Applications of coherent classical communication and the Schur transform to quantum information theory
Aram W. Harrow
备注：
176 pages. Chapters 1 and 4 are a slightly older version of quant-ph/0512015. Chapter 2 is quant-ph/0205057 plus unpublished extensions (slightly outdated by quant-ph/0511219) and chapter 3 is quant-ph/0307091, quant-ph/0412126 and change. Chapters 5-8 are based on quant-ph/0407082, but go much further
发表信息：
Ph.D thesis, Massachusetts Institute of Technology, Cambridge, MA, 2005
Quantum mechanics has led not only to new physical theories, but also a new understanding of information and computation. Quantum information began by yielding new methods for achieving classical tasks such as factoring and key distribution but also suggests a completely new set of quantum problems, such as sending quantum information over quantum channels or efficiently performing particular basis changes on a quantum computer. This thesis contributes two new, purely quantum, tools to quantum information theory--coherent classical communication in the first half and an efficient quantum circuit for the Schur transform in the second half.
Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms
n d维量子系统的舒尔基是总角动量基的推广，它对于利用排列或集体酉旋转下的对称性是有用的。我们提出了有效的（大小poly（n，d，log（1/ε））的精度ε）量子线路的舒尔变换，这是计算和舒尔基地之间的基础的变化。这些电路基于用于Clebsch-Gordan变换的有效电路。我们还提出了一个有效的电路有限版本的舒尔变换，其中只需要投影到不同的舒尔子空间。这第二个电路是基于任何nonabelian有限群，其中存在一个快速量子傅里叶变换的相位估计的推广。
We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of $(\mathbb{C}^d)^{\otimes n}$ and the Schur-Weyl basis of $(\mathbb{C}^d)^{\otimes n}$. We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a maj…
Many quantum algorithms can be represented in a form of a classical circuit positioned between quantum Fourier transformations. Motivated by the search for new quantum algorithms, we turn to circuits where the latter transformation is replaced by the $\textsf{SU}(2)$ quantum Schur Transform -- a global transformation which maps the computational basis to a basis defined by angular momenta. We show that the output distributions of these circuits can be approximately classically sampled in polynomial time if they are sufficiently close to being sparse, thus isolating a regime in which these Quan…
The Schur transform is a unitary operator that block diagonalizes the action of the symmetric and unitary groups on an $n$ fold tensor product $V^{\otimes n}$ of a vector space $V$ of dimension $d$. Bacon, Chuang and Harrow \cite{BCH07} gave a quantum algorithm for this transform that is polynomial in $n$, $d$ and $\logε^{-1}$, where $ε$ is the precision. In a footnote in Harrow's thesis \cite{H05}, a brief description of how to make the algorithm of \cite{BCH07} polynomial in $\log d$ is given using the unitary group representation theory (however, this has not been explained in detail anywhe…
Freie Universität Berlin（柏林自由大学）；Scuola Normale Superiore（比萨高等师范学院）；Università degli Studi di Salerno（萨莱诺大学）；INFN, Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）
The notions of qubits and coherent states correspond to different physical systems and are described by specific formalisms. Qubits are associated with a two-dimensional Hilbert space and can be illustrated on the Bloch sphere. In contrast, the underlying Hilbert space of coherent states is infinite-dimensional and the states are typically represented in phase space. For the particular case of binary coherent state alphabets these otherwise distinct formalisms can equally be applied. We capitalize this formal connection to analyse the properties of optimally cloned binary coherent states. Seve…
Quantum coherence, emerging from the 'superposition' of quantum states, is widely used in various information processing tasks. Recently, the resource theory of multilevel quantum coherence is attracting substantial attention. In this paper, we mainly study the deterministic transformations of resource pure states via free operations in the theoretical framework for multilevel coherence. We prove that any two multilevel coherent resource pure states can be interconverted with a nonzero probability via a completely positive and trace non-increasing $k$-coherence-preserving map. Meanwhile, we pr…
In the paper we made a generalization of the Fourier transform in the complex space, applicable to the pair of Husimi and P-quasi distributions, in the representation of nonlinear coherent states. Implicitly, our result is a generalization similar result of Mehta, but which referred only to the canonical coherent states (associated with the one-dimensional harmonic oscillator). Our result is valid for both types of coherent states (defined in the Barut-Girardello, respectively Klauder-Perelomov manner).
Unitary Schur Sampling of Qudits via Random SWAP Tests: Hunt for Antisymmetry
Duke Quantum Center, Duke University（杜克大学量子中心）；Department of Electrical and Computer Engineering, Duke University（杜克大学电气与计算机工程系）；Department of Physics, Duke University（杜克大学物理系）
本文提出仅用成对 SWAP 测试实现任意置换不变混合态的酉 Schur 采样，通过反复提取最大反对称子系统，将算法转化为 Young 图随机遍历，并应用于量子纯度放大以最优纯化态。
In this work we study the recurrence problem for quantum Markov chains, which are quantum versions of classical Markov chains introduced by S. Gudder and described in terms of completely positive maps. A notion of monitored recurrence for quantum Markov chains is examined in association with Schur functions, which codify information on the first return to some given state or subspace. Such objects possess important factorization and decomposition properties which allow us to obtain probabilistic results based solely on those parts of the graph where the dynamics takes place, the so-called spli…
The Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This generalises the geometric dynamics of a harmonic oscillator in the Fock space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and write down explic…
We describe a family of quantum states of the Schrödinger cat type as superpositions of the harmonic oscillator coherent states with coefficients defined by the quadratic Gauss sums. These states emerge as eigenfunctions of the lowering operators obtained after canonical transformations of the Heisenberg-Weyl algebra associated with the ordinary and fractional Fourier transformation. The first member of this family is given by the well known Yurke-Stoler coherent state.
