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��¢ quantum algorithms with amplitude amplification [Brassard+ 1999]¢ quantum algorithms for element disjointness [Ambainis 2002]¢ quantum algorithms for Gauss sums [van Dam + 2002] ¢ quantum algorithms for solving Pell’s equation [Hallgren 2002]¢ quantum algorithms for quantum simulations [Childs 2004]¢ quantum algorithms for hidden subgroups [Kuperberg 2004]¢ quantum algorithms for finding an unit group [Hallgren 2005]¢ quantum algorithms for triangle finding [Magniez+ 2005]¢ quantum algorithms for computing knot invariants [Aharonov+ 2006]¢ quantum algorithms for data streams [Le Gall 2006]¢ quantum algorithms for hidden nonlinear structures [Childs+ 2007]¢ quantum algorithms for linear equations [Harrow+ 2009]¢ quantum algorithms for group isomorphism [Le Gall 2010]¢ quantum algorithms for matrix multiplication [Le Gall 2011]¢ quantum algorithms using span programs [Belovs 2011]¢ quantum algorithms for matrix inversion [Ta-Shma 2013]¢ quantum algorithms for combinatorial triangle finding [Le Gall 2011]¢ quantum algorithms for pattern matching [Montanaro 2014]¢ quantum algorithms for distributed computation [Le Gall and Magniez 2017]¢ …

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�� ¢ quantum algorithms with amplitude amplification [Brassard+ 1999]¢ quantum algorithms for element disjointness [Ambainis 2002]¢ quantum algorithms for Gauss sums [van Dam + 2002] ¢ quantum algorithms for solving Pell’s equation [Hallgren 2002]¢ quantum algorithms for quantum simulations [Childs 2004]¢ quantum algorithms for hidden subgroups [Kuperberg 2004]¢ quantum algorithms for finding an unit group [Hallgren 2005]¢ quantum algorithms for triangle finding [Magniez+ 2005]¢ quantum algorithms for computing knot invariants [Aharonov+ 2006]¢ quantum algorithms for data streams [Le Gall 2006]¢ quantum algorithms for hidden nonlinear structures [Childs+ 2007]¢ quantum algorithms for linear equations [Harrow+ 2009]¢ quantum algorithms for group isomorphism [Le Gall 2010]¢ quantum algorithms for matrix multiplication [Le Gall 2011]¢ quantum algorithms using span programs [Belovs 2011]¢ quantum algorithms for matrix inversion [Ta-Shma 2013]¢ quantum algorithms for combinatorial triangle finding [Le Gall 2011]¢ quantum algorithms for pattern matching [Montanaro 2014]¢ quantum algorithms for distributed computation [Le Gall and Magniez 2017]¢ …

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,��¢ quantum algorithms with amplitude amplification [Brassard+ 1999]¢ quantum algorithms for element disjointness [Ambainis 2002]¢ quantum algorithms for Gauss sums [van Dam + 2002] ¢ quantum algorithms for solving Pell’s equation [Hallgren 2002]¢ quantum algorithms for quantum simulations [Childs 2004]¢ quantum algorithms for hidden subgroups [Kuperberg 2004]¢ quantum algorithms for finding an unit group [Hallgren 2005]¢ quantum algorithms for triangle finding [Magniez+ 2005]¢ quantum algorithms for computing knot invariants [Aharonov+ 2006]¢ quantum algorithms for data streams [Le Gall 2006]¢ quantum algorithms for hidden nonlinear structures [Childs+ 2007]¢ quantum algorithms for linear equations [Harrow+ 2009]¢ quantum algorithms for group isomorphism [Le Gall 2010]¢ quantum algorithms for matrix multiplication [Le Gall 2011]¢ quantum algorithms using span programs [Belovs 2011]¢ quantum algorithms for matrix inversion [Ta-Shma 2013]¢ quantum algorithms for combinatorial triangle finding [Le Gall 2011]¢ quantum algorithms for pattern matching [Montanaro 2014]¢ quantum algorithms for distributed computation [Le Gall and Magniez 2017]¢ …

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E/�!� ��¢ quantum algorithms with amplitude amplification [Brassard+ 1999]¢ quantum algorithms for element disjointness [Ambainis 2002]¢ quantum algorithms for Gauss sums [van Dam + 2002] ¢ quantum algorithms for solving Pell’s equation [Hallgren 2002]¢ quantum algorithms for quantum simulations [Childs 2004]¢ quantum algorithms for hidden subgroups [Kuperberg 2004]¢ quantum algorithms for finding an unit group [Hallgren 2005]¢ quantum algorithms for triangle finding [Magniez+ 2005]¢ quantum algorithms for computing knot invariants [Aharonov+ 2006]¢ quantum algorithms for data streams [Le Gall 2006]¢ quantum algorithms for hidden nonlinear structures [Childs+ 2007]¢ quantum algorithms for linear equations [Harrow+ 2009]¢ quantum algorithms for group isomorphism [Le Gall 2010]¢ quantum algorithms for matrix multiplication [Le Gall 2011]¢ quantum algorithms using span programs [Belovs 2011]¢ quantum algorithms for matrix inversion [Ta-Shma 2013]¢ quantum algorithms for combinatorial triangle finding [Le Gall 2011]¢ quantum algorithms for pattern matching [Montanaro 2014]¢ quantum algorithms for distributed computation [Le Gall and Magniez 2017]¢ …

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F. Le Gall and F. Magniez. Sublinear-Time Quantum Computation ofthe Diameter in Distributed Networks. Proceedings of the 37th ACMSymposium on Principles of Distributed Computing (PODC 2018).Also arXiv: 1804.02917.

Outline: Quantum Distributed Computingü From the perspective of computability and computational complexity,

quantum distributed computing has mostly been studied in the framework of 2-party communication complexity

ü Relatively few results focusing on more than two parties:

Ø exact quantum protocols for leader election on anonymous networks [Tani, Kobayashi, Matsumoto 2007]

Ø study of quantum distributed algorithms on non-anonymous networks

[Gavoille, Kosowski, Markiewicz 2009][Elkin, Klauck, Nanongkai, Pandurangan 2014]

negative results: show impossibility of quantum distributed computing faster than classical distributed computing for many important problems (shortest paths, minimum spanning tree,…)

Question: can quantum distributed computing be useful? (on non-anonymous networks�

Our result: Yes, we can compute the diameter of the network faster!

Eccentricity and DiameterConsider an undirected and unweighted graph G = (V,E) with n nodes

a

cb d

ef

g

D = max {d(u,v)}u,v ∈ "

The diameter of the graph is the maximum distance between two nodes(��)

d(u,v) = distance between u and v

Eccentricity and DiameterConsider an undirected and unweighted graph G = (V,E) with n nodes

a

cb d

ef

g D = 4

D = max {d(u,v)}u,v ∈ "

The diameter of the graph is the maximum distance between two nodes(��)

The eccentricity of a node u is defined as (��) ecc (u) = max {d(u,v)}

v ∈ "ecc (a) = 3ecc (b) = 3ecc (c) = 2ecc (d) = 3ecc (e) = 3ecc (f ) = 4ecc (g) = 4

d(a,a) = 0d(a,b) = 2d(a,c) = 1d(a,d) = 2d(a,e) = 2d(a, f) = 3d(a,g) = 3

d(u,v) = distance between u and v= max {ecc (u)}u ∈ "

Classical Distributed Computation of the Eccentricity

ü Each node represents a processor (with a unique ID)ü Each edge represents a classical channelü At each round only one short (i.e., O(log n) bits) message sent to each neighbor

CONGEST model (most standard model of synchronous distributed computation)

a

cb d

ef

g

ecc (a) = 3ecc (b) = 3ecc (c) = 2ecc (d) = 3ecc (e) = 3ecc (f ) = 4ecc (g) = 4

Complexity: the number of rounds needed for the computation

example: at each round, a can send one message to c

Let’s write n = number of nodes

Computing eccentricities and the diameter are among the most fundamental tasks

D = 4

b can send one message to cc can send one message to a, one to b, one to d and one to e…..

Classical Distributed Computation of the Eccentricity

ü Each node represents a processor (with a unique ID)ü Each edge represents a classical channelü At each round only one short (i.e., O(log n) bits) message sent to each neighbor


a

cb d

ef

g

“a,0”

d(a,c) = 1

d(a,b) = 2d(a,d) = 2

d(a,e) = 2d(a,f) = 3

d(a,g) = 3

“a,1” “a,1”

“a,1”“a,2”

“a,2”

d(a,a) = 0

Computation of ecc (u):Starting with u, each node broadcasts its distance to u to its neighbors.(Each node knows its distance to u the first time it receives a message.)

The nodes then compute the maximum of their distance (easy)

complexity: ecc (u) rounds

example: computation of ecc (a) first round of communicationsecond round of communicationthird round of communication



Computing eccentricities and the diameter are among the most fundamental tasks

Classical Distributed Computation of the Diameter

ü Each node represents a processor (with a unique ID)ü Each edge represents a classical channel


a

cb d

ef

g

Computation of the diameter D:All the nodes compute simultaneously their eccentricity

Output the maximum eccentricity

“a,0”

first round of communication

“d,0”“b,0”

Congestion: what should c send at the next round? using standard

techniques to handle congestionscomplexity: Θ(n) rounds (even if D is constant)

[Holzer+12, Peleg+12]

ü At each round only one short (i.e., O(log n) bits) message sent to each neighbor


Complexity: the number of rounds needed for the computationComputing eccentricities and the diameter are among the most fundamental tasks

d(a,a) = 0

d(b,b) = 0 d(d,d) = 0

d(e,e) = 0

“e,0”

d(g,g) = 0d(f,f) = 0

d(c,c) = 0

“a,1” “b,1” “d,1” “e,1”

Computation of the Diameter

Classical Quantum (our results)

Exact computation (upper bounds) !(#)[Holzer+12, Peleg+12]

!( #%)

~

3/2-approximation (upper bounds) !( # + %)[Lenzen+13,Holzer+14]

!(' #% + %)

(3/2-ε)-approximation (lower bounds) (Ω(#)[Holzer+12,Abboud+16]

(Ω( # + %) [unconditional]

Exact computation (lower bounds) (Ω(#)[Frischknecht+12]


the tilde notation removes polylog(n) factors condition: holds for algorithms using only polylog(n) qubits of memory per node

main result: sublinear-round quantum computation of the diameter whenever D=o(n)

(Ω( #%) [conditional]

(our algorithm uses O((log n)2) qubits of quantum memory per node)

first gap between classical and quantum for an important problem in the CONGEST model of distributed computing

number of rounds needed to compute the diameter (n: number of nodes, D: diameter)


main result: sublinear-round quantum computation of the diameter whenever D=o(n)

Our Upper Bound



!( #%)

(our algorithm uses O((log n)2) qubits of quantum memory per node)



Quantum Distributed Computation of the Diameter

ü Each node represents a quantum processor (with a unique ID)ü Each edge represents a quantum channel

quantum CONGEST model

ü At each round only one short (O(log n) qubits) message sent to each neighbor



a

cb d

ef

g

Quantum Distributed Computation of the Diameter

ü Each node represents a quantum processor (with a unique ID)ü Each edge represents a quantum channel

quantum CONGEST model

ü At each round only one short (O(log n) qubits) message sent to each neighbor



Given an integer d, decide if diameter ≥ d

Computation of the diameter (decision version)

there is a vertex u such that ecc (u) ≥ d

This is a search problem, so we can try to use Grover search:

Find an element u∈ " such that f(u) = 1 with f(u) = 1 if ecc (u) ≥ d0 otherwise

can be done locally(i.e., without communication)

Usual Grover Algorithm (from Nielsen-Chuang, page 251)

Find an element u∈ " such that f(u) = 1 with f(u) = 1 if ecc (u) ≥ d0 otherwise

m m

O( n) timesm = O(log n)

remember: n = |V|

m mm

G ≡

independent of f

independent of f

depends on f(depends on the graph)

solutionWe compute of the diameter by implementing this circuit in the distributed setting:One arbitrary node (say, node a) will implement this circuit

independent of f



m m


m mm

G ≡

independent of f

independent of f


solution

independent of f

To implement the oracle, the node a needs to communicate with the other nodesTotal number of rounds of communication = O( n x number of rounds to implement the oracle)

We compute of the diameter by implementing this circuit in the distributed setting:One arbitrary node (say, node a) will implement this circuit

Implementation of the Oracle in O(D) rounds

oracle⟩|# ⟩|0 ⟩|# ⟩|%&&(#))*∈,

⟩-*|# ⟩|0 )*∈,

⟩-*|# ⟩|%&&(#)

a

cb d

ef

g

V={a,b,c,d,e,f,g}1. “Broadcast” this state, which gives

)*∈,

-* ⟩|# a ⟩|# b ⟩|# c ⟩|# d ⟩|# e ⟩|# f ⟩|# g

The nodes implements the classical protocol for computing the eccentricity, which gives

)*∈,

-* ⟩|# a ⟩|# b ⟩|# c ⟩|# d ⟩|# e ⟩|# f ⟩|# g ⟩|%&&(#) a

2.

[ecc(a) ≤ D rounds]

[≤ D rounds]

Initially node a owns )*∈,

-* ⟩|# a

a

c

b d

e

g

f

Node a introduces 1 register )*∈,

-* ⟩|# a ⟩|0

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-* ⟩|# a ⟩|#

Node a sends the second register to c )*∈,

-* ⟩|# a ⟩|# c

)*∈,

-* ⟩|# a ⟩|# c ⟩|0 ⟩|0 ⟩|0Node c introduces 3 registers)

*∈,-* ⟩|# a ⟩|# c ⟩|# ⟩|# ⟩|#Node c applies CNOTS

)*∈,

-* ⟩|# a ⟩|# c ⟩|# b ⟩|# e ⟩|# dNode c sends the registers to b,e,d……

Implementation of the Oracle in O(D) rounds

oracle

a

cb d

ef

g

V={a,b,c,d,e,f,g}1. “Broadcast” this state, which gives

!"∈$

%" ⟩|( a ⟩|( b ⟩|( c ⟩|( d ⟩|( e ⟩|( f ⟩|( g

The nodes implements the classical protocol for computing the eccentricity, which gives

!"∈$

%" ⟩|( a ⟩|( b ⟩|( c ⟩|( d ⟩|( e ⟩|( f ⟩|( g ⟩|011(() a

2.

[ecc(a) ≤ D rounds]

3. The nodes revert Step 1 [ecc(a) ≤ D rounds]

!"∈$

%" ⟩|( a ⟩|0 a !"∈$

%" ⟩|( a ⟩|011(() a

Initially node a owns !"∈$

%" ⟩|( a

[≤ D rounds]



m m


m mm

G ≡

independent of f

independent of f


solution

independent of f

To implement the oracle, the node a needs to communicate with the other nodesTotal number of rounds of communication = O( n x number of rounds to implement the oracle)

= O( n x D)

We compute of the diameter by implementing this circuit in the distributed setting:One arbitrary node (say, node a) will implement this circuit

The Upper Bound



!( #%)

ü We have just described a O( # x D)-round quantum distributed algorithm for computing (with high probability) the diameter

ü With further work, the complexity can be reduced to O( #% ) rounds

The Lower Bounds

ü reduce DISJOINTNESS to the distributed computation of diameter [Frischknecht+12]

classical lower bound

ü the (two-party) communication complexity of DISJOINTNESS is Ω(n) bits [Kalyanasundaram+92]

unconditional quantum lower boundü same reduction from DISJOINTNESS to the diameterü the (two-party) communication complexity of DISJOINTNESS is Ω( !) qubits [Razborov03]


Exact computation (upper bounds) "(!)[Holzer+12, Peleg+12]

"( !%)

~

Exact computation (lower bounds) &Ω(!)[Frischknecht+12]

&Ω( ! + %) [unconditional]&Ω( !%) [conditional]

via two-party communication complexity

conditional quantum lower boundü if the quantum distributed algorithm for diameter uses few quantum memory per

node, then the computation of DISJOINTNESS can be done using few messages ü the (two-party) r-message communication complexity of DISJOINTNESS is

Ω() + !/)) qubits [Braverman+15]

Quantum Computation of the Diameter: Summary



!( #%)

~

3/2-approximation (upper bounds) !( # + %)[Lenzen+13,Holzer+14]

!(' #% + %)

(3/2-ε)-approximation (lower bounds) (Ω(#)[Holzer+12,Abboud+16]


Exact computation (lower bounds) (Ω(#)[Frischknecht+12]


main result: sublinear-round quantum computation of the diameter

(Ω( #%) [conditional]



ü Our upper bounds are obtained by showing how to implement quantum search in a distributed setting

ü Interesting research direction: find other applications of this technique

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東京大学¢ quantum algorithms for combinatorial triangle finding [le gall 2011] ¢ quantum...

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