Measurement and sampling

The Born rule

A computational-basis measurement on a pure state returns bitstring x with probability equal to the squared modulus of the amplitude of x. The probabilities over all 2^n strings sum to one. A single shot yields one string. It does not reveal the amplitude, the phase, or the probability, except for the trivial fact that this string’s probability was not zero.

Phases are visible only indirectly, through interference before the measurement. If two paths to the same string cancel, that string becomes rare. If they add, it becomes common. Sampling records the common strings; it does not print the complex vector.

P(x) = |⟨x|ψ⟩|²

Shots and estimators

An expectation value of a diagonal observable is the average of a function of the bitstring, taken over shots. For a Pauli string that is not diagonal in the computational basis, the usual method is to rotate that string into Z-form and then measure. The sample mean converges to the true expectation. The standard error of a bounded outcome falls like one over the square root of the number of shots.

A figure that quotes an expectation without a shot count, or without an error bar, cannot be compared with another figure. Doubling the shots buys a narrower interval, not a different circuit. If two angles differ by less than the shot noise, the optimizer is chasing noise.

standard error ≈ σ / √N_shots

What sampling is not

Sampling is not a listing of the state. Rare strings are missing even when their amplitudes matter for a later interference experiment that you did not run. Sampling is also not a certificate of optimality for a combinatorial cost. The lowest-energy string in a finite sample is the lowest you happened to see. A classical refinement pass, or an exhaustive check on a tiny instance, is how Kryptur separates “best sample” from “best solution.”

Ideal samples and device samples answer different questions. Ideal samples estimate the circuit. Device samples estimate the circuit composed with the device’s noise. Publishing one under the other’s name is a methods error.

Research practice

For every plotted curve in a variational study, fix the shot budget before looking at the ranking of angles. Keep a seed for the random number generator when the study must be rerun. When an exact state vector is available, compare the sampled mean to the exact expectation on at least one point so the estimator itself is audited.