Ap Applications of the Quantum st st-Co Connectivity y Algori - - PowerPoint PPT Presentation

Ap Applications of the Quantum st st-Co Connectivity y Algori rithm

June 3, 2019 University of Maryland

Kai DeLorenzo1, Shelby Kimmel1, and R. Teal Witter1

1Middlebury College

La Layers of

• f A

Abstract ction

• n

Our Algorithms st-Connectivity Span Programs Quantum Gates

[Reichardt, ’09, ‘11] [Belovs, Reichardt, ’12]

Re Recipe

Step 1: Encode a question into a graph structure. Step 2: Analyze worst case effective resistance/capacitance. Step 2b: Look for characteristics of original graph hidden in graph reduction.

Bi Big Qu Question

• n

st-connectivity reduces quantum algorithm design to a simple classical algorithm st-connectivity feels ‘natural’ in the following ways:

• it’s optimal for a wide range of problems (e.g. Boolean formula

evaluation, total connectivity) and

• it’s easy to analyze using effective resistance/capacitance

Does the st-connectivity approach give intuition for optimal underlying quantum algorithms?

st st-Con Connectivity

s v t u

Is there a path between s and t?

st st-Con Connectivity

s v t u

Is there a path between s and t?

st st-Con Connectivity Qu Query Mod Model

s v t u

𝒚𝟏 𝒚𝟐 𝒚𝟑 𝒚𝟑 𝒚𝟏 𝒚𝟐 𝒚𝟑 Input:

• graph skeleton
• hidden bit string

st st-Con Connectivity Qu Query Mod Model

s v t u

𝒚𝟏 𝒚𝟐 𝒚𝟑 1 1 1 𝒚𝟑 𝒚𝟏 𝒚𝟐 𝒚𝟑 Input:

• graph skeleton
• hidden bit string

Output: connected

st st-Con Connectivity Qu Query Mod Model

s v t u

𝒚𝟏 𝒚𝟐 𝒚𝟑 1 1 𝒚𝟑 𝒚𝟏 𝒚𝟐 𝒚𝟑 Input:

• graph skeleton
• hidden bit string

Output: not connected Goal: minimize queries to bits

st st-Con Connectivity Comp Complexity

Space: 𝑃( log #𝑓𝑒𝑕𝑓𝑡 𝑗𝑜 𝑡𝑙𝑓𝑚𝑓𝑢𝑝𝑜 ) Query: Time: query *times* time for quantum walk on skeleton max

9:; <:99=<;=> ? 𝐷A,; 𝐻

𝑃( max

<:99=<;=> ? 𝑆A,; 𝐻

Effective Resistance

[Belovs, Reichardt, ’12]

Effective Capacitance

[Jarret, Jeffery, Kimmel, Piedrafita, ’18] [Jeffery, Kimmel, ’17]

)

[Belovs, Reichardt, ’12] [Belovs, Reichardt, ’12] [Jeffery, Kimmel, ’17]

st st-Con Connectivity Resistance

s v t u

𝑆A,; 𝐻 = min

HI:JA K =>L=A

(𝑔𝑚𝑝𝑥 𝑝𝑜 𝑓𝑒𝑕𝑓)O Bounded by longest path

st st-Con Connectivity Ca Capacitance

s v t u

𝐷A,; 𝐻 = min

P:;=9;QRIA K =>L=A

(𝑞𝑝𝑢𝑓𝑜𝑢𝑗𝑏𝑚 𝑒𝑗𝑔𝑔𝑓𝑠𝑓𝑜𝑑𝑓)O Bounded by biggest cut

𝑗𝑜 𝑡𝑙𝑓𝑚𝑓𝑢𝑝𝑜

Cy Cycle Detection

• n

w v x u

𝒋 Is edge 𝒋 (between u and v) on a cycle? ⟺ Does edge 𝒋 exist? and Is there a path from u to v without 𝒋 ?

u w x T S

v in G

and G 𝒋

G without 𝒋

Cy Cycle Detection

• n

Is there a cycle in G? ⟺ For some edge 𝒋: Does 𝒋 exist? and Is there a path from u to v without 𝒋 ?

u w x T S

• r

G’

Cy Cycle Detection

• n Comp

Complexity

u w x T S

Capacitance: O(n m) Resistance: O(1) Complexity: O(n3/2)

# vertices # edges in skeleton

Cy Cycle Detection

• n Bou

Bounds

Lower Bound: Ω(𝑜Z/O) Previous Bound: \ 𝑃(𝑜Z/O) Our Bound: 𝑃(𝑜Z/O)

[Childs, Kothari, ’12] [Cade, Montenaro, Belovs, ’18]

Ci Circuit Ra Rank Estima mation

• n

w v x u u w x T S

# edges to cut until no cycles 𝑠 = 1/𝑆A,;(𝐻^) G’ G circuit rank

Ci Circuit Ra Rank Estima mation

• n Bou

Bounds

𝑜Z/𝑠 if cactus graph

\ 𝑃(𝜗Z/O 𝑜`/𝑠)

multiplicative error

Lower/Previous Bound: ? Our Bound:

Ev Even Length Cycle Detection

Is edge 𝒋 on an even length cycle? ⟺ Does edge 𝒋 exist? and Is there an odd length path from u to v not including 𝒋?

x T w u x v w u

connected cycles

w v x u

𝒋

S

𝒋 bipartite double

Ev Even Length Cycle Detection Complexity

Complexity: O(n3/2) G’

w T x u w v x u S

Ev Even Length Cycle Detection Bounds

Lower/Previous Bound: Classical Bound: Θ(𝑜O) Our Bound: 𝑃(𝑜Z/O) ?

[Yuster, Zwick, ’97]

Op Open Problems

Full analysis of (highly structured) reductions to show time efficiency Extend cycle length detection to arbitrary modulus Determine if reducing other Symmetric Logarithm problems to st- connectivity always gives optimal algorithm Does the st-connectivity approach give intuition for the optimal underlying quantum algorithms?

Th Thank you!

Kai DeLorenzo Shelby Kimmel