Quantum algorithms for evaluating Boolean formulas

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Quantum algorithms for evaluating Boolean formulas

• Andris Ambainis
• University of Latvia
• (joint work with Andrew Childs, Ben Reichardt,
  Robert Spalek, Shengyu Zhang)

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AND-OR tree
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Evaluating AND-OR trees

• Variables xi accessed by queries to a black box
• Input i
• Black box outputs xi.
• Quantum case
• Evaluate T with the smallest number of queries.

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Motivation

• Vertices chess positions
• Leaves final positions
• xi1 if the 1st player wins
• At internal vertices, AND/OR evaluates whether
  the player who makes the move can win.

How well can we play chess if we only know the
position tree?
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Results

• Full binary tree of depth d.
• N2d leaves.
• Deterministic ?(N).
• Randomized SW,S ?(N.753).
• Quantum?
• Easy q. lower bound ?(?N).

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NAND trees
ANDs, ORs can be replaced by NANDs
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New results

• Farhi, et al.O(?N) time algorithm for NAND
  trees in Hamiltonian query model.

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Farhi, Goldstone, Gutmann
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Farhi, Goldstone, Gutmann

• Basis states v?, v vertices of augmented tree.
• Hamiltonian H, H-adjacency matrix of augmented
  tree.

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Farhi, Goldstone, Gutmann

• Starting state ?? on the infinite line left of
  tree.
• Apply Hamiltonian H for O(?N) time.
• If T1, the new state is on the right
  (transmission).
• If T0, the new state is on the left
  (reflection).

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Improvements to FGG

• A. Childs, B. Reichardt, R. Spalek, S. Zhang.
  arXivquant-ph/0703015.
• A. Ambainis, arXiv0704.3628.

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Improvement I
Quantum algorithm for unbalanced trees.
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Improvement II
Farhi, Goldstone, Gutmann
O(?N) time Hamiltonian quantum algorithm
O(N1/2o(1)) query quantum algorithm
We will design discrete query algorithm directly.
Why do we need that???
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Motivation

• Vertices chess positions
• Leaves final positions
• xi1 if the last player loses
• At internal vertices, NAND evaluates whether the
  player who makes the move can win.

How well can we play chess if we only know the
position tree?
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Motivation

• Hamiltonians
• We can perform H for arbitrarily small time ?gt0.

• Game tree
• Query evaluating a final position, phase shift
  by 1 conditional on the result.
• Performing a smaller phase shift is not easier!
• Evaluating a position takes the same time.

Inherently discrete time problem
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Improvement III

• FGG algorithm seems very different from the
  previous algorithms.
• We relate it to search, amplitude amplification
  and quantization of Markov chains.
• Better understanding of FGG.

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Farhi, Goldstone, Gutmann
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The finite tail Childs et al.

Finite tail in one direction
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Finite tail algorithm

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What happens?

• If T0, the state stays almost unchanged.
• If T1, the state scatters into the tree.

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More formally

• If T0, there is a state ?? with H??0 and
  ????start?.
• If T1, then for every ??, H?????, either
  ?gt1/?N or ????start?.

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Eigenvalue estimation

• Algorithm that, given a state ??, H?????,
  outputs an estimate of ? within ? by running H
  for time O(1/?).
• In our case ? 1/?N.
• Time O(?N) in Hamiltonian model, for the balanced
  NAND tree (same as FGG).

Same result, different intuition
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The next steps

• Discrete query algorithm.
• Algorithm for computing arbitrary NAND formulas.

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From Hamiltonians to unitaries
HH0H1
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From Hamiltonians to unitaries

• Replace H0, H1 by unitary transformations U0, U1.
• Instead of estimating the eigenvalue of HH0H1,
  estimate the eigenvalue of UU1U0.

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Designing U0

• Input-independent part of tree.
• Define U0???? if H0??0.
• Define U0??-?? if H0?????, ??0.
• H0- adjacency matrix.
• 0 queries required.

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Designing U1

• No extra edges.
• Define U1v? -v? if v - leaf with
  xi1.
• Define U1v? v? otherwise.
• 1 query

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Results (balanced case)

• If T0, there is a state ?? with U1U0????
  and ????start?.
• If T1, then for every ??, U1U0??ei???,
  either ?gt1/?N or ????start?.

Eigenvalue estimation, O(?N) repetitions of U1U0
O(?N) queries
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Structure of our algorithm

• Transformation U1U0.
• U0??-?? if H0?????, ??0.
• U1v? -v? if v - leaf with xi1.
• U0, U1 leave vectors unchanged or map them to
  their opposites.

U0, U1 - reflections
Same as in Grovers algorithm
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Two reflections in 2D
Eigenvalues e?i?
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Our algorithm

• The entire state space can be decomposed into two
  dimensional subspaces.
• Need to prove if T1, the angle between the two
  bases is gt1/?N in each subspace.

Eigenvalues e?i?, ?gt1/?N
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Our algorithm

• Angle gt length of projection.

?
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Our algorithm

• Need to prove if T0, there exists ??, H??0,
• S subspace spanned by v?, xi1.

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Our algorithm

• Need to prove if T1, then, for any ??, H??0,
• S subspace spanned by v?, xi1.

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Computing arbitrary NAND formulas
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Arbitrary NAND formulas

• H0- weighted adjacency matrix
• 1s are replaced by positive weights that depend
  on tree structure.
• Define U0??-?? if H0?????, ??0.

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Results (general trees)

• Similar but more complicated analysis.
• O(?Nd) query algorithm.

What if d is large??
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Bshouty, Cleve, Eberly, 91

• Theorem Let F be a formula of size S, depth d.
  There is a formula F, FF,
• Size(F)O(S1?), Depth(F)O(log S).
• Size(F) , Depth(F)

O(N1/2?) quantum algorithm for any formula F
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Summary

• O(?N) query quantum algorithm for balanced NAND
  trees.
• O(?Nd) query quantum algorithm for any depth-d
  NAND tree.
• query quantum algorithm
• for any NAND tree.

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Implications for Boolean formulas

• Any Boolean formula of size S can be evaluated
  with queries.

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Lee et al., 2005

• If quantum adversary method shows that a
  Boolean function requires S queries in quantum
  query model, the classical formula size is S2.

• Not a coincidence smaller formula can be
  converted into a quantum algorithm with less than
  S queries!

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Two reflections strike again

• Aharonov, 98 Analysis of Grovers algorithm
• Other applications
• Amplitude amplification.
• Quantization of Markov chains.
• Now NAND formulas.