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Quantum Science and Quantum Technology Yazhen Wang and Xinyu Song Statistical Science 2020, Vol. 35, No. 1, 51–74 https://doi.org/10.1214/19-STS745 © Institute of Mathematical Statistics, 2020 Quantum Science and Quantum Technology Yazhen Wang and Xinyu Song Abstract. Quantum science and quantum technology are of great current interest in multiple frontiers of many scientific fields ranging from computer science to physics and chemistry, and from engineering to mathematics and statistics. Their developments will likely lead to a new wave of scientific rev- olutions and technological innovations in a wide range of scientific studies and applications. This paper provides a brief review on quantum commu- nication, quantum information, quantum computation, quantum simulation, and quantum metrology. We present essential quantum properties, illustrate relevant concepts of quantum science and quantum technology, and discuss their scientific developments. We point out the need for statistical analysis in their developments, as well as their potential applications to and impacts on statistics and data science. Key words and phrases: Quantum communication, quantum information, quantum computation, quantum simulation, quantum annealing, quantum sensing and quantum metrology, quantum bit (qubit). 1. INTRODUCTION from existing applications of quantum mechanics and in- formation theory, such as lasers, transistors, MRI, and cur- Quantum science and quantum technology arise from a rently used classical computers and classical communica- synthesis of quantum mechanics, information theory, and tion tools, in ways that we utilize distinct quantum phe- computing. They investigate the preparation and control nomena like quantum superposition, entanglement, and of the quantum states of physical systems to generate new tunneling, which do not have classical counterparts. In the knowledge and technologies for information processing past two decades, we have made tremendous progress in and transmission, computation, measurement, and funda- the study of quantum science and quantum technology for mental understanding in ways that classical approaches harnessing quantum phenomena to advance information can only do much less efficiently, or not at all. The fields processing and transmission, computation, and measure- comprise quantum communication, quantum information, ment. quantum computation, quantum simulation, and quantum Quantum science not only establishes a foundation for metrology (also known as quantum sensing), where quan- gaining a deeper understanding of nature, but also makes tum communication utilizes quantum means to transmit it possible to invent new quantum technology for accom- data in a provably secure way; quantum information de- plishing tasks that are impossible to achieve by classical scribes the information of the state of a quantum system techniques. Here quantum technology refers to technolo- and the process of the information by quantum devices; gies that explicitly deal with individual quantum states quantum computation uses quantum effects to speed up and specifically exploit special quantum properties that certain calculations dramatically; quantum simulation re- do not have classical analogue. They enable us to build produces the behavior of hard accessible quantum sys- quantum devices for achieving faster computation, more tems by manipulating well-controlled quantum systems; secure communication, and better physical measurements quantum metrology exploits the high sensitivity of coher- than classical techniques. This article intends to present an ent quantum systems to external perturbations for enhanc- overview of such quantum aspects of science and technol- ing the performance of measurements of physical quan- ogy, particularly in quantum information, quantum com- tities. Quantum science and quantum technology differ munication, and quantum computation. The rest of the paper proceeds as follows. Section 2 Yazhen Wang is Professor, Department of Statistics, University briefly introduces quantum physics. Section 3 reviews ba- of Wisconsin-Madison, Madison, Wisconsin 53706, USA sic quantum concepts used in quantum science and quan- (e-mail: [email protected]). Xinyu Song is Assistant tum technology. Sections 4 and 5 discuss quantum com- Professor, School of Statistics and Management, Shanghai munication and quantum information, respectively. Sec- University of Finance and Economics, Shanghai 200433, tion 6 illustrates quantum computation. It covers univer- China (e-mail: [email protected]). sal quantum computing based on the gate (or circuit) 51 52 Y. WANG AND X. SONG model, adiabatic quantum computing based on quantum 2.2 Quantum Physics annealing, and current development on building quan- Quantum mechanics describes microscopic phenomena tum computers. This section also includes quantum algo- such as the positions and momentums of individual par- rithms, quantum simulation, quantum machine learning, ticles like atoms or electrons, the spins of electrons, the and quantum computational supremacy. Section 7 pro- emissions and absorptions of light by atoms, and the de- vides a short description of quantum metrology. Section 8 tections of light photons. Unlike classical mechanics that features concluding remarks and points out potential ap- can precisely measure physical entities like position and plications of quantum science and quantum technology to momentum, quantum physics is intrinsically stochastic in statistics and data science as well as the need of statis- the sense that only a probabilistic prediction can be made tics in the development of quantum science and quantum about the results of the measurements performed. technology. We may describe a quantum system by its state and the dynamic evolution of the state. A quantum state is of- 2. QUANTUM PHYSICS AND ITS COMPUTATIONAL ten characterized by a unit complex vector with dynamic POTENTIAL unitary evolution, where the unitary evolution means that 2.1 Mathematical Concepts and Notations quantum states are connected by unitary matrices, and the dynamic evolution is governed by a differential equation Unlike the typical literature on quantum mechanics that called the Schrödinger equation. Specifically, let |ψ(t) adopts technically more complicated concepts and nota- be the state of the quantum system at time t (also a wave tions such as operators with a continuous spectrum on function at time t). The states |ψ(t) and |ψ(t + s) at an infinite-dimensional Hilbert space, for simplicity we times t and t + s, respectively, are connected√ through choose to use relatively easy finite-dimensional linear al- |ψ(t + s)=U(s)|ψ(t),whereU(s) = exp(− −1Hs) gebra for the purpose of discussing quantum science and is a unitary matrix, and H is a Hermitian matrix on quantum technology. Since operators correspond to ma- Cd , which is known as the Hamiltonian of the quan- trices in the finite dimensional case, we need to deal tum system.√ Differentiating both sides of |ψ(t + s)= with only matrices and their operations such as eigen- exp(− −1Hs)|ψ(t) with respect to s and letting s go analysis. Denote by R and C, respectively, the sets of all to 0, we obtain the following Schrödinger equation for real numbers and all complex numbers. A simple vector governing the continuous time evolution of |ψ(t): Cd space is comprising all d-tuples of complex numbers √ ∂|ψ(t) (z ,...,z ). We use Dirac notations |· (which is called (2.1) −1 = Hψ(t) . 1 d ∂t ket) and ·| (which is called bra) to show that the objects are column vectors or row vectors in the vector space, re- Note that although the Schrödinger equation is regarded spectively. Denote by superscripts ∗, and † the conjugate as somewhat mysterious when it is first encountered, for a of a complex number, the transpose of a vector or matrix, Markov chain in continuous time with a finite state space, and conjugate transpose operation, respectively. For |u transition probability matrix Pt and Q-matrix Q,weuse = and |v in the vector space, we denote their inner product exactly the same argument: from Ps+t PsPt ,bydiffer- √ ∂Pt = by u|v, which induces a norm u= u|u,andadis- entiation we obtain the Kolmogorov equation ∂t QPt , = tance u − v between |u and |v. For example, Cd has which has the solution Pt exp(Qt)P0. a natural inner product As an alternative, we can describe a quantum system by a so-called density matrix. For a d-dimensional quan- d tum system, its quantum state can be characterized by a | = ∗ = ∗ ∗ u v uj vj u1,...,ud (v1,...,vd ) , density matrix ρ on the d-dimensional complex space Cd , j=1 where ρ satisfies (1) Hermitian; (2) positive semi-definite; (3) unit trace. We often classify a quantum state as a pure where u|=(u1,...,ud ) and |v=(v1,...,vd ) .Givena † state or an ensemble of pure states. A pure state corre- matrix A = (aij ), we say it is Hermitian if A = A ,and sponds to a density matrix ρ =|ψψ|,where|ψ is a denote its trace by tr(A) = k a . A matrix U is said j=1 jj unit vector in Cd . An ensemble of pure states has a den- † = † = to be unitary if UU U U I. For two matrices A1 and sity matrix A2, define their commutator [A1, A2]=A1A2 − A2A1. Denote by ⊗ the tensor product operation of vectors or J = | | matrices. To analyze computer algorithms, we adopt a no- (2.2) ρ pj ψj ψj , = tation O(h(m)) to denote that the asymptotic scaling of j 1 an algorithm is upper-bounded by a function h(m) of the which corresponds to the scenario that the quantum sys- ˜ input size m, with the notation O(h(m)) ignoring loga- tem is in one of states |ψj , j = 1,...,J, with probability rithmic factors. pj beinginthestate|ψj . The quantum evolution in the QUANTUM SCIENCE AND QUANTUM TECHNOLOGY 53 density matrix representation can be described as follows. the states, with an example of the switch being either on Let ρt be the density matrix of the state of the quantum or off. system at time t. With the unitary matrix U(·) and Hamil- In quantum science and quantum technology, the coun- tonian H introduced above, the density matrix evolution is terpart of the classical bit is the quantum bit, which we = † given by ρt+s U(t)ρsU (t), with the Schrödinger equa- call qubit for short.
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