Quantum Science and Quantum Technologypages.stat.wisc.edu/~yzwang/paper/quantum-science.pdfQuantum...

Statistical Science2020, Vol. 35, No. 1, 51–74https://doi.org/10.1214/19-STS745© Institute of Mathematical Statistics, 2020

Quantum Science and Quantum TechnologyYazhen Wang and Xinyu Song

Abstract. Quantum science and quantum technology are of great currentinterest in multiple frontiers of many scientific fields ranging from computerscience to physics and chemistry, and from engineering to mathematics andstatistics. Their developments will likely lead to a new wave of scientific rev-olutions and technological innovations in a wide range of scientific studiesand 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, illustraterelevant concepts of quantum science and quantum technology, and discusstheir scientific developments. We point out the need for statistical analysis intheir developments, as well as their potential applications to and impacts onstatistics and data science.

Key words and phrases: Quantum communication, quantum information,quantum computation, quantum simulation, quantum annealing, quantumsensing and quantum metrology, quantum bit (qubit).

1. INTRODUCTION

Quantum science and quantum technology arise from asynthesis of quantum mechanics, information theory, andcomputing. They investigate the preparation and controlof the quantum states of physical systems to generate newknowledge and technologies for information processingand transmission, computation, measurement, and funda-mental understanding in ways that classical approachescan only do much less efficiently, or not at all. The fieldscomprise quantum communication, quantum information,quantum computation, quantum simulation, and quantummetrology (also known as quantum sensing), where quan-tum communication utilizes quantum means to transmitdata in a provably secure way; quantum information de-scribes the information of the state of a quantum systemand the process of the information by quantum devices;quantum computation uses quantum effects to speed upcertain calculations dramatically; quantum simulation re-produces the behavior of hard accessible quantum sys-tems by manipulating well-controlled quantum systems;quantum metrology exploits the high sensitivity of coher-ent quantum systems to external perturbations for enhanc-ing the performance of measurements of physical quan-tities. Quantum science and quantum technology differ

Yazhen Wang is Professor, Department of Statistics, Universityof Wisconsin-Madison, Madison, Wisconsin 53706, USA(e-mail: [email protected]). Xinyu Song is AssistantProfessor, School of Statistics and Management, ShanghaiUniversity of Finance and Economics, Shanghai 200433,China (e-mail: [email protected]).

from existing applications of quantum mechanics and in-formation theory, such as lasers, transistors, MRI, and cur-rently used classical computers and classical communica-tion tools, in ways that we utilize distinct quantum phe-nomena like quantum superposition, entanglement, andtunneling, which do not have classical counterparts. In thepast two decades, we have made tremendous progress inthe study of quantum science and quantum technology forharnessing quantum phenomena to advance informationprocessing and transmission, computation, and measure-ment.

Quantum science not only establishes a foundation forgaining a deeper understanding of nature, but also makesit possible to invent new quantum technology for accom-plishing tasks that are impossible to achieve by classicaltechniques. Here quantum technology refers to technolo-gies that explicitly deal with individual quantum statesand specifically exploit special quantum properties thatdo not have classical analogue. They enable us to buildquantum devices for achieving faster computation, moresecure communication, and better physical measurementsthan classical techniques. This article intends to present anoverview of such quantum aspects of science and technol-ogy, particularly in quantum information, quantum com-munication, and quantum computation.

The rest of the paper proceeds as follows. Section 2briefly introduces quantum physics. Section 3 reviews ba-sic quantum concepts used in quantum science and quan-tum technology. Sections 4 and 5 discuss quantum com-munication and quantum information, respectively. Sec-tion 6 illustrates quantum computation. It covers univer-sal quantum computing based on the gate (or circuit)

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52 Y. WANG AND X. SONG

model, adiabatic quantum computing based on quantumannealing, and current development on building quan-tum computers. This section also includes quantum algo-rithms, quantum simulation, quantum machine learning,and quantum computational supremacy. Section 7 pro-vides a short description of quantum metrology. Section 8features concluding remarks and points out potential ap-plications of quantum science and quantum technology tostatistics and data science as well as the need of statis-tics in the development of quantum science and quantumtechnology.

2. QUANTUM PHYSICS AND ITS COMPUTATIONALPOTENTIAL

2.1 Mathematical Concepts and Notations

Unlike the typical literature on quantum mechanics thatadopts technically more complicated concepts and nota-tions such as operators with a continuous spectrum onan infinite-dimensional Hilbert space, for simplicity wechoose to use relatively easy finite-dimensional linear al-gebra for the purpose of discussing quantum science andquantum technology. Since operators correspond to ma-trices in the finite dimensional case, we need to dealwith only matrices and their operations such as eigen-analysis. Denote by R and C, respectively, the sets of allreal numbers and all complex numbers. A simple vectorspace is Cd comprising all d-tuples of complex numbers(z1, . . . , zd). We use Dirac notations |·〉 (which is calledket) and 〈·| (which is called bra) to show that the objectsare column vectors or row vectors in the vector space, re-spectively. Denote by superscripts ∗, ′ and † the conjugateof a complex number, the transpose of a vector or matrix,and conjugate transpose operation, respectively. For |u〉and |v〉 in the vector space, we denote their inner productby 〈u|v〉, which induces a norm ‖u‖ = √〈u|u〉, and a dis-tance ‖u − v‖ between |u〉 and |v〉. For example, Cd hasa natural inner product

〈u|v〉 =d∑

j=1

u∗j vj = (

u∗1, . . . , u

∗d

)(v1, . . . , vd)′,

where 〈u| = (u1, . . . , ud) and |v〉 = (v1, . . . , vd)′. Given amatrix A = (aij ), we say it is Hermitian if A = A†, anddenote its trace by tr(A) = ∑k

j=1 ajj . A matrix U is said

to be unitary if UU† = U†U = I. For two matrices A1 andA2, define their commutator [A1,A2] = A1A2 − A2A1.Denote by ⊗ the tensor product operation of vectors ormatrices. To analyze computer algorithms, we adopt a no-tation O(h(m)) to denote that the asymptotic scaling ofan algorithm is upper-bounded by a function h(m) of theinput size m, with the notation O(h(m)) ignoring loga-rithmic factors.

2.2 Quantum Physics

Quantum mechanics describes microscopic phenomenasuch as the positions and momentums of individual par-ticles like atoms or electrons, the spins of electrons, theemissions and absorptions of light by atoms, and the de-tections of light photons. Unlike classical mechanics thatcan precisely measure physical entities like position andmomentum, quantum physics is intrinsically stochastic inthe sense that only a probabilistic prediction can be madeabout the results of the measurements performed.

We may describe a quantum system by its state andthe dynamic evolution of the state. A quantum state is of-ten characterized by a unit complex vector with dynamicunitary evolution, where the unitary evolution means thatquantum states are connected by unitary matrices, and thedynamic evolution is governed by a differential equationcalled the Schrödinger equation. Specifically, let |ψ(t)〉be the state of the quantum system at time t (also a wavefunction at time t). The states |ψ(t)〉 and |ψ(t + s)〉 attimes t and t + s, respectively, are connected through|ψ(t + s)〉 = U(s)|ψ(t)〉, where U(s) = exp(−√−1Hs)

is a unitary matrix, and H is a Hermitian matrix onC

d , which is known as the Hamiltonian of the quan-tum system. Differentiating both sides of |ψ(t + s)〉 =exp(−√−1Hs)|ψ(t)〉 with respect to s and letting s goto 0, we obtain the following Schrödinger equation forgoverning the continuous time evolution of |ψ(t)〉:

(2.1)√−1

∂|ψ(t)〉∂t

= H∣∣ψ(t)

⟩.

Note that although the Schrödinger equation is regardedas somewhat mysterious when it is first encountered, for aMarkov chain in continuous time with a finite state space,transition probability matrix Pt and Q-matrix Q, we useexactly the same argument: from Ps+t = PsPt , by differ-entiation we obtain the Kolmogorov equation ∂Pt

∂t= QPt ,

which has the solution Pt = exp(Qt)P0.As an alternative, we can describe a quantum system

by a so-called density matrix. For a d-dimensional quan-tum system, its quantum state can be characterized by adensity matrix ρ on the d-dimensional complex space Cd ,where ρ satisfies (1) Hermitian; (2) positive semi-definite;(3) unit trace. We often classify a quantum state as a purestate or an ensemble of pure states. A pure state corre-sponds to a density matrix ρ = |ψ〉〈ψ |, where |ψ〉 is aunit vector in Cd . An ensemble of pure states has a den-sity matrix

(2.2) ρ =J∑

j=1

pj |ψj 〉〈ψj |,

which corresponds to the scenario that the quantum sys-tem is in one of states |ψj 〉, j = 1, . . . , J , with probabilitypj being in the state |ψj 〉. The quantum evolution in the

QUANTUM SCIENCE AND QUANTUM TECHNOLOGY 53

density matrix representation can be described as follows.Let ρ t be the density matrix of the state of the quantumsystem at time t . With the unitary matrix U(·) and Hamil-tonian H introduced above, the density matrix evolution isgiven by ρ t+s = U(t)ρsU†(t), with the Schrödinger equa-tion in the form of

ρt = e−√−1Htρ0e√−1Ht or equivalently

(2.3)√−1∂ρt

∂t= [H,ρt ].

See Sakurai and Napolitano (2017) and Shankar (2012)for details.

As we will see in Section 3.1, the number of complexnumbers and the dimensionality of vectors and matricesrequired to describe a quantum state and its evolutionusually increase exponentially in the system size, ratherthan linearly in a classical system. As a result, a quan-tum system can store and manage an exponential num-ber of complex numbers and perform data manipulationsand calculations during the evolution of the system, whileclassical computers find it difficult to cope with the quan-tum system as it requires an exponential number of bitsof memory to store the quantum state. Unlike the clas-sical case where we often need to consider some extrastructural assumptions or approximations when handlinghigh-dimensional objects, quantum systems have poten-tial to deal with exponentially high-dimensional problemswithout imposing additional constraints. Special quantumphenomena are utilized to accomplish quantum commu-nication and computational tasks, and subsequent sec-tions will illustrate that the quantum phenomena are of-ten strange, and counter-intuitive. For example, light canbe particles and waves (wave-particle duality); a cat canbe alive and dead at the same time (quantum superpo-sition); information can transmit instantaneously over along distance without going through the intervening space(quantum teleportation); without sufficient energy, quan-tum particles can pass a barrier that is classically impos-sible (quantum tunneling).

3. QUANTUM BITS AND QUANTUM PROPERTIES

3.1 Quantum Bit and Superposition

In classical information and computation, the most fun-damental entity is the bit, and the information encoded ina bit has two state values, 0 and 1. Classical bits can bematerialized in multiple means, for example, they may berealized mechanically as switches or magnetically as harddrives. An important fact of the classical bit is that its twostate values are mutually exclusive, namely, its state canonly be either 0 or 1. This fact leads to one thing in com-mon for all of the realization means, that is, all classicalphysical devices prevent the simultaneous occurrence of

the states, with an example of the switch being either onor off.

In quantum science and quantum technology, the coun-terpart of the classical bit is the quantum bit, which wecall qubit for short. Similar to a classical bit with twostate values 0 and 1, a qubit has states |0〉 and |1〉, wherewe use the customary Dirac notation |·〉 to denote thequbit state. However, one key difference exists between aclassical bit and a qubit. Specifically, the theory of quan-tum physics allows the description of a quantum physicalsystem through probabilistic combinations of its states,which is referred to as the superposition property. Thesuperposition of states can accommodate all predictionsfor the outcomes of physical measurements, moreover,it bears drastic consequences for the nature of the phys-ical states ascribed to a system. In this regard, besides thestates |0〉 and |1〉, a qubit can be in superposition stateswith the following form:

(3.1) |ψ〉 = α0|0〉 + α1|1〉,where complex numbers α0 and α1 are called amplitudesand satisfy |α0|2 + |α1|2 = 1. As a result, the states of aqubit are unit vectors in a two-dimensional complex vec-tor space C

2. The states |0〉 and |1〉 form an orthonormalbasis for the space and are often referred to as computa-tional basis states. Unlike classical bits that have mutuallyexclusive states, qubits can be one and zero simultane-ously, which is known as the most fundamental aspects ofqubits. In other words, a superposition state is a state ofmatter that can be viewed as simultaneous occurrence ofzero and one at the same time.

Qubits can be realized in various physical systems. Ex-amples of qubits include the two states of an electron or-biting a single atom, the two different polarizations of aphoton, the alignment of a nuclear spin in a uniform mag-netic field, or the two directions of current flows in su-perconducting circuits. Specifically, in the atom model,|0〉 and |1〉 can be treated respectively, as the so-called‘ground’ and ‘excited’ states of the electron; if the atomis shined by light with appropriate energy and for a suit-able amount of time, we may transfer the electron fromthe |0〉 state to the |1〉 state and vice versa. Furthermore,by adjusting the time length for shining the light on theatom, the electron can be moved from the initial state|0〉 into ‘halfway’ between |0〉 and |1〉, for example, intostate |+〉 = (|0〉+|1〉)/√2, or state |−〉 = (|0〉−|1〉)/√2,where |+〉 and |−〉 form a qubit basis that is equiva-lent to the computational qubit basis |0〉 and |1〉. Notethat the quantum state transformations are solutions ofthe Schrödinger equation (2.1) for particular choices ofHamiltonian H and time interval, and it is an interestingexercise for readers to find the appropriate Hamiltonians.

It is easy to examine a classical bit to determine its state,being 0 or 1, however, it is impossible to examine a qubit


|ψ〉 to determine its state or find the values of its ampli-tudes α0 and α1 defined in (3.1). Because of the stochasticnature of quantum theory, performing measurements onthe qubit |ψ〉 will result in measurement outcome 0 withprobability |α0|2, or measurement outcome 1 with prob-ability |α1|2. Moreover, performing measurements on thequbit will change its state.

Like classic bits, we may define multiple qubits. Thestates of one b-qubit are unit vectors in a 2b-dimensionalcomplex vector space. The quantum exponential com-plexity is then shown in the exponential growth of dimen-sionality 2b and the number of 2b amplitudes required tospecify superposition states. For the 2-qubit case, its su-perposition states are unit vectors in a 4-dimensional com-plex vector space, with the following form:

(3.2) |ψ〉 = α00|00〉 + α01|01〉 + α10|10〉 + α11|11〉,where |00〉, |01〉, |10〉, and |11〉 are four computationalbasis states, amplitudes αx are complex numbers satisfy-ing |α00|2 + |α01|2 + |α10|2 + |α11|2 = 1. As in the singlequbit case, when measuring the 2-qubit, we obtain mea-surement outcome x as one of 00,01,10,11, with a corre-sponding probability |αx |2. Furthermore, we may performa measurement just on the first qubit of the 2-qubit sys-tem and obtain either the measurement outcome 0, withprobability |α00|2 + |α01|2, or the outcome 1, with proba-bility |α10|2 + |α11|2. As quantum measuring changes thequantum state, depending on the measurement outcomeobtained for the first qubit, being either 0 or 1, the 2-qubitsystem will be in the state

(3.3)α00|00〉 + α01|01〉√

|α00|2 + |α01|2or

α10|10〉 + α11|11〉√|α10|2 + |α11|2

,

respectively. See Nielsen and Chuang (2010) and Wang(2012) for details.

3.2 Quantum Entanglement

As one of the most mind-bending creatures known toscience, quantum entanglement is often cited as the phe-nomenon that two particles that are connected by an in-visible wave can share each other’s properties regardlessof the distance between them, just like a twin. It leads tothe fact that none of the particles involved in a quantumsystem can be described by quantum states of individualsubsystems. In other words, all information content of anentangled quantum system is fully entailed in the corre-lations between the individual subsystems while none ofthe subsystems on their own convey essential informa-tion of the entangled quantum system. For a multi-qubitsystem, its entangled states are superposition states thatare described by joint properties of the individual qubitsin the multi-qubit system. Consider an entangled 2-qubitsystem, we obtain a completely random outcome whenperforming measurements on only one of its entangled

qubits. The measurement outcome is absolutely random,and it is impossible to gain information about the entan-gled system from the obtained random measurement out-come. As the entangled state involves two qubits, theircorrelation must contain two bits of classical informa-tion, and the classical information can only be gatheredby comparatively examining the outcomes of the individ-ual measurements on the separate subsystems. We as wellpoint out an intriguing feature of entangled states: mea-suring one of the entangled qubits instantaneously caststhe other one into the corresponding perfectly correlatedstate, which immediately destroys the entanglement asqubit measuring changes their quantum state. We take aBell state

(3.4) |ψ〉 = |01〉 − |10〉√2

as an example to demonstrate entanglement, where α00 =α11 = 0, α01 = 1/

√2, and α10 = −1/

√2 in the expres-

sion of (3.2). As described in Section 3.1, measuring thefirst qubit of the Bell state |ψ〉, we obtain measurementoutcome 0 or 1 with probability |α00|2 +|α01|2 = 1/2 and|α10|2 + |α11|2 = 1/2 respectively, which is completelyrandom. According to (3.3), if the measurement outcomeis 0 (or 1), then the state will be |01〉 (or |10〉, respec-tively). This result means that if the first measurementoutcome is 0 (or 1), then the second qubit’s state must be|1〉 (or |0〉, respectively) with measurement always being1 (or 0, respectively), which indicates perfect correlation.Quantum states like the Bell state in (3.4) that cannot beexpressed as products of some single qubits are called en-tangled states, while product states refer to quantum statesthat can be written in the product form of single qubits.Over the past decades many physical experiments havebeen designed and conducted to test quantum entangle-ment through the so-called Bell inequality.

For the case of a 2-qubit system realized by the spinsof two particles, imagine that the two-particle system isfirst prepared in an entangled state, then the two particlesare drifted far away from each other. We now have Aliceand Bob measure the first and second particles, respec-tively, and sequentially. The perfect correlation suggeststhat after Alice obtains her spin measurement result (i.e.,+1 or −1) on the first particle, the system has its stateimmediately plunged into the untangled state. As a re-sult, the second particle now has a definite spin state, andBob’s spin measurement on the second particle alwaysprovides a definite opposite result (i.e., −1 or +1, respec-tively). This phenomenon of perfect correlation is referredto as anti-correlation in entanglement experiments. Wewill show that quantum properties such as superpositionand entanglement play key roles in quantum science andquantum technology. See Horodecki et al. (2009), Nielsenand Chuang (2010) and Wang (2012) for more details.


4. QUANTUM INFORMATION

As its classical analog, quantum information targetsat determining the laws governing any information pro-cess based on quantum theory. The core of classicalinformation theory is Shannon’s two coding theoremson noiseless and noisy channels. The coding theoremsquantify classical bits by Shannon entropy for transmis-sion over a noiseless channel and character the amountof information transmitted over a noisy channel withsome error-correction scheme. On the other hand, thequantum-based theory has been established to apprehendquantum resources such as superposition, entanglement,nonlocality, no-cloning, and quantum randomness. Thequantum counterparts of Shannon entropy and Shannonnoiseless coding theorem are von Neumann entropy andSchumacher’s noiseless channel coding theorem, respec-tively. Schumacher’s noiseless channel coding theoremdescribes quantum information needed to compress quan-tum states by von Neumann entropy (Schumacher, 1995).The quantum analog of Shannon’s noisy channel codingtheorem is Holevo–Schumacher–Westmoreland theorememployed to calculate the product quantum state capacityfor some noisy channels (Holevo, 1998, Schumacher andWestmoreland, 1997).

Despite the resemblance, there exist inherent distinc-tions between classical information and quantum infor-mation. For example, while classical information suchas digital images can be distinguished and copied, quan-tum superposition and no-cloning theorem imply thatunknown quantum states cannot be completely distin-guished or exactly copied. Consider another example, be-sides the computational basis |0〉 and |1〉 for the qubitspace, we have another basis |+〉 = (|0〉 + |1〉)/√2 and|−〉 = (|0〉 − |1〉)/√2 given in Section 3.1, and quantuminformation can be encoded under each of these bases.Different ways of encoding quantum information are em-ployed in quantum error-correction for reliable quantumcomputation and quantum information processing. More-over, the information encoded under one basis cannot beextracted by performing measurement under another ba-sis, which plays an important role in quantum cryptogra-phy. For example, consider encoding one bit of informa-tion in different bases by the polarization of light. Sup-pose that the computational basis formed by |0〉 and |1〉represents the horizontal and vertical basis (correspond-ing to horizontally and vertically polarized photons). Asdiagonally and anti-diagonally polarized photons can beexpressed in the horizontal and vertical basis as coher-ent superpositions of horizontal and vertical parts, the ba-sis formed by |+〉 and |−〉 corresponds to the diagonaland anti-diagonal basis. We may encode a bit of informa-tion in the |0〉 and |1〉 basis by treating 0 to be horizontalpolarization and 1 to be vertical polarization. For a pho-ton encoded in either horizontal or vertical polarization, if

we measure it in the diagonal and anti-diagonal basis, itsinformation cannot be extracted. Indeed, as described inSection 3.1, we have

|0〉 = 1√2|+〉 + 1√

2|−〉, |1〉 = 1√

2|+〉 − 1√

2|−〉,

and thus, when measuring |0〉 or |1〉 in the basis |+〉 and|−〉, we observe + and − with equal probability, that is,we observe a diagonally polarized photon in 50% of thecases and an anti-diagonally polarized photon in the other50% of the cases.

Quantum physics provides new types of resources forinformation processing and transmission such as quan-tum teleportation, superdense coding, quantum key distri-bution, and quantum error-correction. Also, quantum in-formation ideas have effectively been employed in otherscientific studies such as many-body physics, quantumgravity, high-energy physics, quantum chemistry, quan-tum biology, and even for solving conjectures in the fieldsof classical information and computation. See Hayashi(2006), Krenn et al. (2017), Nielsen and Chuang (2010)and Wang (2012) for more details.

5. QUANTUM COMMUNICATION

5.1 Quantum Teleportation

Quantum teleportation is a process through which wetransfer the state of a quantum system (or qubit) to anotherdistant quantum system (or qubit) without ever existing inthe intervening space in between. The phenomenon canbe illustrated by a three-step protocol of quantum telepor-tation as follows. First, Alice (the sender) and Bob (thereceiver) together generated a special pair of entangledqubits, and each took one qubit of the two shared qubitswhen they split. Second, Alice was given a third qubitwhose state was undisclosed to her, and she would liketo teleport the unknown state. Third, Alice interacted thethird qubit with her qubit and performed a special mea-surement on her original qubit so that, while the measure-ment destroyed the entanglement and ruined any infor-mation about the state of her qubit, it would make Bob’squbit instantaneously project onto a new state that Bobcould use to recover the original state of Alice’s qubit.After the three steps, the state of Alice’s qubit was trans-ported to that of Bob’s qubit. A key feature in the quan-tum teleportation protocol is the special entanglement andmeasurement, and the teleportation protocol only workswhen Bob was informed by Alice about her measurementoutcome so that Bob could work accordingly to recoverAlice’s state. That is, a successful teleportation event re-quires classical communication between Alice and Bob,which necessarily restricts the speed of information trans-fer in the teleportation protocol to the speed of the clas-sical communication channel. See Nielsen and Chuang(2010) and Wang (2012) for details.


It is important to note from the entire three-step pro-tocol of quantum teleportation that contrary to what isusually mistakenly cited, quantum teleportation in prin-ciple does not allow faster-than-light communication orany transfer of matter or energy. Quantum teleportationtransfers only the state of Alice’s qubit to Bob’s qubit butdoes not physically move Alice’s qubit (particle) to Bob.Because it is required to send information via the classicalchannel, quantum teleportation is not capable of transmit-ting information faster than the speed of light. Otherwise,if Bob can obtain a copy of Alice’s qubit (in the senseto physically obtain her ‘qubit’), then Bob can make adirect measurement on the copied qubit to obtain the in-formation that was sent over via the classical communica-tion between Alice and Bob. In this way, faster-than-lightcommunication becomes possible, however, the famousno-cloning theorem prevents the teleportation from copy-ing any qubit.

The no-cloning theorem is referred to the fact that quan-tum mechanics prohibits the creation of identical copiesof a general quantum state. Specifically, cloning a quan-tum state |ψ〉 means a procedure with the product state|ψ〉|ψ〉 as an output. We begin by introducing an ancillaquantum system whose state |ϕ〉 is not related to the state|ψ〉 being cloned. The no-cloning theorem means thatthere exists no unitary matrix U such that it evolves theinitial state |ψ〉|ϕ〉 to the desired output state |ψ〉|ψ〉, thatis,

(5.1) U(|ψ〉|ϕ〉) = e

√−1θ(ψ,ϕ)|ψ〉|ψ〉,where e

√−1θ(ψ,ϕ) stands for a phase factor, with phaseθ(ψ,ϕ) being some real number. Indeed, if such U exists,it has a similar effect on any arbitrarily selected state |φ〉since cloning should work for any state. For the pair ofstates |ψ〉 and |φ〉 in Cd , we consider their inner producttogether with the ancilla state |ϕ〉, and use (5.1) to obtain

〈ψ |φ〉 = 〈ψ |φ〉〈ϕ|ϕ〉= 〈ψ |〈ϕ||φ〉|ϕ〉= 〈ψ |〈ϕ|U†U|φ〉|ϕ〉= e−√−1[θ(ψ,ϕ)−θ(φ,ϕ)]〈ψ |〈ψ ||φ〉|φ〉= e−√−1[θ(ψ,ϕ)−θ(φ,ϕ)][〈ψ |φ〉]2

,

which indicates that |〈ψ |φ〉| = |〈ψ |φ〉|2, namely, |〈ψ |φ〉|equals to 0 or 1. By the Cauchy–Schwarz inequality, weconclude that |ψ〉 is either equal to |φ〉 (with a phasefactor) or orthogonal to |φ〉, which is not possible foran arbitrary pair of states |ψ〉 and |φ〉. This shows thenonexistence of such U and thus proves the no-cloningtheorem. Moreover, we may provide a simple illustra-tion to show that no-cloning is a natural consequence ofquantum theory as follows. Consider qubits |0〉, |1〉, and

|+〉 = (|0〉 + |1〉)/√2, along with an ancilla qubit |a〉.Cloning implies there exists a unitary matrix U such that

U(|0〉|a〉) = |0〉|0〉, U

(|1〉|a〉) = |1〉|1〉,(5.2)

U(|+〉|a〉) = |+〉|+〉.

Using the first two equalities in (5.2) and linearity of Uwe immediately obtain

U(|+〉|a〉) = U

(1√2|0〉|a〉 + 1√

2|1〉|a〉

)

= 1√2

U(|0〉|a〉) + 1√

2U

(|1〉|a〉)

= 1√2|0〉|0〉 + 1√

2|1〉|1〉,

which is an entangled state. It is easy to see that the en-tangled state cannot be written as product state

|+〉|+〉 =(

1√2|0〉 + 1√

2|1〉

)(1√2|0〉 + 1√

2|1〉

)

= 1

2

(|0〉|0〉 + |1〉|1〉 + |0〉|1〉 + |1〉|0〉).Therefore, an inconsistency occurs in (5.2), and it is im-possible to have all three equalities in (5.2). See Krennet al. (2017) and Nielsen and Chuang (2010) for more de-tails.

5.2 Communication Components

The classical communication required in quantum tele-portation does not carry complete information about thequbit being teleported. Even if the information communi-cated in the classical channel is intercepted by an eaves-dropper who may have complete knowledge about whatBob is required to do to recover the desired state, the in-formation is futile if the eavesdropper cannot interact withthe entangled qubit held in Bob’s hands. Quantum physicsopens the door to various distinct quantum secret sharingprotocols such as quantum cryptography.

In quantum computation and quantum communication,we need to link qubits in quantum networks and trans-fer their states. As the no-cloning theorem forbids us toperfectly clone a quantum state, we cannot use classicalmethods like amplifiers to carry out any transfer of qubits’states. The solution to this problem is a so-called quan-tum repeater, which allows the end-to-end generation ofquantum entanglement in a way that every two connec-tive particles of independent entangled pairs is combinedso that the entanglement is relayed onto the remainingtwo particles. Thus, we are able to achieve the end-to-endstate transmission of qubits via quantum teleportation. Aquantum repeater is an important building block to inter-connect different nodes in a quantum network, and quan-tum teleportation is one crucial requirement for the quan-tum repeater. This process is referred to as entanglement


swapping and enables us to achieve long-distance quan-tum communication. See Krenn et al. (2017), Nielsen andChuang (2010) and Sangouard et al. (2011) for more dis-cussions.

5.3 Quantum Cryptography

Cryptography allows two parties, the sender Alice andthe receiver Bob, to exchange secret messages in theirprivate communications, while at the same time, keeps itvery hard for the third parties to ‘eavesdrop’ on the con-tent of their communications. Applications of cryptog-raphy include online bank transactions, electronic com-merces, and military communications. We discuss twocryptographic methods adopted in such communications.The first method is a private key cryptosystem, whichcalls for the two parties to share a secret key. Specifically,Alice employs the key to encrypt a message and obtainthe cipher while the cipher can only be understood if thekey is known. Alice sends the cipher to Bob who utilizesthe key to decrypt the received cipher and read her mes-sage. The challenge for the private key cryptosystem liesin guarding the secret key against eavesdropping.

The alternative method is a public key cryptosystem in-vented in the 1970s that does not require the sharing of asecret key. This method is based on the complexity of hardcomputational problems such as finding the prime factorsof very large numbers. Specifically, Bob first generates apair of keys, a public one and a private one. He then an-nounces his ‘public key’ to the general public, everyoneincluding Alice can use the public key to encrypt mes-sages and send him the encrypted messages. The real trickis that the encryption transformation generated by Bob’skeys is specially designed such that with only the publickey, it is extraordinarily hard, though not impossible, toreverse the encryption transformation. When announcingthe public key, Bob retains a corresponding secret key forsimple inversion of the encryption transformation and de-cryption of the received messages. A case in point is theRSA cryptosystem (Rivest, Shamir and Adleman, 1978),one of the most widely used cryptographic protocols.RSA is built on the extreme difficulty of finding primefactors for large composite numbers. Note the mathemat-ical asymmetry of factoring: it is easy to compute a com-posite number from its prime factors by multiplying theprimes, no matter how large they are; however, the reverseprocess can be very hard, in fact, it is extremely difficultto find the prime factors of some very large compositenumbers. RSA encryption retains the large primes as a se-cret key and makes use of their product to design a ‘publickey’. Since the best known classical factoring algorithmshave exponential complexity, and massive computationalattempts to break the RSA system so far have led to nosuccess, it is widely believed that the RSA system is se-cure against any classical computer-based attacks.

On the other hand, Peter Shor in 1994 discovered theso-called Shor’s quantum factoring algorithm that cansolve the factoring problem exponentially faster than thebest known classical algorithms, thus quantum computersmay be able to break the RSA system easily (Shor, 1994).That is, as quantum computers can factor prime numberssignificantly faster than classical computers, an eaves-dropper equipped with a quantum computer can decipherthe encrypted text and read the secret message, with onlythe public information distributed by RSA. An approachto circumventing this difficulty is a quantum procedureknown as quantum cryptography or quantum key distri-bution so that communication security cannot be under-mined. The security of the quantum key distribution isbased on the quantum principle that observing or measur-ing an unknown quantum system will disturb the systemthat is being monitored. When an eavesdropper listens tothe transmission of the quantum key between Alice andBob, the quantum communication channel employed toset up the key will be disturbed by the eavesdropping, andthe disturbance will make eavesdropping noticeable. As aresult, Alice and Bob are able to discard the compromisedkey and retain only the secured key for their communica-tion.

Specifically, quantum key distribution enables two au-thorized parties to create a secret key at a distance intwo stages. In the first stage, the two communicating par-ties, Alice and Bob, obtain a preliminary key by exchang-ing quantum signals over the quantum channel and per-forming measurements. The obtained key is preliminaryin the sense that it has two strongly correlated, but non-identical, and only partly secret strings. In the secondstage, Alice and Bob utilize the classical channel to carryout an interactive post-processing protocol. The protocolpermits them to refine the preliminary key and extracttwo identical and absolutely secret (known only to them-selves) strings as two identical copies of the created se-cret key. During the two-stage process, the quantum chan-nel is open to any possible maneuver from a third person.However, the classical channel communication needs thefollowing authentication: while Alice and Bob recognizethemselves, a third person may listen to their exchange,but cannot engage in it. In particular, the mission of Al-ice and Bob is to guarantee security against an adver-sarial eavesdropper, whom we call Eve, engaging in theconversation over the classical channel and tapping onthe quantum channel. Here we use ‘security’ to conveyprecisely that the authorized parties never use a nonse-cret key, namely, either they can actually generate a se-cret key, or the protocol is aborted. Hence after transmit-ting the quantum signals, Alice and Bob need to evalu-ate the possible amount of information about the prelimi-nary keys may have leaked out to Eve. This is the crucialadvantage of quantum communication where information


leakage in a quantum channel is quantitatively linked to adegradation of the communication. Such degradation andevaluation are not possible in classical communication.For example, when classical communication channels aretapped, such as phone conversations are bugged, the com-munication proceeds without any change, as nothing hap-pens.

Besides taking advantage of quantum property that ob-serving a quantum channel disturbs the quantum commu-nication, we may employ quantum entanglement to fur-ther enhance the security of quantum key distribution bycreating an entanglement-based quantum key distribution.The quantum entanglement property offers secure advan-tages for designing and implementing the protocol of theentanglement-based quantum key distribution. Let us con-sider the case where Alice and Bob share an entangledstate of a qubit (or particle) pair. When they perform mea-surements on the qubits, the obtained random measure-ments will always be opposite due to the perfect correla-tion between entangled qubits. Alice and Bob then needto communicate over a classical channel regarding howthe measurements are performed and obtained in order tosift through the results and obtain a secret key.

The security foundation of quantum key distributioncan be established by the core principles of quantumphysics such as superposition and no-cloning. When Eveis tapping on a quantum communication channel to ex-tract some information, her act is some kind of measure-ment performing on the state of the quantum communica-tion system, and the measurement will generally alter thestate of the system. On the other hand, if Eve wants a cor-rect copy of the state that Alice conveys to Bob, she willnot be successful as the no-cloning theorem shows thatan unknown quantum state cannot be duplicated withoutbeing altered.

To be specific, the essential idea behind quantum keydistribution is that Eve cannot obtain any informationfrom the qubits, whose state is transmitted from Alice toBob, without disturbing their state. Our proof argumentsare as follows. First, the no-cloning theorem describedin Section 5.1 prevents Eve from copying Alice’s qubit.Second, information gain implies disturbance in the sensethat for any try to differentiate between two nonorthog-onal quantum states, gaining information is only possibleat the cost of bringing in disturbance to the signal. Indeed,suppose that |ψ〉 and |φ〉 are two nonorthogonal quantumstates. Eve attempts to gain information about |ψ〉 and|φ〉 by unitarily interacting the states |ψ〉 or |φ〉 with anancilla quantum system prepared in a state |u〉. If Eve’sattempt does not disturb the states, we obtain two unitarymatrices U1 and U2 such that

U1(|ψ〉|u〉) = |ψ〉|v1〉, U2

(|φ〉|u〉) = |φ〉|v2〉,where |v1〉 and |v2〉 are different states so that Eve cangain information about the identity of the states |ψ〉 and

|φ〉. However, since unitary transformations preserve in-ner products, it must be that

〈v1|〈ψ ||φ〉|v2〉 = 〈u|〈ψ |U†1U2|φ〉|u〉 = 〈u|〈ψ ||φ〉|u〉,

that is, 〈v1|v2〉〈ψ |φ〉 = 〈u|u〉〈ψ |φ〉. As |ψ〉 and |φ〉 arenonorthogonal, 〈ψ |φ〉 = 0, and thus we obtain

〈v1|v2〉 = 〈u|u〉 = 1,

which indicates that |v1〉 and |v2〉 have to be equal. Thisleads to a contradiction, as |v1〉 = |v2〉. Therefore, distin-guishing between |ψ〉 and |φ〉 must disturb at least oneof the states, and we can make secure quantum communi-cation by transmitting nonorthogonal qubit states betweenAlice and Bob and checking for disturbance in their trans-mitted states.

Furthermore, the quantum key generation relies on thesame quantum physical principles that quantum computa-tion is based on. Unlike classical cryptography, the quan-tum key distribution does not merely depend on the com-putational difficulty of solving mathematical problemssuch as the factoring problem. Hence, it cannot be bro-ken even by quantum computers. In a nutshell, the funda-mental quantum physical principles allow for the uncon-ditional security of quantum key distribution, namely, thepossibility of guaranteeing security without setting anypower limitation on the eavesdropper. See Bennett andBrassard (2014), Bernstein and Lange (2017), Buhrmanet al. (2010), Krenn et al. (2017), and Nielsen and Chuang(2010) for more details.

Quantum physics was established to describe nature atthe microscopic domain, but many ongoing research en-deavors seek answers to what extent the quantum physicallaws are relevant to the macroscopic realm. In particular,research efforts in quantum science and quantum technol-ogy aim to increase the distance between entangled quan-tum particles, and search for any possible fundamental re-strictions to quantum entanglement, as well as to investi-gate if it is viable to create a global-scale quantum com-munication network in the future. Physical experimentson quantum key distribution have been successfully con-ducted in a long distance with current records of over ahundred kilometers on earth (Krenn et al., 2017) and overa thousand kilometers in space (Yin et al., 2017).

6. QUANTUM COMPUTATION

In contrast to classical computation where transistorsare used to crunch the ones and zeroes individually,the new quantum resources such as quantum superpo-sition and entanglement can allow quantum computa-tion to manage both one and zero at the same timeand do the trick of performing simultaneous calculations.Thus, quantum computers may outperform classical com-puters for solving certain computational problems. SeeBrowne (2014), Campbell, Terhal and Vuillot (2017),


Chong, Franklin and Martonosi (2017), Deutsch (1985),Mohseni et al. (2017), Nielsen and Chuang (2010) andWang (2012).

6.1 Quantum Computers

Classical computers are constructed from electrical cir-cuits containing wires for carrying information around thecircuits and logic gates for executing simple computa-tional tasks. Similarly, quantum computers are built fromquantum circuits with quantum gates to carry out quantumcomputation and process quantum information. In spiteof the similarity, quantum computers are built on the uni-tary evolution of b logical qubits operating on a compu-tational state space of 2b dimensions, and the new quan-tum resources make it possible for quantum computersto outperform classical computers for certain tough tasks.Quantum information and quantum computation investi-gate how to harness the enormous information hidden inthe quantum systems and how to make use of the immensepotential computational power of quantum particles toperform computation and to process information. Inten-sive efforts are underway around the world to explorea number of physical systems and fabrication technolo-gies for constructing quantum computers, where viableconstructions must meet a set of requirements known asthe DiVincenzo criteria (DiVincenzo, 1995). Major sys-tems and technologies include superconducting circuits,ion traps, quantum dots, and other electronic semicon-ductor circuits, impurity spins, and linear optics (Nielsenand Chuang, 2010). Quantum computers of small scalehave been built to demonstrate numerous simple exam-ples of quantum algorithms and protocols. Over the yearsthere are steadily increasing efforts by academics, govern-ment labs, large companies, and startups to reach the chal-lenging goal of large scale quantum computation (DiCarloet al., 2009, Johnson et al., 2011, Mariantoni et al., 2011,Sayrin et al., 2011).

As mentioned above, the physical equipment for thequantum computer fabrication must meet the DiVincenzocriteria including requirements that a quantum system re-alized qubits has to be well isolated to maintain its quan-tum properties and at the same time, the quantum systemneeds to be accessible so that the qubits can be operatedto carry out computations and perform output measure-ments. In reality, there always exists some coupling ofa quantum system to its environment, and the couplingleads to quantum decoherence, where decoherence refersto the loss of coherence between the components of thequantum system or quantum superposition from the in-teraction of the quantum system with its external entities.Therefore, the coupling strength dictates the two oppos-ing requirements stated above. It is very challenging butcritical to manage a quantum system of qubits for con-trolling the coupling strength and rectifying the effects of

decoherence in quantum technology. Given the significantdifficulties to build large-scale quantum computers withpresent technology, it is very important to have scalablearchitectures for building quantum computers with about100 well-behaved logical qubits in the near future. Sucharchitectures may enable us to demonstrate the so-calledquantum (computational) supremacy that is actively pur-sued by academic labs and companies like Google andIBM, where quantum supremacy refers to any major mile-stone achievement in the quest for outperforming classicalcomputers on some tough computational tasks (Aaronsonand Chen, 2017, Boixo et al., 2018, Harrow and Monta-naro, 2017).

The quantum computing approach discussed so far islogic-gate based that has its purpose in developing a quan-tum version of classic logic gate operations and con-structing a universal (or general purpose) quantum com-puters. Since significant technological difficulties presentin the implementation of the gate (or circuit) model forbuilding universal quantum computers, alternative quan-tum computing architectures, such as adiabatic quantumcomputing, are actively being explored to build special-purpose quantum computers for solving specific com-putational problems, though subjected to different chal-lenges (Aharonov and Ta-Shma, 2003, Aharonov et al.,2008, Albash and Lidar, 2018). Examples of special-purpose quantum computers include quantum annealersand quantum simulators for solving tough simulation andoptimization problems. Next, two Sections 6.2 and 6.3will present detailed discussions on quantum annealersand quantum simulators, respectively. Quantum anneal-ers mean physical hardware implementations of quantumannealing. Quantum simulators refer to quantum devicesutilized for simulating one quantum system by using an-other more controllable one, with the aim to solve spe-cial simulation problems that are computationally too de-manding on classical computers.

6.2 Quantum Annealers

Quantum annealing may be considered as adiabaticquantum computing that is based on the quantum adia-batic theorem for building special-purpose quantum com-puters, called quantum annealers, to solve combinatorialoptimization problems. Quantum annealing is the quan-tum analog of classical annealing, with thermodynamicsreplaced by quantum dynamics. Quantum annealers arephysical hardware devices to implement quantum anneal-ing. See Albash and Lidar (2018), McGeoch (2014), andWang, Wu and Zou (2016).

Given an optimization problem, we identify its objec-tive function to be minimized with the energy of a phys-ical system and assign the physical system a temperaturethat serves as an artificially-introduced control parameter.Classical annealing like simulated annealing takes into ac-count the relative configuration energies and a fictitious


time-dependent temperature when exploring the immensesearch space probabilistically. By decreasing the temper-ature slowly from a high value to zero, with certain proba-bility we move the system toward the state with the lowestvalue of the energy and hence arrive at the solution of theoptimization problem.

Specifically, consider a classical Ising model character-ized by a graph G = (V(G),E(G)), where V(G) and E(G)

represent the vertex and edge sets of G, respectively. Eachvertex has a random variable whose value is +1 or −1,and each edge corresponds to the coupling (or interaction)between two vertex variables linked by the edge. Define aconfiguration s to be a set of values assigned to all vertexvariables sj , j ∈ V(G), that is, s = {sj , j ∈ V(G)}. Ver-tices and vertex variables also refer to sites and spins inphysics, respectively, where the values +1 and −1 of aspin stand for spin up and spin down, respectively. A casein point is a 2-dimensional lattice considered as a sim-ple graph, with a magnet put at each lattice site pointingeither up or down. Denote by b the total number of the lat-tice sites. At site j = 1, . . . , b, let sj be a binary randomvariable representing the position of the magnet, wheresj = ±1 means that the j th magnet points up or down,respectively. The classical Ising model has the followingHamiltonian:

(6.1) HcI ≡ Hc

I (s) = − ∑(i,j)∈E(G)

δij sisj − ∑j∈V(G)

γj sj ,

where (i, j) denotes the edge between the sites i and j ,the first sum takes over all pairs of vertices with edge(i, j) ∈ E(G), δij represents the interaction (or coupling)between sites i and j associated with edge (i, j) ∈ E(G),and γj stands for an external magnetic field on vertexj ∈ V(G). We refer to a set of fixed values {δij , γj } asone instance of the Ising model. Hc

I (s) is also called theenergy of the Ising model at configuration s. The probabil-ity of a specific configuration s is given by the followingBoltzmann distribution:

(6.2) PT (s) = e−HcI (s)/T

ZT

, ZT = ∑s

e−HcI (s)/T ,

here T serves as the fundamental temperature of the sys-tem with units of energy. The configuration probabilityPT (s) describes the probability that the physical systemis in a state with configuration s in equilibrium.

When using the Ising model to represent a combina-torial optimization problem, the goal is to find a groundstate of the Ising model, that is, we need to find a con-figuration that can minimize the energy function Hc

I (s).If the Ising model contains b sites, then the configura-tion space is {−1,+1}b and the total number of possi-ble configurations is equal to 2b. We note that the systemcomplexity increases exponentially in b (the number ofsites), and thus, it is very difficult to find a ground state

and solve the minimization problem numerically when b

is large. In fact, the search space that grows exponentiallyprohibits us to solve the minimization problem with de-terministic exhaustive search algorithms. Instead, anneal-ing methods such as simulated annealing are employedto search the space probabilistically. To find a config-uration with minimal energy, simulated annealing usesMarkov chain Monte Carlo (MCMC) methods such as theMetropolitan–Hastings algorithm to generate configura-tion samples from the Boltzmann distribution PT (s) whiledecreasing the temperature T slowly. See Bertsimas andTsitsiklis (19932), Kirkpatrick, Gelatt and Vecchi (1983),and Wang, Wu and Zou (2016).

Quantum annealing utilizes the physical process of aquantum system whose lowest energy, or equivalently,a ground state of the system, renders a solution to theposed optimization problem. It begins by creating a sim-ple quantum system initialized in its ground state and thendrives the simple system slowly to the target complex sys-tem. The quantum adiabatic theorem (Farhi et al., 2000,2001, Farhi, Goldstone and Gutmann, 2002, Kadowakiand Nishimori, 1998) implies that, as the system gradu-ally evolves, it likely stays in a ground state, and thereforewith some probability, we can find a solution to the orig-inal optimization problem by measuring the state of thefinal system. In other words, by replacing thermal fluctu-ations in simulated annealing by quantum fluctuations viaquantum tunneling, quantum annealing manages to keepthe system close to its instantaneous ground state dur-ing the quantum annealing evolution, similar to a quasi-equilibrium state to be maintained during the time evolu-tion of simulated annealing

As in the classical annealing case, graph G is used todescribe the quantum Ising model, where the vertex setV(G) stands for the quantum spins, and the edge set E(G)

denotes the couplings (or interactions) between two quan-tum spins. As qubits can be realized by quantum spins,each vertex is occupied by a qubit. Suppose that G has b

vertices. The quantum system is described by vector spaceC

d (d = 2b), with its quantum state described by a unitvector in C

d , and its dynamic evolution governed by theSchrödinger equation (2.1) via a quantum Hamiltonian,which is a Hermitian matrix of size d . The energies of thequantum system are represented by the eigenvalues of thequantum Hamiltonian, and the ground states are given bythe eigenvectors corresponding to the smallest eigenvalue.Since quantum mechanics is based on mathematics of ma-trices with dimensionality equal to d = 2b (the number ofpossible configurations), we substitute the classical spins(or bits) in (6.1) with quantum spins (or qubits) to obtainthe quantum Hamiltonian. To be specific, define

Ij =(

1 00 1

), σ x

j =(

0 11 0

),

σ zj =

(1 00 −1

), j = 1, . . . , b,


where σ xj and σ z

j are Pauli matrices in x and z axes, re-spectively. For the quantum system, each classical vertexvariable sj = ±1 in (6.1) is replaced by σ z

j for the j thquantum spin (or qubit). The two eigenvalues ±1 of thePauli matrix σ z

j correspond to the eigenstates |+1〉 and|−1〉 which further represent the spin up state |↑〉 and spindown state |↓〉, respectively. In total, there are 2b possiblequantum configurations formed by selecting b eigenstatesfrom the 2b eigenstates of the Pauli matrices {σ z

j }bj=1 andthen putting them in the form | ± 1, . . . ,±1〉.

We replace sj in the classical Ising Hamiltonian HcI (s)

by σ zj to obtain the quantum Hamiltonian,

(6.3) HqI = − ∑

(i,j)∈E(G)

δijσzi σ

zj − ∑

j∈V(G)

γjσzj ,

where δij stands for the Ising coupling along the edge(i, j) ∈ E(G), and γj represents the local field on the ver-tex j ∈ V(G). Here we adopt the convention in quantumliterature that σ z

j and σ zi σ

zj in (6.3) stand for their tensor

products along with identical matrices as follows:

σ zj ≡ I1 ⊗ · · · ⊗ Ij−1

vertex j︷︸︸︷⊗σ z

j⊗ Ij+1 ⊗ · · · ⊗ Ib,(6.4)

σ zi σ

zj ≡ I1 ⊗ · · · ⊗ Ii−1

⊗vertices i and j︷︸︸︷

σ zi ⊗ Ii+1 ⊗ · · · ⊗ Ij−1 ⊗ σ z

j(6.5)

⊗ Ij+1 ⊗ · · · ⊗ Ib.

All elements in (6.4) are identity matrices except for thej th element being a Pauli matrix σ z

j , and σ zi σ

zj in (6.5)

is simply an ordinary matrix multiplication of matricesσ z

i and σ zj treated as tensor products of b matrices in the

sense of (6.4). Each qubit in (6.4) and (6.5) is operated byone matrix, either a Pauli matrix σ z

i or an identity matrixIi . The quantum convention singles out the qubits withPauli matrices for actual actions but leaves out the iden-tity matrices and tensor product signs. To generate quan-tum Hamiltonian Hq

I we in fact replace sj in (6.1) by these2b × 2b matrices, with scalars γj and δij unchanged. Fur-thermore, since each term in (6.3) is a tensor product of b

diagonal matrices of size two, quantum Hamiltonian HqI

is a 2b × 2b diagonal matrix constructed so that its diago-nal elements are completely in agreement with all valuesof classical Hamiltonian Hc

I in (6.1) corresponding to the2b configurations ordered lexicographically.

As a diagonal matrix HqI has eigenvalues equal to its

diagonal entries, which in turn are the 2b possible valuesof classical Hamiltonian Hc

I , thus finding the minimal en-ergy of the classical Ising Hamiltonian Hc

I is equivalent tofinding the minimal energy of the quantum Ising Hamilto-nian Hq

I . That is, we need to find a quantum spin configu-ration with the minimal energy, namely, a ground state ofquantum Hamiltonian Hq

I . Although we have formulated

the original optimization problem in the quantum frame-work, up to now the computational task for solving theoptimization problem is still the same as in the classicalcase.

To carry out quantum annealing for solving the opti-mization problem, it is essential to engineer a transversemagnetic field that is orthogonal to the Ising axis andobtain the corresponding Hamiltonian in the transversefield. The transverse field represents kinetic energy thatdoes not commute with the potential energy Hq

I , there-fore, it induces transitions between the up and down statesof every single spin, and converts the system behaviorfrom classical to quantum. Define the following quantumHamiltonian governing the transverse magnetic field

(6.6) HX = − ∑j∈V(G)

σ xj ,

where again following the quantum convention we denoteby σ x

j the tensor products of b matrices of size 2 as fol-lows:

(6.7) σ xj ≡ I1 ⊗ · · · ⊗ Ij−1

vertex j︷︸︸︷⊗σ x

j⊗ Ij+1 ⊗ · · · ⊗ Ib.

Furthermore, σ xj does not commute with σ z

j in HqI , and

HX is a nondiagonal matrix of size 2b that does not com-mute with diagonal matrix Hq

I . We note that Pauli matrixσ x

j has two eigenvalues +1 and −1 associated with theeigenvectors |uj,+1〉 = (1,1)′ and |uj,−1〉 = (1,−1)′. Asa result, the eigenvector corresponds to the smallest eigen-value of HX is |u+〉 = |u1,+1〉 ⊗ |u2,+1〉 ⊗ · · · ⊗ |ub,+1〉,that is, |u+〉 is the ground state of HX .

We start the quantum annealing procedure with a quan-tum system that is driven by the transverse magnetic fieldHX and initialized in its ground state |u+〉. The systemthen slowly moves from the initial Hamiltonian HX to itsfinal target Hamiltonian Hq

I . During the Hamiltonian evo-lution, according to the adiabatic quantum theorem, thesystem has a tendency to stay in the ground states of theinstantaneous Hamiltonian via quantum tunneling (Farhiet al., 2000, 2001, Farhi, Goldstone and Gutmann, 2002,McGeoch, 2014). At the end of the annealing procedure,if the quantum system stays in a ground state of the finalHamiltonian Hq

I , we can measure the quantum system toobtain a solution of the optimization problem. In specific,quantum annealing is accomplished by the following in-stantaneous Hamiltonian for the Ising model in the trans-verse field:

(6.8) HD(t) = A(t)HX + B(t)HqI , t ∈ [0, tf ],

where A(t) and B(t) are smooth functions that dependon time t and control the annealing schedules, and tf isthe total annealing time. To drive the system from HX toHq

I , we take A(tf ) = B(0) = 0 where A(t) is decreas-ing and B(t) is increasing. It follows that when t = 0,


HD(0) = A(0)HX and when t = tf , HD(tf ) = B(tf )HqI .

Since A(0) and B(tf ) are known scalars, HD(t) has thesame eigenvectors as HX at the initial time t = 0 andas Hq

I at the final time t = tf , where the correspondingeigenvalues differ by factors of A(0) and B(tf ), respec-tively. Therefore, HD(t) moves the system from HX ini-tialized in its ground state to the final target Hq

I . Whenthe control functions A(t) and B(t) are chosen appropri-ately, the quantum adiabatic theorem indicates that the an-nealing procedure driven by (6.8) will have a sufficientlyhigh probability in finding the global minimum of Hc

I (s)and solving the minimization problem at the final anneal-ing time tf . See Brooke, Bitko and Aeppli (1999), Isakovet al. (2016), Jörg et al. (2010), and Wang, Wu and Zou(2016) for details.

While classical annealing depends on thermal fluctua-tions to drive the system to hop from state to state overintermediate energy barriers and find a desired lowest-energy state, quantum annealing substitutes thermal hop-ping by quantum-mechanical fluctuations to search fora ground state. We realize the quantum fluctuations inquantum annealing by quantum tunneling that enablesthe annealing process to explore different states by cross-ing directly through energy barriers, rather than climbingover them thermally. Here quantum tunneling refers to thequantum phenomenon where particles tunnel through abarrier in the situation that is classically infeasible. Wecannot directly observe the tunneling process nor use clas-sical physics to explain it satisfactorily. Quantum tun-neling is generally described by the Heisenberg uncer-tainty principle and the wave-particle duality of matterin quantum physics (Crosson and Harrow, 2016, Das andChakrabarti, 2005, 2008, Denchev et al., 2016, Wang, Wuand Zou, 2016).

Quantum annealing devices are actively pursued by anumber of academic labs and companies such as Googleand D-Wave Systems, with uncertain quantum speedup.In particular, the D-Wave quantum computer is a commer-cially available hardware device that is designed and builtto physically implement quantum annealing. It is an ana-log computing device based on superconducting qubitsto process quantum annealing and solve certain combi-natorial optimization problems. Also although it is ex-tremely difficult to simulate quantum annealing by classi-cal computers, classical Markov chain Monte Carlo sim-ulations have been developed to approximate quantumannealing by path-integral formulation and mean fieldapproximation. See Albash et al. (2015), Boixo et al.(2014, 2016, 2018), Brady and van Dam (2016), Rønnowet al. (2014), and Wang, Wu and Zou (2016) for more dis-cussions.

6.3 Quantum Simulators

Quantum simulation is to intentionally and artificiallymimic interacting quantum systems, which are hard to

access and analyze, by employing other precisely con-trollable quantum systems that are easy to manipulateand investigate. Since the dimensionality of the space de-scribing a quantum system scales exponentially with thesystem size, the classical simulation of quantum systemsdemands exponentially increasing resources. Likewise, ittakes exponentially large resources to solve certain classi-cal optimization problems particularly the NP-hard prob-lems, such as finding the ground-state energy of a classi-cal spin glass and solving the traveling salesman’s prob-lem. Quantum simulation may provide scientific meansto simulate complex biological, chemical or physical sys-tems in order to study and understand certain scien-tific phenomena and solve the related hard computationalproblems. Experimental platforms for quantum simula-tion consist of ultra-cold atomic and molecular quantumgases, ultra-cold trapped ions, polariton condensates insemiconductor nanostructures, circuit-based cavity quan-tum electrodynamics, arrays of quantum dots, photonicquantum technology, and superconducting qubits withcommercial applications in quantum annealers (Aspuru-Guzik and Walther, 2012, Blatt and Roos, 2012, Bloch,Dalibard and Nascimbene, 2012, Boghosian and Taylor,1998, Houck, Türeci and Koch, 2012, Jané et al., 2003,Johnson et al., 2011, Nielsen and Chuang, 2010, Wang,Wu and Zou, 2016).

The essential of quantum simulation is to understandthe dynamic evolution of a quantum system governed bythe Schrodinger equation (2.1). That is, quantum simula-tion needs to describe and solve the Schrodinger equation(2.1) by either digital quantum computers or analog quan-tum machines. The solution of (2.1) has expression

(6.9)∣∣ψ(t)

⟩ = e−iHt∣∣ψ(0)

⟩, i = √−1,

and we need to evaluate e−iHt numerically. It is extremelydifficult to exponentiate the Hamiltonian H because itssize increases exponentially in the system size. Commonnumerical approach often uses the first-order linear ex-pansion 1− iHδ to approximate e−iH(t+δ) −e−iHt , whichoften yields unsatisfactory numerical solutions. Math-ematically, quantum simulation is to explore whetherhigher order approximations are available to provide ef-ficient methods for the evaluation of e−iHt . For exam-ple, consider a system with α particles in a d-dimensionalspace that has the following Hamiltonian:

H =L∑

=1

H,

where L is a polynomial in α + d , and each H acts ona small subsystem of finite size free from α and d . Notethat it is easy to evaluate e−iHδ numerically, but verydifficult to compute e−iHδ . Because H and Hk are non-commutable, e−iHδ = e−∑L

=1 iHδ = e−iH1δ · · · e−iHLδ .


By the Trotter formula (Proposition 3.1 in Wang (2011)),we obtain

e−iHδ = {e−iH1δ/2 · · · e−iHLδ/2}

(6.10)× {

e−iHLδ/2 · · · e−iH1δ/2} +O(δ2)

.

Thus, we obtain a second order approximation of e−iHδ

by the first term on the right-hand side of (6.10), whichonly needs us to evaluate each e−iHδ , = 1, . . . ,L.

Introduced by Feynman (1981/82), quantum simulationitself has been developed into a core field within quan-tum computation. A quantum simulator can be any phys-ical quantum system precisely prepared or manipulatedin a way targeting at studying interesting features of aninteracting complex quantum system, which is compu-tationally intractable or difficult to simulate on classicalcomputers. A quantum simulator can be a digital quan-tum simulator so that the controllable quantum systemis implemented on a universal quantum computer, or ananalog quantum simulator so that the controllable quan-tum system is a quantum physical device to reconstructthe time evolution of an interacting quantum system un-der precisely controlled conditions. Like universal quan-tum computers, digital quantum simulators face signifi-cant challenges in scaling architectures. However, analogquantum simulators can be addressed and experimentedin a relatively large scale with currently available tech-nology, and thus may provide new tools for us to inves-tigate interacting many-particle quantum systems and at-tack optimization problems beyond the reach of classicalcomputers. See Childs et al. (2018), Jiang et al. (2017),Kassal et al. (2008, 2011), Lanyon et al. (2010), Nielsenand Chuang (2010), and Wang (2011, 2012).

It is likely that the first practical application of quantumcomputation is quantum simulation since even moderatequantum simulation devices have the potential to carryout simulations infeasible by classical computers. For ex-ample, in quantum chemistry, molecular energies can becomputed by digital quantum simulation devices of size100 to 150 logical qubits with excellent precision and ac-curacy that considerably exceed the limitations of classi-cal computers. In particular, in the near to medium term,analog quantum simulators may offer us a novel tool tostudy complicated quantum systems and hard optimiza-tion problems that are unreachable by classical comput-ers. Again a computational advantage of quantum simula-tors over classical ones may clearly demonstrate quantumsupremacy (given in Section 6.1) in realistic applications.In the long run, the importance of quantum simulationmay lie in the applications of large-scale quantum sim-ulations to solve fundamental problems in physics, mate-rials science and quantum chemistry (Abrams and Lloyd,1997, Aspuru-Guzik et al., 2005, Boghosian and Taylor,1998, Cirac and Zoller, 2012, Kassal et al., 2011, Lloyd,1996).

6.4 Quantum Algorithms

Quantum algorithms are algorithms that run on quan-tum computation models, such as the most commonlyused quantum gate or circuit model, by taking input qubitsand producing output measurements for the solutions ofspecific computational tasks. While a classical algorithmtakes a step-by-step procedure to solve a given problemon a classical computer, a quantum algorithm is a step-by-step problem-solving procedure, with each step per-formed on a quantum computer. We note that all classi-cal algorithms can be in principle executed on a quan-tum computer, all problems solvable on a quantum com-puter are solvable on a classical computer, and problemsundecidable by classical computers remain undecidableon quantum computers. However, quantum algorithms areessentially different from their classical counterparts inthe sense of being genuine quantum, that is, quantumgate operations are reversible unitary transformations, andquantum algorithms utilize fundamental quantum proper-ties such as quantum superposition and quantum entan-glement. We refer to quantum algorithms as the algo-rithms that are inherently quantum for achieving fasterspeed than classical algorithms in solving some toughproblems. It should be pointed out that while quantum al-gorithms cannot be worse than classical algorithms, weshould not expect quantum algorithms to yield advantagefor every single problem; in fact, they usually do not.As a matter of fact, quantum computers augment, but donot replace classical computers. A continual challenge inquantum science is to invent new quantum algorithms tospeed up the best classical algorithms. For example, quan-tum superposition indicates that we can potentially carryout exponentially many computations in parallel, but itis tricky to extract the solution from such an exponentialsuperposition to achieve some quantum speedup, as ob-serving the qubit system destroys its state. This is wherewe need clever designs of quantum software. Commontechniques employed to create quantum algorithms in-clude quantum Fourier transform, phase estimation, am-plitude amplification, quantum walk, quantum annealingand quantum simulation. The widely known quantum al-gorithms include Shor’s factoring algorithm and Grover’ssearch algorithm, which are, respectively, exponentiallyfaster and quadratically faster than the best known andbest classical algorithms for the same tasks. Many otheralgorithms were created for a wide range of problemsand applications such as searching, sorting, counting,sampling, simulation, and optimization. See Montanaro(2016), Nielsen and Chuang (2010) and Wang (2012) formore discussions.

As a case in point, we consider quantum computationfor the Grover and parity problems. Let f (x) be a functiondefined on the integers from 1 to N and taking the values


±1. Define the parity of f (x) by

Par(f ) =N∏

x=1

f (x).

The parity of f (x) can be either +1 or −1, and alwaysdepends on the values of f (x) at all N points. It hasbeen proved that with no further information about f (x),both classical and quantum algorithms have O(N) time-complexity to determine its parity, and thus quantum com-puters cannot outperform classical computers for the par-ity problem. For the Grover problem, there is a furtherinformation that f (x) is either identically equal to 1 or itis 1 for N − 1 of the x’s and equal to −1 at one unknownvalue of x. For such f (x), its parity indicates its type,and the computational task for the Grover problem is todetermine the type of f (x) and search for the unknownvalue of x (if it exists). With the additional informationabout f (x), the best classical and quantum algorithms forthe Grover problem have complexity O(N) and O(

√N),

respectively, and thus there is an optimal√

N quantumspeedup. It is interesting to note that, although there is aquadratic quantum speedup for the Grover problem, theparity problem has no quantum speedup (see Farhi et al.(1998) and Grover (1997)). This example indicates thatneither classical nor quantum computers are expected tobe best for all computational tasks.

6.5 Quantum Machine Learning

Quantum machine learning extends classical machinelearning to the quantum realm. Classical machine learn-ing and statistical learning often refer to an array of statis-tical approaches to analyzing data, with the goal of in-ferring the future behavior of target variables (such asthe function relationships of variables and their dynamicprocesses) from training data. The learning procedure in-volves inference, which addresses how statistically effi-cient we can learn the functions or processes from givendata, and computation, which handles how much com-putational resources are required to perform a learningtask and how fast algorithms can be designed to carryout the learning task. The learning objective is to searchfor a model that fits well to training data but more im-portantly enjoys good generalization capability, whichrefers to the property of the learned model with goodprediction performance on new observations. Commonlearning approaches rely on regularization-based meth-ods leverage on optimization techniques to solve learn-ing problems. Quantum learning theory investigates howquantum resources can affect the learning efficiency. Thetheory indicates that it is possible for quantum learnersto achieve higher efficiency such as better generaliza-tion errors in learning difficult functions for some par-ticular learning models. However, the major advantagesthat quantum mechanics can provide is largely in terms

of computation. In other words, quantum machine learn-ing can offer advantages over its classical counterpart interms of computational complexity. Therefore, it is rea-sonable to expect quantum computers to be faster thanclassical computers for solving some machine learningproblems, but it is important and challenging to explorequantum softwares that enable quantum machine learn-ing to realize such quantum speedups. Recent develop-ment indeed shows a class of quantum machine learn-ing algorithms exhibit some quantum speedups. For ex-ample, from a computational perspective, solving linearequation systems is almost ubiquitous in machine learn-ing, and finding a learning solution usually comprises asequence of standard linear algebra operations such asmatrix multiplication and inversion. Quantum linear al-gebra algorithms offer quantum speedups over their clas-sical analogs. As a case in point, quantum basic linear al-gebra subroutines (BLAS), which include finding eigen-vectors and eigenvalues and solving linear equations, ex-hibit exponential quantum speedups over their best knownclassical counterparts. The quantum BLAS renders quan-tum speedups for an array of data analysis and ma-chine learning algorithms including linear algebra opera-tion, gradient descent, Newton’s method, linear program-ing, semidefinite and quadratic programming, topologicalanalysis, least-squares, nearest-neighbor, support vectormachines, clustering, and principal component analysis(PCA). Also special-purpose quantum computers, suchas quantum annealers and programmable quantum opti-cal arrays, bear architectures well suited to quantum op-timization and deep learning particularly quantum deeplearning with Boltzmann machines. More discussionscan be found in Adachi and Henderson (2015), Aminet al. (2018), Arodz and Saeedi (2019), Arunachalam andde Wolf (2018), Benedetti et al. (2016), Biamonte et al.(2017), Brandão et al. (2019), Ciliberto et al. (2018),Dunjko, Taylor and Briegel (2016), Dunjko and Briegel(2018), Jordan (2005), Lloyd, Mohseni and Rebentrost(2014), O’Gorman et al. (2015), Rebentrost, Mohseni andLloyd (2014), Salakhutdinov and Hinton (2009), Shenvi,Kempe and Whaley (2003), Svore, Hastings and Freed-man (2014), Wiebe, Kapoor and Svore (2016, 2015),Wiebe and Granade (2016), and Wittek (2014). Belowwe provide short illustrations for some selected topics inquantum machine learning.

6.5.1 Bayesian quantum phase estimation. Quantumphase estimation is the key to achieve quantum speedupsin many well-known quantum algorithms (Nielsen andChuang, 2010, Wang, 2012). Suppose that a unitary op-erator U has an eigenvector |ξ〉 with corresponding eigen-value e

√−12πϕ , ϕ ∈ (0,1). We do not know the phase ϕ ofthe eigenvalue, and our goal is to find ϕ based on a set ofexperiments performed on a quantum circuit. The experi-ments involve preparing the state |ξ〉 and performing mea-surement on U multiple times at some angle, where the


angle θ and the number M of times are randomly chosen.The Bayesian phase estimation procedure is to perform aseries of random measurements and then solve a classicalreconstruction problem using Bayesian inference. Givenϕ, and randomly chosen M and θ , the conditional prob-abilities of obtaining measurement outcomes 1 and 0 areas follows:

P(1|ϕ; θ,M) = 1 − cos(2πMϕ + θ)

2

= 1 − P(0|ϕ; θ,M).

For a given measurement sequence, with a uniform pri-ori distribution on ϕ, we obtain the posterior distributionof ϕ. We may repeat this process for a series of experi-ments, and the Bayesian phase estimation approach pro-vides posterior distributions over the phase ϕ, which offerestimates of the true eigenvalue and the algorithm’s uncer-tainty in that value. More details can be found in Paesaniet al. (2017), Svore, Hastings and Freedman (2014) andWiebe and Granade (2016). More generally, Hamiltonianlearning has been developed to infer quantum Hamilto-nians via Bayesian inference for understanding quantumdynamics. See Granade et al. (2012), Wiebe et al. (2014)and Wiebe, Granade and Cory (2015) for details.

6.5.2 Quantum principal component analysis. PCAdepends on the eigendecomposition of a covariance ma-trix, which, in the quantum context, can be convertedinto simulating a Hamiltonian in quantum simulation de-scribed in Section 6.3. Suppose that data are observed inthe form of vectors vj in a d-dimensional vector space,j = 1, . . . , n. Assume that vj have zero mean and finitevariance. Without loss of generality we further assumevj are unit vectors (otherwise we may normalize eachto be a unit vector and then use their norms to adjustselection probability below). Quantum principal compo-nent analysis randomly selects a vector from v1, . . . , vn

and then maps each selected vector vj into a pure quan-tum state |vj 〉. The random encoding procedure yields aquantum state with b = log2 d qubits and a density ma-trix ρ = 1

n

∑nj=1 |vj 〉〈vj |, which is equal to the sample

covariance matrix up to an overall factor.We first describe a quantum technique called density

matrix exponentiation. Using a simple trick with the par-tial trace over the first variable and the swap operator S,we get

tr1[e−√−1S�tρ ⊗ πe

√−1S�t ]= cos2(�t)π + sin2(�t)ρ − √−1 sin(�t)[ρ,π ]= π − √−1�t[ρ,π ] + O

([�t]2)(6.11)

= e−√−1ρ�tπe√−1ρ�t + O

([�t]2)= e−√−1[ρ,π]�t(π) + O

([�t]2),

where tr1 denotes the partial trace over the first variable,and swap operator S has a matrix representation

S =d∑

j,k=1

|j〉⟨k| ⊗ ∣∣k⟩〈j ∣∣.

S is a sparse matrix of size d2, which can be clearly seenfrom the following explicit expression for the case of d =2:

S =

⎛⎜⎜⎝

1 0 0 00 0 1 00 1 0 00 0 0 1

⎞⎟⎟⎠ .

Thus, for sparse S, e−√−1S�t can be performed effi-ciently. As a result, simply performing infinitesimal swapoperations on ρ ⊗ π allows us to simulate the unitarytime evolution e−√−1ρ�tπe

√−1ρ�t = e−√−1[ρ,π]�t(π)

and thus construct the unitary operator e−√−1ρt via theSchrödinger equation (2.3) in the density matrix form,where we have used the Baker–Campbell–Hausdorff for-mula that for matrices A and B,

AdeAB = eABe−A = eadAB =∞∑

k=0

1

k!(adA)k(B),

and linear transformations adA and AdA with defini-tions adAB = [A,B] = AB − BA, and AdAB = ABA−1.The quantum density matrix exponentiation procedure forconstructing e−√−1ρt comprises the preparation of an en-vironment state π and the application of the global swapoperator S to the combined system and environment stateρ ⊗ π followed by partial trace tr1 to discard the environ-mental degrees of freedom.

Applying the density matrix exponentiation procedureto ρ we construct e−√−1ρ�t . Then we utilize the quan-tum phase estimation algorithm to find eigenvalues andeigenvectors of the density matrix ρ, which renders theprinciple components. The quantum PCA procedure hascomputational complexity O(log2 d) with potential tobe exponentially faster than classical PCA. See Lloyd,Mohseni and Rebentrost (2014) and Rebentrost, Mohseniand Lloyd (2014) for more details.

6.5.3 Quantum support vector machines. Consider thebinary classification problem where we have trainingdata (x1, y1), . . . , (xn, yn), with xi = (xij ) ∈ R

p and yi ∈{−1,1}, and the goal is to use the training data to learnhow to predict classes y for feature vectors x. Definea hyperplane f (x) = xτβ to induce a classification rulesign(xτβ), where β = (βj ) ∈ R

p is a parameter. We needto find a suitable value for parameter β based on the train-ing data. The training of the sparse support vector ma-chines model with the hinge loss is often converted into


solving the following minimization problem:

arg minβ

1

n

n∑i=1

max(0,1 − yiβ

τ xi

) + λ

p∑j=1

|βi |,

where λ > 0 is a tuning parameter. The nonlinear uncon-strained optimization problem can be transformed to anequivalent constrained linear programming problem withn + 2p nonnegative variables and n linear inequality con-straints,

arg minξ,β+,β−

1

n

n∑i=1

ξi + λ

p∑j=1

β+j + λ

p∑j=1

β−j ,

subject to

p∑j=1

yixijβ+j −

p∑j=1

yixijβ+j ≥ 1 − ξi,

ξi, β±j ≥ 0, i = 1, . . . , n, j = 1, . . . , p,

The support vector machines solution is βj = β+j − β−

j .While classical algorithms for solving such linear pro-gramming problems have asymptotic computational com-plexity O(mn) where m = n + 2p, quantum algorithmshave been proposed recently for the optimization prob-lems with time complexity O(

√mn) or even O(

√m +√

n), which offers potential for a quadratic speedup com-pared to the classical algorithms. Furthermore, for theleast squares support vector machines (with the mini-mization problem: minβ

12‖β‖2 + λ

∑ni=1 ξ2

i subject toξi ≥ 0, yix

τi β = 1− ξi, i = 1, . . . , n), quantum algorithms

offer an exponential speedup over classical algorithms.Also nonlinear classification rules can be derived by usingkernels. For a polynomial kernel matrix K , we normal-ize it by its trace to obtain K = K/ tr(K). Using density

matrix exponentiation in (6.11) to construct e−√−1Kt and

then applying quantum phase estimation to e−√−1Kt weperform eigen-analysis and matrix inversion for K andthus efficiently solve the optimization problem for find-ing the support vector machines solution. See Arodz andSaeedi (2019) and Rebentrost, Mohseni and Lloyd (2014)for more details.

6.5.4 Quantum deep learning with Boltzmann ma-chines. Deep learning has been widely explored in quan-tum machine learning. The literature has been mainly con-centrated on speeding up the training of classical modelsand on developing relatively less matured quantum neu-ral networks, which refer to that all their component parts,ranging from the single neurons to the training algorithms,are carried out on quantum computers. Boltzmann ma-chines are stochastic models that enable to produce newdata based on prior observations, which are called gener-ative models in deep learning. Because of their intrinsic

link to the Ising model, Boltzmann machines are espe-cially suitable for learning exploration from a quantumviewpoint.

As a classical machine learning technique, Boltzmannmachines serve as the basis of powerful deep learningmodels. A Boltzmann machine usually consists of visi-ble and hidden binary units that are jointly denoted bysi , i = 1, . . . ,N , where N is the total number of units.We use the notation si = (sv, sh) to distinguish the vis-ible and hidden variables with index v for visible vari-ables and h for hiddens, and reserve vector notations v,h, and s = (v,h) for representing random vectors corre-sponding to visible, hidden, and combined units, respec-tively. A classical Ising model is employed to describethe variables sj . As in Section 6.2, the Hamiltonian (orenergy function) of the Ising model is Hc

I ≡ HcI (s) de-

fined in (6.1), where now we take N = b, and γj and δij

are considered as model parameters to be tuned duringthe training in machine learning. In equilibrium, the prob-ability of observing a state v of the visible variables isdescribed by the Boltzmann distribution summed over allhidden variables,

(6.12) Pv = Z−1∑

h

e−HcI (s), Z = ∑

se−Hc

I (s),

which is called the marginal distribution of the visible ran-dom vector v. We aim to find parameters θ = {γj , δij } inthe Hamiltonian Hc

I such that Pv becomes as close as pos-sible to the corresponding empirical distribution definedby the training data. The common classical approachesto finding the parameters are to maximize likelihoods,and the maximization is often achieved via some com-bination of gradient descent and sampling. Research hasdemonstrated that sampling from Boltzmann machinesand calculating their likelihoods are computationally verydifficult, and MCMC simulations are often employed asstandard techniques to overcome the hard computationaltasks, though MCMC can be be very costly or even impos-sible for models with a large number of neurons. Train-ing with quantum resources can be very helpful in reduc-ing the training cost and offering some quantum speedup.Thus, it makes quantum deep learning more feasible orpreferable than the classical approach. Quantum tech-niques, which include quantum linear algebra, quantumsampling, and quantum annealers, have been developed totrain the classical Boltzmann machines. Special-purposequantum computers such as quantum annealers and pro-grammable photonic circuits are very suitable for trainingBoltzmann machines. In particular the D-Wave devices,quantum annealers with tunable transverse Ising models,have been applied to encode deep quantum learning pro-tocols on over thousands of spins. See Adachi and Hen-derson (2015), Benedetti et al. (2016) and Wiebe, Kapoorand Svore (2016) for details.


Quantum resources can give new, fundamentally quan-tum, models for deep learning. Quantum Boltzmann ma-chines are introduced to cross-breed between training arti-ficial neural networks and fully quantum neural networks.They induce an array of quantum effects such as quantumtunneling. Unlike classical Boltzmann machines, quan-tum Boltzmann machines can yield quantum states as out-puts, and thus deep quantum networks can learn to pro-duce quantum states for representing a broad range ofsystems, which is beyond the capability of classical ma-chine learning. Quantum Boltzmann machines are definedby quantum Ising models in transverse fields. Similar tothe classical case, as described in Section 6.2, the quan-tum Hamiltonian of the quantum Ising model is Hq

I givenby (6.3), where again we take N = b, and γj and δij aremodel parameters to be tuned during the training. Notethat Hq

I is a 2N ×2N diagonal matrix, which is in contrastto vectors with dimensionality equal to N , the number ofvariables, used in classical machine learning.

Denote by |v,h〉 the eigenstates of Hamiltonian HqI ,

where again v and h stand for vectors of visible and hid-den variables, respectively. For diagonal Hamiltonian Hq

I ,

e−HqI is also a diagonal matrix with its 2N diagonal ele-

ments being e−HcI corresponding to all 2N configurations.

Define its partition function Z = tr[e−HqI ] and density ma-

trix

(6.13) ρ = Z−1e−HqI .

Then the diagonal elements of ρ are equal to the Boltz-mann probabilities of all 2N configurations. Given a state|v〉 of the visible variables, we get the following marginalBoltzmann probability Pv by tracing over the hidden vari-ables:

(6.14) Pv = tr[�vρ],where �v is a diagonal matrix whose diagonal elementsare equal to 1 if the visibles are in state v, and 0 other-wise. The use of �v is to limit the trace only to diagonalelements corresponding to the visible variables which arein state v. It is easy to see that definitions (6.12) and (6.14)are identical for the diagonal Hamiltonian and density ma-trix, but (6.14) is still valid for the nondiagonal case.

To add a transverse field to the quantum HamiltonianHq

I , as described in Section 6.2, we need nondiagonal ma-trices σ x

j described in (6.7) to obtain the transverse fieldIsing Hamiltonian as follows:

(6.15) Hq� = −∑

i

�iσxi − ∑

i,j

δijσzi σ

zj − ∑

j

γjσzj ,

where besides the original parameters γj and δij , we haveadditional model parameters �i . A quantum Boltzmannmachine is defined by the quantum Boltzmann distribu-tion of the transverse field Ising Hamiltonian Hq

� . As Hq�

is a nondiagonal matrix, we may express its eigenvectors

by superpositions in the computation basis composed ofthe classical states |v,h〉, and the corresponding quantumBoltzmann machine has quantum probability distributionwith nondiagonal density matrix ρ given by (6.13) withHq

I replaced by Hq� as well as the marginal Boltzmann

probability distribution Pv defined in (6.14). Performingeach measurement on the states of the qubits in the σ z-basis we obtain the outcome ±1, and thus the measure-ment output for the visible variables follows the marginalprobability distribution Pv given by (6.14).

Our learning goal is to train the quantum Boltzmannmachine and find the model parameters θ = {�i, γj , δij }in the Hamiltonian Hq

� such that the probability distri-bution Pv gets as near as possible to the correspondingempirical distribution determined by the input data. Themethod of finding the parameters is to maximize somebounds on relevant likelihoods, due to the likelihood in-tractability for quantum Boltzmann machines. Treating asa class of recurrent quantum neural networks, quantumBoltzmann machines can be trained for machine learningtasks such as discriminative and generative learning, aswell as quantum state tomography and quantum anneal-ing. More details can be found in Amin et al. (2018) andKieferova and Wiebe (2016).

6.5.5 Quantum machine learning for quantum data. Ingeneral quantum machine learning is particularly suitablefor quantum data which are the actual output states gen-erated by quantum systems and processes, and the spe-ciality of quantum data analysis is the capability of usingquantum simulators to probe quantum dynamics. We mayapply quantum machine learning algorithms like quantumPCA and quantum Boltzmann machines to quantum data(such as quantum states of light and matter) for exploit-ing the quantum states and uncovering their hidden fea-tures and patterns. The obtained analysis results in quan-tum modes are often much more efficient and more en-lightening than the the classical analysis of data draw-ing from quantum systems. See Granade et al. (2012),Havlícek et al. (2019), Kieferova and Wiebe (2016),Lloyd, Mohseni and Rebentrost (2014), Marvian andLloyd (2016), Wiebe et al. (2014) and Wiebe, Granadeand Cory (2015) for details.

6.5.6 Quantum sampling. Sampling methods arewidely used techniques to compute some intractablequantities. Examples include the most commonly usedMonte Carlo methods in particular MCMC simulations.While classical Monte Carlo simulations are performedby pseudo-random numbers, quantum computation is ableto generate genuine random numbers and perform trueMonte Carlo simulations. Quantum MCMC algorithmsare developed to offer a quadratic speedup over classi-cal MCMC algorithms in terms of spectral gap, inversetemperature, desired precision or the hitting time. SeeChowdhury and Somma (2017), Richter (2006), Szegedy(2004) and Temme et al. (2011) for details.


6.5.7 Quantum machine learning with noise. Noisecan be potentially beneficial for solving machine learn-ing problems. Research has shown in the classical casethat noise may play a positive role in perturbing gradi-ents for jumping out local optima and improving gen-eralization performance. It becomes promising to ana-lyze noisy learning problems from a quantum perspec-tive and particularly exploit advantageously the effects ofnoise in quantum machine learning. For example, we maystudy whether the kinds of noise occurred in quantum sys-tems have similar distributional and structural behaviorsto those usually seen in classical settings, and if they canplay a beneficial role in quantum machine learning as inthe classical case. See Cross, Smith and Smolin (2015)and Grilo, Kerenidis and Zijlstra (2018) for details.

6.6 Quantum Computational Supremacy

Determining a quantum speedup depends on how wedefine the quantum speedup notation. One approach is totake a formal computational complexity perspective basedon rigorous mathematical proofs. Another realistic per-spective is based on what can be achieved with feasi-ble finite size devices and requires sound statistical evi-dence to confirm a scaling advantage over certain finiterange of problem sizes. For example, it has already beenrigorously proved in terms of computational complexitythat quantum algorithms like Grover’s search algorithmand Shor’s factoring algorithm offer speedups over knownclassical algorithms. Unfortunately, such rigorous proofsare often not available for most cases, and even avail-able, many existing quantum algorithms fail to providereference for any specific implementation such as the ex-act number of qubits needed to implement them; in fact,they often cannot be implemented on about 100 qubit plat-forms available in the near to medium term. We need toresort to the second perspective, and detecting a scalingadvantage of quantum computing over classical comput-ing would hinge on the so-called benchmarking problem,namely, the existence of a quantum computer performedon well designed computational tasks with sound sta-tistical analysis of computing experiments and resultingdata. Such advantages may include improved computa-tional speed, accuracy, and sampling for classically inac-cessible systems. Quantum algorithms and computationalproblems are created for these platforms with a limitednumber of qubits where classical computation is impos-sible. The mission involving both hardware and softwarealong with statistical analysis aims at demonstrating thequantum computational supremacy given in Section 6.1.Quantum scientists are building quantum computers of 50to 100 qubits to demonstrate quantum supremacy. For ex-ample, the Google quantum AI group has achieved quan-tum supremacy by building a quantum processor named‘Sycamore’ of 54 superconducting qubits to sample from

the output distributions of random quantum circuits, whileit is hard for current supercomputers to handle the sam-pling problem beyond around 50 qubits. See Aaronsonand Chen (2017), Arute et al. (2019), Boixo et al. (2018),Bouland et al. (2018), Bravyi, Gosset and König (2018),Harrow and Montanaro (2017), Lund, Bremner and Ralph(2017), Markov et al. (2018), Neill et al. (2018) andRønnow et al. (2014). Below we briefly describe bosonsampling and random quantum circuits.

6.6.1 Boson sampling. Boson sampling is a quantumcomputation model where n identical bosons pass througha network of passive optical elements (beamsplitters andphase-shifters) and then the locations of the bosons are de-tected. Quantum supremacy can be demonstrated by im-plementing boson sampling with a medium size network.A network system with 50 photons (qubits) and 2500paths is currently intractable for classical computers. Toimplement boson sampling all required physical devicesare single-photon sources, beamsplitters, phase-shiftersand photon-detectors. The physical implementation of thescheme encounters a myriad of technicalities such as syn-chronization of pulses, mode-matching, quickly control-lable delay lines, tunable beamsplitters and phase-shifters,single-photon sources, and accurate, fast, single photondetectors.

To define the boson sampling model, we adopt a sta-tistical approach based on the permanents of the subma-trices of a unitary matrix, which requires minimal quan-tum physics and quantum computation terminology. For an × n matrix A = (aij ), we define its permanent by

Perm(A) = ∑π

n∏i=1

aiπ(i),

where the sum is over all permutations π of 1,2, . . . , n.Consider the quantum system involving n identical pho-tons and m modes, where we may loosely interpret‘mode’ as the location of a photon, and we are only inter-ested in the case of m ≥ n. The quantum system has com-putational basis states of the form |s〉 = |s1, s2, . . . , sm〉,where si indicates the number of photons in the ith mode.Denote the set corresponding to all the computational ba-sis states by

�m,n = {s = (s1, s2, . . . , sm) : s1 + s2 + · · · + sm = n

}.

It is easy to see that the total number of elements in �m,n

is equal to M = (m+n−1n

). For a given m × m unitary ma-

trix U and each s ∈ �m,n, we obtain matrix Us from U bykeeping its first n columns and repeating sj times its j throw. Define a discrete probability distribution on �m,n asfollows:

Pr(s) = |Perm(Us)|2s1! · · · sm! .


It can be shown that Pr(s) is a well-defined probability dis-tribution on �m,n and corresponds to the quantum systemwith n photons, m modes and an optical network whoseaction is determined by the unitary matrix U. Boson sam-pling refers to sampling from distribution Pr(s). As clas-sical computers cannot handle the sampling problem evenwith moderate size, we may demonstrate the quantumsupremacy by successfully implementing boson samplingof reasonable size on quantum computing devices. Moredetails can be found in Harrow and Montanaro (2017) andLund, Bremner and Ralph (2017).

6.6.2 Random quantum circuits. Random quantumcircuits are created in a specific way so that when theyare generated with enough ‘complexity’, even the mostpowerful classical supercomputer cannot directly simu-late the generated quantum circuits. However, quantumcomputers can sample from the output distributions cor-responding to the obtained quantum circuits. Here a quan-tum circuit is a sequence of d clock cycles of one- andtwo-qubit gates with gates applied to different qubits inthe same cycle. The number d of cycles is called thedepth of the circuit. We say a random quantum circuithas enough ‘complexity’, if both its qubits and depth arelarge enough. If the gates to be applied are chosen froma universal quantum gate set, the unitary matrix U of thecircuit is a random matrix whose distribution convergesto the Haar measure on the collection of unitary matriceswhen the depth of the circuit goes to infinity. Specificallywhen a quantum circuit contains n qubits, with 2n com-putational basis states |x〉 = |x1x2 · · ·xn〉, xi ∈ {0,1}, aquantum state |ψd〉 produced by the random quantum cir-cuit is a linear combination of the computational basisand thus has 2n amplitudes, each with real and imaginaryparts. Therefore there are 2n+1 parameters in each quan-tum state. As the unitary matrix U of the random quan-tum circuit converges in distribution to the Haar measure,the random vector of the amplitude parameters asymptot-ically follows a uniform distribution on the unit sphere.Define the output distribution of the random quantum cir-cuit to be measurement probability p(x) = |〈x|ψd〉|2. Asthe depth d of the random quantum circuit goes to in-finity, p(x) approaches the Porter–Thomas distribution.The Google research group is working on finding waysto generate random quantum circuits so that their outputdistributions quickly converge to the Porter–Thomas dis-tribution. Based on the asymptotic distribution, we maydevelop a statistical approach to determine if a sampleis generated from the theoretical output distribution ofa desired random quantum circuit. Since it is difficultfor classical supercomputers to deal with the samplingproblem beyond around 50 qubits at the present time,the Google quantum scientists have designed a quantumprocessor named ‘Sycamore’ of 54 transmon qubits and

implemented quantum random circuits in a two dimen-sional lattice to demonstrate quantum supremacy. SeeArute et al. (2019), Boixo et al. (2018), and Neill et al.(2018) for more details.

7. QUANTUM METROLOGY

Measurement is at the heart of science, technology, in-dustry, and commerce. We need measurements and metro-logical standards to quantitatively assess scientific phe-nomena and technological progress, and gauge the ex-change of goods and service including information. Mea-surement devices are physical apparatuses whose func-tions and accuracy are governed by the laws of physics,and transformative improvements in measurement tech-nologies often follow the utilization of a new physical law.Quantum metrology (or quantum sensing) is to exploit thestrange laws of quantum physics to build new and bet-ter sensors and measuring devices. Fueling with quantumlaws, quantum metrology may lead to a game-changingshift in scientific studies, technological progress, as wellas commerce and industry developments.

The basic concept of quantum metrology is that a probedevice interacts with an appropriate system to learn theproperties of the system, where the interaction alters thestate of the probe, and measurements of the probe un-cover the characteristic parameters of the system. Forquantum sensing, the probe is usually prepared in onecertain quantum state, its encounter with the system nor-mally changes its state with both beneficial and adversar-ial effects in the sense that it not only responds to the pa-rameters of interest but also decoheres the probe (whichmeans there is loss of information from the probe into thesystem due to quantum decoherence, illustrated in Sec-tion 6.1). Then appropriately devised measurements canascertain in what way and to what extent such encounterhas changed the state of the probe, which enables quan-tum sensing to evaluate the system parameters. Quantumsensing promises to develop high-resolution and highlysensitive measurement techniques that will provide bet-ter precision than the same measurement performed un-der a classical framework. They include quantum sensors,quantum clocks, and quantum imaging. The applicationsrange from the sub-nano to the galactic scale, while someare in fact close to commercial use. The potential impactof quantum metrology is far-reaching. An array of dis-tinct platforms allow quantum-enhanced measurement oftime, space, rotation, as well as gravitational, electricaland magnetic fields. The technologies are promising tomake fundamental changes in a wide range of fields suchas physics, chemistry, biology, medicine or data storageand processing.

There is a strong link between quantum metrology andquantum information. For example, both quantum in-formation and quantum sensing rely on the same quan-tum properties such as entanglement, in particular, high


level of multipartite entanglement, to achieve better per-formance than their classical counterparts. See Degen,Reinhard and Cappellaro (2017), Kruse et al. (2016), andPezzè et al. (2018) for more details.

Quantum tomography plays an important role in quan-tum sensing. Quantum state tomography refers to recon-struction of a quantum state based on measurements per-formed on the quantum state. Statistically it is a densitymatrix estimation problem based on quantum measure-ments. Common quantum measurements are on observ-able M, which is defined as a Hermitian matrix on C

d .For example, the Pauli matrices as observables are widelyemployed to perform quantum measurements in quantumscience and quantum technology, and we may representmany density matrices through Pauli matrices. Supposethat the observable M has the following spectral decom-position:

(7.1) M =r∑

a=1

λaQa,

where λa are r different real eigenvalues of M, and Qa

are projections onto the eigen-spaces corresponding to λa .Given a quantum system prepared in state ρ , we use aprobability space (�,F,P ) to define measurement out-comes when performing measurements on the observableM. Let R be the measurement outcome of M. The theoryof quantum physics indicates that R is a random variableon (�,F,P ) which takes values in {λ1, λ2, . . . , λr}, andhas probability distribution

P(R = λa) = tr(Qaρ),(7.2)

a = 1,2, . . . , r,E(R) = tr(Mρ).

Quantum state tomography is to reconstruct ρ from in-dependent and identically distributed measurement out-comes R1, . . . ,Rn. See Artiles, Gill and Guta (2005), Caiet al. (2016), Malley and Hornstein (1993) and Wang andXu (2015).

Designing and controlling quantum systems are com-plex and challenging in the development of quantum sci-ence and quantum technology. Statistical methods pro-vide powerful tools for the study of quantum design andquantum control. Examples of successful applications in-clude quantum gate constructions with high fidelity preci-sion in quantum computation and quantum information,extraction of theoretical insights about quantum statesin condensed matter, and quantum control procedures inoptimizing adaptive quantum metrology. Like quantumphase estimation and quantum tomography, many quan-tum problems are in essence statistical problems, and it isour firm belief that statistics and data science have greatpotential to make significant improvement in quantummetrology.

8. CONCLUDING REMARKS

Quantum science and quantum technology gain enor-mous attention in multiple frontiers of many scientificfields. Quantum computation can give rise to an expo-nential speedup over classical counterpart for tacklingcertain computational tasks, quantum information canbring about exponential savings in information transmis-sion for handling computational and communication jobs,and quantum communication can offer more secure cryp-tosystems than classical analogue for solving commu-nication problems. Some of the quantum protocols arealready in practical implementation, such as quantum-fingerprinting, quantum key distribution, quantum anneal-ing, and quantum simulation. This paper reviews quantumscience and quantum technology from a statistical per-spective. We introduce concepts like key quantum proper-ties and qubits. We present quantum communication andquantum information, illustrate quantum computation andquantum metrology, and discuss major quantum technolo-gies associated with them. We show the advantages ofquantum techniques over the available classical counter-parts.

As statistics and machine learning nowadays heav-ily involve computation, it is natural to expect quantumcomputation to play a major role in data science. In-deed, quantum computation and quantum simulation mayhave tremendous potential to revolutionize computationalstatistics and data science. On the other hand, there isgreat demand in studying statistical issues for theoreticalresearch and experimental work in quantum science andquantum technology. As quantum phenomena are intrin-sically stochastic, and data collected in quantum experi-ments become more and more complex, we need to de-velop sophisticated statistical methods for enhancing dataanalysis and improving understanding of quantum events(Paesani et al., 2017, Wang, 2011, 2012, 2013, Wang,Wu and Zou, 2016, Wiebe et al., 2014, Wiebe, Kapoorand Svore, 2016). A great deal of current work is takenplace on creating new protocols and developing novelapproaches to certifying quantum devices such as test-ing and assessing their quantum performances. Clearly,such certification requires efficient and scalable statisti-cal methods for calibrating and validating quantum prop-erties. Moreover, certification needs to take into accountcommercial considerations for compliance with industrystandards by working together with industry, academics,national labs, and government organizations (Acín andMasanes, 2016, Wang, Wu and Zou, 2016, Wiebe et al.,2014).

As indicated in Section 6.1, the bottleneck of quan-tum computing at the present time is primarily on quan-tum hardware, and current quantum computing largely de-pends on what kinds of quantum computers experimental-ists can build. On the other hand, as we have demonstrated


in Section 6.6 for the quantum computational supremacyendeavor, besides hardware quantum computing also re-quires sophisticated mathematical models, sound statisti-cal analysis, and better computational tools. As a matterof fact, in general we call for some combination of newexperimental techniques, better mathematical and statis-tical understanding, and improved computational tools inorder to significantly advance the development of quan-tum science and quantum technology.

ACKNOWLEDGMENTS

The research of Yazhen Wang was supported in part byNSF Grants DMS-1528735, DMS-1707605, and DMS-1913149. The research of Xinyu Song was supportedby the Fundamental Research Funds for the CentralUniversities (2018110128), China Scholarship Council(201806485017) and National Natural Science Founda-tion of China (Grant No. 11871323). The authors thankDavid Siegmund for helpful comments and suggestionswhich led to improvements of the paper.

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