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Quantum walk algorithms: element distinctness and spatial search
Andris AmbainisInstitute for Advanced Study,
Princeton
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Today’s talk
New technique for quantum algorithms.Quantum walks (q. counterparts of random walks).Element distinctness.Spatial search.
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Element distinctness
Determine if x1, x2, ..., xN contains two equal numbers.Classically: N questions.Quantum: O(N2/3).
0 1 0 0...x1 x2 xnx3
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Spatial search
N items on √N*√N grid.Some items marked.Find marked item.Grover: Ω(N).O(√N log N) time in 2D.O(√N) time in 3D.
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Random walk on line
Start in location 0.In every step, move left with probability ½, move right with probability ½.
-2 -1 0 1 2... ...
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Random walk on line
State (x, d), x –location, d-direction.At each step,
Let d=left with prob. ½, d=right w. prob. ½.(x, left) => (x-1, left);(x, right) => (x+1, right).
-2 -1 0 1 2... ...
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Quantum walk on line
States |x, d⟩, x –location, d-direction.
-2 -1 0 1 2... ...
>−>>→
>+>>→
rightleftright
rightleftleft
|2
1|2
1|
|2
1|2
1|
+→−→rightxrightxleftxleftx
,1,,1,
“Coin flip”:
Shift:
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Classical vs. quantum
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
0,18
0,2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0,00E+00
5,00E-02
1,00E-01
1,50E-01
2,00E-01
2,50E-01
3,00E-01
3,50E-01
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Run for t steps, measure the final location.
Distance: Θ(√N) Distance: Θ(N)
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Quantum walks on general graphs
1. Unitary “coin flip” on |e⟩.
2. Shift|v ⟩ |e⟩ → |u ⟩ |e⟩,u – other endpoint of edge e.
States:
,ev
e- edge from v.
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Element distinctness
Numbers x1, x2, ..., xN.
Determine if two of them are equal.Well studied problem in classical CS.Classically: N questions.
7 9 2 1...x1 x2 xNx3
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Element distinctness
[Buhrman et.al., 2001]: O(N3/4) quantum algorithm.[Shi, 2002]: Ω(N2/3) quantum lower bound.This talk: O(N2/3).
7 9 2 1x1 x2 xNx3
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Element distinctness as search on a graph
Vertices: S⊆1, ..., N of size N2/3 or N2/3+1.Edges: (S,T), T=S∪i.Marked: S containsi, j,xi=xj.In one step, we can
Check if vertex marked; orMove to adjacent vertex.
1,2
1,3
1,4
1, 2, 3
1, 2, 4
N2/3 N2/3+1
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Element distinctness as searchon a graph
Finding a marked vertex in M steps => element distinctness in M+N2/3 steps.At the beginning, read all xi
Can check if vertex marked with 0 queries.Can move to neighbour with 1 query.
1,2
1,3
1,4
1, 2, 3
1, 2, 4
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Quantum walk search [Shenvi, Kempe, Whaley, 2003]
Start with a uniform superposition over all S.Apply one transition rule if S marked, another if S not marked.Quantum walk leads to a state in which marked S have higher amplitudes.
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Walk on subsets
1.“Coin flip” unitary on k.
1,2
1,3
1,4
1, 2, 3
1, 2, 4
iSix
∈⊗States
3. “Coin flip” unitary on k.
2. |S⟩ |k⟩ ⇒ |S∪k⟩ |k⟩,
4. |S⟩ |k⟩ ⇒ |S-k⟩ |k⟩,erase xk.
kS
query xk.
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Quantum “coin flip”2 2 21 ...
2 2 21 ...
... ... ... ...2 2 2... 1
M M M
M M M
M M M
− + − +
− +
Restricted to k∈S or k∉S
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Algorithm for element distinctness
Prepare
O(N1/3) times:|S> → -|S> if S contains i, j s.t. xi = xj;O (N1/3) steps of quantum walk.
2 / 3| | ,i S i
S Nk S
S k x∈=
∉
⊗∑
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Analysis of algorithm
Assume unique i, j s.t. xi =xj.1. Simplify analysis by symmetry.2. Analysis of 1 quantum walk step.3. Analysis of entire algorithm.
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Symmetry5 types of states |S>|k>, k ∉ S:
i, j∩S=0, k≠i, k≠j.i, j∩S=0, k=i or k=j. i, j∩S=1, k≠i, k≠j.i, j∩S=1, k=i or k=j. i, j∩S=2.
States of each type have equal amplitudes (symmetry, induction).
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Symmetry
For each of 5 types, take the uniform superposition of all |S⟩|k⟩.At any time, the state of algorithm is a superposition of |Ψ1⟩, |Ψ2⟩, |Ψ3⟩, |Ψ4⟩, |Ψ5⟩.Suffices to analyze 5-dimensional subspace.
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Analysis of quantum walk
One step of q. walk is described by 5*5 matrix.Find eigenvalues and eigenvectors of this matrix.
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Analysis of quantum walk
One eigenvector is a uniform superposition of all |S>|k>, k ∉ S, with eigenvalue 1.The other eigenvalues are eiθ1, e-iθ1, eiθ2, e-iθ2 .
1 1 2 21 21/3 1/3,
| | | |C C C C
N NS Sθ θ= = = =
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N1/3 steps of quantum walk
The uniform superposition of all |S>|k>, k ∉ S with eigenvalue 1.The other eigenvalues are eiθ1, e-iθ1, eiθ2, e-iθ2 .
1 1 2 2,C Cθ θ= =
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Algorithm for element distinctness
Prepare
O(N1/3) times:|S> → -|S> if S contains i, j s.t. xi = xj;O (N1/3) steps of quantum walk.
2 / 3| | ,i S i
S Nk S
S k x∈=
∉
⊗∑
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Analysis of entire algorithm
,all
S kS kψ = ∑
, :,
markS ki j S
S kψ∈
= ∑
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Analysis of entire algorithmStart in |Ψall>.O(N1/3) times repeat:
|Ψmark> → - |Ψmark>.Rotate the subspace orthogonal to |Ψall> by eiθ, |θ|≥const.
| Ψmark>
| Ψall>
Lemma The final state has a constant overlapwith |Ψmark>.
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Main lemmaLemma The final state has a constant overlapwith |Ψmark>.
General statement; applies to any sequence of 2 transformations.
Examples: Grover, element distinctness, other search problems?
Lemma can be used as a black box.
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Handling multiple collisions
What if multiple i, j: xi = xj?Sample part of xi, i∈1, 2, …, N to get unique i, j: xi = xj.
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Element k-distinctness
Numbers x1, x2, ..., xN.
Determine if there are k equal elements.Similar algorithm solves the problem with O(N(k-1)/k) queries.
7 9 2 1...x1 x2 xNx3
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Related work
[Childs, Eisenberg, 2003, Santha 2004]: different analysis.[Magniez, Santha, Szegedy, 2003]: triangle finding.[Buhrman, Spalek, 2003]: testing matrix product.
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Triangle finding [Magniez, Santha, Szegedy, 03]
Graph G with n vertices. We want to know if G contains a triangle.O(n2) time classically.O(n1.3) time quantum algorithm.Uses element distinctness as black box.
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Testing matrix multiplication [Buhrman, Spalek 03]
n*n matrices A, B, C.Does A*B=C?Classically: O(n2) time.Quantum: O(n1.67) time.Uses quantum walk on sets of columns/rows.
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Grover search
Find i for which xi=1. Questions: ask i, get xi.Classically, n questions.Quantum, O(√n) questions [Grover, 1996].
0 1 0 0...x1 x2 xnx3
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Quantum search on grids [Benioff, 2000]
√n* √n grid.Distance between opposite corners = 2√n.Grover’s algorithm takes
steps.No quantum speedup.
nnn =∗
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Quantum search on grids
[Aaronson, A, 2003] non-quantum walk algorithm.O(√N log2 N) time algorithm for 2D grid.O(√N) time algorithm for 3 and more dimensions.
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Quantum search on grids
[Childs, Goldstone, 2003]: continuous-time quantum walk.O(√N log N) time algorithm for 4D grid.O(√N) time algorithm for 5 and more dimensions.
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Quantum walks on grids
This talk: discrete-time quantum walk.O(√N log N) time algorithm for 2D grid.O(√N) time algorithm for 3 and more dimensions.Improves over [Aaronson, A].Shows difference between discrete and continous time quantum walks.
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Quantum walk on grid
Basis states |x,y,←>, |x, y, →>, |x, y, ↑>, |x, y, ↓>.Coin flip on direction:
−
−
−
−
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
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Quantum walk on grid
Shift:|x, y, ←> ⇒ |x-1, y, →>|x, y, →> ⇒ |x+1, y, ←>|x, y, ↑> ⇒ |x, y-1, ↓>|x, y, ↓> ⇒ |x, y+1, ↑>
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Search by quantum walk
Perform a quantum walk with different “coin flip” transformation in marked locations.After steps, measure the state.Gives marked |x, y, d> with prob. 1/log N.In 3 and more dimensions, O(√N) steps, constant probability.
( ) log NNO
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Discrete time quantum walks
State |x, y>|d>, with (x, y) being location, d – direction (←, ↑, →, ↓).
“Coin flip” on |d>;Modify |x, y> dependant on |d>.
Many possible transformations for “coin flip”, with different results.
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Different quantum walkSame coin flip Different shift
|x, y, ↑ ⟩ → |x-1, y, ↑ ⟩|x, y, ↓ ⟩ → |x+1, y, ↓⟩|x, y, ← ⟩ → |x, y-1, ←⟩|x, y, → ⟩ → |x, y+1, →⟩
−
−
−
−
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
Different coin flip for marked locations
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Different quantum walk
Claim The probability of being in marked location never exceeds 2/N.
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Application: set disjointness
Alice has set A⊆1, 2, ..., N, Bob has B⊆1, 2, ..., N.They want to know if there is i: i∈A, i∈B.How many qubits of communication?[Buhrman et.al., 97]: O(√N log N).[Hoyer, de Wolf 02]: O(√N clog*N).[Razborov 02]: Ω(√N).[Aaronson, A, 03]: O(√N).
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Set disjointness
Cube of volume N.Divide in N subcubes.Alice writes 1 in ithsubcube if i∈A.Bob writes 1 in ithsubcube if i∈Β.
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Set disjointness
Is there a location where both Alice and Bob have 1?Alice and Bob run O(√N) algorithm for 3D search.Each step - 5 qubit communication.
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More information
Element distinctness – A, quant-ph/0311001.Search on grid – A, Kempe, Rivosh, Shenvi, coming soon.
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Open problems
What is the complexity of finding if there are k equal items xi1
= … = xik?
Algorithm: O(N(k-1)/k).Lower bound: Ω(N2/3).
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Open problemsOur element distinctness algorithm uses O(N2/3) space.Algorithm with less space?Space restricted to M items:
Quantum: O(N/ √M) queries.Classical: O(N2/M) queries.
Quantum speeds up time but not space.Quantum lower bounds on space?
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Open problems
On 2-D grid, why one coin succeeds and the other fails? Any correspondence to physics?How to handle multiple marked states in quantum walk algorithms?Can we speed up classical Markov chain algorithms (approximating permanent)?