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Xiuwu Zhu

Publications and source records attributed to Xiuwu Zhu.

7 recordsLinked to original sources
arXiv2026

A Single Fixed Shallow Circuit for Classical Shadows of Arbitrary n-Qubit States

Classical shadows extract quantum properties from reusable classical records, typically obtained through randomized measurement settings. Although individual settings may be shallow, switching among them introduces control, calibration, and reconfiguration costs beyond conventional metrics. Here we construct, for every $n$, a single fixed shallow quantum analyzer whose Born outcomes replace externally sampled settings as labels for reusable shadow snapshots. The analyzer combines a freshly prepared $n$-qubit fiducial register $A$ with parallel Bell readout of $A$ and the unknown system $S$, producing one $2n$-bit record per copy. The same circuit realizes a rank-one minimal informationally complete measurement: one fixed setting replaces the $3^n$ local-Pauli settings conventionally used for complete reconstruction while retaining the minimum $d^2$ outcomes with $d=2^n$. Its Pauli-diagonal frame admits an analytic inverse. For fixed Hermitian observables, the Haar-averaged conditional variance has a dimension-independent coefficient, whereas the state-uniform coefficient is dimension dependent. For $n\ge3$, worst-state Pauli variances remain bounded in the axial sector and scale as $Θ(d)$ in the mixed sector. Fiducial preparation before system contact uses $n-1$ arbitrary two-qubit gates, no work qubits beyond $A$, and logarithmic depth under all-to-all connectivity. The unknown system undergoes one parallel system-ancilla entangling layer followed by local Hadamards and readout. Measurement-assisted preparation achieves $O(1)$ adaptive quantum depth using $n+1$ extra qubits in the heralded route, or $O(n\log n)$ auxiliaries for deterministic completion. These tradeoffs show that part of the measurement-setting randomness and control complexity normally supplied shot by shot can instead be compiled into a fixed reusable analyzer.

quant-ph↗
arXiv2026

Uniformly Stable Minimal Weyl--Heisenberg Measurements Approaching the SIC Benchmark

Informational completeness (IC) guarantees that an inverse exists, not that it is statistically well conditioned. For minimal rank-one Weyl--Heisenberg (WH) measurements, covariance makes the nonidentity projector-Gram spectrum proportional to the fiducial's ambiguity intensities, with eigenvalues \(d|χ_ϕ(u)|^2\), turning stability into an explicit worst-direction design problem; write \(λ\) for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while \(\mathbb E[λ^{-1}]=\infty\), and an explicit geometric family used to establish balanced informationally complete measurements in every dimension has a normalized spectral floor bounded above by an exponentially decaying envelope. We then construct a hierarchy of minimal measurements. A cyclic family with exactly \(d^2\) outcomes in every integer dimension has floors \(Θ(d^{-3})\) for odd \(d\) and \(Θ(d^{-5})\) for even \(d\); a finite-field family for \(q=2^m\) obeys the uniform bound \(λ\ge4/9\). Our main result treats every prime-power dimension of characteristic \(p\ge5\). A balanced one-coordinate perturbation repairs the zero ambiguity axis of a cubic Alltop state, gives an attained floor uniformly bounded below by a positive constant, and confines the entire nonidentity spectrum to \([L_q,U_q]\) with \(U_q/L_q\to1\). Its SIC-normalized minimum tends to one, and \(λ(ϕ_q)/Λ_q^\star\to1\) for the global finite-field WH max--min optimum \(Λ_q^\star\), without assuming SIC existence. The complete spectrum determines the exact finite-sample Hilbert--Schmidt error of canonical linear inversion at \(I/d\), while its lower edge controls local Fisher efficiency and canonical-shadow bounds.

quant-ph↗
arXiv2026

Asymptotic independence of class-group 4-ranks in correlated pairs of imaginary quadratic fields

Fix a squarefree integer $d_0>1$, and let $d$ range over the positive squarefree integers coprime to $2d_0$. Although $\mathbb{Q}(\sqrt{-d})$ and $\mathbb{Q}(\sqrt{-d_0d})$ share all variable ramified primes, we prove that their class-group $4$-ranks are asymptotically independent. Over the subfamily $d\le X$, their joint distribution converges in total variation to the product of two copies of the Cohen--Lenstra--Gerth distribution, with error bounded by a negative power of $\log\log X$. We further conjecture that the corrected $2$-primary groups $2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d})}[2^\infty]$ and $2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d_0d})}[2^\infty]$ are asymptotically independent, each with the Cohen--Lenstra distribution. Suppose in addition that the class number of $\mathbb{Q}(\sqrt{d_0})$ is odd. For a density-one subset of this family, we prove that extension of ideals to $K(d)=\mathbb{Q}(\sqrt{d_0},\sqrt{-d})$ induces $4\operatorname{Cl}_{K(d)}[2^\infty]\cong 2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d})}[2^\infty]\oplus 2\operatorname{Cl}_{\mathbb{Q}(\sqrt{-d_0d})}[2^\infty]$. Together with this decomposition, the group-valued conjecture predicts that $4\operatorname{Cl}_{K(d)}[2^\infty]$ is distributed as the direct sum of two independent Cohen--Lenstra $2$-groups, giving a corrected Cohen--Lenstra--Martinet distribution for the biquadratic family. Unconditionally, the $8$-rank of $\operatorname{Cl}_{K(d)}$ has limiting distribution given by the convolution of two copies of the Cohen--Lenstra--Gerth distribution. The proof combines Smith's box method with quantitative truncated Gaussian-binomial moment inversion for diagonally coupled, fixed-width bordered Rédei matrices.

math.NT↗
arXiv2026

Main conjectures for non-CM elliptic curves at good ordinary primes

Let $E/\mathbb{Q}$ be an elliptic curve and $p > 2$ be a prime of good ordinary reduction for $E$. Assume that the residue representation associated with $(E, p)$ is irreducible. In this paper, we prove more cases on several Iwasawa main conjectures for $E$. As applications, we prove more general cases of $p$-converse theorem and $p$-part BSD formula when the rank is less than or equal to $1$.

math.NT↗
arXiv2025

New optimal function field towers over finite fields of quartic power

We introduce two new types of towers of Drinfeld modular curves. These towers originate from a specific domain $\mathcal{A} $ and are analogous to the towers of rank-two Drinfeld modular curves over the polynomial ring. Specifically, the domain $\mathcal{A} $ corresponds to the projective line over the finite field $ \mathbb{F}_q $, equipped with an infinite place of degree two. We select an arbitrary non-zero principal $\mathcal{A} $-ideal $ I_η $ of degree two. Notably, the $ I_η $-reduction of the tower of minimal Drinfeld modular curves is asymptotically optimal over the finite field $ \mathbb{F}_{q^4} $.

math.NT↗
arXiv2025

Error term in the Cohen-Lenstra heuristic via random matrix approach

The Cohen-Lenstra heuristic predicts the distribution of ideal class groups over number fields. Random matrix models provide a natural framework for explaining this heuristic, and recent results demonstrate the effectiveness of these tools. In this paper, we extend the analysis of the random matrix model to examine the error term in the Cohen-Lenstra heuristic. Additionally, we derive the asymptotic distribution of the corank of random matrices over finite fields, which can be modeled as a special class of Markov chains.

math.PR↗