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A clear explanation on QPU alignment

$1 A formal definition of QPU-alignment To begin, consider product of intervals [-L, L]n-1 C Rn-1 where L where S is the surface measure on R, and the kernel B、R × R → R is a 32768 R be a continuous function in which there exists finite such that (i) each Tk is a simplex (i.e. an unction with compact, path-connected support (i.e. a bump function moo whose support is also path-connected), is called a Mario bump operator partition {TRA-1 of [-L, n-1 (n - 1)-dimensional analogue of a triangle) and (ii) f r, is linear for each k. We note that graph of f ' denoted「(f) , can be thought of as a (n-1)-dimensional manifold composed of simplices and that ▽f is well-defined and constant in the interior of each simplex. Extend f to a function f : 32768, 32768]n-1 → We often consider φ E C (R) such that 110112-1: these may be though of as wave functions corresponding to positions of objects in thePU-space (though note that in the rigorous sense,"o are determined by continuous wave functions defined on the principle「(f)-see may only refer those whose positions in and no e entire Wi onal res when paired with appropriatehER, comprise the Scuttlebug space of R 0 otherwise Definition 3 Consider pair (φ, ho), where φ E C (R) with llo12-1, and E Endow「 ) with the usual subspace topology inherited from Rn, then identify boundary points (..., -32768,.. .) ~ (...,32768,...). We then define the PU R. Suppose (φ,ho) have the additional properties that (i) there is unique for given φ, (ii) φ (x) = 0 for all x E R such that |Pan-1x-Pan-1hal > 1000 (here, PRn-1 : Rn → Rn-1 denotes the projection, and 1-1 the usual Euclidean norm), and (ii) for every Mario bump operator M, there exists unique hMo E R such that Mo(x) = 0 for all x E R satisfying \Pan-1x-Pan-1hMol > 1000 We call the set of all such pairs (φ,ho) the Scuttlebug space of R, denoted SB (R); such φ are called Scuttlebugs, and the corresponding h·is known as the Scuttlebug's home h the quotient topolo gy. We are now ready for a formal definition of QPU-alignment ion ~ be a precewise 4 65536 Remark 4 It is easy to see that, given Mario bump operator M, the pair M<p, hMo) is also in the Scuttlebug space. The corresponding hMo may be thought of the Scuttlebug's new home under the transformation M for all s in the domain of r such that r' (s) exists Definition 5 Given fixed xER, e > 0, the collection of Scuttlebugs Note that we often consider the case in which the last coordinate of r ean inne is COnstrain magnitude s called a jamboree. We say that an arbitrary collection of Scuttlebugs is ε-jamboree-capable if there exists x E R and finite collection {Mk} Mario bump operators such that M1 . . . Mmox E Jx.e for each p x E {φλ} where n is a non-negative integer and A is some constant that is dependent only In the special case in which n is furthe onstrain o be even, we ca ion path- connected room within X Conjecture 7 Suppose each R1, R2 are rooms such that Ri U R2 is not path connected B. (y) C R1 or B. (y) C R2, then the collection {φ : (φ, ho) E SB (RI U R2)) is not ε-jamboree-capable 2 Scuttlebug space y) with center y and radius ε such that Let X be a PU-space, as defined in Section 1. Fix some compact subset R C X and consider the space C (R) of continuous functions φ : R → R, with L2-norm We first define the Mario bump operatoı It is likely that a proof of the above conjecture requires significant advanc tion n untary operator of the jorm er$