Quantum Lower Bounds by Quantum Arguments

Ronald de Wolf

Stefan Heinrich

Xavier Coiteux-Roy, Maxime Flin, Carlos de Gois +3 more

We prove that any distributed quantum algorithm that finds a $3$-coloring with probability $1$ in a cycle of anonymous identical computers has to be global, that is, it needs $Ω(n)$ communication rounds. It follows that quantum computation and communication does not help with this problem. All prior lower bounds on quantum advantage in distributed graph algorithms use arguments related to physical causality. However, it is known that such arguments cannot rule out fast quantum advantage for $3$-coloring cycles. In particular, any causality-based argument would rule out the existence of finitely dependent coloring, but Holroyd and Liggett (2016) showed that such colorings do exist. Hence to tackle this problem, we need a ``genuinely quantum'' lower-bound technique that can distinguish between (1) distributions that do not violate physical causality vs. (2) distributions that can be realized with a quantum strategy. We present the first such lower-bound technique in this context. First, we show that $1$-round quantum algorithms cannot break symmetry with probability $1$. Second, we present a wishful teleportation strategy that can be used to turn $T$-round quantum 3-coloring algorithms into $1$-round quantum algorithms breaking symmetry, while preserving success probability $1$. Put together, the lower bound follows.

Andris Ambainis

Andris Ambainis

We propose a new method for proving lower bounds on quantum query algorithms.Instead of a classical adversary that runs the algorithm with one input and then modifies the input we use a quantum adversary that runs the input with a superposition of inputs.Using this method, we prove two new ~2(x/-N) lower bounds on AND of ORs and inverting a permutation and also provide more uniform proofs for some known lower bounds which have been previously proven via variety of different techniques.

Todd A. Oliynyk