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. Similar to a classical bit with two tionintheformof state values 0 and 1, a qubit has states |0 and |1,where √ √ we use the customary Dirac notation |· to denote the = − −1Ht −1Ht ρt e ρ0e or equivalently qubit state. However, one key difference exists between a (2.3) √ ∂ρ classical bit and a qubit. Specifically, the theory of quan- −1 t =[H, ρ ]. ∂t t tum physics allows the description of a quantum physical system through probabilistic combinations of its states, See Sakurai and Napolitano (2017)andShankar (2012) which is referred to as the superposition property. The for details. superposition of states can accommodate all predictions As we will see in Section 3.1, the number of complex for the outcomes of physical measurements, moreover, numbers and the dimensionality of vectors and matrices it bears drastic consequences for the nature of the phys- required to describe a quantum state and its evolution ical states ascribed to a system. In this regard, besides the usually increase exponentially in the system size, rather states |0 and |1, a qubit can be in superposition states than linearly in a classical system. As a result, a quan- with the following form: tum system can store and manage an exponential num- ber of complex numbers and perform data manipulations (3.1) |ψ=α0|0+α1|1, and calculations during the evolution of the system, while where complex numbers α0 and α1 are called amplitudes classical computers find it difficult to cope with the quan- 2 2 and satisfy |α0| +|α1| = 1. As a result, the states of a tum system as it requires an exponential number of bits qubit are unit vectors in a two-dimensional complex vec- of memory to store the quantum state. Unlike the clas- tor space C2. The states |0 and |1 form an orthonormal sical case where we often need to consider some extra basis for the space and are often referred to as computa- structural assumptions or approximations when handling tional basis states. Unlike classical bits that have mutually high-dimensional objects, quantum systems have poten- exclusive states, qubits can be one and zero simultane- tial to deal with exponentially high-dimensional problems ously, which is known as the most fundamental aspects of without imposing additional constraints. Special quantum qubits. In other words, a superposition state is a state of phenomena are utilized to accomplish quantum commu- matter that can be viewed as simultaneous occurrence of nication and computational tasks, and subsequent sec- zero and one at the same time. tions will illustrate that the quantum phenomena are of- Qubits can be realized in various physical systems. Ex- ten strange, and counter-intuitive. For example, light can amples of qubits include the two states of an electron or- be particles and waves (wave-particle duality); a cat can biting a single atom, the two different polarizations of a be alive and dead at the same time (quantum superpo- photon, the alignment of a nuclear spin in a uniform mag- sition); information can transmit instantaneously over a netic field, or the two directions of current flows in su- long distance without going through the intervening space perconducting circuits. Specifically, in the atom model, (quantum teleportation); without sufficient energy, quan- |0 and |1 can be treated respectively, as the so-called tum particles can pass a barrier that is classically impos- ‘ground’ and ‘excited’ states of the electron; if the atom sible (quantum tunneling). is shined by light with appropriate energy and for a suit- able amount of time, we may transfer the electron from 3. QUANTUM BITS AND QUANTUM PROPERTIES the |0 state to the |1 state and vice versa. Furthermore, 3.1 Quantum Bit and Superposition by adjusting the time length for shining the light on the atom, the electron can be moved from the initial state In classical information and computation, the most fun- |0 into ‘halfway’ between√ |0 and |1, for example,√ into damental entity is the bit, and the information encoded in state |+ = (|0+|1)/ 2, or state |− = (|0−|1)/ 2, a bit has two state values, 0 and 1. Classical bits can be where |+ and |− form a qubit basis that is equiva- materialized in multiple means, for example, they may be lent to the computational qubit basis |0 and |1.Note realized mechanically as switches or magnetically as hard that the quantum state transformations are solutions of drives. An important fact of the classical bit is that its two the Schrödinger equation (2.1) for particular choices of state values are mutually exclusive, namely, its state can Hamiltonian H and time interval, and it is an interesting only be either 0 or 1. This fact leads to one thing in com- exercise for readers to find the appropriate Hamiltonians. mon for all of the realization means, that is, all classical It is easy to examine a classical bit to determine its state, physical devices prevent the simultaneous occurrence of being 0 or 1, however, it is impossible to examine a qubit 54 Y. WANG AND X. SONG

|ψ to determine its state or find the values of its ampli- qubits. The measurement outcome is absolutely random, tudes α0 and α1 defined in (3.1). Because of the stochastic and it is impossible to gain information about the entan- nature of quantum theory, performing measurements on gled system from the obtained random measurement out- the qubit |ψ will result in measurement outcome 0 with come. As the entangled state involves two qubits, their 2 probability |α0| , or measurement outcome 1 with prob- correlation must contain two bits of classical informa- 2 ability |α1| . Moreover, performing measurements on the tion, and the classical information can only be gathered qubit will change its state. by comparatively examining the outcomes of the individ- Like classic bits, we may define multiple qubits. The ual measurements on the separate subsystems. We as well states of one b-qubit are unit vectors in a 2b-dimensional point out an intriguing feature of entangled states: mea- complex vector space. The quantum exponential com- suring one of the entangled qubits instantaneously casts plexity is then shown in the exponential growth of dimen- the other one into the corresponding perfectly correlated sionality 2b and the number of 2b amplitudes required to state, which immediately destroys the entanglement as specify superposition states. For the 2-qubit case, its su- qubit measuring changes their quantum state. We take a perposition states are unit vectors in a 4-dimensional com- Bell state plex vector space, with the following form: |01−|10 (3.4) |ψ= √ (3.2) |ψ=α00|00+α01|01+α10|10+α11|11, 2 where |00, |01, |10,and|11 are four computational as an example to demonstrate√ entanglement,√ where α00 = basis states, amplitudes αx are complex numbers satisfy- α11 = 0, α01 = 1/ 2, and α10 =−1/ 2 in the expres- 2 2 2 2 ing |α00| +|α01| +|α10| +|α11| = 1. As in the single sion of (3.2). As described in Section 3.1, measuring the qubit case, when measuring the 2-qubit, we obtain mea- first qubit of the Bell state |ψ, we obtain measurement 2 2 surement outcome x as one of 00, 01, 10, 11, with a corre- outcome 0 or 1 with probability |α00| +|α01| = 1/2and | |2 2 2 sponding probability αx . Furthermore, we may perform |α10| +|α11| = 1/2 respectively, which is completely a measurement just on the first qubit of the 2-qubit sys- random. According to (3.3), if the measurement outcome tem and obtain either the measurement outcome 0, with is 0 (or 1), then the state will be |01 (or |10, respec- 2 2 probability |α00| +|α01| , or the outcome 1, with proba- tively). This result means that if the first measurement 2 2 bility |α10| +|α11| . As quantum measuring changes the outcome is 0 (or 1), then the second qubit’s state must be quantum state, depending on the measurement outcome |1 (or |0, respectively) with measurement always being obtained for the first qubit, being either 0 or 1, the 2-qubit 1 (or 0, respectively), which indicates perfect correlation. system will be in the state Quantum states like the Bell state in (3.4) that cannot be

α00|00+α01|01 α10|10+α11|11 expressed as products of some single qubits are called en- (3.3) or , tangled states, while product states refer to quantum states | |2 +| |2 | |2 +| |2 α00 α01 α10 α11 that can be written in the product form of single qubits. respectively. See Nielsen and Chuang (2010)andWang Over the past decades many physical experiments have (2012) for details. been designed and conducted to test quantum entangle- ment through the so-called Bell inequality. 3.2 Quantum Entanglement For the case of a 2-qubit system realized by the spins As one of the most mind-bending creatures known to of two particles, imagine that the two-particle system is science, quantum entanglement is often cited as the phe- first prepared in an entangled state, then the two particles nomenon that two particles that are connected by an in- are drifted far away from each other. We now have Alice visible wave can share each other’s properties regardless and Bob measure the first and second particles, respec- of the distance between them, just like a twin. It leads to tively, and sequentially. The perfect correlation suggests the fact that none of the particles involved in a quantum that after Alice obtains her spin measurement result (i.e., system can be described by quantum states of individual +1or−1) on the first particle, the system has its state subsystems. In other words, all information content of an immediately plunged into the untangled state. As a re- entangled quantum system is fully entailed in the corre- sult, the second particle now has a definite spin state, and lations between the individual subsystems while none of Bob’s spin measurement on the second particle always the subsystems on their own convey essential informa- provides a definite opposite result (i.e., −1or+1, respec- tion of the entangled quantum system. For a multi-qubit tively). This phenomenon of perfect correlation is referred system, its entangled states are superposition states that to as anti-correlation in entanglement experiments. We are described by joint properties of the individual qubits will show that quantum properties such as superposition in the multi-qubit system. Consider an entangled 2-qubit and entanglement play key roles in quantum science and system, we obtain a completely random outcome when quantum technology. See Horodecki et al. (2009), Nielsen performing measurements on only one of its entangled and Chuang (2010)andWang (2012) for more details. QUANTUM SCIENCE AND QUANTUM TECHNOLOGY 55

4. QUANTUM INFORMATION we measure it in the diagonal and anti-diagonal basis, its information cannot be extracted. Indeed, as described in As its classical analog, quantum information targets Section 3.1,wehave at determining the laws governing any information pro- 1 1 1 1 cess based on quantum theory. The core of classical |0=√ |+ + √ |−, |1=√ |+ − √ |−, information theory is Shannon’s two coding theorems 2 2 2 2 on noiseless and noisy channels. The coding theorems and thus, when measuring |0 or |1 in the basis |+ and quantify classical bits by Shannon entropy for transmis- |−, we observe + and − with equal probability, that is, sion over a noiseless channel and character the amount we observe a diagonally polarized photon in 50% of the of information transmitted over a noisy channel with cases and an anti-diagonally polarized photon in the other some error-correction scheme. On the other hand, the 50% of the cases. quantum-based theory has been established to apprehend Quantum physics provides new types of resources for quantum resources such as superposition, entanglement, information processing and transmission such as quan- nonlocality, no-cloning, and quantum randomness. The tum teleportation, superdense coding, quantum key distri- quantum counterparts of Shannon entropy and Shannon bution, and quantum error-correction. Also, quantum in- noiseless coding theorem are von Neumann entropy and formation ideas have effectively been employed in other Schumacher’s noiseless channel coding theorem, respec- scientific studies such as many-body physics, quantum tively. Schumacher’s noiseless channel coding theorem gravity, high-energy physics, quantum chemistry, quan- describes quantum information needed to compress quan- tum biology, and even for solving conjectures in the fields tum states by von Neumann entropy (Schumacher, 1995). of classical information and computation. See Hayashi The quantum analog of Shannon’s noisy channel coding (2006), Krenn et al. (2017), Nielsen and Chuang (2010) theorem is Holevo–Schumacher–Westmoreland theorem and Wang (2012) for more details. employed to calculate the product quantum state capacity for some noisy channels (Holevo, 1998, Schumacher and 5. QUANTUM COMMUNICATION Westmoreland, 1997). Despite the resemblance, there exist inherent distinc- 5.1 Quantum Teleportation tions between classical information and quantum infor- Quantum teleportation is a process through which we mation. For example, while classical information such transfer the state of a quantum system (or qubit) to another as digital images can be distinguished and copied, quan- distant quantum system (or qubit) without ever existing in tum superposition and no-cloning theorem imply that the intervening space in between. The phenomenon can unknown quantum states cannot be completely distin- be illustrated by a three-step protocol of quantum telepor- guished or exactly copied. Consider another example, be- tation as follows. First, Alice (the sender) and Bob (the sides the computational basis |0 and |1 for the√ qubit receiver) together generated a special pair of entangled space, we have another√ basis |+ = (|0+|1)/ 2and qubits, and each took one qubit of the two shared qubits |− = (|0−|1)/ 2 given in Section 3.1, and quantum when they split. Second, Alice was given a third qubit information can be encoded under each of these bases. whose state was undisclosed to her, and she would like Different ways of encoding quantum information are em- to teleport the unknown state. Third, Alice interacted the ployed in quantum error-correction for reliable quantum third qubit with her qubit and performed a special mea- computation and quantum information processing. More- surement on her original qubit so that, while the measure- over, the information encoded under one basis cannot be ment destroyed the entanglement and ruined any infor- extracted by performing measurement under another ba- mation about the state of her qubit, it would make Bob’s sis, which plays an important role in quantum cryptogra- qubit instantaneously project onto a new state that Bob phy. For example, consider encoding one bit of informa- could use to recover the original state of Alice’s qubit. tion in different bases by the polarization of light. Sup- After the three steps, the state of Alice’s qubit was trans- pose that the computational basis formed by |0 and |1 ported to that of Bob’s qubit. A key feature in the quan- represents the horizontal and vertical basis (correspond- tum teleportation protocol is the special entanglement and ing to horizontally and vertically polarized photons). As measurement, and the teleportation protocol only works diagonally and anti-diagonally polarized photons can be when Bob was informed by Alice about her measurement expressed in the horizontal and vertical basis as coher- outcome so that Bob could work accordingly to recover ent superpositions of horizontal and vertical parts, the ba- Alice’s state. That is, a successful teleportation event re- sis formed by |+ and |− corresponds to the diagonal quires classical communication between Alice and Bob, and anti-diagonal basis. We may encode a bit of informa- which necessarily restricts the speed of information trans- tion in the |0 and |1 basis by treating 0 to be horizontal fer in the teleportation protocol to the speed of the clas- polarization and 1 to be vertical polarization. For a pho- sical communication channel. See Nielsen and Chuang ton encoded in either horizontal or vertical polarization, if (2010)andWang (2012) for details. 56 Y. WANG AND X. SONG √ It is important to note from the entire three-step pro- |+ = (|0+|1)/ 2, along with an ancilla qubit |a. tocol of quantum teleportation that contrary to what is Cloning implies there exists a unitary matrix U such that usually mistakenly cited, quantum teleportation in prin- U |0|a =|0|0, U |1|a =|1|1, ciple does not allow faster-than-light communication or (5.2) any transfer of matter or energy. Quantum teleportation U |+|a =|+|+. transfers only the state of Alice’s qubit to Bob’s qubit but U does not physically move Alice’s qubit (particle) to Bob. Using the first two equalities in (5.2) and linearity of we immediately obtain Because it is required to send information via the classical channel, quantum teleportation is not capable of transmit- 1 1 U |+|a = U √ |0|a+√ |1|a ting information faster than the speed of light. Otherwise, 2 2 if Bob can obtain a copy of Alice’s qubit (in the sense 1 1 to physically obtain her ‘qubit’), then Bob can make a = √ U |0|a + √ U |1|a direct measurement on the copied qubit to obtain the in- 2 2 formation that was sent over via the classical communica- 1 1 = √ |0|0+√ |1|1, tion between Alice and Bob. In this way, faster-than-light 2 2 communication becomes possible, however, the famous which is an entangled state. It is easy to see that the en- no-cloning theorem prevents the teleportation from copy- tangled state cannot be written as product state ing any qubit. 1 1 1 1 The no-cloning theorem is referred to the fact that quan- |+|+ = √ |0+√ |1 √ |0+√ |1 tum mechanics prohibits the creation of identical copies 2 2 2 2 of a general quantum state. Specifically, cloning a quan- 1 | = |0|0+|1|1+|0|1+|1|0 . tum state ψ means a procedure with the product state 2 |ψ|ψ as an output. We begin by introducing an ancilla | Therefore, an inconsistency occurs in (5.2), and it is im- quantum system whose state ϕ is not related to the state possible to have all three equalities in (5.2). See Krenn | ψ being cloned. The no-cloning theorem means that et al. (2017)andNielsen and Chuang (2010) for more de- there exists no unitary matrix U such that it evolves the tails. initial state |ψ|ϕ to the desired output state |ψ|ψ,that is, 5.2 Communication Components √ − The classical communication required in quantum tele- (5.1) U |ψ|ϕ = e 1θ(ψ,ϕ)|ψ|ψ, √ portation does not carry complete information about the where e −1θ(ψ,ϕ) stands for a phase factor, with phase qubit being teleported. Even if the information communi- θ(ψ,ϕ) being some real number. Indeed, if such U exists, cated in the classical channel is intercepted by an eaves- it has a similar effect on any arbitrarily selected state |φ dropper who may have complete knowledge about what since cloning should work for any state. For the pair of Bob is required to do to recover the desired state, the in- states |ψ and |φ in Cd , we consider their inner product formation is futile if the eavesdropper cannot interact with together with the ancilla state |ϕ, and use (5.1) to obtain the entangled qubit held in Bob’s hands. Quantum physics opens the door to various distinct quantum secret sharing ψ|φ=ψ|φϕ|ϕ protocols such as quantum cryptography. In quantum computation and quantum communication, =ψ|ϕ||φ|ϕ we need to link qubits in quantum networks and trans- =ψ|ϕ|U†U|φ|ϕ fer their states. As the no-cloning theorem forbids us to √ − − [ − ] perfectly clone a quantum state, we cannot use classical = e 1 θ(ψ,ϕ) θ(φ,ϕ) ψ|ψ||φ|φ √ methods like amplifiers to carry out any transfer of qubits’ − − [ − ] = e 1 θ(ψ,ϕ) θ(φ,ϕ) ψ|φ 2, states. The solution to this problem is a so-called quan- tum repeater, which allows the end-to-end generation of which indicates that |ψ|φ| = |ψ|φ|2, namely, |ψ|φ| quantum entanglement in a way that every two connec- equals to 0 or 1. By the Cauchy–Schwarz inequality, we tive particles of independent entangled pairs is combined conclude that |ψ is either equal to |φ (with a phase so that the entanglement is relayed onto the remaining factor) or orthogonal to |φ, which is not possible for two particles. Thus, we are able to achieve the end-to-end an arbitrary pair of states |ψ and |φ. This shows the state transmission of qubits via quantum teleportation. A nonexistence of such U and thus proves the no-cloning quantum repeater is an important building block to inter- theorem. Moreover, we may provide a simple illustra- connect different nodes in a quantum network, and quan- tion to show that no-cloning is a natural consequence of tum teleportation is one crucial requirement for the quan- quantum theory as follows. Consider qubits |0, |1,and tum repeater. This process is referred to as entanglement QUANTUM SCIENCE AND QUANTUM TECHNOLOGY 57 swapping and enables us to achieve long-distance quan- On the other hand, Peter Shor in 1994 discovered the tum communication. See Krenn et al. (2017), Nielsen and so-called Shor’s quantum factoring algorithm that can Chuang (2010)andSangouard et al. (2011) for more dis- solve the factoring problem exponentially faster than the cussions. best known classical algorithms, thus quantum computers may be able to break the RSA system easily (Shor, 1994). 5.3 Quantum Cryptography That is, as quantum computers can factor prime numbers Cryptography allows two parties, the sender Alice and significantly faster than classical computers, an eaves- the receiver Bob, to exchange secret messages in their dropper equipped with a quantum computer can decipher private communications, while at the same time, keeps it the encrypted text and read the secret message, with only very hard for the third parties to ‘eavesdrop’ on the con- the public information distributed by RSA. An approach tent of their communications. Applications of cryptog- to circumventing this difficulty is a quantum procedure raphy include online bank transactions, electronic com- known as quantum cryptography or quantum key distri- merces, and military communications. We discuss two bution so that communication security cannot be under- cryptographic methods adopted in such communications. mined. The security of the quantum key distribution is The first method is a private key cryptosystem, which based on the quantum principle that observing or measur- calls for the two parties to share a secret key. Specifically, ing an unknown quantum system will disturb the system Alice employs the key to encrypt a message and obtain that is being monitored. When an eavesdropper listens to the cipher while the cipher can only be understood if the the transmission of the quantum key between Alice and key is known. Alice sends the cipher to Bob who utilizes Bob, the quantum communication channel employed to the key to decrypt the received cipher and read her mes- set up the key will be disturbed by the eavesdropping, and sage. The challenge for the private key cryptosystem lies the disturbance will make eavesdropping noticeable. As a in guarding the secret key against eavesdropping. result, Alice and Bob are able to discard the compromised The alternative method is a public key cryptosystem in- key and retain only the secured key for their communica- vented in the 1970s that does not require the sharing of a tion. secret key. This method is based on the complexity of hard Specifically, quantum key distribution enables two au- computational problems such as finding the prime factors thorized parties to create a secret key at a distance in of very large numbers. Specifically, Bob first generates a two stages. In the first stage, the two communicating par- pair of keys, a public one and a private one. He then an- ties, Alice and Bob, obtain a preliminary key by exchang- nounces his ‘public key’ to the general public, everyone ing quantum signals over the quantum channel and per- including Alice can use the public key to encrypt mes- forming measurements. The obtained key is preliminary sages and send him the encrypted messages. The real trick in the sense that it has two strongly correlated, but non- is that the encryption transformation generated by Bob’s identical, and only partly secret strings. In the second keys is specially designed such that with only the public stage, Alice and Bob utilize the classical channel to carry key, it is extraordinarily hard, though not impossible, to out an interactive post-processing protocol. The protocol reverse the encryption transformation. When announcing permits them to refine the preliminary key and extract the public key, Bob retains a corresponding secret key for two identical and absolutely secret (known only to them- simple inversion of the encryption transformation and de- selves) strings as two identical copies of the created se- cryption of the received messages. A case in point is the cret key. During the two-stage process, the quantum chan- RSA cryptosystem (Rivest, Shamir and Adleman, 1978), nel is open to any possible maneuver from a third person. one of the most widely used cryptographic protocols. However, the classical channel communication needs the RSA is built on the extreme difficulty of finding prime following authentication: while Alice and Bob recognize factors for large composite numbers. Note the mathemat- themselves, a third person may listen to their exchange, ical asymmetry of factoring: it is easy to compute a com- but cannot engage in it. In particular, the mission of Al- posite number from its prime factors by multiplying the ice and Bob is to guarantee security against an adver- primes, no matter how large they are; however, the reverse sarial eavesdropper, whom we call Eve, engaging in the process can be very hard, in fact, it is extremely difficult conversation over the classical channel and tapping on to find the prime factors of some very large composite the quantum channel. Here we use ‘security’ to convey numbers. RSA encryption retains the large primes as a se- precisely that the authorized parties never use a nonse- cret key and makes use of their product to design a ‘public cret key, namely, either they can actually generate a se- key’. Since the best known classical factoring algorithms cret key, or the protocol is aborted. Hence after transmit- have exponential complexity, and massive computational ting the quantum signals, Alice and Bob need to evalu- attempts to break the RSA system so far have led to no ate the possible amount of information about the prelimi- success, it is widely believed that the RSA system is se- nary keys may have leaked out to Eve. This is the crucial cure against any classical computer-based attacks. advantage of quantum communication where information 58 Y. WANG AND X. SONG leakage in a quantum channel is quantitatively linked to a |φ. However, since unitary transformations preserve in- degradation of the communication. Such degradation and ner products, it must be that evaluation are not possible in classical communication. | || | = | | † | | = | || | For example, when classical communication channels are v1 ψ φ v2 u ψ U1U2 φ u u ψ φ u , tapped, such as phone conversations are bugged, the com- that is, v1|v2ψ|φ=u|uψ|φ.As|ψ and |φ are munication proceeds without any change, as nothing hap- nonorthogonal, ψ|φ =0, and thus we obtain pens. | = | = Besides taking advantage of quantum property that ob- v1 v2 u u 1, serving a quantum channel disturbs the quantum commu- which indicates that |v1 and |v2 have to be equal. This nication, we may employ quantum entanglement to fur- leads to a contradiction, as |v1 =|v2. Therefore, distin- ther enhance the security of quantum key distribution by guishing between |ψ and |φ must disturb at least one creating an entanglement-based quantum key distribution. of the states, and we can make secure quantum communi- The quantum entanglement property offers secure advan- cation by transmitting nonorthogonal qubit states between tages for designing and implementing the protocol of the Alice and Bob and checking for disturbance in their trans- entanglement-based quantum key distribution. Let us con- mitted states. sider the case where Alice and Bob share an entangled Furthermore, the quantum key generation relies on the state of a qubit (or particle) pair. When they perform mea- same quantum physical principles that quantum computa- surements on the qubits, the obtained random measure- tion is based on. Unlike classical cryptography, the quan- ments will always be opposite due to the perfect correla- tum key distribution does not merely depend on the com- tion between entangled qubits. Alice and Bob then need putational difficulty of solving mathematical problems to communicate over a classical channel regarding how such as the factoring problem. Hence, it cannot be bro- the measurements are performed and obtained in order to ken even by quantum computers. In a nutshell, the funda- sift through the results and obtain a secret key. mental quantum physical principles allow for the uncon- The security foundation of quantum key distribution ditional security of quantum key distribution, namely, the can be established by the core principles of quantum possibility of guaranteeing security without setting any physics such as superposition and no-cloning. When Eve power limitation on the eavesdropper. See Bennett and is tapping on a quantum communication channel to ex- Brassard (2014), Bernstein and Lange (2017), Buhrman tract some information, her act is some kind of measure- et al. (2010), Krenn et al. (2017), and Nielsen and Chuang ment performing on the state of the quantum communica- (2010) for more details. tion system, and the measurement will generally alter the Quantum physics was established to describe nature at state of the system. On the other hand, if Eve wants a cor- the microscopic domain, but many ongoing research en- rect copy of the state that Alice conveys to Bob, she will deavors seek answers to what extent the quantum physical not be successful as the no-cloning theorem shows that laws are relevant to the macroscopic realm. In particular, an unknown quantum state cannot be duplicated without research efforts in quantum science and quantum technol- being altered. ogy aim to increase the distance between entangled quan- To be specific, the essential idea behind quantum key tum particles, and search for any possible fundamental re- distribution is that Eve cannot obtain any information strictions to quantum entanglement, as well as to investi- from the qubits, whose state is transmitted from Alice to gate if it is viable to create a global-scale quantum com- Bob, without disturbing their state. Our proof arguments munication network in the future. Physical experiments are as follows. First, the no-cloning theorem described on quantum key distribution have been successfully con- in Section 5.1 prevents Eve from copying Alice’s qubit. ducted in a long distance with current records of over a Second, information gain implies disturbance in the sense hundred kilometers on earth (Krenn et al., 2017) and over that for any try to differentiate between two nonorthog- a thousand kilometers in space (Yin et al., 2017). onal quantum states, gaining information is only possible at the cost of bringing in disturbance to the signal. Indeed, 6. QUANTUM COMPUTATION suppose that |ψ and |φ are two nonorthogonal quantum states. Eve attempts to gain information about |ψ and In contrast to classical computation where transistors |φ by unitarily interacting the states |ψ or |φ with an are used to crunch the ones and zeroes individually, ancilla quantum system prepared in a state |u.IfEve’s the new quantum resources such as quantum superpo- attempt does not disturb the states, we obtain two unitary sition and entanglement can allow quantum computa- matrices U1 and U2 such that tion to manage both one and zero at the same time and do the trick of performing simultaneous calculations. U |ψ|u =|ψ|v , U |φ|u =|φ|v , 1 1 2 2 Thus, quantum computers may outperform classical com- where |v1 and |v2 are different states so that Eve can puters for solving certain computational problems. See gain information about the identity of the states |ψ and Browne (2014), Campbell, Terhal and Vuillot (2017), QUANTUM SCIENCE AND QUANTUM TECHNOLOGY 59

Chong, Franklin and Martonosi (2017), Deutsch (1985), decoherence in quantum technology. Given the significant Mohseni et al. (2017), Nielsen and Chuang (2010)and difficulties to build large-scale quantum computers with Wang (2012). present technology, it is very important to have scalable architectures for building quantum computers with about 6.1 Quantum Computers 100 well-behaved logical qubits in the near future. Such Classical computers are constructed from electrical cir- architectures may enable us to demonstrate the so-called cuits containing wires for carrying information around the quantum (computational) supremacy that is actively pur- circuits and logic gates for executing simple computa- sued by academic labs and companies like Google and tional tasks. Similarly, quantum computers are built from IBM, where quantum supremacy refers to any major mile- quantum circuits with quantum gates to carry out quantum stone achievement in the quest for outperforming classical computation and process quantum information. In spite computers on some tough computational tasks (Aaronson of the similarity, quantum computers are built on the uni- and Chen, 2017, Boixo et al., 2018, Harrow and Monta- tary evolution of b logical qubits operating on a compu- naro, 2017). tational state space of 2b dimensions, and the new quan- The quantum computing approach discussed so far is tum resources make it possible for quantum computers logic-gate based that has its purpose in developing a quan- to outperform classical computers for certain tough tasks. tum version of classic logic gate operations and con- Quantum information and quantum computation investi- structing a universal (or general purpose) quantum com- gate how to harness the enormous information hidden in puters. Since significant technological difficulties present the quantum systems and how to make use of the immense in the implementation of the gate (or circuit) model for potential computational power of quantum particles to building universal quantum computers, alternative quan- perform computation and to process information. Inten- tum computing architectures, such as adiabatic quantum sive efforts are underway around the world to explore computing, are actively being explored to build special- a number of physical systems and fabrication technolo- purpose quantum computers for solving specific com- putational problems, though subjected to different chal- gies for constructing quantum computers, where viable lenges (Aharonov and Ta-Shma, 2003, Aharonov et al., constructions must meet a set of requirements known as 2008, Albash and Lidar, 2018). Examples of special- the DiVincenzo criteria (DiVincenzo, 1995). Major sys- purpose quantum computers include quantum annealers tems and technologies include superconducting circuits, and quantum simulators for solving tough simulation and ion traps, quantum dots, and other electronic semicon- optimization problems. Next, two Sections 6.2 and 6.3 ductor circuits, impurity spins, and linear optics (Nielsen will present detailed discussions on quantum annealers and Chuang, 2010). Quantum computers of small scale and quantum simulators, respectively. Quantum anneal- have been built to demonstrate numerous simple exam- ers mean physical hardware implementations of quantum ples of quantum algorithms and protocols. Over the years annealing. Quantum simulators refer to quantum devices there are steadily increasing efforts by academics, govern- utilized for simulating one quantum system by using an- ment labs, large companies, and startups to reach the chal- other more controllable one, with the aim to solve spe- lenging goal of large scale quantum computation (DiCarlo cial simulation problems that are computationally too de- et al., 2009, Johnson et al., 2011, Mariantoni et al., 2011, manding on classical computers. Sayrin et al., 2011). As mentioned above, the physical equipment for the 6.2 Quantum Annealers quantum computer fabrication must meet the DiVincenzo Quantum annealing may be considered as adiabatic criteria including requirements that a quantum system re- quantum computing that is based on the quantum adia- alized qubits has to be well isolated to maintain its quan- batic theorem for building special-purpose quantum com- tum properties and at the same time, the quantum system puters, called quantum annealers, to solve combinatorial needs to be accessible so that the qubits can be operated optimization problems. Quantum annealing is the quan- to carry out computations and perform output measure- tum analog of classical annealing, with thermodynamics ments. In reality, there always exists some coupling of replaced by quantum dynamics. Quantum annealers are a quantum system to its environment, and the coupling physical hardware devices to implement quantum anneal- leads to quantum decoherence, where decoherence refers ing. See Albash and Lidar (2018), McGeoch (2014), and to the loss of coherence between the components of the Wang, Wu and Zou (2016). quantum system or quantum superposition from the in- Given an optimization problem, we identify its objec- teraction of the quantum system with its external entities. tive function to be minimized with the energy of a phys- Therefore, the coupling strength dictates the two oppos- ical system and assign the physical system a temperature ing requirements stated above. It is very challenging but that serves as an artificially-introduced control parameter. critical to manage a quantum system of qubits for con- Classical annealing like simulated annealing takes into ac- trolling the coupling strength and rectifying the effects of count the relative configuration energies and a fictitious 60 Y. WANG AND X. SONG time-dependent temperature when exploring the immense and solve the minimization problem numerically when b search space probabilistically. By decreasing the temper- is large. In fact, the search space that grows exponentially ature slowly from a high value to zero, with certain proba- prohibits us to solve the minimization problem with de- bility we move the system toward the state with the lowest terministic exhaustive search algorithms. Instead, anneal- value of the energy and hence arrive at the solution of the ing methods such as simulated annealing are employed optimization problem. to search the space probabilistically. To find a config- Specifically, consider a classical Ising model character- uration with minimal energy, simulated annealing uses ized by a graph G = (V(G), E(G)),whereV(G) and E(G) Markov chain Monte Carlo (MCMC) methods such as the represent the vertex and edge sets of G, respectively. Each Metropolitan–Hastings algorithm to generate configura- vertex has a random variable whose value is +1or−1, tion samples from the Boltzmann distribution PT (s) while and each edge corresponds to the coupling (or interaction) decreasing the temperature T slowly. See Bertsimas and Tsitsiklis (19932), Kirkpatrick, Gelatt and Vecchi (1983), between two vertex variables linked by the edge. Define a and Wang, Wu and Zou (2016). configuration s to be a set of values assigned to all vertex Quantum annealing utilizes the physical process of a variables s , j ∈ V(G),thatis,s ={s ,j ∈ V(G)}.Ver- j j quantum system whose lowest energy, or equivalently, tices and vertex variables also refer to sites and spins in + − a ground state of the system, renders a solution to the physics, respectively, where the values 1and 1ofa posed optimization problem. It begins by creating a sim- spin stand for spin up and spin down, respectively. A case ple quantum system initialized in its ground state and then in point is a 2-dimensional lattice considered as a sim- drives the simple system slowly to the target complex sys- ple graph, with a magnet put at each lattice site pointing tem. The quantum adiabatic theorem (Farhietal., 2000, either up or down. Denote by b the total number of the lat- 2001, Farhi, Goldstone and Gutmann, 2002, Kadowaki tice sites. At site j = 1,...,b,letsj be a binary random and Nishimori, 1998) implies that, as the system gradu- variable representing the position of the magnet, where ally evolves, it likely stays in a ground state, and therefore sj =±1 means that the jth magnet points up or down, with some probability, we can find a solution to the orig- respectively. The classical Ising model has the following inal optimization problem by measuring the state of the Hamiltonian: final system. In other words, by replacing thermal fluctu- c ≡ c =− − ations in simulated annealing by quantum fluctuations via (6.1) HI HI (s) δij sisj γj sj , quantum tunneling, quantum annealing manages to keep ∈E G ∈V G (i,j) ( ) j ( ) the system close to its instantaneous ground state dur- where (i, j) denotes the edge between the sites i and j, ing the quantum annealing evolution, similar to a quasi- the first sum takes over all pairs of vertices with edge equilibrium state to be maintained during the time evolu- (i, j) ∈ E(G), δij represents the interaction (or coupling) tion of simulated annealing between sites i and j associated with edge (i, j) ∈ E(G), As in the classical annealing case, graph G is used to describe the quantum Ising model, where the vertex set and γj stands for an external magnetic field on vertex V(G) stands for the quantum spins, and the edge set E(G) j ∈ V(G). We refer to a set of fixed values {δij ,γj } as c denotes the couplings (or interactions) between two quan- one instance of the Ising model. HI (s) is also called the energy of the Ising model at configuration s. The probabil- tum spins. As qubits can be realized by quantum spins, G b ity of a specific configuration s is given by the following each vertex is occupied by a qubit. Suppose that has vertices. The quantum system is described by vector space Boltzmann distribution: Cd (d = 2b), with its quantum state described by a unit −Hc (s)/T d e I − c vector in C , and its dynamic evolution governed by the = = HI (s)/T (6.2) PT (s) ,ZT e , Schrödinger equation (2.1) via a quantum Hamiltonian, ZT s which is a Hermitian matrix of size d. The energies of the here T serves as the fundamental temperature of the sys- quantum system are represented by the eigenvalues of the tem with units of energy. The configuration probability quantum Hamiltonian, and the ground states are given by PT (s) describes the probability that the physical system the eigenvectors corresponding to the smallest eigenvalue. is in a state with configuration s in equilibrium. Since quantum mechanics is based on mathematics of ma- When using the Ising model to represent a combina- trices with dimensionality equal to d = 2b (the number of torial optimization problem, the goal is to find a ground possible configurations), we substitute the classical spins state of the Ising model, that is, we need to find a con- (or bits) in (6.1) with quantum spins (or qubits) to obtain figuration that can minimize the energy function Hc (s). the quantum Hamiltonian. To be specific, define I If the Ising model contains b sites, then the configura- 10 x 01 b I = , σ = , tion space is {−1, +1} and the total number of possi- j 01 j 10 ble configurations is equal to 2b. We note that the system complexity increases exponentially in b (the number of z = 10 = σ j − ,j 1,...,b, sites), and thus, it is very difficult to find a ground state 0 1 QUANTUM SCIENCE AND QUANTUM TECHNOLOGY 61

x z where σ j and σ j are Pauli matrices in x and z axes, re- the original optimization problem in the quantum frame- spectively. For the quantum system, each classical vertex work, up to now the computational task for solving the =± z variable sj 1in(6.1) is replaced by σ j for the jth optimization problem is still the same as in the classical quantum spin (or qubit). The two eigenvalues ±1ofthe case. z |+ Pauli matrix σ j correspond to the eigenstates 1 and To carry out quantum annealing for solving the opti- |−1 which further represent the spin up state |↑ and spin mization problem, it is essential to engineer a transverse down state |↓, respectively. In total, there are 2b possible magnetic field that is orthogonal to the Ising axis and quantum configurations formed by selecting b eigenstates obtain the corresponding Hamiltonian in the transverse { z }b from the 2b eigenstates of the Pauli matrices σ j j=1 and field. The transverse field represents kinetic energy that |± ± q then putting them in the form 1,..., 1 . does not commute with the potential energy HI ,there- c We replace sj in the classical Ising Hamiltonian HI (s) fore, it induces transitions between the up and down states z by σ j to obtain the quantum Hamiltonian, of every single spin, and converts the system behavior from classical to quantum. Define the following quantum q =− z z − z (6.3) HI δij σ i σ j γj σ j , Hamiltonian governing the transverse magnetic field (i,j)∈E(G) j∈V(G) (6.6) H =− σ x, where δ stands for the Ising coupling along the edge X j ij j∈V(G) (i, j) ∈ E(G),andγj represents the local field on the ver- tex j ∈ V(G). Here we adopt the convention in quantum where again following the quantum convention we denote z z z by σ x the tensor products of b matrices of size 2 as fol- literature that σ j and σ i σ j in (6.3) stand for their tensor j products along with identical matrices as follows: lows: vertex j vertex j x ≡ ⊗···⊗ ⊗ x⊗ ⊗···⊗ z ≡ ⊗···⊗ ⊗ z ⊗ ⊗···⊗ (6.7) σ j I1 Ij−1 σ j Ij+1 Ib. (6.4) σ j I1 Ij−1 σ j Ij+1 Ib, z q z z Furthermore, σ x does not commute with σ in H ,and σ σ ≡ I1 ⊗···⊗Ii−1 j j I i j b HX is a nondiagonal matrix of size 2 that does not com- vertices i and j q mute with diagonal matrix HI . We note that Pauli matrix ⊗ z ⊗ ⊗···⊗ ⊗ z x + − (6.5) σ i Ii+1 Ij−1 σ j σ j has two eigenvalues 1and 1 associated with the   eigenvectors |uj,+1=(1, 1) and |uj,−1=(1, −1) .As ⊗ I + ⊗···⊗I . j 1 b a result, the eigenvector corresponds to the smallest eigen- All elements in (6.4) are identity matrices except for the value of HX is |u+=|u1,+1⊗|u2,+1⊗···⊗|ub,+1, z z z | jth element being a Pauli matrix σ j ,andσ i σ j in (6.5) that is, u+ is the ground state of HX. is simply an ordinary matrix multiplication of matrices We start the quantum annealing procedure with a quan- z z σ i and σ j treated as tensor products of b matrices in the tum system that is driven by the transverse magnetic field sense of (6.4). Each qubit in (6.4)and(6.5) is operated by HX and initialized in its ground state |u+. The system z one matrix, either a Pauli matrix σ or an identity matrix then slowly moves from the initial Hamiltonian HX to its i q Ii . The quantum convention singles out the qubits with final target Hamiltonian HI . During the Hamiltonian evo- Pauli matrices for actual actions but leaves out the iden- lution, according to the adiabatic quantum theorem, the tity matrices and tensor product signs. To generate quan- system has a tendency to stay in the ground states of the q tum Hamiltonian HI we in fact replace sj in (6.1)bythese instantaneous Hamiltonian via quantum tunneling (Farhi b b 2 × 2 matrices, with scalars γj and δij unchanged. Fur- et al., 2000, 2001, Farhi, Goldstone and Gutmann, 2002, thermore, since each term in (6.3) is a tensor product of b McGeoch, 2014). At the end of the annealing procedure, q diagonal matrices of size two, quantum Hamiltonian HI if the quantum system stays in a ground state of the final b × b q is a 2 2 diagonal matrix constructed so that its diago- Hamiltonian HI , we can measure the quantum system to nal elements are completely in agreement with all values obtain a solution of the optimization problem. In specific, c of classical Hamiltonian HI in (6.1) corresponding to the quantum annealing is accomplished by the following in- 2b configurations ordered lexicographically. stantaneous Hamiltonian for the Ising model in the trans- q As a diagonal matrix HI has eigenvalues equal to its verse field: b diagonal entries, which in turn are the 2 possible values = + q ∈[ ] c (6.8) HD(t) A(t)HX B(t)HI ,t 0,tf , of classical Hamiltonian HI , thus finding the minimal en- c ergy of the classical Ising Hamiltonian HI is equivalent to where A(t) and B(t) are smooth functions that depend finding the minimal energy of the quantum Ising Hamilto- on time t and control the annealing schedules, and tf is q nian HI . That is, we need to find a quantum spin configu- the total annealing time. To drive the system from HX to q = = ration with the minimal energy, namely, a ground state of HI ,wetakeA(tf ) B(0) 0whereA(t) is decreas- q = quantum Hamiltonian HI . Although we have formulated ing and B(t) is increasing. It follows that when t 0, 62 Y. WANG AND X. SONG = = = q HD(0) A(0)HX and when t tf , HD(tf ) B(tf )HI . access and analyze, by employing other precisely con- Since A(0) and B(tf ) are known scalars, HD(t) has the trollable quantum systems that are easy to manipulate same eigenvectors as HX at the initial time t = 0and and investigate. Since the dimensionality of the space de- q = as HI at the final time t tf , where the corresponding scribing a quantum system scales exponentially with the eigenvalues differ by factors of A(0) and B(tf ), respec- system size, the classical simulation of quantum systems tively. Therefore, HD(t) moves the system from HX ini- demands exponentially increasing resources. Likewise, it q tialized in its ground state to the final target HI .When takes exponentially large resources to solve certain classi- the control functions A(t) and B(t) are chosen appropri- cal optimization problems particularly the NP-hard prob- ately, the quantum adiabatic theorem indicates that the an- lems, such as finding the ground-state energy of a classi- nealing procedure driven by (6.8) will have a sufficiently cal spin glass and solving the traveling salesman’s prob- c high probability in finding the global minimum of HI (s) lem. Quantum simulation may provide scientific means and solving the minimization problem at the final anneal- to simulate complex biological, chemical or physical sys- ing time tf .SeeBrooke, Bitko and Aeppli (1999), Isakov tems in order to study and understand certain scien- et al. (2016), Jörg et al. (2010), and Wang, Wu and Zou tific phenomena and solve the related hard computational (2016) for details. problems. Experimental platforms for quantum simula- While classical annealing depends on thermal fluctua- tion consist of ultra-cold atomic and molecular quantum tions to drive the system to hop from state to state over gases, ultra-cold trapped ions, polariton condensates in intermediate energy barriers and find a desired lowest- semiconductor nanostructures, circuit-based cavity quan- energy state, quantum annealing substitutes thermal hop- tum electrodynamics, arrays of quantum dots, photonic ping by quantum-mechanical fluctuations to search for quantum technology, and superconducting qubits with a ground state. We realize the quantum fluctuations in commercial applications in quantum annealers (Aspuru- quantum annealing by quantum tunneling that enables Guzik and Walther, 2012, Blatt and Roos, 2012, Bloch, the annealing process to explore different states by cross- Dalibard and Nascimbene, 2012, Boghosian and Taylor, ing directly through energy barriers, rather than climbing 1998, Houck, Türeci and Koch, 2012, Jané et al., 2003, over them thermally. Here quantum tunneling refers to the Johnson et al., 2011, Nielsen and Chuang, 2010, Wang, quantum phenomenon where particles tunnel through a Wu and Zou, 2016). barrier in the situation that is classically infeasible. We The essential of quantum simulation is to understand cannot directly observe the tunneling process nor use clas- the dynamic evolution of a quantum system governed by sical physics to explain it satisfactorily. Quantum tun- the Schrodinger equation (2.1). That is, quantum simula- neling is generally described by the Heisenberg uncer- tainty principle and the wave-particle duality of matter tion needs to describe and solve the Schrodinger equation in quantum physics (Crosson and Harrow, 2016, Das and (2.1) by either digital quantum computers or analog quan- tum machines. The solution of (2.1) has expression Chakrabarti, 2005, 2008, Denchev et al., 2016, Wang, Wu √ − and Zou, 2016). (6.9) ψ(t) = e iHt ψ(0) ,i= −1, Quantum annealing devices are actively pursued by a −iHt number of academic labs and companies such as Google and we need to evaluate e numerically. It is extremely and D-Wave Systems, with uncertain quantum speedup. difficult to exponentiate the Hamiltonian H because its In particular, the D-Wave quantum computer is a commer- size increases exponentially in the system size. Common cially available hardware device that is designed and built numerical approach often uses the first-order linear ex- − + − to physically implement quantum annealing. It is an ana- pansion 1 − iHδ to approximate e iH(t δ) − e iHt ,which log computing device based on superconducting qubits often yields unsatisfactory numerical solutions. Math- to process quantum annealing and solve certain combi- ematically, quantum simulation is to explore whether natorial optimization problems. Also although it is ex- higher order approximations are available to provide ef- tremely difficult to simulate quantum annealing by classi- ficient methods for the evaluation of e−iHt .Forexam- cal computers, classical Markov chain Monte Carlo sim- ple, consider a system with α particles in a d-dimensional ulations have been developed to approximate quantum space that has the following Hamiltonian: annealing by path-integral formulation and mean field L approximation. See Albash et al. (2015), Boixo et al. H = H , (2014, 2016, 2018), BradyandvanDam(2016), Rønnow =1 et al. (2014), and Wang, Wu and Zou (2016) for more dis- + cussions. where L is a polynomial in α d, and each H acts on a small subsystem of finite size free from α and d.Note − 6.3 Quantum Simulators that it is easy to evaluate e iH δ numerically, but very −iHδ Quantum simulation is to intentionally and artificially difficult to compute e . Because H and Hk are non- − − L − − mimic interacting quantum systems, which are hard to commutable, e iHδ = e =1 iH δ = e iH1δ ···e iHLδ. QUANTUM SCIENCE AND QUANTUM TECHNOLOGY 63

By the Trotter formula (Proposition 3.1 in Wang (2011)), 6.4 Quantum Algorithms we obtain Quantum algorithms are algorithms that run on quan- − − − e iHδ = e iH1δ/2 ···e iHLδ/2 tum computation models, such as the most commonly (6.10) − − used quantum gate or circuit model, by taking input qubits × e iHLδ/2 ···e iH1δ/2 + O δ2 . and producing output measurements for the solutions of Thus, we obtain a second order approximation of e−iHδ specific computational tasks. While a classical algorithm by the first term on the right-hand side of (6.10), which takes a step-by-step procedure to solve a given problem − only needs us to evaluate each e iH δ, = 1,...,L. on a classical computer, a quantum algorithm is a step- Introduced by Feynman (1981/82), quantum simulation by-step problem-solving procedure, with each step per- itself has been developed into a core field within quan- formed on a quantum computer. We note that all classi- tum computation. A quantum simulator can be any phys- cal algorithms can be in principle executed on a quan- ical quantum system precisely prepared or manipulated tum computer, all problems solvable on a quantum com- in a way targeting at studying interesting features of an puter are solvable on a classical computer, and problems interacting complex quantum system, which is compu- undecidable by classical computers remain undecidable tationally intractable or difficult to simulate on classical on quantum computers. However, quantum algorithms are computers. A quantum simulator can be a digital quan- essentially different from their classical counterparts in tum simulator so that the controllable quantum system the sense of being genuine quantum, that is, quantum is implemented on a universal quantum computer, or an gate operations are reversible unitary transformations, and analog quantum simulator so that the controllable quan- quantum algorithms utilize fundamental quantum proper- tum system is a quantum physical device to reconstruct ties such as quantum superposition and quantum entan- the time evolution of an interacting quantum system un- glement. We refer to quantum algorithms as the algo- der precisely controlled conditions. Like universal quan- rithms that are inherently quantum for achieving faster tum computers, digital quantum simulators face signifi- speed than classical algorithms in solving some tough cant challenges in scaling architectures. However, analog problems. It should be pointed out that while quantum al- quantum simulators can be addressed and experimented gorithms cannot be worse than classical algorithms, we in a relatively large scale with currently available tech- should not expect quantum algorithms to yield advantage nology, and thus may provide new tools for us to inves- for every single problem; in fact, they usually do not. tigate interacting many-particle quantum systems and at- As a matter of fact, quantum computers augment, but do tack optimization problems beyond the reach of classical not replace classical computers. A continual challenge in computers. See Childs et al. (2018), Jiang et al. (2017), quantum science is to invent new quantum algorithms to Kassaletal.(2008, 2011), Lanyon et al. (2010), Nielsen speed up the best classical algorithms. For example, quan- and Chuang (2010), and Wang (2011, 2012). tum superposition indicates that we can potentially carry It is likely that the first practical application of quantum out exponentially many computations in parallel, but it computation is quantum simulation since even moderate quantum simulation devices have the potential to carry is tricky to extract the solution from such an exponential out simulations infeasible by classical computers. For ex- superposition to achieve some quantum speedup, as ob- ample, in quantum chemistry, molecular energies can be serving the qubit system destroys its state. This is where computed by digital quantum simulation devices of size we need clever designs of quantum software. Common 100 to 150 logical qubits with excellent precision and ac- techniques employed to create quantum algorithms in- curacy that considerably exceed the limitations of classi- clude quantum Fourier transform, phase estimation, am- cal computers. In particular, in the near to medium term, plitude amplification, quantum walk, quantum annealing analog quantum simulators may offer us a novel tool to and quantum simulation. The widely known quantum al- study complicated quantum systems and hard optimiza- gorithms include Shor’s factoring algorithm and Grover’s tion problems that are unreachable by classical comput- search algorithm, which are, respectively, exponentially ers. Again a computational advantage of quantum simula- faster and quadratically faster than the best known and tors over classical ones may clearly demonstrate quantum best classical algorithms for the same tasks. Many other supremacy (given in Section 6.1) in realistic applications. algorithms were created for a wide range of problems In the long run, the importance of quantum simulation and applications such as searching, sorting, counting, may lie in the applications of large-scale quantum sim- sampling, simulation, and optimization. See Montanaro ulations to solve fundamental problems in physics, mate- (2016), Nielsen and Chuang (2010)andWang (2012)for rials science and quantum chemistry (Abrams and Lloyd, more discussions. 1997, Aspuru-Guzik et al., 2005, Boghosian and Taylor, As a case in point, we consider quantum computation 1998, Cirac and Zoller, 2012, Kassaletal., 2011, Lloyd, for the Grover and parity problems. Let f(x)be a function 1996). defined on the integers from 1 to N and taking the values 64 Y. WANG AND X. SONG

±1. Define the parity of f(x)by of computation. In other words, quantum machine learn- N ing can offer advantages over its classical counterpart in Par(f ) = f(x). terms of computational complexity. Therefore, it is rea- x=1 sonable to expect quantum computers to be faster than + − classical computers for solving some machine learning The parity of f(x) can be either 1or 1, and always problems, but it is important and challenging to explore depends on the values of f(x) at all N points. It has quantum softwares that enable quantum machine learn- been proved that with no further information about f(x), ing to realize such quantum speedups. Recent develop- both classical and quantum algorithms have O(N) time- ment indeed shows a class of quantum machine learn- complexity to determine its parity, and thus quantum com- ing algorithms exhibit some quantum speedups. For ex- puters cannot outperform classical computers for the par- ample, from a computational perspective, solving linear ity problem. For the Grover problem, there is a further equation systems is almost ubiquitous in machine learn- information that f(x)is either identically equal to 1 or it ing, and finding a learning solution usually comprises a is 1 for N − 1ofthex’s and equal to −1 at one unknown sequence of standard linear algebra operations such as value of x.Forsuchf(x), its parity indicates its type, matrix multiplication and inversion. Quantum linear al- and the computational task for the Grover problem is to gebra algorithms offer quantum speedups over their clas- determine the type of f(x) and search for the unknown sical analogs. As a case in point, quantum basic linear al- value of x (if it exists). With the additional information gebra subroutines (BLAS), which include finding eigen- about f(x), the best classical and quantum algorithms√ for vectors and eigenvalues and solving linear equations, ex- the Grover problem have complexity O(N)√and O( N), hibit exponential quantum speedups over their best known respectively, and thus there is an optimal N quantum classical counterparts. The quantum BLAS renders quan- speedup. It is interesting to note that, although there is a tum speedups for an array of data analysis and ma- quadratic quantum speedup for the Grover problem, the chine learning algorithms including linear algebra opera- parity problem has no quantum speedup (see Farhietal. tion, gradient descent, Newton’s method, linear program- (1998)andGrover (1997)). This example indicates that ing, semidefinite and quadratic programming, topological neither classical nor quantum computers are expected to analysis, least-squares, nearest-neighbor, support vector be best for all computational tasks. machines, clustering, and principal component analysis (PCA). Also special-purpose quantum computers, such 6.5 Quantum Machine Learning as quantum annealers and programmable quantum opti- Quantum machine learning extends classical machine cal arrays, bear architectures well suited to quantum op- learning to the quantum realm. Classical machine learn- timization and deep learning particularly quantum deep ing and statistical learning often refer to an array of statis- learning with Boltzmann machines. More discussions tical approaches to analyzing data, with the goal of in- can be found in Adachi and Henderson (2015), Amin ferring the future behavior of target variables (such as et al. (2018), Arodz and Saeedi (2019), Arunachalam and the function relationships of variables and their dynamic de Wolf (2018), Benedetti et al. (2016), Biamonte et al. processes) from training data. The learning procedure in- (2017), Brandão et al. (2019), Ciliberto et al. (2018), volves inference, which addresses how statistically effi- Dunjko, Taylor and Briegel (2016), Dunjko and Briegel cient we can learn the functions or processes from given (2018), Jordan (2005), Lloyd, Mohseni and Rebentrost data, and computation, which handles how much com- (2014), O’Gorman et al. (2015), Rebentrost, Mohseni and putational resources are required to perform a learning Lloyd (2014), Salakhutdinov and Hinton ( 2009), Shenvi, task and how fast algorithms can be designed to carry Kempe and Whaley (2003), Svore, Hastings and Freed- out the learning task. The learning objective is to search man (2014), Wiebe, Kapoor and Svore (2016, 2015), for a model that fits well to training data but more im- Wiebe and Granade (2016), and Wittek (2014). Below portantly enjoys good generalization capability, which we provide short illustrations for some selected topics in refers to the property of the learned model with good quantum machine learning. prediction performance on new observations. Common 6.5.1 Bayesian quantum phase estimation. Quantum learning approaches rely on regularization-based meth- phase estimation is the key to achieve quantum speedups ods leverage on optimization techniques to solve learn- in many well-known quantum algorithms (Nielsen and ing problems. Quantum learning theory investigates how Chuang, 2010, Wang, 2012). Suppose that a unitary op- | quantum resources can affect the learning efficiency. The erator U√has an eigenvector ξ with corresponding eigen- theory indicates that it is possible for quantum learners value e −12πϕ, ϕ ∈ (0, 1). We do not know the phase ϕ of to achieve higher efficiency such as better generaliza- the eigenvalue, and our goal is to find ϕ based on a set of tion errors in learning difficult functions for some par- experiments performed on a quantum circuit. The experi- ticular learning models. However, the major advantages ments involve preparing the state |ξ and performing mea- that quantum mechanics can provide is largely in terms surement on U multiple times at some angle, where the QUANTUM SCIENCE AND QUANTUM TECHNOLOGY 65 angle θ and the number M of times are randomly chosen. where tr1 denotes the partial trace over the first variable, The Bayesian phase estimation procedure is to perform a andswapoperatorS has a matrix representation series of random measurements and then solve a classical d reconstruction problem using Bayesian inference. Given S = |j k|⊗k j. ϕ, and randomly chosen M and θ, the conditional prob- j,k=1 abilities of obtaining measurement outcomes 1 and 0 are as follows: S isasparsematrixofsized2, which can be clearly seen = 1 − cos(2πMϕ + θ) from the following explicit expression for the case of d P(1|ϕ; θ,M)= 2: 2 ⎛ ⎞ = 1 − P(0|ϕ; θ,M). 1000 ⎜0010⎟ S = ⎜ ⎟ . For a given measurement sequence, with a uniform pri- ⎝0100⎠ ori distribution on ϕ, we obtain the posterior distribution 0001 of ϕ. We may repeat this process for a series of experi- √ ments, and the Bayesian phase estimation approach pro- Thus, for sparse S, e− −1St can be performed effi- vides posterior distributions over the phase ϕ, which offer ciently. As a result, simply performing infinitesimal swap ⊗ estimates of the true eigenvalue and the algorithm’s uncer- operations on ρ √π allows√ us to simulate√ the unitary − −1ρt −1ρt − −1[ρ,π]t tainty in that value. More details can be found in Paesani time evolution e πe = e √ (π) et al. (2017), Svore, Hastings and Freedman (2014)and and thus construct the unitary operator e− −1ρt via the Wiebe and Granade (2016). More generally, Hamiltonian Schrödinger equation (2.3) in the density matrix form, learning has been developed to infer quantum Hamilto- where we have used the Baker–Campbell–Hausdorff for- nians via Bayesian inference for understanding quantum mula that for matrices A and B, dynamics. See Granade et al. (2012), Wiebe et al. (2014) ∞ 1 and Wiebe, Granade and Cory (2015) for details. A −A adA k Ad A B = e Be = e B = (ad ) (B), e k! A 6.5.2 Quantum principal component analysis.PCA k=0 depends on the eigendecomposition of a covariance ma- and linear transformations adA and AdA with defini- −1 trix, which, in the quantum context, can be converted tions adAB =[A, B]=AB − BA,andAdAB = ABA . into simulating a Hamiltonian in quantum simulation de- The quantum density√ matrix exponentiation procedure for scribedinSection6.3. Suppose that data are observed in constructing e− −1ρt comprises the preparation of an en- the form of vectors vj in a d-dimensional vector space, vironment state π and the application of the global swap = j 1,...,n. Assume that vj have zero mean and finite operator S to the combined system and environment state variance. Without loss of generality we further assume ρ ⊗ π followed by partial trace tr1 to discard the environ- vj are unit vectors (otherwise we may normalize each mental degrees of freedom. to be a unit vector and then use their norms to adjust Applying the density matrix exponentiation procedure selection probability below). Quantum principal compo- √ to ρˆ we construct e− −1ρtˆ . Then we utilize the quan- nent analysis randomly selects a vector from v ,...,v 1 n tum phase estimation algorithm to find eigenvalues and and then maps each selected vector v into a pure quan- j eigenvectors of the density matrix ρˆ, which renders the tum state |vj . The random encoding procedure yields a = principle components. The quantum PCA procedure has quantum state with b log2 d qubits and a density ma- 2 1 n computational complexity O(log d) with potential to trix ρˆ = = |v v |, which is equal to the sample n j 1 j j be exponentially faster than classical PCA. See Lloyd, covariance matrix up to an overall factor. We first describe a quantum technique called density Mohseni and Rebentrost (2014)andRebentrost, Mohseni matrix exponentiation. Using a simple trick with the par- and Lloyd (2014) for more details. tial trace over the first variable and the swap operator S, 6.5.3 Quantum support vector machines. Consider the we get binary classification problem where we have training √ √ data (x ,y ),...,(x ,y ), with x = (x ) ∈ Rp and y ∈ − −1St ⊗ −1St 1 1 n n i ij i tr1 e ρ πe {− } √ 1, 1 , and the goal is to use the training data to learn = cos2(t)π + sin2(t)ρ − −1sin(t)[ρ,π] how to predict classes y for feature vectors x.Define √ = τ 2 a hyperplane f(x) x β to induce a classification rule (6.11) = π − −1t[ρ,π]+O [t] τ = ∈ Rp √ √ sign(x β),whereβ (βj ) is a parameter. We need − − − = e 1ρtπe 1ρt + O [t]2 to find a suitable value for parameter β based on the train- √ ing data. The training of the sparse support vector ma- − − [ ] = e 1 ρ,π t (π) + O [t]2 , chines model with the hinge loss is often converted into 66 Y. WANG AND X. SONG solving the following minimization problem: link to the Ising model, Boltzmann machines are espe- cially suitable for learning exploration from a quantum n p 1 τ viewpoint. arg min max 0, 1 − yiβ xi + λ |βi|, β n i=1 j=1 As a classical machine learning technique, Boltzmann machines serve as the basis of powerful deep learning where λ>0 is a tuning parameter. The nonlinear uncon- models. A Boltzmann machine usually consists of visi- strained optimization problem can be transformed to an ble and hidden binary units that are jointly denoted by equivalent constrained linear programming problem with si , i = 1,...,N,whereN is the total number of units. n + 2p nonnegative variables and n linear inequality con- We use the notation si = (sv,sh) to distinguish the vis- straints, ible and hidden variables with index v for visible vari- n p p ables and h for hiddens, and reserve vector notations v, 1 + + + − arg min ξi λ βj λ βj , h,ands = (v, h) for representing random vectors corre- ξ,β+,β− n i=1 j=1 j=1 sponding to visible, hidden, and combined units, respec- subject to tively. A classical Ising model is employed to describe the variables sj . As in Section 6.2, the Hamiltonian (or p p c ≡ c + + energy function) of the Ising model is HI HI (s) de- yixij β − yixij β ≥ 1 − ξi, j j fined in (6.1), where now we take N = b,andγj and δij = = j 1 j 1 are considered as model parameters to be tuned during ± ≥ = = ξi,βj 0,i 1,...,n,j 1,...,p, the training in machine learning. In equilibrium, the prob- ability of observing a state v of the visible variables is = + − − The support vector machines solution is βj βj βj . described by the Boltzmann distribution summed over all While classical algorithms for solving such linear pro- hidden variables, gramming problems have asymptotic computational com- − − c − c = 1 HI (s) = HI (s) plexity O(mn)˜ where m = n + 2p, quantum algorithms (6.12) Pv Z e ,Z e , h s have been proposed recently for√ the optimization√ prob- ˜ ˜ lems√ with time complexity O( mn) or even O( m + which is called the marginal distribution of the visible ran- n), which offers potential for a quadratic speedup com- dom vector v. We aim to find parameters θ ={γj ,δij } in c pared to the classical algorithms. Furthermore, for the the Hamiltonian HI such that Pv becomes as close as pos- least squares support vector machines (with the mini- sible to the corresponding empirical distribution defined 1  2 + n 2 mization problem: minβ 2 β λ i=1 ξi subject to by the training data. The common classical approaches ≥ τ = − = ξi 0,yixi β 1 ξi,i 1,...,n), quantum algorithms to finding the parameters are to maximize likelihoods, offer an exponential speedup over classical algorithms. and the maximization is often achieved via some com- Also nonlinear classification rules can be derived by using bination of gradient descent and sampling. Research has kernels. For a polynomial kernel matrix K, we normal- demonstrated that sampling from Boltzmann machines ˆ = ize it by its trace to obtain K K/tr(K). Using√ density and calculating their likelihoods are computationally very − −1Ktˆ difficult, and MCMC simulations are often employed as matrix exponentiation in (6.11) to construct e √ and ˆ then applying quantum phase estimation to e− −1Kt we standard techniques to overcome the hard computational perform eigen-analysis and matrix inversion for K and tasks, though MCMC can be be very costly or even impos- sible for models with a large number of neurons. Train- thus efficiently solve the optimization problem for find- ing with quantum resources can be very helpful in reduc- ing the support vector machines solution. See Arodz and ing the training cost and offering some quantum speedup. Saeedi (2019)andRebentrost, Mohseni and Lloyd (2014) Thus, it makes quantum deep learning more feasible or for more details. preferable than the classical approach. Quantum tech- 6.5.4 Quantum deep learning with Boltzmann ma- niques, which include quantum linear algebra, quantum chines. Deep learning has been widely explored in quan- sampling, and quantum annealers, have been developed to tum machine learning. The literature has been mainly con- train the classical Boltzmann machines. Special-purpose centrated on speeding up the training of classical models quantum computers such as quantum annealers and pro- and on developing relatively less matured quantum neu- grammable photonic circuits are very suitable for training ral networks, which refer to that all their component parts, Boltzmann machines. In particular the D-Wave devices, ranging from the single neurons to the training algorithms, quantum annealers with tunable transverse Ising models, are carried out on quantum computers. Boltzmann ma- have been applied to encode deep quantum learning pro- chines are stochastic models that enable to produce new tocols on over thousands of spins. See Adachi and Hen- data based on prior observations, which are called gener- derson (2015), Benedetti et al. (2016)andWiebe, Kapoor ative models in deep learning. Because of their intrinsic and Svore (2016) for details. QUANTUM SCIENCE AND QUANTUM TECHNOLOGY 67

Quantum resources can give new, fundamentally quan- by superpositions in the computation basis composed of tum, models for deep learning. Quantum Boltzmann ma- the classical states |v, h, and the corresponding quantum chines are introduced to cross-breed between training arti- Boltzmann machine has quantum probability distribution ficial neural networks and fully quantum neural networks. with nondiagonal density matrix ρ given by (6.13) with q q They induce an array of quantum effects such as quantum HI replaced by H as well as the marginal Boltzmann tunneling. Unlike classical Boltzmann machines, quan- probability distribution Pv defined in (6.14). Performing tum Boltzmann machines can yield quantum states as out- each measurement on the states of the qubits in the σ z- puts, and thus deep quantum networks can learn to pro- basis we obtain the outcome ±1, and thus the measure- duce quantum states for representing a broad range of ment output for the visible variables follows the marginal systems, which is beyond the capability of classical ma- probability distribution Pv givenby(6.14). chine learning. Quantum Boltzmann machines are defined Our learning goal is to train the quantum Boltzmann machine and find the model parameters θ ={i,γj ,δij } by quantum Ising models in transverse fields. Similar to q the classical case, as described in Section 6.2, the quan- in the Hamiltonian H such that the probability distri- tum Hamiltonian of the quantum Ising model is Hq given bution Pv gets as near as possible to the corresponding I empirical distribution determined by the input data. The by (6.3), where again we take N = b,andγj and δij are model parameters to be tuned during the training. Note method of finding the parameters is to maximize some q bounds on relevant likelihoods, due to the likelihood in- that H is a 2N × 2N diagonal matrix, which is in contrast I tractability for quantum Boltzmann machines. Treating as to vectors with dimensionality equal to N, the number of a class of recurrent quantum neural networks, quantum variables, used in classical machine learning. Boltzmann machines can be trained for machine learning Denote by |v, h the eigenstates of Hamiltonian Hq , I tasks such as discriminative and generative learning, as where again v and h stand for vectors of visible and hid- q well as quantum state tomography and quantum anneal- den variables, respectively. For diagonal Hamiltonian HI , − q ing. More details can be found in Amin et al. (2018)and e HI is also a diagonal matrix with its 2N diagonal ele- Kieferova and Wiebe (2016). − c ments being e HI corresponding to all 2N configurations. = [ −Hq ] 6.5.5 Quantum machine learning for quantum data.In Define its partition function Z tr e I and density ma- general quantum machine learning is particularly suitable trix for quantum data which are the actual output states gen- − − q (6.13) ρ = Z 1e HI . erated by quantum systems and processes, and the spe- ciality of quantum data analysis is the capability of using Then the diagonal elements of ρ are equal to the Boltz- quantum simulators to probe quantum dynamics. We may N mann probabilities of all 2 configurations. Given a state apply quantum machine learning algorithms like quantum |v of the visible variables, we get the following marginal PCA and quantum Boltzmann machines to quantum data Boltzmann probability Pv by tracing over the hidden vari- (such as quantum states of light and matter) for exploit- ables: ing the quantum states and uncovering their hidden fea- (6.14) P = tr[ ρ], tures and patterns. The obtained analysis results in quan- v v tum modes are often much more efficient and more en- where v is a diagonal matrix whose diagonal elements lightening than the the classical analysis of data draw- are equal to 1 if the visibles are in state v,and0other- ing from quantum systems. See Granade et al. (2012), wise. The use of v is to limit the trace only to diagonal Havlícekˇ et al. (2019), Kieferova and Wiebe (2016), elements corresponding to the visible variables which are Lloyd, Mohseni and Rebentrost (2014), Marvian and in state v. It is easy to see that definitions (6.12)and(6.14) Lloyd (2016), Wiebe et al. (2014)andWiebe, Granade are identical for the diagonal Hamiltonian and density ma- and Cory (2015) for details. trix, but (6.14) is still valid for the nondiagonal case. 6.5.6 Quantum sampling. Sampling methods are To add a transverse field to the quantum Hamiltonian q widely used techniques to compute some intractable HI , as described in Section 6.2, we need nondiagonal ma- quantities. Examples include the most commonly used x trices σ j described in (6.7) to obtain the transverse field Monte Carlo methods in particular MCMC simulations. Ising Hamiltonian as follows: While classical Monte Carlo simulations are performed q by pseudo-random numbers, quantum computation is able (6.15) H =−  σ x − δ σ zσ z − γ σ z ,  i i ij i j j j to generate genuine random numbers and perform true i i,j j Monte Carlo simulations. Quantum MCMC algorithms where besides the original parameters γj and δij ,wehave are developed to offer a quadratic speedup over classi- additional model parameters i . A quantum Boltzmann cal MCMC algorithms in terms of spectral gap, inverse machine is defined by the quantum Boltzmann distribu- temperature, desired precision or the hitting time. See q q tion of the transverse field Ising Hamiltonian H.AsH Chowdhury and Somma (2017), Richter (2006), Szegedy is a nondiagonal matrix, we may express its eigenvectors (2004)andTemme et al. (2011) for details. 68 Y. WANG AND X. SONG

6.5.7 Quantum machine learning with noise.Noise the output distributions of random quantum circuits, while can be potentially beneficial for solving machine learn- it is hard for current supercomputers to handle the sam- ing problems. Research has shown in the classical case pling problem beyond around 50 qubits. See Aaronson that noise may play a positive role in perturbing gradi- and Chen (2017), Arute et al. (2019), Boixo et al. (2018), ents for jumping out local optima and improving gen- Bouland et al. (2018), Bravyi, Gosset and König (2018), eralization performance. It becomes promising to ana- Harrow and Montanaro (2017), Lund, Bremner and Ralph lyze noisy learning problems from a quantum perspec- (2017), Markov et al. (2018), Neill et al. (2018)and tive and particularly exploit advantageously the effects of Rønnow et al. (2014). Below we briefly describe boson noise in quantum machine learning. For example, we may sampling and random quantum circuits. study whether the kinds of noise occurred in quantum sys- 6.6.1 Boson sampling. Boson sampling is a quantum tems have similar distributional and structural behaviors computation model where n identical bosons pass through to those usually seen in classical settings, and if they can a network of passive optical elements (beamsplitters and play a beneficial role in quantum machine learning as in phase-shifters) and then the locations of the bosons are de- the classical case. See Cross, Smith and Smolin (2015) and Grilo, Kerenidis and Zijlstra (2018) for details. tected. Quantum supremacy can be demonstrated by im- plementing boson sampling with a medium size network. 6.6 Quantum Computational Supremacy A network system with 50 photons (qubits) and 2500 paths is currently intractable for classical computers. To Determining a quantum speedup depends on how we define the quantum speedup notation. One approach is to implement boson sampling all required physical devices take a formal computational complexity perspective based are single-photon sources, beamsplitters, phase-shifters on rigorous mathematical proofs. Another realistic per- and photon-detectors. The physical implementation of the spective is based on what can be achieved with feasi- scheme encounters a myriad of technicalities such as syn- ble finite size devices and requires sound statistical evi- chronization of pulses, mode-matching, quickly control- dence to confirm a scaling advantage over certain finite lable delay lines, tunable beamsplitters and phase-shifters, range of problem sizes. For example, it has already been single-photon sources, and accurate, fast, single photon rigorously proved in terms of computational complexity detectors. that quantum algorithms like Grover’s search algorithm To define the boson sampling model, we adopt a sta- and Shor’s factoring algorithm offer speedups over known tistical approach based on the permanents of the subma- classical algorithms. Unfortunately, such rigorous proofs trices of a unitary matrix, which requires minimal quan- are often not available for most cases, and even avail- tum physics and quantum computation terminology. For a × = able, many existing quantum algorithms fail to provide n n matrix A (aij ), we define its permanent by reference for any specific implementation such as the ex- n act number of qubits needed to implement them; in fact, Perm(A) = aiπ(i), they often cannot be implemented on about 100 qubit plat- π i=1 forms available in the near to medium term. We need to where the sum is over all permutations π of 1, 2,...,n. resort to the second perspective, and detecting a scaling Consider the quantum system involving n identical pho- advantage of quantum computing over classical comput- tons and m modes, where we may loosely interpret ing would hinge on the so-called benchmarking problem, ‘mode’ as the location of a photon, and we are only inter- namely, the existence of a quantum computer performed ested in the case of m ≥ n. The quantum system has com- on well designed computational tasks with sound sta- putational basis states of the form |s=|s1,s2,...,sm, tistical analysis of computing experiments and resulting where si indicates the number of photons in the ith mode. data. Such advantages may include improved computa- Denote the set corresponding to all the computational ba- tional speed, accuracy, and sampling for classically inac- sis states by cessible systems. Quantum algorithms and computational problems are created for these platforms with a limited m,n = s = (s1,s2,...,sm) : s1 + s2 +···+sm = n . number of qubits where classical computation is impos- It is easy to see that the total number of elements in m,n sible. The mission involving both hardware and software + − M = m n 1 m × m along with statistical analysis aims at demonstrating the is equal to n .Foragiven unitary ma- ∈ quantum computational supremacy given in Section 6.1. trix U and each s m,n, we obtain matrix Us from U by Quantum scientists are building quantum computers of 50 keeping its first n columns and repeating sj times its jth to 100 qubits to demonstrate quantum supremacy. For ex- row. Define a discrete probability distribution on m,n as ample, the Google quantum AI group has achieved quan- follows: tum supremacy by building a quantum processor named | |2 = Perm(Us) ‘Sycamore’ of 54 superconducting qubits to sample from Pr(s) . s1!···sm! QUANTUM SCIENCE AND QUANTUM TECHNOLOGY 69

It can be shown that Pr(s) is a well-defined probability dis- implemented quantum random circuits in a two dimen- tribution on m,n and corresponds to the quantum system sional lattice to demonstrate quantum supremacy. See with n photons, m modes and an optical network whose Arute et al. (2019), Boixo et al. (2018), and Neill et al. action is determined by the unitary matrix U. Boson sam- (2018) for more details. pling refers to sampling from distribution Pr(s).Asclas- sical computers cannot handle the sampling problem even 7. QUANTUM METROLOGY with moderate size, we may demonstrate the quantum Measurement is at the heart of science, technology, in- supremacy by successfully implementing boson sampling dustry, and commerce. We need measurements and metro- of reasonable size on quantum computing devices. More logical standards to quantitatively assess scientific phe- details can be found in Harrow and Montanaro (2017)and nomena and technological progress, and gauge the ex- Lund, Bremner and Ralph (2017). change of goods and service including information. Mea- surement devices are physical apparatuses whose func- 6.6.2 Random quantum circuits. Random quantum tions and accuracy are governed by the laws of physics, circuits are created in a specific way so that when they and transformative improvements in measurement tech- are generated with enough ‘complexity’, even the most nologies often follow the utilization of a new physical law. powerful classical supercomputer cannot directly simu- Quantum metrology (or quantum sensing) is to exploit the late the generated quantum circuits. However, quantum strange laws of quantum physics to build new and bet- computers can sample from the output distributions cor- ter sensors and measuring devices. Fueling with quantum responding to the obtained quantum circuits. Here a quan- laws, quantum metrology may lead to a game-changing tum circuit is a sequence of d clock cycles of one- and shift in scientific studies, technological progress, as well two-qubit gates with gates applied to different qubits in as commerce and industry developments. the same cycle. The number d of cycles is called the The basic concept of quantum metrology is that a probe depth of the circuit. We say a random quantum circuit device interacts with an appropriate system to learn the has enough ‘complexity’, if both its qubits and depth are properties of the system, where the interaction alters the large enough. If the gates to be applied are chosen from state of the probe, and measurements of the probe un- a universal quantum gate set, the unitary matrix U of the cover the characteristic parameters of the system. For circuit is a random matrix whose distribution converges quantum sensing, the probe is usually prepared in one to the Haar measure on the collection of unitary matrices certain quantum state, its encounter with the system nor- when the depth of the circuit goes to infinity. Specifically mally changes its state with both beneficial and adversar- when a quantum circuit contains n qubits, with 2n com- ial effects in the sense that it not only responds to the pa- putational basis states |x=|x1x2 ···xn, xi ∈{0, 1},a rameters of interest but also decoheres the probe (which quantum state |ψd produced by the random quantum cir- means there is loss of information from the probe into the cuit is a linear combination of the computational basis system due to quantum decoherence, illustrated in Sec- and thus has 2n amplitudes, each with real and imaginary tion 6.1). Then appropriately devised measurements can parts. Therefore there are 2n+1 parameters in each quan- ascertain in what way and to what extent such encounter tum state. As the unitary matrix U of the random quan- has changed the state of the probe, which enables quan- tum circuit converges in distribution to the Haar measure, tum sensing to evaluate the system parameters. Quantum sensing promises to develop high-resolution and highly the random vector of the amplitude parameters asymptot- sensitive measurement techniques that will provide bet- ically follows a uniform distribution on the unit sphere. ter precision than the same measurement performed un- Define the output distribution of the random quantum cir- der a classical framework. They include quantum sensors, cuit to be measurement probability p(x) =|x|ψ |2.As d quantum clocks, and quantum imaging. The applications d the depth of the random quantum circuit goes to in- range from the sub-nano to the galactic scale, while some finity, p(x) approaches the Porter–Thomas distribution. are in fact close to commercial use. The potential impact The Google research group is working on finding ways of quantum metrology is far-reaching. An array of dis- to generate random quantum circuits so that their output tinct platforms allow quantum-enhanced measurement of distributions quickly converge to the Porter–Thomas dis- time, space, rotation, as well as gravitational, electrical tribution. Based on the asymptotic distribution, we may and magnetic fields. The technologies are promising to develop a statistical approach to determine if a sample make fundamental changes in a wide range of fields such is generated from the theoretical output distribution of as physics, chemistry, biology, medicine or data storage a desired random quantum circuit. Since it is difficult and processing. for classical supercomputers to deal with the sampling There is a strong link between quantum metrology and problem beyond around 50 qubits at the present time, quantum information. For example, both quantum in- the Google quantum scientists have designed a quantum formation and quantum sensing rely on the same quan- processor named ‘Sycamore’ of 54 transmon qubits and tum properties such as entanglement, in particular, high 70 Y. WANG AND X. SONG level of multipartite entanglement, to achieve better per- 8. CONCLUDING REMARKS formance than their classical counterparts. See Degen, Quantum science and quantum technology gain enor- Reinhard and Cappellaro (2017), Kruse et al. (2016), and mous attention in multiple frontiers of many scientific Pezzè et al. (2018) for more details. fields. Quantum computation can give rise to an expo- Quantum tomography plays an important role in quan- nential speedup over classical counterpart for tackling tum sensing. Quantum state tomography refers to recon- certain computational tasks, quantum information can struction of a quantum state based on measurements per- bring about exponential savings in information transmis- formed on the quantum state. Statistically it is a density sion for handling computational and communication jobs, matrix estimation problem based on quantum measure- and quantum communication can offer more secure cryp- ments. Common quantum measurements are on observ- tosystems than classical analogue for solving commu- Cd able M, which is defined as a Hermitian matrix on . nication problems. Some of the quantum protocols are For example, the Pauli matrices as observables are widely already in practical implementation, such as quantum- employed to perform quantum measurements in quantum fingerprinting, quantum key distribution, quantum anneal- science and quantum technology, and we may represent ing, and quantum simulation. This paper reviews quantum many density matrices through Pauli matrices. Suppose science and quantum technology from a statistical per- that the observable M has the following spectral decom- spective. We introduce concepts like key quantum proper- position: ties and qubits. We present quantum communication and r quantum information, illustrate quantum computation and (7.1) M = λaQa, quantum metrology, and discuss major quantum technolo- a=1 gies associated with them. We show the advantages of quantum techniques over the available classical counter- where λ are r different real eigenvalues of M,andQ a a parts. are projections onto the eigen-spaces corresponding to λ . a As statistics and machine learning nowadays heav- Given a quantum system prepared in state ρ,weusea ily involve computation, it is natural to expect quantum probability space (, F,P) to define measurement out- computation to play a major role in data science. In- comes when performing measurements on the observable deed, quantum computation and quantum simulation may M.LetR be the measurement outcome of M.Thetheory have tremendous potential to revolutionize computational of quantum physics indicates that R is a random variable statistics and data science. On the other hand, there is on (, F,P) which takes values in {λ ,λ ,...,λ },and 1 2 r great demand in studying statistical issues for theoretical has probability distribution research and experimental work in quantum science and

P(R= λa) = tr(Qaρ), quantum technology. As quantum phenomena are intrin- (7.2) sically stochastic, and data collected in quantum experi- a = 1, 2,...,r,E(R)= tr(Mρ). ments become more and more complex, we need to de- velop sophisticated statistical methods for enhancing data Quantum state tomography is to reconstruct ρ from in- analysis and improving understanding of quantum events dependent and identically distributed measurement out- (Paesani et al., 2017, Wang, 2011, 2012, 2013, Wang, comes R1,...,Rn.SeeArtiles, Gill and Gu¸ta˘ (2005), Cai Wu and Zou, 2016, Wiebe et al., 2014, Wiebe, Kapoor et al. (2016), Malley and Hornstein (1993)andWang and and Svore, 2016). A great deal of current work is taken Xu (2015). place on creating new protocols and developing novel Designing and controlling quantum systems are com- approaches to certifying quantum devices such as test- plex and challenging in the development of quantum sci- ing and assessing their quantum performances. Clearly, ence and quantum technology. Statistical methods pro- such certification requires efficient and scalable statisti- vide powerful tools for the study of quantum design and cal methods for calibrating and validating quantum prop- quantum control. Examples of successful applications in- erties. Moreover, certification needs to take into account clude quantum gate constructions with high fidelity preci- commercial considerations for compliance with industry sion in quantum computation and quantum information, standards by working together with industry, academics, extraction of theoretical insights about quantum states national labs, and government organizations (Acín and in condensed matter, and quantum control procedures in Masanes, 2016, Wang, Wu and Zou, 2016, Wiebe et al., optimizing adaptive quantum metrology. Like quantum 2014). phase estimation and quantum tomography, many quan- As indicated in Section 6.1, the bottleneck of quan- tum problems are in essence statistical problems, and it is tum computing at the present time is primarily on quan- our firm belief that statistics and data science have great tum hardware, and current quantum computing largely de- potential to make significant improvement in quantum pends on what kinds of quantum computers experimental- metrology. ists can build. On the other hand, as we have demonstrated QUANTUM SCIENCE AND QUANTUM TECHNOLOGY 71 in Section 6.6 for the quantum computational supremacy ARUNACHALAM,S.andDE WOLF, R. (2018). Optimal quantum sam- endeavor, besides hardware quantum computing also re- ple complexity of learning algorithms. J. Mach. Learn. Res. 19 Pa- quires sophisticated mathematical models, sound statisti- per No. 71, 36. MR3899773 cal analysis, and better computational tools. As a matter ARUTE,F.,ARYA,K.,BABBUSH,R.,BACON,D.,BARDIN,J.C., BARENDS,R.,BISWAS,R.,BOIXO,S.,BRANDAO,F.G.S.L. of fact, in general we call for some combination of new et al. (2019). Quantum supremacy using a programmable supercon- experimental techniques, better mathematical and statis- ducting processor. Nature 574 505–510. tical understanding, and improved computational tools in ASPURU-GUZIK,A.andWALTHER, P. (2012). Photonic quantum order to significantly advance the development of quan- simulators. Nature Physics 8 285. tum science and quantum technology. ASPURU-GUZIK,A.,DUTOI,A.D.,LOVE,P.J.andHEAD- GORDON, M. (2005). Simulated quantum computation of molecu- lar energies. Science 309 1704–1707. ACKNOWLEDGMENTS BENEDETTI,M.,REALPE-GÓMEZ,J.,BISWAS,R.andPERDOMO- The research of Yazhen Wang was supported in part by ORTIZ, A. (2016). Estimation of effective temperatures in quantum NSF Grants DMS-1528735, DMS-1707605, and DMS- annealers for sampling applications: A case study with possible ap- plications in deep learning. Phys. Rev. A 94 022308. 1913149. The research of Xinyu Song was supported BENNETT,C.H.andBRASSARD, G. (2014). Quantum cryptography: by the Fundamental Research Funds for the Central Public key distribution and coin tossing. Theoret. Comput. Sci. 560 Universities (2018110128), China Scholarship Council 7–11. MR3283256 https://doi.org/10.1016/j.tcs.2014.05.025 (201806485017) and National Natural Science Founda- BERNSTEIN,D.J.andLANGE, T. (2017). Post-quantum tion of China (Grant No. 11871323). The authors thank cryptography-dealing with the fallout of physics success. IACR David Siegmund for helpful comments and suggestions Cryptology ePrint Archive 2017 314. BERTSIMAS,D.andTSITSIKLIS, J. (1993). Simulated annealing. which led to improvements of the paper. Statist. Sci. 8 10–15. 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