RtU:
- need to run U twice and U† once, with same errors.
- need to be able to do Rs & Rt
Quantum search
• Database search & function inversion
• Scheduling Problems
• Collision problem & Element Distinctness
•
•
•
•
Precision Measurements
Pendulum Modes
Moving Particles in a Harmonic oscillator
Confocal Resonator Design.
“A good idea finds application in contexts beyond where it was
originally conceived.”