bibkey: boixoknillsomma2009randomization
title: Quantum state preparation by phase randomization
authors: Sergio Boixo, Emanuel Knill, Rolando D. Somma
year: 2009
doi: null
url: https://arxiv.org/abs/0903.1652v2
claim: Random evolution times multiply spectral coherences by their characteristic function; independent sums multiply those functions, and fair two-point waits cancel specified known gaps exactly.
strata_touched: []
license: citation-only
triage: anchor

Random evolution and known-gap cancellation

Boixo, Knill and Somma, arXiv:0903.1652v2, Quantum Information and Computation 9, 833–855, develop phase randomization. The primary HTML text carries the heading Eigenpath traversal by phase randomization.

Section II.2, Theorem 1 and equation (4), express the action on a spectral coherence with gap $\omega$ through the characteristic function $\Phi (\omega )=\mathbb{E}(e^{i\omega T})$. Equation (6) multiplies the characteristic functions for independent sums of evolution times. Section II.3, Example 1 and equation (7), choose $T=0$ or $T=\pi /{\omega }_1$, each with probability $1/2$, for a positive known gap ${\omega }_1$. Its characteristic function vanishes at that gap. The same example explicitly extends this construction by summing independent two-point times over finitely many known gaps. All summands are nonnegative. This finite construction is an established result.

The source’s bounds for uniform suppression over an interval of unknown gaps have different hypotheses from exact cancellation at finitely many known gaps. A finite sum need not suppress other gaps; its value there must be checked directly. Recording the selected evolution time also retains conditional information that is absent from the averaged channel. Neither averaging nor a coherence multiplier identifies the instrument produced by a subsequent rank-one read.

The AURIC reconstruction continuation, §§37–40 uses this machinery at the fixed five-mode frequencies $2a,2b,a+b$. Its source-relative deductions retain $a-b$, obtain noncommuting polar ray effects, and compute the distinct seed-resolved records and actual post-read dark leakage. The cited paper is the supplier of the random-time mechanism, not of those five-mode deductions. This note contains bibliographic description and no reproduced source text.

The repository’s existing first-read suppliers are in RankOneFeedbackProtocol: the private branch_is_lueders and suffix_law, and public luders_law_eq and actual_protocol_effect_correspondence. MeasurableRankOneControl supplies arbitrary_protocol_tv_supremum for independent measurable control. The continuation uses five original modes, wait $U(t)$, first outcome $j$, independent eight-valued seed and immediate Stop. FiveModeRealResponse contains the private first_effect_real, deterministic_response_fiber and actual_effect_span_eq, and public five_mode_real_response_fiber; these concern the fixed real-frame response and do not by themselves supply a positive randomized event. The explicit positive filter in the continuation is additional to those response/span facts.