arXiv subjects

Daniela Falco-Pomares

Publications and source records attributed to Daniela Falco-Pomares.

3 recordsLinked to original sources

arXiv2026

On the encoding complexity of quantum numerical integration: an angle-structure characterization

We study numerical integration on $[0,1]$ by quantum amplitude estimation (QAE), with emphasis on the cost of constructing the amplitude oracle. We introduce a hierarchy of grid functions $\mathcal{G}_n^{(d)}$ whose angle map $Θ_g:\{0,1\}^n\to[0,π]$ is multilinear of degree at most $d$. Membership is classically checkable in $O(n2^n)$ time by the Walsh--Hadamard transform, and each $g\in\mathcal{G}_n^{(d)}$ admits a canonical encoding circuit with $\sum_{k=0}^d\binom{n}{k}$ multi-controlled $R_Y$ gates. Combining this circuit bound with classical discretisation estimates, we obtain a depth-versus-accuracy trade-off: for $g\in C^α[0,1]$, total gate count $O((\log(1/\varepsilon))^d\varepsilon^{-1})$ suffices for $\varepsilon$-accuracy with constant probability; in the affine case $d=1$ this is $O(\varepsilon^{-1}\log(1/\varepsilon))$ at fixed discretisation. We also show that encoding degree and Sobolev smoothness are independent: for every $s\in(0,1/2)$, $\mathcal{G}_n^{(1)}$ contains restrictions of functions in $W^{s',2}(0,1)$ for all $s'<s$ but not in $W^{s,2}(0,1)$. Experiments on the SpinQ Triangulum (NMR) and IBM Kingston (superconducting) processors at $n=2$ validate the predicted hierarchy: affine encodings run reliably on both platforms, while quadratic encodings exceed the Triangulum coherence budget but execute on Kingston.

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arXiv2026

A Rigorous and Self--Contained Proof of the Grover--Rudolph State Preparation Algorithm

We give a rigorous and self-contained analysis of the Grover--Rudolph quantum state-preparation algorithm, which encodes a probability distribution $\{p_k\}$ as an $n$-qubit amplitude state $\sum_k\sqrt{p_k}\ket{k}$ via a hierarchy of controlled $\RY$ rotations determined by a dyadic refinement of the target. We formalize the dyadic probability tree, derive the trigonometric factorization of conditional masses, and prove by induction that the circuit prepares exactly the desired measurement law. We further prove that perturbing each rotation angle by at most $η$ changes the output distribution by at most $\min(1,nη)$ in total variation, and combine this with a Hoeffding concentration bound to obtain an explicit design rule: $b\ge\log_2(2nπ/\varepsilon)$ bits and $S\ge 2^{n+1}\log(2/δ)/\varepsilon^2$ shots suffice to achieve accuracy $\varepsilon$ with confidence $1-δ$. As a circuit-theoretic complement, we provide an ancilla-free transpilation of each stage into $\{\RY(\cdot),X,\CNOT\}$ via Gray-code ladders and a Walsh--Hadamard angle transform.

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arXiv2026

Elementary Quantum Gates from Lie Group Embeddings in $U(2^n)$: Geometry, Universality, and Discretization

In the standard circuit model, elementary gates are defined relative to a chosen tensor factorization and are therefore extrinsic to the ambient group $U(2^n)$. Writing $N=2^n$, we introduce an \emph{intrinsic descriptor layer} in $U(N)$ by declaring as primitive the motions inside faithful embedded copies of $SU(2)$ (phase-free), together with a phase-inclusive $U(2)$ variant. We describe the embedding landscape $\Emb(SU(2),U(N))$ as a finite union of $U(N)$-homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers, and we isolate a canonical \emph{two-level sector} parameterized by $\Gr_2(\C^N)$ up to a $PSU(2)$ gauge. Equipping $U(N)$ with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic, yielding a variational characterization of elementary motions via minimal-norm logarithms. On the constructive side, we prove phase-free universality in $SU(N)$ from two-level primitives using QR/Givens factorizations together with explicit diagonal generation, and we obtain full universality in $U(N)$ by explicit abelian phase bookkeeping (equivalently, via the $U(2)$ two-level dictionary). Finally, we formalize a modular finite-alphabet compilation interface: any approximation routine in $SU(2)$ (e.g.\ Solovay--Kitaev) can be lifted through two-level embeddings to yield $U(N)$-level synthesis with global operator-norm error control.

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