Quantum Combinatorial Games: Structures and Computational Complexity

11/07/2020
by   Kyle Burke, et al.
18

Recently, a standardized framework was proposed for introducing quantum-inspired moves in mathematical games with perfect information and no chance. The beauty of quantum games-succinct in representation, rich in structures, explosive in complexity, dazzling for visualization, and sophisticated for strategic reasoning-has drawn us to play concrete games full of subtleties and to characterize abstract properties pertinent to complexity consequence. Going beyond individual games, we explore the tractability of quantum combinatorial games as whole, and address fundamental questions including: Quantum Leap in Complexity: Are there polynomial-time solvable games whose quantum extensions are intractable? Quantum Collapses in Complexity: Are there PSPACE-complete games whose quantum extensions fall to the lower levels of the polynomial-time hierarchy? Quantumness Matters: How do outcome classes and strategies change under quantum moves? Under what conditions doesn't quantumness matter? PSPACE Barrier for Quantum Leap: Can quantum moves launch PSPACE games into outer polynomial space We show that quantum moves not only enrich the game structure, but also impact their computational complexity. In settling some of these basic questions, we characterize both the powers and limitations of quantum moves as well as the superposition of game configurations that they create. Our constructive proofs-both on the leap of complexity in concrete Quantum Nim and Quantum Undirected Geography and on the continuous collapses, in the quantum setting, of complexity in abstract PSPACE-complete games to each level of the polynomial-time hierarchy-illustrate the striking computational landscape over quantum games and highlight surprising turns with unexpected quantum impact. Our studies also enable us to identify several elegant open questions fundamental to quantum combinatorial game theory (QCGT).

READ FULL TEXT

page 1

page 2

page 3

page 4

research
04/02/2023

A complete solution for a nontrivial ruleset with entailing moves

Combinatorial Game Theory typically studies sequential rulesets with per...
research
01/18/2021

Transverse Wave: an impartial color-propagation game inspired by Social Influence and Quantum Nim

In this paper, we study a colorful, impartial combinatorial game played ...
research
06/03/2021

Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography

We settle two long-standing complexity-theoretical questions-open since ...
research
10/30/2020

3XOR Games with Perfect Commuting Operator Strategies Have Perfect Tensor Product Strategies and are Decidable in Polynomial Time

We consider 3XOR games with perfect commuting operator strategies. Given...
research
01/24/2023

PSPACE ⊆ BQP

The complexity class PSPACE includes all computational problems that can...
research
01/30/2020

The Hidden Subgroup Problem for Universal Algebras

The Hidden Subgroup Problem (HSP) is a computational problem which inclu...
research
08/30/2023

Efficient Approximation of Quantum Channel Fidelity Exploiting Symmetry

Determining the optimal fidelity for the transmission of quantum informa...

Please sign up or login with your details

Forgot password? Click here to reset