[Paper Review] Quantum algorithms for shortest paths problems in structured instances

Aran Nayebi, Virginia Vassilevska Williams|arXiv (Cornell University)|Oct 23, 2014

Quantum Computing Algorithms and Architecture22 references3 citations

TL;DR

This paper presents the first nontrivial quantum algorithms for structured all-pairs shortest paths (APSP) problems, achieving sub-2.5 exponent runtime by combining quantum minima finding and quantum Dijkstra’s algorithm with classical techniques. It obtains an $ ilde{O}(n^{2.5 - (2.5 - heta)/(4 heta + 2)})$ time algorithm for geometrically weighted graphs and $ ilde{O}(n^{2.487})$ for the all-pairs nondecreasing path problem, marking the first quantum speedups beyond Grover’s $ ilde{O}(n^{2.5})$ bound for these problems.

ABSTRACT

We consider the quantum time complexity of the all pairs shortest paths (APSP) problem and some of its variants. The trivial classical algorithm for APSP and most all pairs path problems runs in $O(n^3)$ time, while the trivial algorithm in the quantum setting runs in $ ilde{O}(n^{2.5})$ time, using Grover search. A major open problem in classical algorithms is to obtain a truly subcubic time algorithm for APSP, i.e. an algorithm running in $O(n^{3-\varepsilon})$ time for constant $\varepsilon>0$. To approach this problem, many truly subcubic time classical algorithms have been devised for APSP and its variants for structured inputs. Some examples of such problems are APSP in geometrically weighted graphs, graphs with small integer edge weights or a small number of weights incident to each vertex, and the all pairs earliest arrivals problem. In this paper we revisit these problems in the quantum setting and obtain the first nontrivial (i.e. $O(n^{2.5-\varepsilon})$ time) quantum algorithms for the problems.

Motivation & Objective

• To address the open question of whether APSP admits a nontrivial quantum algorithm running in $O(n^{2.5 - heta})$ time for $ heta > 0$.
• To extend known classical truly subcubic algorithms for structured APSP variants into the quantum setting with improved runtime.
• To develop quantum algorithms for geometrically weighted graphs, node-weighted graphs, and graphs with bounded distinct edge weights per vertex.
• To provide the first quantum speedup for the all-pairs nondecreasing path (APNP) problem, a variant of APSP with nondecreasing path constraints.
• To explore quantum complexity equivalence between minimum triangle and APSP, and to identify new quantum speedup frontiers.

Proposed method

• Adapt Chan’s classical truly subcubic algorithm for geometric graphs to the quantum setting using quantum minima finding and quantum Dijkstra’s algorithm.
• Use the $( ext{min}, ext{sum})$ product computation via quantum algorithms with $O(n^{2.473})$ time to compute short nondecreasing paths.
• Apply a hitting set argument to reduce long-path computation by selecting $\tilde{O}(n/s)$ representative nodes for quantum Dijkstra’s computation.
• Replace classical Dijkstra’s steps with a quantum variant that uses amplitude amplification to achieve $O(n^{1.5})$ cost per node in the hitting set.
• Combine matrix multiplication techniques with quantum search to accelerate distance product computation in structured instances.
• Leverage the equivalence between minimum triangle and APSP in the classical setting to derive quantum implications for APSP if triangle can be solved faster.

Experimental results

Research questions

• RQ1 Can a nontrivial quantum algorithm for APSP be constructed in structured instances where classical truly subcubic algorithms exist?
• RQ2 What is the quantum time complexity of APSP in geometrically weighted graphs with constant $κ$?
• RQ3 Can quantum speedups be achieved for the all-pairs nondecreasing path (APNP) problem, which models scheduling applications?
• RQ4 Is there a quantum analog of the classical subcubic equivalence between minimum triangle and APSP?
• RQ5 Can quantum algorithms for APSP in structured graphs achieve $O(n^{2.5 - ε})$ time for $ε > 0$?

Key findings

• An $ ilde{O}(n^{2.5 - (2.5 - ω)/(4κ + 2)})$ time quantum algorithm is developed for APSP in geometrically weighted graphs, with $ω < 2.373$ and $κ$ a constant depending on the weight function family.
• For the all-pairs nondecreasing path (APNP) problem, a $ ilde{O}(n^{2.487})$ time quantum algorithm is achieved, improving over the classical $ ilde{O}(n^{2.843})$ bound.
• An $ ilde{O}(√n^{2.5 - (2.5 - ω)/6})$ time quantum algorithm is obtained for graphs with at most $L$ distinct edge weights per vertex, improving on the classical $ ilde{O}(√Ln^{(9+ω)/4})$ runtime.
• The paper establishes a quantum analog of the classical subcubic equivalence: if minimum triangle can be solved in $O(n^{1.5 - ε})$ time, then APSP can be solved in $O(n^{2.5 - δ})$ time for some $δ > 0$.
• The results demonstrate that quantum algorithms can achieve nontrivial speedups beyond Grover’s $ ilde{O}(n^{2.5})$ bound for structured APSP variants.
• The approach generalizes to all APSP variants with known classical truly subcubic algorithms, including node-weighted and small-integer-weight graphs.