DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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Absolute spectral gap of a finite reversible chain

Definition

For a finite reversible chain $(X, P, π)$ , let $H_{0} = {f : \sum_{x} π_{x} f_{x} = 0}$ with the weighted inner product from A finite reversible transition operator is self-adjoint on weighted L2. Define its absolute spectral gap by

$δ_{a b s} (P) : = 1 - ∥ P ∣_{H_{0}} ∥_{2, π} .$

The norm of the restriction to the zero subspace is $0$ by The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||, so a one-state chain has absolute gap $1$ . No irreducibility or aperiodicity is built into this definition: a reducible chain with a nonconstant stationary vector in $H_{0}$ can have gap $0$ , and a periodic chain can have gap $0$ because of an eigenvalue $- 1$ . The eigenvalue form of the definition is proved in Absolute gap equals one minus the largest nonconstant eigenvalue modulus.

Depends on

Used by

MNRS finite reversible-walk finding Algorithm
The two-vertex deterministic swap has absolute gap zero Example
Spectrum of the lazy Johnson walk Lemma
Absolute gap equals one minus the largest nonconstant eigenvalue modulus Theorem
Finite reversible-walk finding cost Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Andrew M. Childs, Lecture Notes on Quantum Algorithms, §18.4 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §8.1 (standard reference, not scraped)