how to make quantum query algorithms · yelement distinctness ygraph collision ytriangle finding...
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by Rajat Mittal
IQC, University of Waterloo.
How to make quantum query algorithms
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x j
Least queries needed to evaluate f(x)
Given: Function )
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x |
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= Least quantum queries needed to evaluate f(x)
Given: Function )
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Query Algorithm
Unitary operators are free Cost of algorithm = number of queries
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Query algorithms Grover search
unordered search in O( ) queries AND/OR trees Element distinctness Graph collision Triangle finding
Many algorithms were given using
quantum walks.
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Lower bounds on quantum query complexity
Adversary bound : SDP
Equivalence ]
Multiplicative ]
(f)
Polynomial bound ]: Separation ]
Element Distinctness ]
Direct products [
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Lower bounds on quantum query complexity
Adversary bound : SDP
Equivalence ]
Multiplicative ]
(f)
Polynomial bound ]: Separation ]
Element Distinctness ]
Direct products [
For any boolean function f:
[
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Vector set
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Vector set : construction Query algorithm for f : Construct vectors for every element of and x y . . . Z 1 . . . . . . n .
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Vector set : Example Query algorithm for f : 0,1} 000 001 . . . 111 1 . . . . . . n .
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Query algorithm for f : Follow constraint for . x y 1 . . n
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Query algorithm for f : Follow constraint for . x y product 1 < , . + . < , . + n < ,
.
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The dual of adversary bound
Vector set : solution of the dual of adversary bound The value of the solution :
The value of best construction : Q(f)
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Algorithmic applications
Q. Can we develop algorithms using solution of dual? A. Not easy, because of the great number of constraints in SDP.
1. (optimal formula evaluation algorithms
for any read-once formula)
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Algorithmic applications
Q. Can we develop algorithms using filtered factorization norm? A. Not easy, because of the great number of constraints in SDP.
2. Learning graphs [Bel11] (Element Distinctness and algorithm for Triangle Finding )
s t
Cut 1-certificate
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Outline
Span programs
Learning graphs
Comparison
Open problems
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Span Programs
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Span program
For a function f : 1,0 1,1 2,0 . . n,0 n,1 free t 6 5 . . . 3 3 4 1 5 2 . . . 8 2 1 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 . . . 4 2 3 1
target
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Span program
1,0 1,1 2,0 . . n,0 n,1 free t 6 5 . . . 3 3 4 1 5 2 . . . 8 2 1 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 . . . 4 2 3 1
target
Useful vectors for free vectors + vectors in column
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Useful vectors for free vectors + vectors in column
t : as linear combination of useful vectors for y Witness: The coefficients of linear combination Witness size : The length of witness vector
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Useful vectors for : free vectors + vectors in column
t : NOT a linear combination of useful vectors for x Witness: w: (<w,t> =1) & (<w,v> = 0, if v useful). Witness size: The length of Aw.
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Equivalence to dual solution Every span program can be converted to a canonical span program
Fixed vector space for columns No free vectors
Canonical span program is equivalent to a solution of dual adversary. Complexity of best span program is the query complexity of f.
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Features Easy to manipulate span programs
Complementation Composition
Optimal formula evaluation algorithms These were used to show query algorithms using adversary bound.
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Learning Graph
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Learning graph For a function f : Need to construct a graph
{2} {5}
{2,7} {5,2}
{5,7}
{2,7,5}
Vertex: Edge:
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Learning graph For a function f : Need to construct a graph
{2} {5}
{2,7} {5,2}
{5,7}
{2,7,5}
Edges: Every edge has weight
j
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Flow for 1-input
For every , there is a flow of value 1 Source: The empty vertex Sink: - Flow in edge e:
{2} {5}
{2,7} {5,2}
{5,7}
{2,7,9}
1-certificate (depends on y)
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Complexity of learning graph
{2} {5}
{2,7} {5,2}
{5,7}
{2,7,9}
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Reduction Convert learning graph into vector construction
need coordinates in j, e, (S)
Assignment on S
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Constraint for dual adversary Constraint:
=
= = 1 (value of the flow)
{2} {5}
{2,7} {5,2}
{5,7}
{2,7,9}
Cut
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Important results Element distinctness:
Showed the previous bound of
Triangle finding: Improved the upper bound to
k-element distinctness: Improved under certain conditions
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Limitations and improvement Certificate complexity barrier
Learning graph complexity 1-certificate complexity
Alphabet size barrier Only depends on certificate structure
Improved learning graph : overcomes both barriers
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Comparison
Span Program Learning graph Easy to manipulate Equivalent to dual ?????
Weaker than dual Easy to construct
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Solution of dual adversary
Symmetric Tight
Learning graph / Span programs
Intuitive
Separation of constraints really help Need more combinatorial constructions to make
query algorithms
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Open problems
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Constructing dual solution Need other techniques to construct dual solution.
Easy to manipulate Easy to construct Tight
When is positive adversary tight ?
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Other query algorithms
Graph collision Important in triangle finding.
Triangle finding k-element distinctness
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Thank you