The Qudit ZH Calculus for Arbitrary Finite Fields: Universality and Application

• URL: http://arxiv.org/abs/2406.02219v1
• Date: Tue, 4 Jun 2024 11:21:10 GMT
• Title: The Qudit ZH Calculus for Arbitrary Finite Fields: Universality and Application
• Authors: Dichuan, Gao,
• Abstract summary: We propose a generalization of the graphical ZH calculus to qudits of prime-power dimensions $q = pt$. We show this calculus to be universal over matrices $mathbb Cqn to mathbb Cqm$ with entries in the ring $mathbb Z[omega]$ where $omega$ is a $p$th root of unity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a generalization of the graphical ZH calculus to qudits of prime-power dimensions $q = p^t$, implementing field arithmetic in arbitrary finite fields. This is an extension of a previous result by Roy which implemented arithmetic of prime-sized fields; and an alternative to a result by de Beaudrap which extended the ZH to implement cyclic ring arithmetic in $\mathbb Z / q\mathbb Z$ rather than field arithmetic in $\mathbb F_q$. We show this generalized ZH calculus to be universal over matrices $\mathbb C^{q^n} \to \mathbb C^{q^m}$ with entries in the ring $\mathbb Z[\omega]$ where $\omega$ is a $p$th root of unity. As an illustration of the necessity of such an extension of ZH for field rather than cyclic ring arithmetic, we offer a graphical description and proof for a quantum algorithm for polynomial interpolation. This algorithm relies on the invertibility of multiplication, and therefore can only be described in a graphical language that implements field, rather than ring, multiplication.

Related papers

• Provably learning a multi-head attention layer [55.2904547651831]
  Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. In this work, we initiate the study of provably learning a multi-head attention layer from random examples. We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
  arXiv  Detail & Related papers  (2024-02-06T15:39:09Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Quantum and classical low-degree learning via a dimension-free Remez inequality [52.12931955662553]
  We show a new way to relate functions on the hypergrid to their harmonic extensions over the polytorus. We show the supremum of a function $f$ over products of the cyclic group $exp(2pi i k/K)_k=1K$. We extend to new spaces a recent line of work citeEI22, CHP, VZ22 that gave similarly efficient methods for learning low-degrees on hypercubes and observables on qubits.
  arXiv  Detail & Related papers  (2023-01-04T04:15:40Z)
• Optimal Quantum Algorithm for Vector Interpolation [0.0]
  We study the functions that can be learned through the functions that can be learned through the In quantum algorithm designed by Childs et al. We show that the success probability approaches 1 for large $q$ and large domain order $|mathcalV|. We provide a conservative formula for the number of queries required to achieve this success probability.
  arXiv  Detail & Related papers  (2022-12-07T20:08:24Z)
• Quantum teleportation in the commuting operator framework [63.69764116066747]
  We present unbiased teleportation schemes for relative commutants $N'cap M$ of a large class of finite-index inclusions $Nsubseteq M$ of tracial von Neumann algebras. We show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(mathbbC)$ over $N'$.
  arXiv  Detail & Related papers  (2022-08-02T00:20:46Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Qudit lattice surgery [91.3755431537592]
  We observe that lattice surgery, a model of fault-tolerant qubit computation, generalises straightforwardly to arbitrary finite-dimensional qudits. We relate the model to the ZX-calculus, a diagrammatic language based on Hopf-Frobenius algebras.
  arXiv  Detail & Related papers  (2022-04-27T23:41:04Z)
• Metric Hypertransformers are Universal Adapted Maps [4.83420384410068]
  metric hypertransformers (MHTs) are capable of approxing any adapted map $F:mathscrXmathbbZrightarrow mathscrYmathbbZ$ with approximable complexity. Our results provide the first (quantimat) universal approximation theorem compatible with any such $mathscrX$ and $mathscrY$.
  arXiv  Detail & Related papers  (2022-01-31T10:03:46Z)
• The decoherence-free subalgebra of Gaussian Quantum Markov Semigroups [0.0]
  We show that $mathcalN(mathcalT)$ is a type I von Neumann algebra $Linfty(mathbbRd_c;mathbbC)barotimesmathcalB(Gamma(Gamma(mathbbCd))$ determined, up to unitary equivalence, by two natural numbers $d_c,d_fleq d$.
  arXiv  Detail & Related papers  (2021-12-27T16:50:53Z)
• A quantum number theory [0.0]
  We build our QNT by defining pure quantum number operators ($q$-numbers) of a Hilbert space that generate classical numbers ($c$-numbers) belonging to discrete Euclidean spaces. The eigenvalues of each $textbfZ$ component generate a set of classical integers $m in mathbbZcup frac12mathbbZ*$, $mathbbZ* = mathbbZ*$, albeit all components do not generate $mathbbZ3
  arXiv  Detail & Related papers  (2021-08-18T17:26:03Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.