Applications of the Quantum st-Connectivity Algorithm June 3, 2019 University of Maryland Kai DeLorenzo1, Shelby Kimmel1, and R. Teal Witter1 1Middlebury College Layers of Abstraction Our Algorithms st-Connectivity [Belovs, Reichardt, ’12] Span Programs [Reichardt, ’09, ‘11] Quantum Gates 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. Big Question 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-Connectivity s t Is there a path between s and t? v u st-Connectivity s t Is there a path between s and t? v u st-Connectivity Query Model Input: � � � � � • graph skeleton s t � � � • hidden bit string �� �� �� v u st-Connectivity Query Model Input: � � � � � • graph skeleton s t � � � • hidden bit string 1 1 1 �� �� Output: �� connected v u st-Connectivity Query Model Input: � � � � � • graph skeleton s t � � � • hidden bit string 0 1 1 �� �� Output: �� not connected Goal: v u minimize queries to bits st-Connectivity Complexity Space: �( log #����� �� �������� ) [Belovs, Reichardt, ’12] [Jeffery, Kimmel, ’17] Effective Resistance [Belovs, Reichardt, ’12] Query: �( max �A,; � max �A,; � ) <:99=<;=> ? 9:; <:99=<;=> ? Effective Capacitance [Jarret, Jeffery, Kimmel, Piedrafita, ’18] Time: query *times* time for quantum walk on skeleton [Belovs, Reichardt, ’12] [Jeffery, Kimmel, ’17] st-Connectivity Resistance s t Bounded by longest path v u O �A,; � = min K (���� �� ����) HI:JA =>L=A st-Connectivity Capacitance s t Bounded by biggest cut v u O �A,; � = min K (��������� ����������) P:;=9;QRIA =>L=A �� �������� Cycle Detection S G Is edge � (between w x � u and v) on a cycle? G without � u and ⟺ Does edge � exist? w x and Is there a path from u to v without � ? v u T � v in G Cycle Detection S G’ or Is there a cycle in G? ⟺ For some edge �: u Does � exist? and w x Is there a path from u to v without � ? T Cycle Detection Complexity # vertices S Resistance: O(1) Capacitance: O(n m) u # edges in skeleton w x T Complexity: O(n3/2) Cycle Detection Bounds Lower Bound: Ω(�Z/O) [Childs, Kothari, ’12] Previous Bound: �\(�Z/O) [Cade, Montenaro, Belovs, ’18] Our Bound: �(�Z/O) Circuit Rank Estimation # edges to cut until no cycles S w x G’ ^ � = 1/�A,;(� ) u circuit rank w x v u G T Circuit Rank Estimation Bounds Lower/Previous Bound: ? �Z/� if cactus graph Our Bound: �\(�Z/O �`/�) multiplicative error Even Length Cycle Detection S connected cycles � Is edge � on an even w x u u length cycle? ⟺ x x Does edge � exist? and w w Is there an odd length path from u to v not v u v T including �? � bipartite double Even Length Cycle Detection Complexity S G’ u u w w 3/2 x x Complexity: O(n ) v T Even Length Cycle Detection Bounds Lower/Previous Bound: ? Classical Bound: Θ(�O) [Yuster, Zwick, ’97] Our Bound: �(�Z/O) 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? Thank you! Kai DeLorenzo Shelby Kimmel.