faster quantum algorithm for evaluating game treesbreichar/talks/faster...faster quantum algorithm...
TRANSCRIPT
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Faster quantum algorithm for evaluating game treesBen Reichardt x9x5x6x1 x1x9
x8x7
x5
x4x2 x3
AND OR AND
AND
AND
OR
OR
!(x)
x1x1
University of Waterloo
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x9x5x6x1 x1x9
x8x7
x5
x4x2 x3
AND OR AND
AND
AND
OR
OR
!(x)
x1x1
Problem: Evaluate the AND-OR formula with minimal queries to the input bits xi.
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x9x5x6x1 x1x9
x8x7
x5
x4x2 x3
AND OR AND
AND
AND
OR
OR
!(x)
x1x1
Motivations:
• Two-player games (Chess, Go, …)- Nodes ↔ game histories
- White wins iff ∃ move s.t. ∀ responses, ∃ move s.t. …
• Decision version of min-max tree evaluation
- inputs are real numbers
- want to decide if minimax is ≥10 or not
• Model for studying effects of composition on complexity
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For balanced, binary formulas
α-β pruning is optimal ⇒ Randomized complexity N0.754
OR
AND
OR
AND AND AND AND
x1 x2 x3 x4 x5 x7x6 x8
!(x)
[Snir ‘85, Saks & Wigderson ’86, Santha ’95]
Deterministic decision-tree complexity = NAny deterministic algorithm for evaluating a read-once AND-OR formula must examine every leaf
N0.51 ≤ Randomized complexity ≤ N [Heiman, Wigderson ’91]
(see also K. Amano, Session 12B Tuesday)
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Deterministic decision-tree complexity = NN0.51 ≤ Randomized complexity ≤ N
Quantum query complexity = √N (very special case of the next talk)
This talk: What is the time complexity for quantum algorithms?
U0qu
ery x U1
quer
y x … UT f(x)
w/ prob. ≥2/3
|1!+ |2! "# ($1)x1|1!+ ($1)x2 |2!|x !
{0,1}
n ||j! (!1)xj |j"
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OR
AND
OR
AND AND AND AND
x1 x2 x3 x4 x5 x7x6 x8
!(x)
Farhi, Goldstone, Gutmann ’07 algorithm
• Theorem ([FGG ’07]): A balanced binary AND-OR formula can be evaluated in time N½+o(1).
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• Theorem ([FGG ’07]): A balanced binary AND-OR formula can be evaluated in time N½+o(1).
• Convert formula to a tree, and attach a line to the root
• Add edges above leaf nodes evaluating to one
=0
=1
Farhi, Goldstone, Gutmann ’07 algorithm
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OR
AND
OR
AND AND AND AND
x1 x2 x3 x4 x5 x7x6 x8
!(x)
Farhi, Goldstone, Gutmann ’07 algorithm
• Theorem ([FGG ’07]): A balanced binary AND-OR formula can be evaluated in time N½+o(1).
• Convert formula to a tree, and attach a line to the root
• Add edges above leaf nodes evaluating to one
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x11 = 0x11 = 1
=0
=1
!(x) = 0 !(x) = 1
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|!t! = eiAGt|!0!
!(x) = 0 !(x) = 1
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|!t! = eiAGt|!0!
Wave transmits!Wave reflects!
!(x) = 0 !(x) = 1
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!(x) = 0
What’s going on?
!""""""""""""#
""""""""""""$
Observe: State inside tree converges to energy-zero eigenstate of the graph
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!(x) = 0
What’s going on?
=0
=1!""""""""""""#
""""""""""""$
Observe: State inside tree converges to energy-zero eigenstate of the graph(supported on vertices that witness the formula’s value)
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Energy-zero eigenvectors for AND & OR gadgets
• Identify the output edge of one gadget with an input edge of the next.
• (Note: all I/O edge weights are 1.)
• Corollary: Gφ(x) has an eigenvalue-zero eigenstate supported on root aO ⇔
φ(x)=1
1
2
OR:AND:
1
2
+1+1
-1
-1
-1
-1
Together in a formula:
+1
-1
-1
+1
+1 Input adds constraints via dangling edges:
G0
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Balanced AND-OR formula evaluation in O(√n) time
Squared norm =
+1
-1
-1
+1
+1
+1
+1
+1
+1
1 + 2 + 2 + 4 + 4 + 8 + 8 + · · · + 212 log2 n = O(
!n)
· · ·
1
2
2
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Effective spectral gap lemma
If M u ≠ 0, then M u ⊥ Kernel(M✝)! !!
by the SVD M =!
!
! |v!!"u!| !
!!!!! !projection of M u onto the
span of the left singular vectors of M with singular values ≤λ
≤ λ u
!!!!!
!!!
!since !!!M!u!2 =
!
!!"
"2|"u!|u#|2
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Squared norm = O(√n)
1
2
2
n¼
-1
-1
+1
+1
+1
+1
+1
+1
· · ·1/n¼
1/n¼
Case φ(x)=1
Constant overlap on root vertex
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Case φ(x)=1
Eigenvalue-zero eigenvector with constant overlap on root vertex
Case φ(x)=0
!!!!! !projection of M u onto the
span of the left singular vectors of M with singular values ≤λ
≤ λ u
!!!!!
!!!
n!
-1
-1
+1
+1
+1
+1
+1
+1
· · ·1/n!
1/n!
· · ·1/n¼
1/n¼
n¼-n¼
n¼
n¼
u!
1M u!
root√n
Root vertex has Ω(1/√n) effective spectral gap
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Case φ(x)=1
Eigenvalue-zero eigenvector with constant overlap on root vertex
Case φ(x)=0
n!
-1
-1
+1
+1
+1
+1
+1
+1
· · ·1/n!
1/n!
Root vertex has Ω(1/√n) effective spectral gap
· · ·1/n!
1/n!
n!-n!
n!
n!
u!
1
M u!
Quantum algorithm: Run a quantum walk on the graph, for √n steps from the root.
• φ(x)=1 ⇒ walk is stationary
• φ(x)=0 ⇒ walk mixes
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Evaluating unbalanced formulas[Ambainis, Childs, Reichardt, Špalek, Zhang ’10]
Proper edge weights on an unbalanced formula give √(n·depth) queries
depth n, spectral gap 1/n
“Rebalancing” Theorem: For any AND-OR formula with n leaves, there is an equivalent formula with n e√log n leaves, and depth e√log n
[Bshouty, Cleve, Eberly ’91, Bonet, Buss ’94]
n e√log n leaves, and depth (log n) e√log n
O(√n e√log n)query algorithm⇒
Today: O(√n log n)
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Tensor-product composition
OR: AND:
Direct-sum composition
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! " !
Tensor-product compositionDirect-sum composition
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Tensor-product composition
! "
Properties• Depth from root stays ≤2
— 1/√n spectral gap
• Graph stays sparse—provided composition is along the maximally unbalanced formula
• Middle vertices Maximal false inputs
↔
010100
00101100
01010
10010
00100
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Final algorithm
∧ ∨ ∧∧
sort
sub
form
ulas
by
size
• Identify the output edge of one gadget with an input edge of the next.
• (Note: all I/O edge weights are 1.)
• Corollary: Gφ(x) has an eigenvalue-zero eigenstate supported on root aO ⇔
φ(x)=1
• With direct-sum composition, large depth implies small spectral gap
• Tensor-product composition gives √n-query algorithm (optimal), but graph is dense and norm too large for efficient implementation of quantum walk
• Hybrid approach:• Decompose the formula into paths,
longer in less balanced areas• Along each path, tensor-product• Between paths, direct-sum
• Tradeoff gives 1/(√n log n) spectral gap, while maintaining sparsity and small norm⇒ Quantum walk has efficient implementation
(poly-log n after preprocessing)
ACRŠZ ’10
√n e√log n
today
√n log n