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 DeLorenzo 1 , Shelby Kimmel 1 , and R. Teal Witter 1 1 Middlebury College

La Layers of of A Abstract ction on Our Algorithms st-Connectivity [Belovs, Reichardt, ’12] Span Programs [Reichardt, ’09, ‘11] Quantum Gates

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 on 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 t Is there a path between s and t? v u

st st-Con Connectivity s t Is there a path between s and t? v u

st st-Con Connectivity Qu Query Mod Model 𝒚 𝟑 Input: 𝒚 𝟏 𝒚 𝟐 𝒚 𝟑 s t • graph skeleton • hidden bit string 𝒚 𝟐 𝒚 𝟑 𝒚 𝟏 v u

st-Con st Connectivity Qu Query Mod Model 𝒚 𝟑 Input: 𝒚 𝟏 𝒚 𝟐 𝒚 𝟑 s t • graph skeleton 1 1 1 • hidden bit string 𝒚 𝟐 𝒚 𝟑 𝒚 𝟏 Output: connected v u

st st-Con Connectivity Qu Query Mod Model 𝒚 𝟑 Input: 𝒚 𝟏 𝒚 𝟐 𝒚 𝟑 s t • graph skeleton 0 1 1 • hidden bit string 𝒚 𝟐 𝒚 𝟑 𝒚 𝟏 Output: not connected Goal: v u minimize queries to bits

st st-Con Connectivity Comp Complexity Space: 𝑃( log #𝑓𝑒𝑕𝑓𝑡 𝑗𝑜 𝑡𝑙𝑓𝑚𝑓𝑢𝑝𝑜 ) [Belovs, Reichardt, ’12] [Jeffery, Kimmel, ’17] Effective Resistance [Belovs, Reichardt, ’12] Query: 9:; <:99=<;=> ? 𝐷 A,; 𝐻 max ) 𝑃( <:99=<;=> ? 𝑆 A,; 𝐻 max Effective Capacitance [Jarret, Jeffery, Kimmel, Piedrafita, ’18] Time: query *times* time for quantum walk on skeleton [Belovs, Reichardt, ’12] [Jeffery, Kimmel, ’17]

st st-Con Connectivity Resistance s t Bounded by longest path v u (𝑔𝑚𝑝𝑥 𝑝𝑜 𝑓𝑒𝑕𝑓) O 𝑆 A,; 𝐻 = min HI:JA K =>L=A

st st-Con Connectivity Ca Capacitance s t Bounded by biggest cut v u (𝑞𝑝𝑢𝑓𝑜𝑢𝑗𝑏𝑚 𝑒𝑗𝑔𝑔𝑓𝑠𝑓𝑜𝑑𝑓) O 𝐷 A,; 𝐻 = P:;=9;QRIA K min =>L=A 𝑗𝑜 𝑡𝑙𝑓𝑚𝑓𝑢𝑝𝑜

Cy Cycle Detection on S G 𝒋 Is edge 𝒋 (between w x u and v) on a cycle? and u G without 𝒋 ⟺ Does edge 𝒋 exist? w x and Is there a path from u to v without 𝒋 ? T v u 𝒋 v in G

Cy Cycle Detection on G’ S or Is there a cycle in G? ⟺ u For some edge 𝒋 : Does 𝒋 exist? and w x Is there a path from u to v without 𝒋 ? T

Cy Cycle Detection on Comp Complexity # vertices S Capacitance: O(n m) Resistance: O(1) # edges in u skeleton w x T Complexity: O(n 3/2 )

Cy Cycle Detection on Bou Bounds Ω(𝑜 Z/O ) Lower Bound: [Childs, Kothari, ’12] \ 𝑃(𝑜 Z/O ) Previous Bound: [Cade, Montenaro, Belovs, ’18] 𝑃(𝑜 Z/O ) Our Bound:

Ci Circuit Ra Rank Estima mation on # edges to cut until no cycles S G’ w x 𝑠 = 1/𝑆 A,; (𝐻 ^ ) u circuit rank w x v u G T

Ci Circuit Ra Rank Estima mation on Bou Bounds Lower/Previous Bound: ? 𝑜 Z /𝑠 if cactus graph 𝑃(𝜗 Z/O 𝑜 ` /𝑠) \ Our Bound: multiplicative error

Ev Even Length Cycle Detection S connected cycles 𝒋 Is edge 𝒋 on an even u u w x length cycle? ⟺ x x Does edge 𝒋 exist? and w w Is there an odd length path from u to v not v u v T 𝒋 including 𝒋 ? bipartite double

Ev Even Length Cycle Detection Complexity S G’ u u w w Complexity: O(n 3/2 ) x x v T

Ev Even Length Cycle Detection Bounds Lower/Previous Bound: ? Θ(𝑜 O ) Classical Bound: [Yuster, Zwick, ’97] 𝑃(𝑜 Z/O ) Our Bound:

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