A quantum-walk-inspired adiabatic algorithm for graph isomorphism

We present a 2-local quantum algorithm for graph isomorphism GI based on an adiabatic protocol. By exploiting continuous-time quantum-walks, we are able to avoid a mere diffusion over all possible configurations and to significantly reduce the dimensionality of the visited space. Within this restricted space, the graph isomorphism problem can be translated into the search of a satisfying assignment to a 2-SAT formula without resorting to perturbation gadgets or projective techniques. We present an analysis of the execution time of the algorithm on small instances of the graph isomorphism problem and discuss the issue of an implementation of the proposed adiabatic scheme on current quantum computing hardware.Comment: 10 pages, 5 figure

Grounding and Auto-abstraction

Abstraction principles and grounding can be combined in a natural way (Rosen 2010, 117; Schwartzkopff 2011, 362]). However, some ground-theoretic abstraction principles entail that there are circles of partial ground (Donaldson 17, 793). I call this problem auto-abstraction. In this paper I sketch a solution. Sections 1 and 2 are introductory. In section 3 I start comparing different solutions to the problem. In section 4 I contend that the thesis that the right-hand side of an abstraction principle is (metaphysically) prior to its left-hand side motivates an independence constraint, and that this constraint leads to predicative restric- tions on the acceptable instances of ground-theoretic abstraction principles. In section 5 I argue that auto-abstraction is acceptable unless the left-hand side is essentially grounded by the right-hand side. In section 6 I highlight sev- eral parallelisms between auto-abstraction and the puzzles of ground. I finally compare my solution with the strategies listed in section 3

The Quest for Certainty

Abstract The aim of this paper is to vindicate the Cartesian quest for certainty by arguing that to aim at certainty is a constitutive feature of cognition. My argument hinges on three observations concerning the nature of doubt and judgment: first, it is always possible to have a doubt as to whether p in so far as one takes the truth of p to be uncertain; second, in so far as one takes the truth of p to be certain, one is no longer able to genuinely wonder whether p is true; third, to ask the question whether p is to desire to receive a true answer. On this ground I clarify in what sense certainty is the aim of cognition. I then argue that in judging that p we commit ourselves to p's being certain and that certainty is the constitutive norm of judgment. The paper as a whole provides a picture of the interplay between doubt and judgment that aims at vindicating the traditional insight that our ability to doubt testifies our aspiration to know with absolute certainty

Grounding and Auto-abstraction

Abstraction principles and grounding can be combined in a natural way (Rosen 2010, 117; Schwartzkopff 2011, 362]). However, some ground-theoretic abstraction principles entail that there are circles of partial ground (Donaldson 17, 793). I call this problem auto-abstraction. In this paper I sketch a solution. Sections 1 and 2 are introductory. In section 3 I start comparing different solutions to the problem. In section 4 I contend that the thesis that the right-hand side of an abstraction principle is (metaphysically) prior to its left-hand side motivates an independence constraint, and that this constraint leads to predicative restric- tions on the acceptable instances of ground-theoretic abstraction principles. In section 5 I argue that auto-abstraction is acceptable unless the left-hand side is essentially grounded by the right-hand side. In section 6 I highlight sev- eral parallelisms between auto-abstraction and the puzzles of ground. I finally compare my solution with the strategies listed in section 3

Determinism and Judgment. A Critique of the Indirect Epistemic Transcendental Argument for Freedom

In a recent book entitled Free Will and Epistemology. A Defence of the Transcendental Argument for Freedom, Robert Lockie argues that the belief in determinism is self-defeating. Lockie’s argument hinges on the contention that we are bound to assess whether our beliefs are justified by relying on an internalist deontological conception of justification. However, the determinist denies the existence of the free will that is required in order to form justified beliefs according to such deontological conception of justification. As a result, by the determinist’s own lights, the very belief in determinism cannot count as justified. On this ground Lockie argues that we are bound to act and believe on the presupposition that we are free. In this paper I discuss and reject Lockie’s transcendental argument for freedom. Lockie’s argument relies on the assumption that in judging that determinism is true the determinist is committed to take it that there are epistemic obligations – e.g., the obligation to believe that determinism is true, or the obligation to aim to believe the truth about determinism. I argue that this assumption rests on a wrong conception of the interplay between judgments and commitments

Geometric Bounds on the Fastest Mixing Markov Chain

In the Fastest Mixing Markov Chain problem, we are given a graph $G = (V, E)$ and desire the discrete-time Markov chain with smallest mixing time $\tau$ subject to having equilibrium distribution uniform on $V$ and non-zero transition probabilities only across edges of the graph. It is well-known that the mixing time $\tau_\textsf{RW}$ of the lazy random walk on $G$ is characterised by the edge conductance $\Phi$ of $G$ via Cheeger's inequality: $\Phi^{-1} \lesssim \tau_\textsf{RW} \lesssim \Phi^{-2} \log |V|$. Analogously, we characterise the fastest mixing time $\tau^\star$ via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance $\Psi$ of $G$: $\Psi^{-1} \lesssim \tau^\star \lesssim \Psi^{-2} (\log |V|)^2$. This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on $G$ with equilibrium distribution which need not be uniform, but rather only $\varepsilon$-close to uniform in total variation. We show that it is always possible to construct such a chain with mixing time $\tau \lesssim \varepsilon^{-1} (\operatorname{diam} G)^2 \log |V|$. Finally, we discuss analogous questions for continuous-time and time-inhomogeneous chains.Comment: 31 page