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Zeqian Chen

Publications and source records attributed to Zeqian Chen.

At least 19 recordsLinked to original sources
arXiv2025

Physical models realizing the transformer architecture of large language models

The introduction of the transformer architecture in 2017 marked the most striking advancement in natural language processing. The transformer is a model architecture relying entirely on an attention mechanism to draw global dependencies between input and output. However, we believe there is a gap in our theoretical understanding of what the transformer is, and how it works physically. From a physical perspective on modern chips, such as those chips under 28nm, modern intelligent machines should be regarded as open quantum systems beyond conventional statistical systems. Thereby, in this paper, we construct physical models realizing large language models based on a transformer architecture as open quantum systems in the Fock space over the Hilbert space of tokens. Our physical models underlie the transformer architecture for large language models.

cs.LG↗
arXiv2024

Non-Abelian observable-geometric phases and the Riemann zeros

The Hilbert-Pólya conjecture asserts that the imaginary parts of the nontrivial zeros of the Riemann zeta function (the Riemann zeros) are the eigenvalues of a self-adjoint operator (a quantum mechanical Hamiltonian, in the physical sense), as a promising approach to prove the Riemann hypothesis (cf.\cite{SH2011}). Instead of the eigenvalues, in this paper we consider observable-geometric phases as the realization of the Riemann zeros in a periodically driven quantum system, which were introduced in \cite{Chen2020} for the study of geometric quantum computation. To this end, we further introduce the notion of non-Abelian observable-geometric phases, involving which we give an approach to finding a physical system to study the Riemann zeros. Since the observable-geometric phases are connected with the geometry of the observable space according to the evolution of the Heisenberg equation, this sheds some light on the investigation of the Riemann hypothesis.

quant-ph↗
arXiv2023

Banach space formalism of quantum mechanics

This paper presents a generalization of quantum mechanics from conventional Hilbert space formalism to Banach space one. We construct quantum theory starting with any complex Banach space beyond a complex Hilbert space, through using a basic fact that a complex Banach space always admits a semi-inner product. Precisely, in a complex Banach space $\mathbb{X}$ with a given semi-inner product, a pure state is defined by Lumer \cite{Lumer1961} to be a bounded linear functional on the space of bounded operators determined by a normalized element of $\mathbb{X}$ under the semi-inner product, and then the state space $\mathcal{S} (\mathbb{X})$ of the system is the weakly closed convex set spanned by all pure states. Based on Lumer's notion of the state, we associate a quantum system with a complex Banach space $\mathbb{X}$ equipped with a fixed semi-inner product, and then define a physical event at a quantum state $ω\in \mathcal{S}(\mathbb{X})$ to be a projection $P$ (bounded operator such that $P^2 =P$) in $\mathbb{X}$ satisfying the positivity condition $0 \le ω(P) \le 1,$ and a physical quantity at a quantum state $ω$ to be a spectral operator of scalar type with real spectrum so that the associated spectral projections are all physical events at $ω.$ The Born formula for measurement of a physical quantity is the natural pairing of operators with linear functionals satisfying the probability conservation law. A time evolution of the system is governed by a one-parameter group of invertible spectral operators determined by a scalar type operator with the real spectrum, which satisfies the Schrödinger equation. Our formulation is just a generalization of the Dirac-von Neumann formalism of quantum mechanics to the Banach space setting. We include some examples for illustration.

quant-ph↗
arXiv2022

A mathematical formalism of non-Hermitian quantum mechanics and observable-geometric phases

We present a mathematical formalism of non-Hermitian quantum mechanics, following the Dirac-von Neumann formalism of quantum mechanics. In this formalism, the state postulate is the same as in the Dirac-von Neumann formalism, but the observable postulate should be changed to include para-Hermitian operators (spectral operators of scalar type with real spectrum) representing observable, as such both the measurement postulate and the evolution postulate must be modified accordingly. This is based on a Stone type theorem as proved here that the dynamics of non-Hermitian quantum systems is governed by para-unitary time evolution. The Born formula on the expectation of an observable at a certain state is given in the non-Hermitian setting, which is proved to be equal to the usual Born rule for every Hermitian observable, but for a non-Hermitian one it may depend on measurement via the choice of a metric operator associated with the non-Hermitian observable under measurement. Our formalism is nether Hamiltonian-dependent nor basis-dependent, but can recover both PT-symmetric and biorthogonal quantum mechanics, and it reduces to the Dirac-von Neumann formalism of quantum mechanics in the Hermitian setting. As application, we study observable-geometric phases for non-Hermitian quantum systems.

quant-ph↗
arXiv2020

Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommutative martingale Hardy space $\h_p$ for all $0<p<2$ by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of $\h_p$ for all $0< p < 1,$ and provide a constructive proof of the atomic decomposition for $p=1$. We also study $(p,\8)_c$-atoms, and show that every $(p,2)_c$-atom can be decomposed into a sum of $(p,\8)_c$-atoms; consequently, for every $0<p\le 1$, the $(p,q)_c$-atoms lead to the same atomic space for all $2\le q\le\8$. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space $\h_p$ ($0<p<1$) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to proving some sharp martingale inequalities.

math.OA↗
arXiv2019

Observable-geometric phases and quantum computation

This paper presents an alternative approach to geometric phases from the observable point of view. Precisely, we introduce the notion of observable-geometric phases, which is defined as a sequence of phases associated with a complete set of eigenstates of the observable. The observable-geometric phases are shown to be connected with the quantum geometry of the observable space evolving according to the Heisenberg equation. They are indeed distinct from Berry's phase \cite{Berry1984, Simon1983} as the system evolves adiabatically. It is shown that the observable-geometric phases can be used to realize a universal set of quantum gates in quantum computation. This scheme leads to the same gates as the Abelian geometric gates of Zhu and Wang \cite{ZW2002,ZW2003}, but based on the quantum geometry of the observable space beyond the state space.

quant-ph↗
arXiv2017

An infinite linear hierarchy for the incompressible Navier-Stokes equation and application

This paper introduces an infinite linear hierarchy for the homogeneous, incompressible three-dimensional Navier-Stokes equation. The Cauchy problem of the hierarchy with a factorized divergence-free initial datum is shown to be equivalent to that of the incompressible Navier-Stokes equation in $\mathcal{H}^1.$ This allows us to present an explicit formula for solutions to the incompressible Navier-Stokes equation under consideration. The obtained formula is an expansion in terms of binary trees encoding the collision histories of the "particles" in a concise form. Precisely, each term in the summation of $n$ "particles" collision is expressed by a $n$-parameter singular integral operator with an explicit kernel in Fourier space, describing a kind of processes of two-body interaction of $n$ "particles". Therefore, this formula is a physical expression for the solutions of the incompressible Navier-Stokes equation.

math-ph↗
arXiv2015

Noncommutative martingale inequalities associated with convex functions

We report recent advances on noncommutative martingale inequalities associated with convex functions. These include noncommutative Burkholder-Gundy inequalities associated with convex functions due to the present authors and Dirksen and Ricard, noncommutative maximal inequalities associated with convex functions due to Osȩkowski and the present authors, and noncommutative Burkholder and Junge-Xu inequalities associated with convex functions due to Randrianantoanina and Lian Wu. Some open problems for noncommutative martingales are also included.

math.OA↗
arXiv2015

Mathematical formalism of many-worlds quantum mechanics

We combine the ideas of Dirac's orthonormal representation, Everett's relative state, and 't Hooft's ontological basis to define the notion of a world for quantum mechanics. Mathematically, for a quantum system $\mathcal{Q}$ with an associated Hilbert space $\mathbb{H},$ a world of $\mathcal{Q}$ is defined to be an orthonormal basis of $\mathbb{H}.$ The evolution of the system is governed by Schrödinger's equation for the worlds of it. An observable in a certain world is a self-adjoint operator diagonal under the corresponding basis. Moreover, a state is defined in an associated world but can be uniquely extended to the whole system as proved recently by Marcus, Spielman, and Srivastava. Although the states described by unit vectors in $\mathbb{H}$ may be determined in different worlds, there are the so-called topology-compact states which must be determined by the totality of a world. We can apply the Copenhagen interpretation to a world for regarding a quantum state as an external observation, and obtain the Born rule of random outcomes. Therefore, we present a mathematical formalism of quantum mechanics based on the notion of a world instead of a quantum state.

quant-ph↗
arXiv2014

Noncommutative maximal inequalities associated with convex functions

We prove several noncommutative maximal inequalities associated with convex functions, including a Doob type inequality for a convex function of maximal operators on noncommutative martingales, noncommutative Dunford-Schwartz and Stein maximal ergodic inequalities for a convex function of positive and symmetric positive contractions. The key ingredient in our proofs is a Marcinkiewicz type interpolation theorem for a convex function of maximal operators in the noncommutative setting, which we establish in this paper. These generalize the results of Junge and Xu in the $L^p$ case to the case of convex functions.

math.OA↗
arXiv2014

Local well-posedness for Gross-Pitaevskii hierarchies

We consider the Cauchy problem for the Gross-Pitaevskii infinite linear hierarchy of equations on $\mathbb{R}^n.$ By introducing a (F)-norm in certain Sobolev type spaces of sequences of marginal density matrices, we establish local existence, uniqueness and stability of solutions. Explicit space-time type estimates for the solutions are obtained as well. In particular, this (F)-norm is compatible with the usual Sobolev space norm whenever the initial data is factorized.

math-ph↗
arXiv2013

Geometrical perspective on quantum states and quantum computation

We interpret quantum computing as a geometric evolution process by reformulating finite quantum systems via Connes' noncommutative geometry. In this formulation, quantum states are represented as noncommutative connections, while gauge transformations on the connections play a role of unitary quantum operations. Thereby, a geometrical model for quantum computation is presented, which is equivalent to the quantum circuit model. This result shows a geometric way of realizing quantum computing and as such, provides an alternative proposal of building a quantum computer.

quant-ph↗
arXiv2013

Maximal and area integral characterizations of Bergman spaces in the unit ball of $\mathbb{C}^n$

In this paper, we present maximal and area integral characterizations of Bergman spaces in the unit ball of $\mathbb{C}^n.$ The characterizations are in terms of maximal functions and area integral functions on Bergman balls involving the radial derivative, the complex gradient, and the invariant gradient. As an application, we obtain new maximal and area integral characterizations of Besov spaces. Moreover, we give an atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces.

math.FA↗
arXiv2013

Atomic decomposition of real-variable type for Bergman spaces in the unit ball of $\mathbb{C}^n$

In this paper, we show that every (weighted) Bergman space $\mathcal{A}^p_α (\mathbb{B}_n)$ in the complex ball admits an atomic decomposition of real-variable type for any $0 < p \le 1$ and $α> -1.$ More precisely, for each $f \in \mathcal{A}^p_α (\mathbb{B}_n)$ there exist a sequence of real-variable $(p, \8)_α$-atoms $a_k$ and a scalar sequence $\{λ_k \}$ with $\sum_k | λ_k |^p < \8$ such that $f = \sum_k λ_k P_α (a_k),$ where $P_α$ is the Bergman projection from $L^2_α (\mathbb{B}_n)$ onto $\mathcal{A}^2_α (\mathbb{B}_n).$ The proof is constructive, and our construction is based on some sharp estimates about Bergman metric and Bergman kernel functions in $\mathbb{B}_n.$

math.FA↗
arXiv2012

Harmonic analysis on quantum tori

This paper is devoted to the study of harmonic analysis on quantum tori. We consider several summation methods on these tori, including the square Fejér means, square and circular Poisson means, and Bochner-Riesz means. We first establish the maximal inequalities for these means, then obtain the corresponding pointwise convergence theorems. In particular, we prove the noncommutative analogue of the classical Stein theorem on Bochner-Riesz means. The second part of the paper deals with Fourier multipliers on quantum tori. We prove that the completely bounded $L_p$ Fourier multipliers on a quantum torus are exactly those on the classical torus of the same dimension. Finally, we present the Littlewood-Paley theory associated with the circular Poisson semigroup on quantum tori. We show that the Hardy spaces in this setting possess the usual properties of Hardy spaces, as one can expect. These include the quantum torus analogue of Fefferman's $\mathrm{H}_1$-BMO duality theorem and interpolation theorems. Our analysis is based on the recent developments of noncommutative martingale/ergodic inequalities and Littlewood-Paley-Stein theory.

math.OA↗
arXiv2012

Blowup alternative for Gross-Pitaevskii hierarchies

In this paper, we prove the blowup alternative for Gross-Pitaevskii hierarchies on $\mathbb{R}^n$ and give the associated lower bounds on the blowup rate. In particular, we prove that any solution of density operators to the focusing Gross-Pitaevskii hierarchy blow up in finite time for $n \ge 3$ if the energy per some $k$ particles in the initial condition is negative. All of these results hold without the assumption of factorized conditions for initial values as well as the admissible ones. Our analysis is based on use of a quasi-Banach space of sequences of marginal density matrices.

math-ph↗
arXiv2011

Tent spaces and Littlewood-Paley $g$-functions associated with Bergman spaces in the unit ball of $\mathbb{C}^n$

In this paper, a family of holomorphic spaces of tent type in the unit ball of $\mathbb{C}^n$ is introduced, which is closely related to maximal and area integral functions in terms of the Bergman metric. It is shown that these spaces coincide with Bergman spaces. Furthermore, Littlewood-Paley type $g$-functions for the Bergman spaces are introduced in terms of the radial derivative, the complex gradient, and the invariant gradient. The corresponding characterizations for Bergman spaces are obtained as well. As an application, we obtain new maximal and area integral characterizations for Hardy-Sobolev spaces.

math.FA↗