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 � , � max � , � )
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
� , � = min (���� �� ����) st-Connectivity Capacitance
s t
Bounded by biggest cut
v u
� , � = min (��������� ����������) �� �������� 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(� / ) [Childs, Kothari, ’12]
Previous Bound: � (� / ) [Cade, Montenaro, Belovs, ’18]
Our Bound: �(� / ) Circuit Rank Estimation # edges to cut until no cycles
S w x G’ � = 1/� , (� ) u circuit rank
w x
v u G T Circuit Rank Estimation Bounds
Lower/Previous Bound: ?
� /� if cactus graph
Our Bound: � (� / � /�)
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(� ) [Yuster, Zwick, ’97]
Our Bound: �(� / ) 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