PYTHON · NUMPY

The order it visits

Eight p-bits in a 4 x 2 ferromagnet at beta = 1, and both schedules keep the same Boltzmann law. Counted in site updates, how many times as many updates as the fixed order does a random scan need for the same precision on the magnetisation?

Compare a random scan with a fixed-order sweep exactly and at equal work, derive the uncoupled ratio 2 - 1/n, and separate the property the mixing theory relies on from the one the measurement rewards.

01Challenge

A fabric updates its p-bits in a fixed order, and most of the mixing results you would cite were proved for a random scan, which picks each site with a random number. The literature calls the fixed order the systematic scan, and Blanca and Rafid's 2026 abstract states the imbalance: "The systematic scan is often favored in practice because it exhibits strong empirical performance, but its theoretical analysis remains far less developed than that of Glauber dynamics." Measure that performance: at equal work, counted in site updates, decide which schedule gives more independent samples of the magnetisation and by what factor, and hold the answer to a closed form you derive.

Predict firstEight p-bits in a 4 x 2 ferromagnet at beta = 1, and both schedules keep the same Boltzmann law. Counted in site updates, how many times as many updates as the fixed order does a random scan need for the same precision on the magnetisation?

02Model

Lesson 10 showed that a fabric's schedule is part of its model, because changing which p-bit moves when can change the law it samples. This lesson holds the law fixed and asks what the schedule costs. A fabric visits its p-bits in a fixed order, and a fabric that colours its graph, as Lesson 16 will show, does too. Most of the mixing theory was written for the other choice, Glauber dynamics, which picks the next site with a random number. That chain is reversible: the probability flowing from one state to another equals the flow back, and reversibility is what lets the theory read a chain's speed off its spectrum. Results for the fixed order go back at least to 2006, and in 2026 Blanca and Rafid took what they call a step toward closing the gap, because the fixed order is favoured in practice for its performance. Measure that performance exactly, at equal work counted in site updates. The integrated autocorrelation time of the magnetisation sets how many updates one independent sample costs, and with no coupling it has a closed form. A random scan leaves a given site alone with probability one minus one over n at every step, so from any moment a site waits n updates on average for its next refresh, however long it has already waited, and the autocorrelation time is n minus a half. A fixed order refreshes every site once per sweep, and its time is n over two. The random scan needs two minus one over n times the work. An uncoupled sweep is reversible too, so reversibility is not what sets the factor. On eight-spin ladders, ferromagnet and glass, coupling holds it between one point seven four and one point nine seven, from infinite temperature down to inverse temperature three. What the measurement rewards is a bounded wait. Reversibility makes a chain easy to prove things about. It does not make it fast.

03Guided practice

Derive the autocorrelation time of the magnetisation of uncoupled spins under each schedule, in site updates, and check it against the exact computation from the transition matrices at every size from two to eight. Then switch the couplings on, compare the schedules across temperature for the magnetisation and for the energy, and find when each of the two chains is reversible.

04Feedback

PASS WHEN You compared two schedules at equal work and derived the factor between them. Both refresh a site once every n updates on average; the random scan's wait has no memory, and on uncoupled spins that alone costs it 2 - 1/n times the updates for the magnetisation. What the measurement rewards is a bounded wait: a fixed order refreshes every site exactly once every n updates. Reversibility, the property the theory leans on, is not what it pays for.

If your run fails, check:

An uncoupled spin's correlation with itself k updates later is the probability that its site has not been refreshed in between, because a refreshed site keeps nothing of its old value.
Under a random scan that probability is (1 - 1/n)**k. The convention adds one half, for lag zero, to the sum from k = 1, and that sum is a geometric series.
A fixed-order sweep refreshes every site before the next reading, so its tau is one half of a sweep; the comparison is per site update, and a sweep is n of them.

05Retrieve & space

Compare two update schedules at equal work: the integrated autocorrelation time per site update, not per step and not per sweep.
A random scan leaves a site untouched with probability 1 - 1/n per step, so for uncoupled spins the magnetisation's tau is n - 1/2 updates against n/2 for a fixed-order sweep, a ratio of exactly 2 - 1/n.
On 4 x 2 ladders, ferromagnet and glass, coupling keeps the magnetisation's ratio between 1.74 and 1.97 from beta = 0 to 3, falling slowly as the chain gets colder.
The ratio belongs to the observable: in an energy made only of pair terms, a pair term is refreshed when either of its two sites is picked, so at infinite temperature the energy's ratio is 1 - 1/n and the random scan leads, while cold the fixed order wins by up to 1.87 on these ladders.
On coupled spins the random scan is reversible and the fixed order is not; uncoupled, a sweep draws a fresh sample and is reversible too, and the ratio is still 2 - 1/n. The theory is furthest along for the reversible chain, and the measurement rewards the other: for the magnetisation it needs about half the updates.

06Mastery & project

Given two update schedules for the same sampler, state the unit in which their cost is compared, name the observable the comparison holds for, and say which property the theory relies on and which the measurement rewards.

Take a sampler whose update order you control, on a model small enough to enumerate. Compute tau per site update exactly for a random scan and for the order your code or hardware runs, for the magnetisation and for the energy across temperature, and report both ratios, with 2 - 1/n for the magnetisation at beta = 0 as the control.

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PYTHON · NUMPY

The order it visits

Two ways of choosing the next p-bit, and every autocorrelation time computed exactly from the transition matrices. One closed form, the uncoupled one, is yours to derive.

You read

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You change

the code you write in each cell

Fixed

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