path detection: a quantum algorithmic primitive

Path Detection: A Quantum Algorithmic Primitive

Shelby KimmelMiddlebury College

Based on work with Stacey Jeffery: arXiv: 1704.00765 (Quantum vol 1 p 26)Michael Jarret, Stacey Jeffery, Alvaro Piedrafita, arXiv:1804.10591 (ESA 2018)Kai DeLorenzo, Teal Witter, arXiv:1904.05995 (TQC 2019)

Primitives!

• Quantum algorithmic primitives1. Widely applicable2. Can be used in a black box manner (with easily

analyzable behavior)

Primitives!



– Ex: Searching unordered list of 𝑛 items– Classically, takes Ω(𝑛) time– Quantumly, takes 𝑂( 𝑛) time

Primitives!



– Ex: Searching unordered list of 𝑛 items– Classically, takes Ω(𝑛) time– Quantumly, takes 𝑂( 𝑛) time

Good primitive: 𝒔𝒕-connectivity

Outline:

A. Introduction to st-connectivityB. st-connectivity makes a good algorithmic primitive

1. Widely applicable2. Easy to analyze

Outline:



Applications:• Read-once Boolean formulas (query optimal) [JK] • Total connectivity (query optimal) [JJKP]• Cycle detection (query optimal) [DKW]• Even length cycle detection [DKW]• Bipartiteness (query optimal) [DKW]• Directed st-connectivity (query optimal) (Beigi et al ‘19)• Directed smallest cycle (query optimal) (Beigi et al ‘19)

Outline:

Applications:• Topological sort (Beigi et al ‘19)• Connected components (Beigi et al ‘19)• Strongly connected components (Beigi et al ‘19)• k-cycle at vertex v (Beigi et al ‘19)• st-connectivity (Reichardt, Belovs ‘12)

𝑠𝑡-connectivity

𝑠

𝑡

𝒔𝒕 − 𝒄𝒐𝒏𝒏𝒆𝒄𝒕𝒊𝒗𝒊𝒕𝒚:is there a path from 𝑠 to 𝑡?

Input to Algorithm

Bit String:

𝑥! 𝑥" 𝑥#…

Input to Algorithm

Bit String:

𝑥! 𝑥" 𝑥#101

…

Input to Algorithm

𝑥! 𝑥" 𝑥$

Skeleton Graph

Bit String:2

83 1

3

2

13

𝑠

𝑡

Input to Algorithm

2

1

3

2

13

𝑠

𝑡

𝑥! 𝑥" 𝑥$101

Skeleton Graph

Bit String:

83

Input to Algorithm

2

1

3

2

13

𝑠

𝑡

𝑥! 𝑥" 𝑥$100

Skeleton Graph

Bit String:

83

Input to Algorithm

2

1

3

2

13

𝑠

𝑡

𝑥! 𝑥" 𝑥$1

Skeleton Graph

Bit String:

• Bit string initially hidden, can query value of string at each bit.

83

Input to Algorithm

2

1

3

2

13

𝑠

𝑡

𝑥! 𝑥" 𝑥$1

Skeleton Graph

Bit String:

• Bit string initially hidden, can query value of string at each bit.

83

𝑂! 𝑖 𝑏 = 𝑖 𝑏 + 𝑥"

Algorithm

𝑥! 𝑥" 𝑥$

Skeleton Graph

𝑂%for Bit String:

2

83 1

3

2

13

𝑠

𝑡

Q. Algorithm to Solve

connectivityYes/No

Outline:



Cycle Detection

Is there a cycle?

Yes

Cycle Detection

Is there a cycle?

No

Cycle Detection

1

2 3

4

5

Is there a cycle?

Input to Cycle Detection: • Skeleton graph• Hidden bit string

𝑥!𝑥"𝑥$𝑥+𝑥,

𝑥- = 1 ↔edge 𝑖 is present

Cycle Detection

Is there a cycle through edge 1?

1

2 3

4

5

Cycle Detection

There is a cycle through Edge 1 iff • Edge 1 is present• Path between the

endpoints of Edge 1not using Edge 1

1

2 3

4

5


Cycle Detection

There is a cycle through Edge 1 iff • Edge 1 is present• Path between the

endpoints of Edge 1not using Edge 1

1

2 3

4

5


1

2

3

4

5

𝑠

𝑡

Edge 1 is present

Path between the endpoints of Edge 1 not using Edge 1

Cycle Detection

Is there a cycle?

There is a cycle if there is a cycle through some edge

1

2 3

4

5

Cycle Detection

Is there a cycle?

There is a cycle if there is a cycle through some edge

1

2 3

4

5

1

2

3

4

5

𝑠

𝑡

12

3

4 5

2

3

4

5

1

1

2

3

4

51

2

3

5

4

Boolean Formulas

⋀

⋁

⋀ ⋀ ⋀

⋁ ⋁

𝑥#$

𝑥# 𝑥% 𝑥& 𝑥'𝑥( 𝑥) 𝑥* 𝑥+ 𝑥,

𝑓(𝑥)

1

0 0 1 0

0 1 0 1 1

𝒔

𝒕𝑥!0

𝑥!

𝑥"

𝑥$

𝑥+

𝑥,

𝑥1

𝑥2𝑥3

𝑥4

Outline:



Algorithm Analysis:

Space Complexity: 𝑂 log(# 𝑒𝑑𝑔𝑒𝑠 𝑖𝑛 𝑠𝑘𝑒𝑙𝑒𝑡𝑜𝑛 𝑔𝑟𝑎𝑝ℎ)

Algorithm Analysis:

Query Complexity:

• Bit string initially unknown• Minimum # of oracle uses to

determine w.h.p. on worst input

Algorithm Analysis:

Query Complexity:

• Bit string initially unknown• Minimum # of oracle uses to

determine w.h.p. on worst input

≈Time Complexity

Algorithm Analysis:

Query Complexity:

𝑂 max!"##$!%$& '

𝑅(,% 𝐺 max#"% !"##$!%$& '

𝐶(,% 𝐺

Effective resistance Effective capacitance

Algorithm Analysis:

Query Complexity:

𝑂 max!"##$!%$& '

𝑅(,% 𝐺 max#"% !"##$!%$& '

𝐶(,% 𝐺

Effective resistance Effective capacitance[Belovs, Reichardt, ’12]

[Jarret, Jeffery, Kimmel, Piedrafita, ’18]

Effective Resistance

𝑠

𝑡

1 unit of flow

1 unit of flow


Valid flow:• 1 unit in at 𝑠• 1 unit out at 𝑡• At all other nodes, zero

net flow

𝑠

𝑡

1 unit of flow

1 unit of flow

1

1

𝑓 1 − 𝑓

0


Flow energy:

N56758

𝑓𝑙𝑜𝑤 𝑜𝑛 𝑒𝑑𝑔𝑒 "

𝑠

𝑡

1 unit of flow

1 unit of flow

1

1

𝑓 1 − 𝑓

0


Flow energy:

N56758


Effective Resistance: 𝑅8,:(𝐺)• Smallest energy of any valid

flow from 𝑠 to 𝑡 on 𝐺.

𝑠

𝑡

1 unit of flow

1 unit of flow

1

1

𝑓 1 − 𝑓

0


𝑠

𝑡

1 unit of flow

1 unit of flow

1

1

1/2 1/2

0Flow energy:

N56758


Effective Resistance: 𝑅8,:(𝐺)• Smallest energy of any valid

flow from 𝑠 to 𝑡 on 𝐺.

Algorithm Analysis:

Query Complexity:

𝑂 max!"##$!%$& '

𝑅(,% 𝐺 max#"% !"##$!%$& '

𝐶(,% 𝐺


Effective Capacitance

Generalized cut:• 1 at 𝑠• 0 at 𝑡• Difference is 0 across edge

𝑠

𝑡

11 1

0𝑣


𝑠

𝑡

11 1

0𝑣

Potential energy:

N56758 -#

8;5<5:=# 7>?@A

𝑐𝑢𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 "


Potential energy:

N56758 -#

8;5<5:=# 7>?@A

𝑐𝑢𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 "

Effective Capacitance: 𝐶8,:(𝐺)• Smallest potential energy of

any valid generalized cut between 𝑠 and 𝑡 on 𝐺.

𝑠

𝑡

11 1

02/3

Algorithm Analysis:

Query Complexity:

𝑂 max!"##$!%$& '

𝑅(,% 𝐺 max#"% !"##$!%$& '

𝐶(,% 𝐺


Example

1

2

3

4

5

𝑠

𝑡

12

3

4 5

2

3

4

51

12

3

4

51

2

3

5

4

Cycle Detection 𝑂 maxB=##5B:56 C

𝑅8,: 𝐺 max#=: B=##5B:56 C

𝐶8,: 𝐺

Example

Cycle Detection

1

2

3

4

5

𝑠

𝑡

12

3

4 5

2

3

4

51

12

3

4

51

2

3

5

4

𝑂 maxB=##5B:56 C

𝑅8,: 𝐺 max#=: B=##5B:56 C

𝐶8,: 𝐺

Example

Cycle Detection

1

2

3

4

5

𝑠

𝑡

12

3

4 5

2

3

4

51

12

3

4

51

2

3

5

4

𝑂 maxB=##5B:56 C

𝑅8,: 𝐺 max#=: B=##5B:56 C

𝐶8,: 𝐺

Length k cycle• k parallel paths of length k• 1/𝑘 flow on each path• Effective resistance is 1

Example

Cycle Detection

1

2

3

4

5

𝑠

𝑡

12

3

4 5

2

3

4

51

12

3

4

51

2

3

5

4

𝑂 maxB=##5B:56 C

𝑅8,: 𝐺 max#=: B=##5B:56 C

𝐶8,: 𝐺

Multiple cycles Effective resistance is < 1

Example

Cycle Detection

1

2

3

4

5

𝑠

𝑡

12

3

4 5

2

3

4

51

12

3

4

51

2

3

5

4

𝑂 maxB=##5B:56 C

𝑅8,: 𝐺 max#=: B=##5B:56 C

𝐶8,: 𝐺

Example

Cycle Detection

1

2

3

4

5

𝑠

𝑡

12

3

4 5

2

3

4

51

12

3

4

51

2

3

5

4

𝑂 maxB=##5B:56 C

𝑅8,: 𝐺 max#=: B=##5B:56 C

𝐶8,: 𝐺

• 𝑂 𝑛% subgraphs corresponding to non-present edges: cut at top

• 𝑂 𝑛 subgraphs corresponding to present edges, cut could have 𝑂(𝑛%) edges in cut

Example


𝑅8,: 𝐺 max#=: B=##5B:56 C

𝐶8,: 𝐺

𝐶8,: 𝐺 = 𝑂(𝑛$)𝑅8,: 𝐺 = 𝑂(1)

Query complexity: 𝑂 𝑛$/"

(optimal – logarithmic improvement over previous algorithm)

Example


𝑅8,: 𝐺 max#=: B=##5B:56 C

𝐶8,: 𝐺

𝑅8,: 𝐺 = 𝑐𝑖𝑟𝑐𝑢𝑖𝑡 𝑟𝑎𝑛𝑘 E!

Circuit rank = min # of edges to cut to create a cycle free graph

• Quantum algorithm picks out critical topological parameter• If promised either large circuit rank or no cycle, then cycle

detection algorithm runs faster• Proved by 2nd year undergrads

Estimation Algorithm:

Quantum query algorithm to estimate effective resistance or effective capacitance of 𝐺. (Jeffery, Ito ’15)

Estimation Algorithm:

Quantum query algorithm to estimate effective resistance or effective capacitance of 𝐺. (Jeffery, Ito ’15)

Because effective resistance depends directly on circuit rank, we now have a quantum algorithm to estimate circuit rank.

Umm…Algorithm?

Umm…Algorithm?

Theorem 14. Let U(P, x) = (2Πker A − I)(2ΠH(x)− I). Fix X ⊆ {0, 1}N and f : X → {0, 1}, and let P be a span

program on {0, 1}N that decides f . Let W+( f , P) = maxx∈ f −1(1) w+(x, P) and W−( f , P) = maxx∈ f −1(0) w−(x, P).

Then there is a bounded error quantum algorithm that decides f by making O(!

W+( f , P)W−( f , P)) calls to U(P, x),and elementary gates. In particular, this algorithm has quantum query complexity O(

!W+( f , P)W−( f , P)).

Ref. [IJ16] defines the approximate positive and negative witness sizes, w̃+(x, P) and w̃−(x, P). These aresimilar to the positive and negative witness sizes, but with the conditions |w⟩ ∈ H(x) and ωAΠH(x) = 0relaxed.

Definition 15 (Approximate Positive Witness). For any span program P on {0, 1}N and x ∈ {0, 1}N, we definethe positive error of x in P as:

e+(x) = e+(x, P) := min"###ΠH(x)⊥|w⟩

###2

: A|w⟩ = τ

$. (14)

We say |w⟩ is an approximate positive witness for x in P if###ΠH(x)⊥ |w⟩

###2= e+(x) and A|w⟩ = τ. We define

the approximate positive witness size as

w̃+(x) = w̃+(x, P) := min"∥|w⟩∥2 : A|w⟩ = τ,

###ΠH(x)⊥ |w⟩###

2= e+(x)

$. (15)

If x ∈ P1, then e+(x) = 0. In that case, an approximate positive witness for x is a positive witness, andw̃+(x) = w+(x). For negative inputs, the positive error is larger than 0.

We can define a similar notion of approximate negative witnesses (see [IJ16]).

Theorem 16 ([IJ16]). Let U(P, x) = (2Πker A − I)(2ΠH(x) − I). Fix X ⊆ {0, 1}N and f : X → R≥0. Let P be a

span program on {0, 1}N such that for all x ∈ X, f (x) = w−(x, P) and define %W+ = %W+(P) = maxx∈X w̃+(x, P).

Then there exists a quantum algorithm that estimates f to accuracy ϵ and that uses %O&

1ϵ3/2

'w−(x) %W+

(calls to

U(P, x) and elementary gates.

A span program for st-connectivity An important example of a span program is one for st-connectivity,first introduced in [KW93], and used in [BR12] to give a new quantum algorithm for st-connectivity. Westate this span program below, somewhat generalized to include weighted graphs, and to allow the inputto be specified as a subgraph of some parent graph G that is not necessarily the complete graph. We allowa string x ∈ {0, 1}N to specify a subgraph G(x) of G in a fairly general way, as described in Section 2.2.In particular, for i ∈ [N], let

−→E i,1 ⊆ −→

E (G) denote the set of (directed) edges associated with the literal xi,and

−→E i,0 the set of edges associated with the literal xi. Note that if (u, v, ℓ) ∈ −→

E i,b then we must also have(v, u, ℓ) ∈ −→

E i,b, since G(x) is an undirected graph. We assume G has some implicit weighting function c.Then we refer to the following span program as PG:

∀i ∈ [N], b ∈ {0, 1} : Hi,b = span{|e⟩ : e ∈ −→E i,b}

U = span{|v⟩ : v ∈ V(G)}τ = |s⟩ − |t⟩

∀e = (u, v, ℓ) ∈ −→E (G) : A|u, v, ℓ⟩ =

'c(u, v, ℓ)(|u⟩ − |v⟩) (16)

One can check that if s and t are connected, then if |w⟩ represents a weighted st-path or linear combinationof weighted st-paths in G(x), then |w⟩ is a positive witness for x. Furthermore, this is the only possibilityfor a positive witness, so x is a positive input for PG if and only if G(x) is st-connected, and in particular,w+(x, PG) = 1

2 Rs,t(G(x)) [BR12]. Since the weights c(e) are positive, the set of positive inputs of PG areindependent of the choice of c, however, the witness sizes will depend on c.

10

Span Program

Span Program->Unitary U = (reflection about space that depends on skeleton graph)(reflection about a space that depends on input)

Do phase estimation on U to precision 𝑂 max!"##$!%$& '

𝑅(,% 𝐺 max#"% !"##$!%$& '

𝐶(,% 𝐺

Open Questions and Current Directions

• How to choose edge weights? (Beigi et al ‘19)• Conditions when st-connectivity reduction optimal? • What is the classical time/query complexity of st-

connectivity in the black box model? Under the promise of small capacitance/resistance?

• Better estimation algorithm for st-connectivity effective resistance/capacitance

• Primitives/Pedagogical Problems?

Thank you!

Andrew Zhao

Teal Witter

Kai De Lorenzo

Stacey Jeffery

Michael Jarret

Alvaro Piedrafita

path detection: a quantum algorithmic primitive

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