Computer games for

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Computer games for quantum computers Are they
harder or easier?
Work in progresscollaboration welcome
HP Labs talkPalo Alto, CA 9/22/03
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Quantum Information Science
011010110101101110100101 -------------------- 110
100010000
Computation
Teleportation
Cryptography
Simulation
Communication
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Quantum Games
Economic Games

• Prisoners Dilemma
• Fair Auctions
• Public Goods

Combinatorial Games

• Tetris
• Minesweeper
• Quake

• Chess
• Checkers
• Tic-Tac-Toe

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How computers play games
AND-OR
NAND
Game
Tree
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Game Tree Evaluation Known Results
For a game tree with leaves
Best possible deterministic classical
algorithm all leaves must be queried in
worst case.
Best possible randomized classical algorithm
expected queries in worst case.
Saks Wigderson, 1986
Algorithm can make random choices averaging
over instances is NOT considered
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Nondeterministic algorithms
0-sided Las Vegas algorithms
ZPP

• Always succeed, stochastic runtime
• Complexity Runtime

1-sided Monte Carlo algorithms
RP

• If ans 0, then , else
• Complexity Worst-case runtime

2-sided Monte Carlo algorithms
BPP

• Complexity Worst-case runtime

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Santhas Theorem
Santha, 1991
Any q-query Monte Carlo algorithm for AND-OR tree
evaluation with error probability pobeys
1-sided error
2-sided error
Where is the runtime of the optimal Las
Vegasalgorithm for AND-OR tree evaluation.
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Randomized Classical Algorithm
Depth-First Search (DFS) algorithm
Depth-First Search (DFS) algorithm Snir, 1985

• Evaluate left subtree L of root.
• If L determines root value, stop.
• Otherwise, evaluate right subtree R of root.

Key concept Recursion!
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Randomized Classical Algorithm
complexity when result is 0.
complexity when result is 1.
complexity
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Quantum Game Tree Evaluation
For a game tree with leaves
Best possible deterministic quantum
algorithm all leaves must be queried in
worst case.
Best known quantum algorithm
expected queries in worst case.
Best known quantum lower bound
expected queries in worst case.
Barnum Saks, 2002 Ambainis, 2003
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Polynomial Method
Boolean functions can be represented
bypolynomials
AND
OR
NAND
NAND(n )
Thm Any quantum circuit computing f mustmake at
least (deg f ) queries.
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Adversary Method
(Prove by Ambainis iterated fn.method?
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Recursive Idea I Improve base case
By Santhas Theorem, we seek a q-query Monte
Carlo quantum algorithm with failureprobability
1-sided error
2-sided error
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Quantum Recursion Ordered Search
Query Complexity
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Better base cases
Blue Classical
NAND(1)
Black Quantum
Queries Success Probability
Queries Success Probability
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Better base cases
Blue Classical
NAND(2)
Black Quantum
Queries Success Probability
Queries Success Probability
Circuit found by computer search.
Barnum, Bernstein, Spector, 2000
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Recursive Idea II Recurse Smarter
Success probability drops rapidly by majority
voting
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Noisy Quantum Recursion
HØyer et al., 2003 (quant-ph/0304052)
Perfect oracle
Noisy oracle
Naïve recursive Grover

• 1 query in size-4 list
• Recursion, majority voting ?

Adaptive Grover step
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Noisy Quantum Recursion
HØyer et al., 2003 (quant-ph/0304052)
Usual Grover algorithm
Adaptive trick
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Summary

• Quantum economic game theory ideas may aid
  quantum combinatorial game theory.
• Game tree evaluation can be mapped to the query
  complexity of NAND.
• Quantum query complexity of NAND is between
  and . WHAT IS IT?
• Quantum outperforms classical for small NAND
  instances.
• Clever recursive step may be the key.

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Summary

• Color scheme Muted chessboard background?
  Tetris-like background? Some random computer
  game background? Pong? Deep blue natural
  background with yellow/mustard text? Beige
  background with dark text (for math)?
• Slide transitions fade? Alternate?