Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.38340

Einstein Metrics and Quantum Control on Symmetric Tensor Product Quotients

Honghuai Fang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $M_n=(\mathrm{PU}(2)^n\rtimes S_n)\backslash\mathrm{PU}(2^n)$ be the space of projective $n$-qubit unitaries modulo local unitaries and permutations of the qubits. We prove that for all sufficiently large $n$ the manifold $M_n$ carries exactly two $\mathrm{PU}(2^n)$-invariant Einstein metrics up to scaling. Both Einstein metrics have uniformly bounded diameter after normalizing two-qubit costs, although approximating a Haar-random unitary requires exponentially many two-qubit gates. In contrast, we construct non-Einstein metrics with nonnegative Ricci curvature and exponentially growing typical control distances, and determine the sharp curvature threshold for this family.

Explore related subjects

quant-phmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Honghuai Fang. 2026-10-28. Einstein Metrics and Quantum Control on Symmetric Tensor Product Quotients. https://arxiv.org/abs/2610.38340

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A quantum central path algorithm for linear optimization

We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a simulation working directly with the nonlinear complementarity equations. Using a two-stage quantum eigenpath traversal schedule, this approach yields an algorithm for solving linear optimization problems involving $m$ constraints and $n$ variables to $\varepsilon$-optimality using $\widetilde{O}\!\left((m+n)^{3/2}\,\varepsilon^{-1}+(m+n)^{1/2}\,\varepsilon^{-3/2}\right)$ queries to an oracle that evaluates a potential function. In the standard gate model (i.e., without access to quantum RAM), our algorithm can obtain $\varepsilon$-precise solutions to LO problems using at most $$\widetilde{O}\!\left(\left[\frac{(m+n)^{3/2}}{\varepsilon} + \frac{(m+n)^{1/2}}{\varepsilon^{3/2}}\right] \cdot \mathrm{nnz}(A)\right)$$ elementary gates, where $\mathrm{nnz}(A)$ is the total number of non-zero elements found in the constraint matrix.

quant-ph↗
arXiv2026

Entanglement Scaling of Motzkin Chains under Spiral Hamiltonian-path Embeddings

We investigate how a two-dimensional spatial embedding changes the entanglement observed across geometrically local bipartitions of a one-dimensional quantum spin chain. A $q$-deformed colored Motzkin chain is placed along a spiral-inspired Hamiltonian path that visits $N(L)=Θ(L^2)$ sites of a square-lattice region while retaining a path interaction graph. Straight ambient-space cuts then correspond to unions of $Θ(L)$ disconnected intervals in the chain ordering. We derive the scaling of a height-variation proxy for the leading color-pairing contribution and predict ambient-space entropy scaling proportional to $L^2$, $L$, and $O(1)$ for $q>1$, $q=1$, and $q<1$, respectively. We evaluate this proxy numerically using the matrix-product-state representation and verify its predicted finite-size scaling. The embedded and original Hamiltonians have exactly identical spectra and spectral gaps at fixed chain length. We also describe a sparse tensor-network representation and propose a two-chain junction extension. These results isolate the roles of interaction connectivity, spatial embedding, and subsystem geometry in entanglement scaling.

quant-ph↗
arXiv2026

Brace for impact: ECDLP challenges for quantum cryptanalysis

Precise suites of benchmarks are required to assess the progress of early fault-tolerant quantum computers at economically impactful applications such as cryptanalysis. Appropriate challenges exist for factoring but those for elliptic curve cryptography are either too sparse or inadequate for standard applications of Shor's algorithm. We introduce a difficulty-graded suite of elliptic curve discrete logarithm (ECDLP) challenges that use Bitcoin's curve y^2=x^3+7 mod p while incrementally lowering the prime field from 256 down to 6 bits. For each bit-length, we provide the prime, the prime group order, and two deterministic nothing-up-my-sleeve (NUMS) points in compressed SEC1 form. All challenges are generated by a deterministic, reproducible procedure, and no private challenge scalar is chosen in advance. We calibrate classical cost against Pollard's rho records and quantum cost against resource estimation results for Shor's algorithm. We compile Shor's ECDLP circuit to logical counts and map them to physical resources for various parameters of the surface code, the repetition cat code and the LDPC cat codes. Under explicit and testable assumptions on physical error rates, code distances, and non-Clifford supply, our scenarios place the full 256-bit instance within a 2027--2033 window. The challenge ladder thus offers a transparent ruler to track fault-tolerant progress on a cryptanalytic target of immediate relevance, and it motivates proactive migration of digital assets to post-quantum signatures.

quant-ph↗