Full-slice averages and index equivalences #
Zero-family definition, shared average-reindexing helpers, and averaged scalar and tensor quantities for the full-slice outcome space, together with data-reindexing equivalences.
The tensor-form averages are internal to the scalar/tensor comparison recorded
in docs/decisions/713-scalar-tensor-decision.md; downstream code should use
the scalar public API.
References #
references/ldt-paper/commutativity-G.texblueprint/src/chapter/ch08_commutativity.tex
Swapping the full-slice question and outcome identifies the averaged
BAB/ABA terms and the averaged BABA/ABAB terms.
The zero operator family on the full-slice outcome space.
Equations
- MIPStarRE.LDT.Commutativity.zeroFullSliceOpFamily params = { outcome := fun (x : MIPStarRE.LDT.Commutativity.FullSliceOutcome params) => 0, total := 0 }
Instances For
Full-slice ABA scalar average: E_{x,y} ∑_{g,h} ⟨ψ| G^x_g G^y_h G^x_g ⊗ I |ψ⟩.
Full-polynomial analog of the evaluated evaluatedSliceABATerm (line 664);
obtained from it by replacing the evaluated outcomes a,b with polynomial
outcomes g,h summed over FullSliceOutcome.
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Full-slice ABAB scalar average:
E_{x,y} ∑_{g,h} ⟨ψ| G^x_g G^y_h G^x_g G^y_h ⊗ I |ψ⟩.
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Evaluated-slice ABA scalar average:
E_{u,v,x,y} ∑_{a,b} ⟨ψ| G^x_[g(u)=a] G^y_[h(v)=b] G^x_[g(u)=a] ⊗ I |ψ⟩.
Averaged analog of evaluatedSliceABATerm (line 664) over the full slice
question.
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Evaluated-slice ABAB scalar average:
E_{u,v,x,y} ∑_{a,b} ⟨ψ| G^x_[g(u)=a] G^y_[h(v)=b] G^x_[g(u)=a] G^y_[h(v)=b] ⊗ I |ψ⟩.
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Full-slice BAB ⊗ A tensor average
(paper eq:gcom4 RHS, commutativity-G.tex line 334):
E_{x,y} ∑_{g,h} ⟨ψ| G^y_h G^x_g G^y_h ⊗ G^x_g |ψ⟩.
This is the manifestly-PSD tensor-form partner of fullSliceABAAvg used by the
marginalization step: each summand factors as V† V with
V = (G^x_g G^y_h) ⊗ √(G^x_g), so the outer absolute value drops and the
Schwartz–Zippel collision bound applies per outcome. This tensor-form average
is internal to the scalar public API recorded by decision #713.
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Full-slice ABA ⊗ B tensor average (y-side analogue):
E_{x,y} ∑_{g,h} ⟨ψ| G^x_g G^y_h G^x_g ⊗ G^y_h |ψ⟩.
Naming convention (consistent with the sibling fullSliceBABAtensorAvg for
BAB ⊗ A): the four-letter operator string ABAB decomposes as left register
ABA followed by right register B. This is not the same operator as the
scalar fullSliceABABAvg, whose left register is the full quartic
G^x_g G^y_h G^x_g G^y_h; the tensorAvg suffix marks the tensor split.
The manifestly-PSD tensor-form partner of fullSliceABABAvg reached from it by
closenessOfIP (moving the trailing G^y_h factor from the left register to
the right). Each summand factors as V† V with
V = (G^y_h G^x_g) ⊗ √(G^y_h). Internal per #713.
The evaluated-side analogue is evaluatedSliceABABtensorAvg below.
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Evaluated-slice ABA ⊗ B tensor average (evaluated-side analogue of
fullSliceABABtensorAvg):
E_{u,v,x,y} ∑_{a,b} ⟨ψ| G^x_[g(u)=a] G^y_[h(v)=b] G^x_[g(u)=a] ⊗ G^y_[h(v)=b] |ψ⟩.
This is the second tensor-form endpoint in paper commutativity-G.tex lines
356-360. The scalar-to-tensor comparison
evaluatedSliceABAB_scalar_to_ABABtensor reaches it by moving the trailing
G^y_[h(v)=b] factor from the left register to the right register.
The shared tensor endpoint is defined here so the evaluated-side transport lemmas can use the same notation as the full-slice transport lemmas.
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X-evaluated BAB ⊗ A tensor average.
This is the intermediate obtained from fullSliceBABAtensorAvg after
postprocessing only the first/full-x polynomial outcome by a sampled point
u : Point params; the second/y polynomial outcome remains full. The
x-side tensor marginalization lemma below identifies its difference from the
full tensor average with fullSliceBABAxCollisionFactored.
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X-evaluated, y-full ABAB scalar average.
This is the scalar endpoint in the display from eq:evaluate-gcom-at-points to
eq:don't-understand-the-numbering-system: the x polynomial outcome has been
postprocessed at u, but the second closenessOfIP move has not yet transferred
the trailing G^y_h to the right register.
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X-evaluated, y-full ABA ⊗ B tensor average.
This is the y-side intermediate in paper eq:evaluate-gcom-at-points-part-dos:
the first/x family has already been postprocessed at u, while the second/y
family still ranges over full polynomial outcomes. The y-side tensor
marginalization lemma below compares this to evaluatedSliceABABtensorAvg.
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Averaging mixed x-evaluated data and ignoring the full-y coordinate gives the
uniform average over Point params.next.
Reindex xEvaluatedSliceBABAtensorAvg into the mixed (u,x,y) data order.
Reindex evaluated-slice questions as ((u, (x, y)), v).
This product order is tailored to the y-marginalization expansion: the residual
is indexed by the already x-evaluated data (u, x, y), and the remaining uniform
average is over the y-evaluation point v.
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