Quantum Computation and Quantum Information - Project Euclid

Statistical Science2012, Vol. 27, No. 3, 373–394DOI: 10.1214/11-STS378© Institute of Mathematical Statistics, 2012

Quantum Computation and QuantumInformationYazhen Wang

Abstract. Quantum computation and quantum information are of great cur-rent interest in computer science, mathematics, physical sciences and engi-neering. They will likely lead to a new wave of technological innovations incommunication, computation and cryptography. As the theory of quantumphysics is fundamentally stochastic, randomness and uncertainty are deeplyrooted in quantum computation, quantum simulation and quantum informa-tion. Consequently quantum algorithms are random in nature, and quantumsimulation utilizes Monte Carlo techniques extensively. Thus statistics canplay an important role in quantum computation and quantum simulation,which in turn offer great potential to revolutionize computational statistics.While only pseudo-random numbers can be generated by classical comput-ers, quantum computers are able to produce genuine random numbers; quan-tum computers can exponentially or quadratically speed up median evalua-tion, Monte Carlo integration and Markov chain simulation. This paper givesa brief review on quantum computation, quantum simulation and quantuminformation. We introduce the basic concepts of quantum computation andquantum simulation and present quantum algorithms that are known to bemuch faster than the available classic algorithms. We provide a statisticalframework for the analysis of quantum algorithms and quantum simulation.

Key words and phrases: Quantum algorithm, quantum bit (qubit), quan-tum Fourier transform, quantum information, quantum mechanics, quantumMonte Carlo, quantum probability, quantum simulation, quantum statistics.

1. INTRODUCTION

For decades computer hardware has grown in powerapproximately according to Moore’s law, which statesthat the computer power doubles for constant costroughly once every two years. However, because of thefundamental difficulties of size in conventional com-puter technology, this dream run is ending. The con-ventional approaches to the fabrication of computertechnology are to make electronic devices smaller andsmaller in order to increase the computer power. Asthe sizes of the electronic devices get close to theatomic scale, quantum effects are starting to inter-fere in their functioning, and thus the conventional

Yazhen Wang is Professor, Department of Statistics,University of Wisconsin–Madison, Madison, Wisconsin53706, USA (e-mail: [email protected]).

approaches run up against the size limit. One possi-ble way to get around the difficulties is to move to anew computing paradigm provided by quantum infor-mation science. Quantum information science is basedon the idea of using quantum devices to perform com-putation and manipulate and transmit information, in-stead of electronic devices following the laws of clas-sical physics, see Deutsch (1985), DiVincenzo (1995),Feynman (1981/82). Quantum mechanics and infor-mation theory are two of the great scientific develop-ments and technological revolutions in the 20th cen-tury, and quantum information science is to marry thetwo previously disparate fields and form a single uni-fying viewpoint. Quantum information science studiesthe preparation and control of the quantum states ofphysical systems for the purposes of information trans-mission and manipulation. It includes quantum compu-tation, quantum communication and quantum cryptog-raphy. This revolutionary field will enable a range of

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374 Y. WANG

exotic new devices to be possible. There is now a gen-eral agreement that quantum information science willlikely lead to the creation of a quantum computer tosolve problems that could not be efficiently solved ona classical computer. Already scientists have built rudi-mentary quantum computers in the research laboratoryto run quantum algorithms and perform certain calcu-lations. Intensive research efforts are under way aroundthe world to investigate a number of technologies thatcould lead to more powerful and more prevalent quan-tum computers in the near future. It is believed thatquantum information and quantum bits are to lead toa 21st century technological revolution much as clas-sic information and classic bits did to the 20th century.Since the theory of quantum mechanics is fundamen-tally stochastic, randomness and uncertainty are deeplyrooted in quantum computation and quantum informa-tion. As a result, quantum algorithms are of random na-ture in the sense that they yield correct solutions onlywith some probabilities, and Monte Carlo methods arewidely employed in quantum simulation. Thus statis-tics has an important role to play in quantum compu-tation, quantum simulation and quantum information.On the other hand, quantum computation and quantumsimulation have tremendous potential to revolutionizecomputational statistics.

A quantum system is generally described by its state,and the state is mathematically defined to be a unitvector in some complex Hilbert space. The number ofcomplex numbers required to characterize the quan-tum state usually grows exponentially with the size ofthe system, rather than linearly, as occurs in classi-cal systems. As a consequence, it takes an exponen-tial number of bits of memory on a classical com-puter to store the quantum state, which puts classicalcomputers in a difficult position to simulate a quan-tum system. On the other hand, nature quantum sys-tems are able to store and keep track of an exponentialnumber of complex numbers and perform data manip-ulations and calculations as the systems evolve. Quan-tum information science is to grapple with understand-ing how to take advantage of the enormous informa-tion hidden in the quantum systems and to harnessthe immense potential computational power of atomsand molecules for the purpose of performing compu-tation and processing information. Already it has beenshown that quantum algorithms like Grover’s search al-gorithm and Shor’s factoring algorithm provide greatadvantage over known classical algorithms.

Contemporary scientific studies often rely on un-derstanding complex quantum systems, such as those

in biochemistry and nanotechnology for the design ofbiomolecules and nano-materials. Quantum simulationis to use computers to simulate a quantum system andits time evolution. Classical computers are being usedfor quantum simulation in designing novel moleculesand creating innovative nano-products. Quantum com-puters built upon quantum systems may excel in sim-ulating naturally occurring quantum systems, whilelarge quantum systems may be impossible to simulatein an efficient manner by classical computers. A quan-tum system with b distinct components may be de-scribed with b quantum bits in a quantum computer,while a classical computer requires 2b bits of memoryto store its quantum state. This advantage allows quan-tum computers to efficiently simulate general quantumsystems that are not efficiently simulatable on classicalcomputers.

In this article we review the concepts of quan-tum computation and introduce quantum algorithmsand quantum simulation. The quantum algorithms areknown to be much faster than the available classi-cal algorithms. Statistical analyses of quantum algo-rithms and quantum simulation are provided. We givea brief description on quantum information. The ar-ticle sections start with presentations in broad brush-strokes, followed by specific discussions along withsome mathematical derivations if necessary. The inten-tion is to give each topic first an overview and thena general description and a precise characterization.It is recommended to focus on the qualitative discus-sions but skip the derivations for the readers who wouldlike to get a quick picture of quantum computation andquantum simulation.

The rest of the paper proceeds as follows. Section 2briefly introduces quantum mechanics, quantum prob-ability and quantum statistics. Section 3 reviews basicconcepts of quantum computation and entanglement.Section 4 illustrates some widely known quantum al-gorithms and provides a statistical framework for thestudy of quantum algorithms. Section 5 presents quan-tum simulation and discusses its statistical analysis.Section 6 gives a short description on quantum infor-mation theory. Section 7 features concluding remarksand lists some open research problems.

2. BRIEF BACKGROUND REVIEW ON QUANTUMTHEORY

Quantum mechanics has been applied to everythingunder and inside the Sun, from chemical reaction andsuperconductor to the structure of DNA and nuclearfusion in stars. Although the significant difference be-

QUANTUM COMPUTATION AND QUANTUM INFORMATION 375

tween classical physics and quantum physics lies in thequantum prediction of physical entity when the scaleof observations becomes comparable to the atomic orsub-atomic scale, many macroscopic properties of sys-tems can only be fully explained and understood byquantum physics. The quantum world is extremelystrange, and quantum theory is completely counterin-tuitive. Light waves behave like particles and particlesbehave like waves (wave particle duality); matter cango from one spot to another without moving throughthe intermediate space (quantum tunneling); informa-tion can be moved across a vast distance without trans-mitting it through the intervening space (quantum tele-portation). Quantum theory provides a mathematicaldescription of wave particle duality and interaction ofmatter and energy. It describes the time evolutions ofphysical systems via wave functions. The wave func-tions encapsulate the probabilities that particles are tobe found in a given state at a given time. For exam-ple, the probability of finding a photon in some regionis the square of the modulus of a wave function, and,since at some point the sum of two wave functions canbe zero but neither wave function is zero, probabili-ties appear to cancel out each other in a way totallyunexpected from classical probability. The intrinsicstochastic nature of quantum theory indicates a deepconnection between quantum mechanics and probabil-ity. Since the main focus of this paper is on quan-tum computation and quantum information, we givea brief description of quantum theory in this sectionto provide some quantum background for the purposeof reviewing quantum computation and quantum sim-ulation in subsequent sections. For further reading onthe subjects we recommend textbooks by Sakurai andNapolitano (2010) at the graduate level and Griffiths(2004) at the undergraduate level for quantum mechan-ics, Holevo (1982), Parthasarathy (1992) and Wang(1994) for quantum probability and quantum stochas-tic processes, and Artiles, Gill and Guta (2005) andBarndorff-Nielsen, Gill and Jupp (2003) for quantumstatistics.

2.1 Hilbert Space and Operator

For the sake of simplicity we choose to work withcomparatively easy finite-dimensional situations. De-note by C the set of all complex numbers. We startwith vector space in linear algebra. A simple exam-ple of vector space is C

k consisting of all k-tuples ofcomplex numbers (z1, . . . , zk). The elements of a vec-tor space are called vectors. As in quantum mechanicsand quantum computation, we use Dirac notations |·〉(which is called ket) and 〈·| (which is called bra) to in-

dicate that the objects are column vectors or row vec-tors in the vector space, respectively. Denote by super-scripts ∗, ′ and † the conjugate of a complex number,the transpose of a vector or matrix, and conjugate trans-pose operation, respectively. We define an inner prod-uct on the vector space to be a function that takes asinput two vectors from the vector space and producesa complex number as output. For |u〉 and |v〉 in thevector space, we denote their inner product by 〈u|v〉.The inner product must satisfy (i) conjugate symmetry,〈u|v〉 = (〈v|u〉)∗; (ii) linearity in the second argument,〈u|v + w〉 = 〈u|v〉 + 〈u|w〉; (iii) positive-definiteness,〈u|u〉 ≥ 0 with equality only for u = 0. For example,C

k has a natural inner product

〈u|v〉 =k∑

j=1

u∗j vj = (u∗

1, . . . , u∗k)(v1, . . . , vk)

′,

where 〈u| = (u1, . . . , uk) and |v〉 = (v1, . . . , vk)′. An

inner product induces a norm ‖u‖ = √〈u|u〉, and adistance ‖u − v‖ between |u〉 and |v〉. For the finite-dimensional case, a Hilbert space H is simply a vectorspace with an inner product.

An operator A on H, denoted by A(|u〉) for |u〉 ∈H, is a function mapping from H to H that satisfiesA(a|u〉 + b|v〉) = aA(|u〉) + bA(|v〉) for any |u〉, |v〉 ∈H and a, b ∈ C. We can represent an operator througha matrix. Suppose that A is an operator on H ande1, . . . , ek form an orthonormal basis in H. Then eachA(|ej 〉) ∈ H and there exists a unique k × k matrix(aj�) such that A(|ej 〉) = ∑k

�=1 |e�〉a�j , j = 1, . . . , k.We will identify operator A with matrix (aj�) and useA for both operator and matrix (aj�). An operator Aon H is said to be self-adjoint if its corresponding ma-trix A is Hermitian, that is, A = A†. We also refer toself-adjoint operators as Hermitian operators. An oper-ator U is said to be unitary if its corresponding matrixU is unitary, that is, UU† = U†U = I. We say an oper-ator A is semi-positive (or positive) definite if its cor-responding matrix A is semi-positive (or positive) def-inite, that is, 〈u|ρ|u〉 ≥ 0 for |u〉 ∈ H (or 〈u|ρ|u〉 ≥ 0for |u〉 ∈ H with equality only for |u〉 = 0). The traceof an operator A, denoted by Tr(A), is defined to bethe trace of its corresponding matrix A = (aj�), that is,Tr(A) = ∑k

j=1 ajj .

2.2 Quantum System

Quantum mechanics depicts phenomena at micro-scopic level such as position and momentum of an indi-vidual particle like an atom or electron, spin of an elec-tron, detection of light photons, and the emission andabsorption of light by atoms. Unlike classical mechan-ics where physical entities like position and momentum

376 Y. WANG

can be measured precisely, the theory of quantum me-chanics is intrinsically stochastic in a sense that we canonly make probabilistic prediction about the results ofthe measurements performed.

Quantum mechanics is mathematically described bya Hilbert space H and self-adjoint operators on H.A quantum system is completely characterized by itsstate and the time evolution of the state. A state isdefined to be a unit vector in H. Let |ψ(t)〉 be thestate of the quantum system at time t , which is alsoreferred to as a wave function. The states |ψ(t1)〉 and|ψ(t2)〉 at t1 and t2 are connected through |ψ(t2)〉 =U(t1, t2)|ψ(t1)〉, where U(t1, t2) is a unitary operatordepending only on time t1 and t2. In fact, there exists aself-adjoint operator H, which is known as the Hamil-tonian of the quantum system, such that U(t1, t2) =exp[−iH(t2 − t1)]. With Hamiltonian H, we may de-scribe the continuous time evolution of |ψ(t)〉 by theSchrödinger equation

i∂|ψ(t)〉

∂t= H|ψ(t)〉, i = √−1.(1)

Alternatively a quantum system can be described bya density operator (or density matrix). A density oper-ator ρ is an operator on H which (1) is self-adjoint;(2) is semi-positive definite; (3) has unit trace [i.e.,Tr(ρ) = 1]. Following the convention in quantum in-formation science, we reserve notation ρ for state, den-sity operator or density matrix. A state is often classi-fied as a pure state or an ensemble of pure states. A purestate is a unit vector |ψ〉 in H, which corresponds to adensity operator ρ = |ψ〉〈ψ |, and an ensemble of purestates corresponds to the case that the quantum systemis in one of states |ψj 〉, j = 1, . . . , J , with probabilitypj being in state |ψj 〉, and the corresponding densityoperator

ρ =J∑

j=1

pj |ψj 〉〈ψj |.(2)

See Griffiths (2004), Sakurai and Napolitano (2010)and Shankar (1994).

2.3 Quantum Probability

We can test the theory of quantum mechanics bychecking its predictions with experiments of perform-ing measurements on quantum systems in the labora-tory. The usual quantum measurements are on observ-ables such as position, momentum, spin, and so on,where an observable X is defined as a self-adjoint op-erator on Hilbert space H. The observable definition

is motivated from the fact that the eigenvalues of self-adjoint operators are real. Assume that an observableX has a discrete spectrum with the following diagonalform

X =p∑

a=1

xaQa,(3)

where xa are real eigenvalues of X and Qa are thecorresponding one-dimensional projections onto theorthogonal eigenvectors of X. Consider such an ob-servable in the quantum system prepared in state ρ.Measure space (�, F ) is used to describe possiblemeasurement outcomes of the observable, and the re-sult of the measurement is a random variable on (�, F )

with probability distribution Pρ . We denote by X theresult of the measurement of observable X given by(3). Then X is a random variable taking values in{x1, x2, . . . , }, and under pure state |ψ〉, the probabil-ity that measurement outcome xa occurs is defined tobe

P(a) = Pρ(X = xa)

= 〈ψ |Qa|ψ〉 = Tr(Qa|ψ〉〈ψ |), a = 1,2, . . . .

With the probability we derive the expectation underpure state |ψ〉,

Eψ(X) = ∑a

xaP (a) = ∑a

xa〈ψ |Qa|ψ〉

= 〈ψ |X|ψ〉 = Tr(X|ψ〉〈ψ |).Note the difference between an observable X which is aHermitian matrix and its measurement result X whichis a real-valued random variable.

Measuring observable X will alter the state of thequantum system (Kiefer, 2004; von Neumann, 1955).If the quantum system is prepared with initial state |ψ〉,the state of the system after the measurement result xa

is defined to beQa|ψ〉√

P(a).(4)

For an ensemble state with density operator ρ givenby (2), if the quantum state is |ψj 〉, the probability thatresult xa occurs is

P(a|j) = 〈ψj |Qa|ψj 〉 = Tr(Qa|ψj 〉〈ψj |).Applying the law of total probability, we obtain thatunder state ρ, the probability that xa occurs is equal to

P(a) = Pρ(X = xa) =J∑

j=1

pjP (a|j)

=J∑

j=1

pj Tr(Qa|ψj 〉〈ψj |) = Tr(Qaρ).


The expectation of X under state ρ,

Eρ[X] =p∑

a=1

xaPρ[X = xa] =p∑

a=1

xa Tr(Qaρ)

= tr(Xρ),

and variance

Varρ[X] = tr[X2ρ] − (tr[Xρ])2.

We may derive the density operator of the quan-tum system after obtaining the measurement result xa

by conditional probability arguments as follows. If thequantum system is in pure state |ψj 〉 before the mea-surement, the quantum state after measurement resultxa has occurred is

|ψaj 〉 = Qa|ψj 〉√

P(a|j).

If the quantum state is ρ before the measurement, afterobserving measurement outcome xa we have the fol-lowing ensemble of states: the quantum system is inpure state |ψa

j 〉 with probability P(j |a), where Bayes’stheorem shows

P(j |a) = pjP (a|j)/P (a).

Thus after measurement xa the density operator for theensemble state is given by

ρa =J∑

j=1

P(j |a)|ψaj 〉〈ψa

j |

=J∑

j=1

P(j |a)Qa|ψj 〉〈ψj |Qa

P (a|j)

=J∑

j=1

pj

Qa|ψj 〉〈ψj |Qa

P (a)= QaρQa

Tr(Qaρ).

See Holevo (1982), Parthasarathy (1992) and Sakuraiand Napolitano (2010).

2.4 Quantum Statistics

For a given quantum system, it is very important butdifficult to know its state. If we do not know in ad-vance the state of the quantum system, we may in-fer the quantum state by the measurement results ofsome observables obtained from the quantum systemand show that a certain state has been created. In statis-tical terminology, we want to estimate density matrixρ based on measurements on an often large numberof systems which are identically prepared in the stateρ. That is, after measuring observables on some identi-cal quantum systems, we can make statistical inference

about probability distribution Pρ of the measurementsand thus indirectly about density matrix ρ. In the liter-ature of quantum physics, quantum tomography is re-ferred to as the reconstruction of the underlying den-sity matrix ρ by probing identically prepared quantumsystems from some different angles. Specifically, sup-pose that we perform measurements of observables onidentically prepared quantum systems in an unknownstate ρ and obtain measurement results X1, . . . ,Xn.Assume that ρ is known up to some unknown parame-ter θ ; then X1, . . . ,Xn are i.i.d. observations with dis-tributions Pρ which depend on θ . This gives a quan-tum parametric statistical model. We may then definequantum likelihood and Fisher quantum informationand establish quantum point estimation and quantumhypothesis testing theory. Alternatively we may modelρ nonparametrically by assuming that ρ is an infinitematrix and then use nonparametric methods to estimatethe density matrix. For details see Artiles, Gill andGuta (2005), Barndorff-Nielsen, Gill and Jupp (2003),Butucea, Guta and Artiles (2007) and Nussbaum andSzkoła (2009).

3. QUANTUM COMPUTING CONCEPTS

Unlike classical computers using transistors tocrunch the ones and zeroes individually, quantum com-puters can handle both one and zero simultaneously viawhat are known as superposition quantum states. A su-perposition state is a state of matter which we maythink of as both one and zero at the same time. Quan-tum computers use the strange superposition states andquantum entanglements to do the trick of performingsimultaneous calculations and extracting the calculatedresults. The spooky phenomena of quantum entangle-ment and superposition are the key that enables quan-tum computers to be superfast and vastly outperformclassical computers.

3.1 Quantum Bit

Analogous to the fundamental concept of bit in clas-sical computation and classical information, we haveits counterpart, quantum bit, in quantum computationand quantum information. Quantum bit is called qubitfor short. Just like a classical bit with state either 0 or1, a qubit has states |0〉 and |1〉. However, the real dif-ference between a bit and a qubit is that besides states|0〉 and |1〉, a qubit may take the superposition states,

|ψ〉 = α0|0〉 + α1|1〉,where α0 and α1 are complex numbers and called am-plitudes satisfying |α0|2 + |α1|2 = 1. That is, the statesof a qubit are unit vectors in a two-dimensional com-plex vector space, and states |0〉 and |1〉 consist of an

378 Y. WANG

orthonormal basis for the space and are often referredto as computational basis states. For a classical bit wecan examine it to determine whether it is in the state0 or 1. However, for a qubit we cannot determine itsstate and find the values of α0 and α1 by examining it.The stochastic nature of quantum theory shows that wecan measure a qubit and obtain either the result 0, withprobability |α0|2, or the result 1, with probability |α1|2.Physical experiments have realized qubits as physicalobjects in different physical systems, such as the twostates of an electron orbiting a single atom, the two dif-ferent polarizations of a photon, or the alignment of anuclear spin in a uniform magnetic field. Consider thecase of atom model by corresponding |0〉 and |1〉 withthe so-called “ground” and “excited” states of the elec-tron, respectively. As the atom is shined by light withsuitable energy and for a proper amount of time, theelectron can be moved from the |0〉 state to the |1〉 stateand vice versa. Furthermore, by shortening the lengthof time shining the light on the atom, we may move theelectron initially in the state |0〉 to “halfway” between|0〉 and |1〉, say, into a state (|0〉 + |1〉)/√2.

Note that for qubit state |ψ〉 the only measurablequantities are the probabilities |α0|2 and |α1|2; since|eiθαx |2 = |αx |2, where x = 0,1, i = √−1, and θ isa real number, from the viewpoint of the qubit mea-surements, states eiθ |ψ〉 and |ψ〉 are identical. That is,multiplying a qubit state by a global phase factor eiθ

bears no observational consequence.Note the distinction between superposition states

and probability mixtures (or ensemble of pure statesdefined in Section 2.1). Consider superposition (|0〉 +|1〉)/√2 as a pure state. Its density matrix is given by

12(|0〉 + |1〉)(〈0| + 〈1|)

= 12(|0〉〈0| + |1〉〈1|) + 1

2(|0〉〈1| + |1〉〈0|),while the first term on the right-hand side of the aboveequation corresponds to the ensemble of pure states |0〉and |1〉, that is, a probabilistic mixture of states |0〉 and|1〉 with equal probability.

Similar to classic bits, we can define multiplequbits. The states of b qubits are unit vectors in a 2b-dimensional complex vector space with 2b computa-tional basis states of the form |x1x2 · · ·xb〉, xj = 0 or1, j = 1, . . . , b. For example, the states of two qubitsare unit vectors in a four-dimensional complex vec-tor space, with four computational basis states labeledby |00〉, |01〉, |10〉 and |11〉. The computational ba-sis states |00〉, |01〉, |10〉 and |11〉 generate the four-dimensional complex vector space, and the superposi-tion states are all unit vector in the space with the forms

|ψ〉 = α00|00〉 + α01|01〉 + α10|10〉 + α11|11〉,

where amplitudes αx are complex numbers satisfying|α00|2 + |α01|2 + |α10|2 + |α11|2 = 1. As in the singlequbit case, when two qubits are measured we get resultx being one of 00,01,10,11, with probability |αx |2.Moreover, we may measure just the first qubit of thetwo-qubit system and obtain either the result 0, withprobability |α00|2 + |α01|2, or the result 1, with proba-bility |α10|2 + |α11|2. As quantum measuring changesthe quantum state, if the measurement result on the firstqubit is 0, after the measurement the qubits are in thestate

α00|00〉 + α01|01〉√|α00|2 + |α01|2

.(5)

A qubit is the simplest quantum system. The quan-tum system of b qubits is described by a 2b-dimension-al complex vector space with each superposition statespecified by 2b amplitudes. As 2b increases exponen-tially in b, it is very easy for such a system to havean enormously big vector space. A quantum systemwith even a few dozens of “qubits” will strain the re-sources of even the largest supercomputers. Considera quantum system of 50 qubits. 250 ≈ 1015 complexamplitudes are needed to specify its quantum states.With 128 bits of precision, it requires approximately32 thousand terabytes of information to store all 1015

complex amplitudes. Such storage capacity may beavailable in future supercomputers. For a quantum sys-tem with b = 500 qubits we need to specify 2500 com-plex amplitudes for its states. It is unimaginable tostore all 2500 complex numbers in any classical com-puters. In principle, a quantum system with only a fewhundred atoms can manage such an enormous amountof data and execute calculations as the system evolves.Quantum computation and quantum information are tofind ways to utilize the immense potential computa-tional power in quantum systems.

3.2 Quantum Circuit Model

As a classical computer is built from an electri-cal circuit consisting of wires for carrying informa-tion around the circuit and logic gates for perform-ing simple computational tasks, a quantum computercan be created from a quantum circuit with quantumgates to perform quantum computation and manipulatequantum information. A number of physical systemsare being investigated for building quantum comput-ers. These include optical photon, optical cavity quan-tum electrodynamics, ion traps, nuclear magnetic res-onance with molecules, quantum dots, and supercon-ductors (Nielsen and Chuang, 2000). In fact, prim-


itive solid-state quantum processors have been cre-ated in research laboratories to run quantum algorithms(DiCarlo et al., 2009; Johnson et al., 2011; Mariantoniet al., 2011; Sayrin et al., 2011). The circuit modelis particularly important in quantum computation andquantum information, and a quantum computer is oftensynonymous with the quantum circuit model. A quan-tum circuit operates on b qubits for some integer b.The state takes a form of |x1 · · ·xb〉, with state spacebeing a 2b-dimensional complex Hilbert space. Whenxi = 0 or 1, states |x1 · · ·xb〉 are the computational ba-sis states of the quantum computer and often written as|x〉, where x is the integer with binary representationx1 · · ·xb.

As a classical logic gate converts classical bits fromone form to another such as 0 → 1 and 1 → 0, a quan-tum gate operates on qubits. Quantum mechanics dic-tates that quantum gates operating on b qubits are 2b

by 2b unitary matrices on the 2b-dimensional Hilbertspace. For example, a Hadamard gate on one qubitis the 2 × 2 unitary matrix that realizes the followingtransformation:

|0〉 → |0〉 + |1〉√2

, |1〉 → |0〉 − |1〉√2

.

Consider another important gate on two qubits which iscalled control-NOT gate. It takes the two input qubitsas control qubit and target qubit, respectively, and theoutput target qubit of the gate retains the input targetqubit if the control qubit is |0〉 and is flipped if the con-trol qubit is |1〉, that is,

|00〉 → |00〉, |01〉 → |01〉,|10〉 → |11〉, |11〉 → |10〉.

Generally for any single qubit unitary operation U, acontrol-U gate is a two-qubit gate, with one controlqubit and one target qubit. If the control qubit is |1〉,U is applied to the target qubit; if the control qubit is|0〉, the target qubit is left alone, that is,

|0〉|0〉 → |0〉|0〉, |0〉|1〉 → |0〉|1〉,|1〉|0〉 → |1〉U|0〉, |1〉|1〉 → |1〉U|1〉.

If f (x) maps {0,1}b onto {0,1}, we define a unitarytransformation Uf that operates on b + 1 qubit state

|x, y〉 → |x, y ⊕ f (x)〉,(6)

where x = x1 · · ·xb with xj = 0 or 1 is the data register,y = 0 or 1 is the target register, ⊕ denotes additionalmodulo 2. If y = 0, after the transformation Uf , thestate of the last qubit is the value of f (x).

3.3 Entanglement

Quantum entanglement is one of the most mind-bending creatures known to science. It is referred toas the phenomenon that two qubits behave like twinsthat are connected by an invisible wave to share eachother’s properties.

3.3.1 Bell states. Consider a quantum gate on two-qubit basis states |00〉, |01〉, |10〉 and |11〉 that is com-posed of a Hadamard gate on the first qubit and then isfollowed by a control-NOT gate. The output states ofthe gate are as follows:

|00〉 → |00〉 + |11〉√2

, |01〉 → |01〉 + |10〉√2

,

|10〉 → |00〉 − |11〉√2

, |11〉 → |01〉 − |10〉√2

.

Physicists Bell, Einstein, Podolsky and Rosen discov-ered the amazing properties of these four states, whichare often referred to as the Bell states, EPR states orEPR pairs (Bell, 1964; Einstein, Podolsky and Rosen,1935). In general states such as these four states thatcannot be expressed as products of some single qubitsare called entangled states. Entangled states, which arenot fully understood in quantum physics, have remark-able properties.

For the two-qubit system consider a Bell state

|ψ〉 = |01〉 − |10〉√2

,

and an observable

M = axσ x + ayσ y + azσ z,

where (ax, ay, az) is a real unit vector (i.e., a2z + a2

y +a2z = 1), and σ x , σ y and σ z are Pauli matrices given by

σ x =(

0 11 0

), σ y =

(0 −i

i 0

),

(7)

σ z =(

1 00 −1

), i = √−1.

It is easy to show that M has eigenvalues ±1 for anyreal unit vector (ax, ay, az). If measuring observableM on each qubit of |ψ〉, we will obtain a measure-ment result of +1 or −1. Surprisingly, no matter whatchoice of (ax, ay, az), the measurement results on thetwo qubits are always opposite of each other, that is,when the first qubit measurement is −1, then the sec-ond qubit measurement will be +1, and vice versa.

The two-qubit system can be realized by the spins oftwo particles, and the measurement of M is referred to

380 Y. WANG

as a measurement of spin along the (ax, ay, az) axis.After the two-particle system is prepared in the Bellstate |ψ〉, the two particles drift far apart. Alice mea-sures the spin of the first particle and Bob measuresthe spin of the second particle. The above oppositemeasurement phenomenon corresponds to that due tothe entangled state |ψ〉, if Alice gets a result +1 fromher spin measurement on the first particle, then thestate of the system immediately jumps to the untangledstate so that the second particle now has definite spinstate and Bob’s spin measurement on the second parti-cle gives definite result −1. This phenomenon is oftenreferred to as anti-correlation in entanglement experi-ments (Neumann et al., 2008; Sakurai and Napolitano,2010).

The mathematical arguments for the anti-correlationphenomenon are as follows. The measurement of M onthe first (or second) qubit of |ψ〉 corresponds to the spinmeasurement of Alice’s (or Bob’s) particle along the(ax, ay, az) axis in the above two-particle spin model.From Sections 2.3 and 3.1 we have that the two-qubitsystem is described by the Bell state |ψ〉 in C

4; mea-suring M on the first qubit of |ψ〉 means performingmeasurement on observable M⊗I in the Bell state |ψ〉,which alters the quantum state of the two-qubit system;measuring M on the second qubit corresponds to mea-suring observable I ⊗ M in the altered quantum state,where I is the 2 by 2 identity matrix, and M ⊗ I andI ⊗ M are matrix tensor products.

Denote by |ϕ±〉 the two orthonormal eigenvectors ofM corresponding to eigenvalues ±1, respectively, andlet Q± be the respective projections onto the eigenvec-tors |ϕ±〉. Following (3)–(5) we have a diagonal repre-sentation M = Q+−Q−; when we measure observableM on each qubit, the possible measurement results are±1; measuring M on the first qubit changes the state ofthe two-qubit system, and after the measurement result±1 on the first qubit, the post-measurement state of thetwo-qubit system is Q± ⊗ I|ψ〉/‖Q± ⊗ I|ψ〉‖. Belowwe will evaluate the post-measurement state and showthat measuring I⊗M in the post-measurement state al-ways yields measurement results opposite to the mea-surement results on the first qubit.

Since (|0〉, |1〉) and (|ϕ+〉, |ϕ−〉) are two bases forthe one-qubit system in C

2, then( |0〉|1〉

)=

[α11 α12α21 α22

]( |ϕ+〉|ϕ−〉

),

where (αj�) forms a 2 × 2 unitary matrix with deter-minant equal to a phase factor eiθ (i = √−1) for some

real θ . Substituting the above expressions into the en-tangled state and ignoring a global phase factor eiθ

(which has no effects on measurement results; see Sec-tion 3.1), we obtain

|ψ〉 = |01〉 − |10〉√2

= eiθ |ϕ+ϕ−〉 − |ϕ−ϕ+〉√2

(8)

∼ |ϕ+ϕ−〉 − |ϕ−ϕ+〉√2

.

From the definitions of |ϕ±〉 and Q±, Q+ ⊗ I|ϕ+ϕ−〉 =|ϕ+ϕ−〉, Q− ⊗ I|ϕ−ϕ+〉 = −|ϕ−ϕ+〉, Q+ ⊗ I|ϕ−ϕ+〉 =0, and Q− ⊗ I|ϕ+ϕ−〉 = 0. If the measurement result ofM on the first qubit is +1 (or −1), from (8) we obtainthe post-measurement state of the two-qubit system asfollows:

Q+ ⊗ I|ψ〉‖Q+ ⊗ I|ψ〉‖

= eiθ Q+ ⊗ I|ϕ+ϕ−〉 − Q+ ⊗ I|ϕ−ϕ+〉√2‖Q+ ⊗ I|ψ〉‖

= eiθ |ϕ+ϕ−〉‖|ϕ+ϕ−〉‖ = eiθ |ϕ+ϕ−〉 ∼ |ϕ+ϕ−〉

(or Q− ⊗ I|ψ〉/‖Q− ⊗ I|ψ〉‖ = eiθ |ϕ−ϕ+〉 ∼ |ϕ−ϕ+〉).Since

I ⊗ M|ϕ+ϕ−〉 = I ⊗ Q+|ϕ+ϕ−〉 − I ⊗ Q−|ϕ+ϕ−〉= −|ϕ+ϕ−〉,

I ⊗ M|ϕ−ϕ+〉 = |ϕ−ϕ+〉,that is, the post-measurement state |ϕ+ϕ−〉 (or |ϕ−ϕ+〉)is the eigenvector of I ⊗ M corresponding to eigen-value −1 (or +1), performing measurement on I ⊗ Min the post-measurement state must always yield mea-surement result −1 (or +1). Thus, the measurement re-sults of M on the two qubits of |ψ〉 are always oppositeto each other.

3.3.2 Quantum teleportation. Quantum teleporta-tion is a process by which we can transfer the stateof a qubit from one location to another, without trans-mitting it through the intervening space. We illustratethe phenomenon as follows. Alice and Bob togethergenerated a Bell state long ago. Each took one qubit ofthe Bell state when they split. Now they are far awayfrom each other. The mission for Alice is to delivera qubit |ψ〉 to Bob, while he is hiding, and she canonly send classical information to Bob but does notknow the state of the qubit |ψ〉. Quantum teleporta-tion is a way that Alice utilizes the entangled Bell stateto send a qubit of unknown state to Bob, with only a


small overhead of classical communication. Recently abreakthrough in quantum teleportation has been madeby successfully transferring complex quantum data in-stantaneously from one place to another, paving theway for real-world applications of quantum communi-cations (Lee et al., 2011).

Here is how it works. Alice interacts the qubit |ψ〉 tobe teleported with her half of the Bell state, and thenperforms a measurement on the two interacted qubitsto obtain one of four possible two-classical-bit results:00,01,10 and 11. She sends the two-bit informationvia classical communication to Bob. Depending on Al-ice’s classical message, Bob performs one of four op-erations on his half of the Bell state. Surprisingly, thedescribed procedure allows Bob to recover the originalstate |ψ〉.

Specifically assume that the state to be teleported is|ψ〉 = α0|0〉 + α1|1〉, where α0 and α1 are unknownamplitudes. First, consider a three-qubit state

|ϕ0〉 = |ψ〉 |00〉 + |11〉√2

= 1√2[α0|0〉(|00〉 + |11〉) + α1|1〉(|00〉 + |11〉)],

where the first two qubits (on the left) belong to Alice,and the third qubit to Bob. Note that Alice’s secondqubit and Bob’s third qubit are from the entangled Bellstate. Second, Alice applies a control-NOT gate to herqubits in |ϕ0〉 and obtains

|ϕ1〉 = 1√2[α0|0〉(|00〉 + |11〉) + α1|1〉(|10〉 + |01〉)].

Third, she applies a Hadamard gate to the first qubit in|ϕ1〉 and gets

|ϕ2〉 = 12 [α0(|0〉 + |1〉)(|00〉 + |11〉)

+ α1(|0〉 − |1〉)(|10〉 + |01〉)].We regroup the terms of |ϕ2〉 and rewrite it as follows:

|ϕ2〉 = 12 [|00〉(α0|0〉 + α1|1〉) + |01〉(α0|1〉 + α1|0〉)

+ |10〉(α0|0〉 − α1|1〉)+ |11〉(α0|1〉 − α1|0〉)].

The new expression has four terms, and each term hasAlice’s qubits in one of four possible states |00〉, |01〉,|10〉 and |11〉, and Bob’s qubit is in the state related tothe original state |ψ〉. If Alice performs a measurementon her qubits and informs Bob of the measurement re-sult, then his post-measurement state is completely de-termined. For example, the first term has Alice’s qubits

in the state |00〉 and Bob’s qubit in state |ψ〉. There-fore, if Alice’s measurement result on her qubits is 00,then Bob’s qubit will be in state |ψ〉. Below is a list ofBob’s four post-measurement states corresponding tothe results of Alice’s measurements:

00 → α0|0〉 + α1|1〉, 01 → α0|1〉 + α1|0〉,10 → α0|0〉 − α1|1〉, 11 → α0|1〉 − α1|0〉.

As Alice’s measurement outcome on her qubits is oneof 00,01,10 and 11, depending on her measurementoutcome Bob’s qubit will be one of the above four pos-sible states. Once Alice sends to Bob her two-classical-bit measurement outcome through a classical chan-nel, he applies appropriate quantum gates to his stateand recovers |ψ〉. For example, if her measurement is00, Bob’s state is |ψ〉, and he does not need to applyany quantum gate. If her measurement is 01, then Bobneeds to apply a σ x gate to his state α0|1〉 + α1|0〉 andyields |ψ〉. If her measurement is 10, then applying aσ z gate to his state α0|0〉 − α1|1〉 Bob recovers |ψ〉. Ifher measurement is 11, then Bob can fix up his stateα0|1〉 − α1|0〉 to recover |ψ〉 by applying first a σ x

gate and then a σ z gate. Here the σ x and σ z gates aredefined by Pauli matrices σ x and σ z given by (7). Insummary, according to Alice’s measurement outcome,applying some appropriate quantum gates to his qubitBob will recover the state |ψ〉.

A few important remarks about quantum teleporta-tion are in the line. First, quantum teleportation doesnot involve any transfer of matter or energy. Alice’sparticle has not been physically moved to Bob; onlyits state has been transferred. Second, after the tele-portation Bob’s qubit will be on the teleported state,while Alice’s qubit will become some undefined partof an entangled state. In other words, what the telepor-tation does is that a qubit was destroyed in one placebut instantaneously resurrected in another. Teleporta-tion does not copy any qubits, and hence is consistentwith the no-cloning theorem (which forbids the cre-ation of identical copies of an arbitrary unknown quan-tum state; see Wootters and Zurek, 1982). Third, in or-der to teleportate a qubit, Alice has to inform Bob ofher measurement by sending him two classical bits ofinformation. These two classical bits do not carry com-plete information about the qubit being teleported. Ifthe two bits are intercepted by an eavesdropper, he orshe may know exactly what Bob needs to do in order torecover the desired state. However, this information isuseless if the eavesdropper cannot interact with the en-tangled particle in Bob’s possession. Also the require-ment of sending two bits of information via classical

382 Y. WANG

channel prevents quantum teleportation from transmit-ting information faster than the speed of light.

3.3.3 Bell’s inequality. The Bell test experimentsare designed to investigate the validity of the entan-glement effect in quantum mechanics through Bell’sinequality. Over the past four decades many physi-cal experiments on quantum systems were conductedto check the validity of Bell’s inequality and resultedin some violation of the inequality. For example, As-pect, Grangier and Roger (1981, 1982a, 1982b) pro-vided overwhelming support to the violation of Bell’sinequality. The experimental results are often invokedas the proof of quantum non-locality and lack of real-ism that no particle has definite form until it is mea-sured and measuring a quantum entity can instanta-neously influence another far away. See Aspect, Grang-ier and Roger (1981, 1982a, 1982b), Bohm (1951),Bell (1964), Clauser et al. (1969) and Einstein, Podol-sky and Rosen (1935). Below we describe the CHSHversion of the Bell’s inequality (Clauser et al., 1969).

Suppose Xi , i = 1,2,3,4, are four random variablestaking values ±1. Consider an ordinary experimentwith two people, Alice and Bob. In the experiment Al-ice observes X1 or X2 while Bob measures X3 or X4.Consider the quantity X1X3 +X2X3 +X2X4 −X1X4.It is equal to

(X1 + X2)X3 + (X2 − X1)X4 = ±2 ≤ 2.

Regardless of the distributions of Xi , taking expecta-tion on both sides of the above inequality we arrive atthe famous Bell inequality,

E(X1X3) + E(X2X3) + E(X2X4)(9)

− E(X1X4) ≤ 2.

The violation of Bell’s inequality demonstrates en-tanglement effect in quantum mechanics. In fact, quan-tum experiments yield a quantum version of the in-equality. Consider that a quantum system of two qubitsis prepared in a Bell state

|ψ〉 = |01〉 − |10〉√2

.

Alice takes the first qubit of |ψ〉 while Bob gets itssecond qubit. Define four observables with eigenvalues±1,

X1 = σ z, X2 = σ x,

on the first qubit and

X3 = −σ z + σ x√2

, X4 = σ z − σ x√2

,

on the second qubit, where σ x and σ z are Pauli matri-ces given by (7). Again Alice performs measurementson X1 or X2 while Bob measures X3 or X4. The quan-tum expectations of X1X3, X2X3, X2X4, X1X4 in thestate |ψ〉 are calculated below:

Eψ(X1X3) = 1√2, Eψ(X2X3) = 1√

2,

Eψ(X2X4) = 1√2, Eψ(X1X4) = − 1√

2.

Here the observable product is in the sense of tensorproduct. Thus we obtain a value in the quantum frame-work for the analog quantity on the left-hand side ofthe Bell’s inequality (9)

Eψ(X1X3) + Eψ(X2X3) + Eψ(X2X4)

− Eψ(X1X4) = 2√

2,

which exceeds 2 and hence violates the Bell’s inequal-ity. In fact, the quantum version of the Bell’s inequal-ity is the Tsirelson’s inequality (Tsirelson, 1980) whichshows that in any quantum state ρ,

Eρ(X1X3) + Eρ(X2X3) + Eρ(X2X4)(10)

− Eρ(X1X4) ≤ 2√

2.

3.4 Quantum Parallelism

Quantum computation has an amazing featuretermed as quantum parallelism, which may be heuristi-cally explained by the following oversimplifying de-scription: a quantum computer can simultaneouslyevaluate the whole range of a function f (x) at manydifferent values of x.

For function f (x) with b bit input x = x1 · · ·xb and1 bit output f (x), we illustrate quantum parallel eval-uation of its values at many different x simultaneouslyas follows. First we apply b Hadamard gates to the firstb qubits of |0 · · ·0〉|0〉 to obtain

|0〉 + |1〉√2

· · · |0〉 + |1〉√2

|0〉 = 1√2b

∑x

|x〉|0〉,

x = x1 · · ·xb, xj = 0,1,

where the sum is over all possible 2b values of x. Sec-ond, apply quantum circuit Uf defined in (6) to theobtained b + 1 qubit state to yield

1√2b

∑x

|x〉|f (x)〉.

The quantum circuit with b Hadamard gates is ex-tremely efficient in producing an equal superposition


of all 2b computational basis states with only b gates;and quantum parallelism enables simultaneous evalua-tion of the whole range of the function f , although weevidently evaluate f just once with single quantum cir-cuit Uf applied to the superposition state. To make itmore clear we consider the case of b = 1. Apply circuitUf to a superposition state as follows:

Uf

( |0〉 + |1〉√2

|0〉)

= Uf

( |00〉 + |10〉√2

)

= |0f (0)〉 + |1f (1)〉√2

.

One application of a single circuit Uf results in a su-perposition state whose two components contain infor-mation about both f (0) and f (1), as if we have evalu-ated f (x) at values 0 and 1 simultaneously. The quan-tum parallelism is in contrast with classical parallelism,where multiple circuits each built to compute one valueof f (x) are executed simultaneously. Quantum paral-lelism arises from superposition states. A superposi-tion state has many components, each of which maybe thought of as a single argument to function f (x).Because of quantum nature, a single circuit Uf appliedonce to the superposition state is actually performed oneach of the components of the superposition, and thewhole range of the values of function f (x) is stored inthe resulted outcome superposition state.

The quantum parallelism can be a potentially pow-erful tool for computational statistics. For example,Bayesian analysis often encounters the problems ofevaluating sums over 2b quantities, with b proportionalto sample size or the number of variables. For moder-ate to large b, the evaluation of such sums is computa-tionally prohibitive by classical computers (Vidakovic,1999). Because of the quantum parallelism, it is possi-ble for quantum computers to perform such computingtasks.

4. QUANTUM ALGORITHMS

Quantum algorithms are described by quantum cir-cuits that take input qubits and yield output measure-ments for the solutions of the given problems. As aclassical algorithm is a step-by-step problem-solvingprocedure, with each step performed on a classicalcomputer, a quantum algorithm is a step-by-step pro-cedure to solve a problem, with each step executed bya quantum computer. Although all classical algorithmscan also be carried out on a quantum computer, we re-fer to quantum algorithms as the algorithms that uti-lize essential quantum features such as quantum super-position and quantum entanglement. While it is true

that all problems solvable on a quantum computer aresolvable on a classical computer, and problems unde-cidable by classical computers remain undecidable onquantum computers, what makes quantum algorithmsexciting is the faster speed that quantum algorithmsmight be able to achieve, compared to classical algo-rithms, for solving some tough problems. The well-known quantum algorithms are Shor’s factoring algo-rithm and Grover’s search algorithm. Shor’s algorithmand Grover’s algorithm run, respectively, exponentiallyfaster and quadratically faster than the best known clas-sical algorithms for the same tasks. Common tech-niques used in quantum algorithms include quantumFourier transform, phase estimation and quantum walk.

4.1 Quantum Fourier Transform

The quantum Fourier transform is defined to be a lin-ear transformation on n qubits that maps the computa-tional basis states |j〉, j = 0,1, . . . ,2n − 1, to superpo-sition states as follows:

|j〉 −→ 1√2n

2n−1∑k=0

e2πijk/2n |k〉, i = √−1.

The inverse of quantum Fourier transform is given by

|k〉 −→ 1√2n

2n−1∑j=0

e−2πijk/2n |j〉.

We use the binary representation to express the statej = j12n−1 + j22n−2 + · · · + jn20 as j = j1j2 · · · jn

and represent binary fraction j�/2 + j�+1/22 + · · · +jm/2m−�+1 as 0.j�j�+1 · · · jm, where 1 ≤ � ≤ m ≤ 2n.Then the quantum Fourier transform of state |j〉 =|j1j2 · · · jn〉 has the following useful product represen-tation:

|j1j2 · · · jn〉 → 1

2n/2 (|0〉 + e2πi0.jn |1〉)· (|0〉 + e2πi0.jn−1jn |1〉) · · ·· (|0〉 + e2πi0.j1j2···jn |1〉).

It can be easily checked from the product represen-tation that with quantum parallelism the quantumFourier transform can be realized as a quantum circuitwith only O(n2) operations, while classically the fastFourier transform requires O(n2n) operations for pro-cessing 2n data, which indicates an exponential speed-up (Nielsen and Chuang, 2000). Realizing such anexponential saving accommodated by quantum paral-lelism requires clever measurement schemes. Success-ful examples include quantum phase estimation andShor’s algorithms for factoring and discrete logarithm.

384 Y. WANG

4.2 Phase Estimation

Quantum algorithms are of random nature in thesense that they are able to produce correct answers onlywith some probabilities. Consider quantum phase esti-mation which provides the key to many quantum algo-rithms. Assume that a unitary operator U has an eigen-vector |x〉 with eigenvalue e2πiϕ . The phase ϕ of theeigenvalue is unknown and the goal of the phase es-timation algorithm is to estimate ϕ based on the as-sumption that the state |x〉 can be prepared and thecontrolled-U2j

operations [see Section 3.2 for controlgate] can be performed for suitable nonnegative inte-gers j .

The registers are used in phase estimation. The firstregister consists of b qubits initially in the state |0〉.The second register starts in the state |x〉 and involvesenough qubits to store |x〉. The phase estimation pro-cedure is performed in two stages.

First, we apply Hadamard transform to the first reg-ister and then controlled-U operations on the secondregister, with U raised to successive powers of 2, to ob-tain the final state with the second register unchangedand the first register given by

1

2b/2 (|0〉 + e2πi2b−1ϕ|1〉)

· (|0〉 + e2πi2b−2ϕ|1〉) · · · (|0〉 + e2πi20ϕ|1〉)(11)

= 1

2b/2

2b−1∑k=0

e2πiϕk|k〉.

If ϕ is expressed exactly in b bits as ϕ = 0.ϕ1 · · ·ϕb,(11) becomes

1

2b/2 (|0〉 + e2πi0.ϕb |1〉)· (|0〉 + e2πi0.ϕb−1ϕb |1〉) · · ·(12)

· (|0〉 + e2πi0.ϕ1ϕ1···ϕb |1〉),which is the quantum Fourier transform of the productstate |ϕ1ϕ2 · · ·ϕb〉.

The second stage of phase estimation is to take theinverse quantum Fourier transform on the first register.For ϕ = 0.ϕ1 · · ·ϕb, the output state from the secondstage is |ϕ1ϕ2 · · ·ϕb〉, and a measurement in the com-putational basis yields ϕ1 · · ·ϕb and dividing the mea-surement by 2b gives ϕ1 · · ·ϕb/2b = 0.ϕ1 · · ·ϕb = ϕ.We obtain a perfect estimate of ϕ.

Now we consider the case that ϕ cannot be expressedexactly with a b bit binary expansion. Take 0 ≤ η <

2b to be the integer that its binary fraction η/2b =

0.η1η2 · · ·ηb is the first b bit representation in the bi-nary expansion of ϕ, which satisfies 0 ≤ ϕ − η/2b ≤2−b.

Perform the inverse quantum Fourier transform onthe first register given by (11), which is obtained in thefirst stage results, and get

1

2b

2b−1∑k,�=0

e−2πik�/2b

e2πiϕk|�〉 =2b−1∑�=0

β�|�〉,

where amplitudes of |(η + �)(mod 2b)〉 are

β� = 1

2b

2b−1∑k=0

{e2πi[ϕ−(η+�)/2b]}k

= 1

2b

(1 − e2πi(2bϕ−η−�)

1 − e2πi(ϕ−η/2b−�/2b)

).

Assume that the result of the final measurement fromphase estimation is η and dividing the result by 2b givesϕ = η/2b. Let ζ be the specified accuracy for the phaseestimation procedure. By adding up |β�|2 with � beingwithin ζ2b, we bound the probability that the obtainedϕ is within ζ from ϕ:

P(|ϕ − ϕ| ≤ ζ ) ≥ P(|η − η| ≤ ζ2b − 1)

≥ 1 − 1

2(ζ2b − 2).

For ε > 0, set

b =[log2

(1

ζ

)]+

[log2

(2 + 1

2ε

)].(13)

Then P(|ϕ − ϕ| ≤ ζ ) ≥ 1 − ε, that is, with probabil-ity at least 1 − ε the phase estimation procedure cansuccessfully produce ϕ within ζ from the true ϕ. SeeNielsen and Chuang (2000).

4.3 Statistical Analysis

The phase estimation algorithm requires b qubits forthe first register to achieve [− log2 ζ ] bit accuracy andsuccess probability 1 − ε. With accuracy fixed, to in-crease the success probability the required qubits

b ∼ 1 − log2 ζ + 1

4ε log 2,

which grows at a very fast rate. For example, an in-crease in success probability from 90% to 99% re-quires eighteen times of qubit increase compared to thechange from 80% to 90%.

Quantum algorithms are of random nature in thesense that they often produce correct answers only with


certain probabilities. The success probabilities dependupon the schemes of the algorithms as well as the con-text of applications. Given a quantum algorithm forsolving a problem, a common practice is to repeatedlyrun the quantum algorithm to achieve high probabilityof successfully obtaining a correct answer. Considerthat the phase estimation procedure is repeatedly runn times to obtain results ϕ1, . . . , ϕn. Then ϕ1, . . . , ϕn

may be treated as i.i.d. random variables with each ϕj

satisfying

P(|ϕj − ϕ| > ζ) ≤ ε.

We may statistically model ϕj by the gross error model(Huber and Ronchetti, 2009) as follows. Assume thatϕj are independently and identically generated from(1 − ε)F (x) + εH(x), where F(x) is the distributionof the correct answers that are within ζ from true ϕ,and H(x) is the distribution of wrong answers that areat least ζ away from true ϕ. Then ϕj are correct withprobability 1 − ε and incorrect with probability ε.

If the outcome result of the algorithm is verifiable tobe a correct answer or not [as in the case of Shor’s al-gorithms for factoring and order-finding in Section 4.4below], the obtained result from each run is checked tobe a correct answer or not. Then the number of timesrequired to run the algorithm for obtaining a correct an-swer follows a geometric distribution. Thus the proba-bility that we obtain a correct answer in n repetitions isequal to

P(obtain a correct answer in n trials)

= 1 − P(no success in the n trials) = 1 − εn.

Since εn goes to zero geometrically fast, we maychoose a moderate ε with fewer qubits to achieve veryhigh probability of successfully obtaining a correctanswer by repeatedly running the algorithm enoughtimes.

On the other hand, if the outcome result is not ver-ifiable to be a correct answer or not [as in the caseof phase estimation], careful analysis is needed to de-sign ways for obtaining a correct answer with veryhigh probability. As wrong answers are far away fromtrue ϕ, estimators like sample average of ϕ1, . . . , ϕn

may not estimate ϕ well. We adopt a robust statisti-cal method to estimate ϕ by α-trimmed mean ϕ, whichis defined as follows. Ordering ϕ1, . . . , ϕn and then re-moving [nα] largest and [nα] smallest ones, we takethe average of the remaining ϕj as α-trimmed mean,where α is chosen to be greater than ε/2. One exam-ple is the sample median of ϕ1, . . . , ϕn. The probability

that ϕ is within ζ from ϕ can be calculated from the bi-nomial probability as follows:

P(|ϕ − ϕ| ≤ ζ ) ≥ P(more than n(1 − 2α) number

of ϕj are within ζ from ϕ)

=n∑

k=[n(1−2α)]−1

(n

k

)(1 − ε)kεn−k.

As n → ∞, nε approaches to infinity. The binomialprobability can be approximated by resorting to a nor-mal approximation, yielding

n∑k=[n(1−2α)]−1

(n

k

)(1 − ε)kεn−k

∼ 1 − �

(√n(ε − 2α)√ε(1 − ε)

)= �

(√n(2α − ε)√ε(1 − ε)

),

where �(·) is the standard normal distribution func-tion. Since 2α − ε > 0, as n increases, P(|ϕ − ϕ| ≤ ζ )

approaches to 1 exponentially fast. Combining the twocases together, we arrive at the following theorem.

THEOREM 1. Suppose that the outcome ϕ of aquantum algorithm obeys the gross error model thatwith probability 1− ε it produces a correct answer andprobability ε it gives a wrong answer. Then by repeat-edly running the quantum algorithm we will obtain acorrect answer with probability approaching 1 expo-nentially fast in the number of repetitive runs.

For a quantum algorithm that produces a correct an-swer with probability 70% and α = 0.2, in order to ob-tain a correct answer with 0.999 probability we need torun the algorithm five times and 20 times, respectively,for the cases that the outcome results are verifiable andnot verifiable.

4.4 Factoring and Order-Finding Algorithms

The factoring problem is to find all prime factors ofa given positive composite number such that the prod-uct of these prime numbers is equal to the compositenumber. Factoring is known to be a very hard problemfor classical computers. Shor (1994, 1997) developeda quantum algorithm for the factoring problem that isexponentially faster than the most efficient known clas-sical factoring algorithm.

Shor’s quantum algorithms work as follows. Math-ematically the factoring problem is equivalent to theorder-finding problem that for two positive integers x

and N , x < N , with no common factors, find the small-est integer r such that dividing xr by N we obtain a

386 Y. WANG

reminder 1 (Shor, 1997; Nielsen and Chuang, 2000).The quantum algorithm for factoring is reduced to aquantum algorithm for order-finding. The quantum al-gorithm for order-finding is to apply the phase estima-tion algorithm to the unitary operator

U|y〉 = |xy(modN)〉.The eigenvectors of U are

|us〉 = 1√r

r−1∑k=0

exp(−2πisk

r

)|xk modN〉,

s = 0,1, . . . , r − 1, i = √−1,

with corresponding eigenvalues exp(2πis/r). Usingthe phase estimation algorithm we can obtain theeigenvalues exp(2πis/r) with high accuracy and thusfind the order r with certain probability.

While the quantum factoring algorithm can accom-plish the task of factoring an n-bit integer with op-erations of order n2 logn log logn, the current bestknown classical algorithm requires operations of orderexp(n1/3 log2/3 n) to factor an n-bit composite number(Crandall and Pomerance, 2001). Note that the num-ber of operations required in the best classical algo-rithm grows exponentially in the size of the numberbeing factored. Because of the exponential complex-ity, the factoring problem is generally regarded as anintractable problem on classical computers.

The factoring problem plays an important role incryptography. Cryptography is to enable two parties,Alice and Bob, to communicate privately, while itis very difficult for the third parties to “eavesdrop”on the contents of the communications. Examples in-clude ATM cards, computer passwords, internet com-mences, clandestine meetings and military communi-cations. Two cryptographic protocols used in the com-munications are private key cryptosystem and publickey cryptosystem. A private key cryptosystem requiresthe two communicating parties to share a private key.Alice uses the key to encrypt the information, sendsthe encrypted information to Bob who uses the keyto decrypt the received information. The severe draw-back of the private key cryptosystem is that the par-ties have to safeguard the key transmission from beingeavesdropped. A public key cryptosystem invented inthe 1970s requires no sharing secret key in advance.Bob publishes a “public key” available to the gen-eral public, and Alice uses the public key to encryptinformation and sends the encrypted information toBob. The encryption transformation is specially cre-ated such that with only the public key, it is extremely

difficult, though not impossible, to invert the encryp-tion transformation. When publishing the public keyBob keeps a matched secret key for easy inversion ofthe encryption transformation and decryption of the re-ceived information. One of the most widely used pub-lic key cryptosystems is the RSA cryptosystem, whichis named after its creator Rivest, Shamir and Adleman(Menezes, van Oorschot and Vanstone, 1996; Rivest,Shamir and Adleman, 1978). RSA is built on the math-ematical asymmetry of factoring: it is easy to multi-ply large prime numbers and obtain their product as acomposite number but hard to find the prime factorsof a given large composite number. RSA encryptionkeeps the large primes as a secret key and uses theirproduct to make a “public key.” Because of its expo-nential complexity, tremendous efforts tried to breakthe RSA system so far have resulted in vain, and thereis a widespread belief that the RSA system is secureagainst any classical computer based attacks. As thefactoring problem can be efficiently solved by Shor’squantum factoring algorithm, a quantum computer canbreak the RSA system easily. Fortunately, while quan-tum mechanics takes away with one hand, it gives backwith the other. A quantum procedure known as quan-tum cryptography or quantum key distribution can dokey distribution so that the communication securitycannot be compromised. The idea is based on the quan-tum principle that observing a quantum system willdisturb the system being observed. If there is an eaves-dropper during the transmission of the quantum key be-tween Alice and Bob, eavesdropping will disturb thequantum communication channel that is used to estab-lish the key, and the disturbance will make eavesdrop-ping visible. Alice and Bob will throw away the com-promised key and keep only the secured key for theircommunication.

4.5 Quantum Search Algorithm

Suppose that you would like to find the name corre-sponding to a given phone number in a telephone direc-tory; or suppose that there are some locations in a givencity you would like to visit and wish to find the shortestroute passing through all the locations. If there are N

names in the telephone directory or N possible routesto pass through all the locations, search algorithms byclassical computers usually require operations of or-der N . One such simple classical algorithm is to checkexhaustively all names to find a name matching withthe given phone number or to search all possible routesand then find the shortest route among all routes. How-ever, Grover (1996, 1997) developed a quantum search


algorithm that needs only operations of order√

N tofind a solution to the search problem.

The quantum search algorithm works as follows.Suppose that the search space has N elements andthe search problem has exactly M solutions. AssumeM ≤ N/2. (For the silly case of M > N/2, we eithersearch for the solution by doing random selection fromthe search space or double the number of the elementsin the search space by adding N extra non-solution el-ements to the search space.) The algorithm works bycreating superposition state with Hadamard gate,

|ψ〉 = 1

N1/2

N−1∑x=0

|x〉,

and then applying a so-called Grover iteration (or op-erator) repeatedly. Set

|φ〉 = 1√N − M

∑x′

|x′〉, |ϕ〉 = 1√M

∑x′′

|x′′〉,

where the summations over x′ and x′′ denote sums overall non-solutions and solutions, respectively. Then wecan express |ψ〉 as follows:

|ψ〉 =√

N − M

N|φ〉 +

√M

N|ϕ〉.

The Grover operator is to perform two reflections, oneabout the vector |φ〉 and another about the vector |ψ〉.The two reflections together are a rotation with angle θ

in the two-dimensional space spanned by |φ〉 and |ϕ〉,where

cos(θ/2) =√

N − M

N.

After the rotation, the initial state |ψ〉 = cos(θ/2)|φ〉+sin(θ/2)|ϕ〉 becomes state

cos(3θ/2)|φ〉 + sin(3θ/2)|ϕ〉.Thus each application of the Grover operator is a ro-tation with angle θ . The initial state |ψ〉 has angleπ/2 − θ/2 with |ϕ〉; after the first rotation, the resultedstate has angle π/2 − 3θ/2 with |ϕ〉; and in general af-ter the r th rotation, the resulted state has angle π/2 −(2r + 1)θ/2 with |ϕ〉. Repeatedly applying the Groveroperator, we rotate the state vector near |ϕ〉. With theinitial state |ψ〉 = cos(θ/2)|φ〉 + sin(θ/2)|ϕ〉, we needto rotate through arccos

√M/N radians to transform

the state vector to |ϕ〉. After R = arccos(√

M/N)/θ =O(

√N/M) times of applications of the Grover opera-

tor, we rotate the state vector |ψ〉 to within an angle θ/2of |ϕ〉. Performing measurements of the state yields a

solution to the search problem with probability at leastcos2(θ/2) ≥ 1 − M/N .

The number of iterations R depends on M , thenumber of solutions. Since R ≤ π/(2θ) and θ/2 ≥sin(θ/2) = √

M/N , R ≤ (π/4)√

N/M . Typically,M � N , θ ≈ sin θ ≈ 2

√M/N , thus R ≈ (π/4) ·√

N/M . We estimate the number of solutions by quan-tum counting, which is to combine the Grover operatorwith the phase estimation method. Under the basis |φ〉and |ϕ〉 the Grover operator has eigenvalues eiθ andei(2π−θ). Applying the phase estimation method wecan estimate the eigenvalues and thus θ with prescribedprecision and probability, which in turn yields M . Thecombination of the quantum counting and search pro-cedure will find a solution of the search problem withcertain probability. Repeating the quantum search al-gorithm will boost the probability and enable us to ob-tain a solution to the search problem.

Quantum walk and quantum Markov chain are cur-rently being investigated for new quantum search algo-rithms and quantum speed-up of Markov chain basedalgorithms (Aharonov and Ta-Shma, 2003, Childs etal., 2003; Childs, 2010; Tulsi, 2008; Shenvi, Kempeand Whaley, 2003 and Szegedy, 2004). In Section 5 weshow that the quantum search algorithm can also beviewed as a quantum simulation procedure.

5. QUANTUM SIMULATION

Quantum simulation is to intentionally and artifi-cially mimic a natural quantum dynamics, which ishard to access, and analyze, by a computer-generatedquantum system, which is easy to manipulate and in-vestigate. It provides scientific means for simulatingcomplex biological, chemical or physical systems inorder to study and understand certain scientific phe-nomena and evaluate hard-to-obtain quantities in thesystems. Examples in modern scientific studies includethe estimation of dielectric constant, proton mass, andprecise energy of molecular hydrogen, the study of su-perconductivity, the test of novel nano-materials, andthe design of new biomolecules.

To simulate a quantum system we need to solve theSchrödinger equation (1) which governs the dynamicevolution of the system. For a typical Hamiltonian withreal particles the Schrödinger equation usually consistsof elliptical differential equations, each of which canbe easily simulated by a classical computer. However,the real challenge in simulating a quantum system isto solve the exponential number of such differentialequations. For a quantum system of b qubits, its states

388 Y. WANG

have 2b amplitudes. To simulate the dynamic behav-ior of b qubits evolving according to the Schrödingerequation, we need to solve a system of 2b differentialequations. Due to the exponential growth in the num-ber of differential equations, the simulation of generalquantum systems by classical computers is very in-efficient. Classical simulation of quantum systems isfeasible for the cases where insightful approximationsare available to dramatically reduce the effective num-ber of equations involved. Quantum computers mayexcel in simulating physically interesting and impor-tant quantum systems for which efficient simulation byclassical computers may not be available.

5.1 Simulate a Quantum System

The key of quantum simulation is to solve theSchrödinger equation (1) which has solution

|ψ(t)〉 = e−iHt |ψ(0)〉, i = √−1.(14)

Numerical evaluation of e−iHt is needed. The Hamil-tonian H is usually exponentially large and extremelydifficult to exponentiate. The common approach in nu-merical analysis is to use the first-order linear approx-imation, 1 − iHδ, of e−iH(t+δ) − e−iHt , which oftenyields unsatisfactory numerical solutions.

Many classes of Hamiltonians have sparse represen-tations. For such sparse Hamiltonians we can find ef-ficient evaluation of the solutions (14) with high-orderapproximation. For example, the Hamiltonians in manyphysical systems involve only local interactions, whichoriginate from the fact that most interactions fall offwith increasing distance in location or increasing dif-ference in energy. In the local Hamiltonian case, theHamiltonian of a quantum system with α particles in ad-dimensional space has the form

H = 2L∑

�=1

H�,(15)

where L is a polynomial in α+d , and each H� acts on asmall subsystem of size free from α and d . For exam-ple, the terms H� are typically two-body interactionsand one-body Hamiltonians. Hence e−iH�δ are easyto approximate, although e−iHδ is very hard to eval-uate. Since H� and Hk are non-commuting, e−iHδ �=e−iH1δ · · · e−iHLδ . Applying a modification of the Trot-ter formula (Kato, 1978; Trotter, 1959; Yu, 2001) weobtain

e−iHδ = Uδ + O(δ2),

where

Uδ = [e−iH1δ · · · e−iHLδ][e−iHLδ · · · e−iH1δ].(16)

Thus we can approximate e−iHδ by Uδ which needs toevaluate only each e−iH�δ .

Assume that the quantum system starts at t = 0 withinitial state |ψ(0)〉 and ends at final time t = 1. For aninteger m, set tj = j/m, j = 0,1, . . . ,m. The quan-tum simulation is to apply approximation Uδ of e−2iHδ

to evaluate (14) at tj iteratively and generate approxi-mate solutions to |ψ(tj )〉. Denote by |ψ(tj )〉 the stateat tj obtained from the quantum simulation as an ap-proximation of the true state |ψ(tj )〉 at tj . Then forj = 1, . . . ,m,

|ψ(tj )〉 = e−2iHδ|ψ(tj−1)〉 = e−2iHjδ|ψ(t0)〉,(17)

|ψ(tj )〉 = Uδ|ψ(tj−1)〉 = Ujδ |ψ(t0)〉.

While classical computers are inefficient in simulat-ing general quantum systems, quantum computers canefficiently carry out the quantum simulation proce-dure and provide an exponential speedup for the quan-tum simulation on classical computers. In spite of theinefficiency, classical computers are currently beingused to simulate quantum systems in biochemistry andmaterial science. Quantum simulation will be amongthe important applications of quantum computers. SeeAbrams and Lloyd (1997), Aspuru-Guzik, Dutoi andHead-Gordon (2005), Bennett et al. (2002), Berry etal. (2007), Freedman, Kitaev and Wang (2002), Jané etal. (2003), Boghosian and Taylor (1998), Lloyd (1996),Nielsen and Chuang (2000), and Zalka (1998).

5.2 Recast Quantum Search Algorithm asQuantum Simulation

Grover’s search algorithm discussed in Section 4.5is an important finding in quantum computation. It canbe heuristically sketched as a quantum simulation bywriting down an explicit Hamiltonian H such that aquantum system evolves from its initial state |ψ〉 to |x〉after some specified time, where x is a solution of thesearch problem. Of course the Hamiltonian H dependson the initial state |ψ〉 and solution x. Suppose that |y〉is another state such that |x〉 and |y〉 form an compu-tational basis, and |ψ〉 = α|x〉 + β|y〉 for real α and β

with α2 + β2 = 1. Define Hamiltonian

H = |x〉〈x| + |ψ〉〈ψ | = I + α(βσ x + ασ z),

where σ x and σ z are Pauli matrices defined in (7).Then

exp(−iHt)|ψ〉= e−it [cos(αt)|ψ〉 − i sin(αt)(βσ x + ασ z)|ψ〉]= e−it [cos(αt)|ψ〉 − i sin(αt)|x〉].


Measuring the system at time t = π/(2α) yields thesolution state |x〉.5.3 Quantum Monte Carlo Simulation

Quantum theory is intrinsically stochastic and quan-tum measurement outcome is random. As many natu-rally occurring quantum systems involve a large num-ber of interacting particles, due to the computationalcomplexity we are forced to utilize Monte Carlo tech-niques in the simulations of such quantum systems.The combination of Monte Carlo methods with quan-tum simulation makes it possible to obtain reliablequantifications of quantum phenomena and estimatesof quantum quantities. Such combination proceduresare often referred to as quantum Monte Carlo sim-ulation (Nightingale and Umrigar, 1999; Rousseau,2008). Consider the problem of estimating the follow-ing quantity:

θ = Tr(Xρ) = E(X),(18)

where X is an observable, X is its measurement result,and ρ is the state of the quantum system under whichwe perform the measurements and evaluate the quan-tity θ .

As ρ is the true final state of the quantum system, wedenote by ρ the final state of the quantum system ob-tained via quantum simulation. The quantum systemsare prepared in initial state ρ0, and we use the quan-tum simulation procedure described above to simulatethe evolutions of the systems from initial state ρ0 tofinal state ρ according to Schrödinger’s equation (14)with Hamiltonian H given by (15). We repeatedly per-form the measurements of such n identically simulatedquantum systems at the final state and obtain measure-ment results X1, . . . ,Xn. We estimate θ defined in (18)by

θ = 1

n

n∑j=1

Xj .(19)

The target θ given by (18) is defined under the truestate ρ, while the simulated quantum system is underapproximate final state ρ which is close to ρ. The mea-surement results X1, . . . ,Xn are obtained via quantumsimulation from the quantum systems in the simulatedstate ρ. Therefore, the Monte Carlo quantum estimatorθ in (19) involves both bias and variance. Wang (2011)studied the quantum simulation procedure and investi-gated the bias and variance of θ . The derived bias andvariance results can be used to design optimal strategyfor the best utilization of computational resources toobtain the quantum Monte Carlo estimator.

6. QUANTUM INFORMATION

Classical information theory is centered on Shan-non’s two coding theorems on noiseless and noisychannels. The noiseless channel coding theorem quan-tifies the number of classical bits required to storeinformation for transmission by Shannon entropy,while the noisy channel coding theorem quantifies theamount of information that can be reliably transmittedthrough a noisy channel by an error-correction codingscheme. The quantum analogs of Shannon entropy andShannon noiseless coding theorem are von Neumannentropy and Schumacher’s noiseless channel codingtheorem, respectively. The von Neumann entropy is de-fined to be S(ρ) = − tr(ρ logρ). Schumacher’s noise-less channel coding theorem quantifies quantum re-sources required to compress quantum states by vonNeumann entropy (Schumacher, 1995). Analogous toShannon’s noisy channel coding theorem, a theoremknown as Holevo–Schumacher–Westmoreland theo-rem can be used to compute the product quantumstate capacity for some noisy channels (Holevo, 1998;Schumacher and Westmoreland, 1997). However, com-munications over noisy quantum channels are muchless understood than the classical counterpart. It is anunsolved problem to determine quantum channel ca-pacity or the amount of quantum information that canbe reliably transmitted over noisy quantum channels.See Hayashi (2006) and Nielsen and Chuang (2000).

In spite of the above similarity, there are intrinsicdifferences between classical information and quan-tum information. Classical information can be distin-guished and copied. For example, we can identify dif-ferent letters and produce an identical version of adigital image for back-up. However, quantum mechan-ics does not allow unknown quantum states to be dis-tinguished or copied exactly. For example, we cannotreliably distinguish between quantum states |0〉 and(|0〉 + |1〉)/√2. If we perform measurement for quan-tum state |0〉, the measurement result will be 0 withprobability 1, while measuring quantum state (|0〉 +|1〉)/√2 yields measurements 0 or 1 with equal proba-bility. A measurement result of 0 cannot tell the iden-tity of the quantum state being measured. A theoremknown as a no-cloning theorem states that unknownquantum states cannot be copied exactly (Wootters andZurek, 1982; Nielsen and Chuang, 2000).

As we discussed in Section 3.3, quantum entan-glement plays a crucial role in strange quantum ef-fects such as quantum teleportation, violation of Bell’sinequality, and superdense coding (Hayashi, 2006;

390 Y. WANG

Nielsen and Chuang, 2000). Entanglement is a newtype of resource that differs vastly from the tradi-tional resources in classical information theory. We arefar from having a general theory to understand quan-tum entanglement but encouraging progress made sofar reveals the amazing property and intriguing struc-ture of entangled states and remarkable connectionsbetween noisy quantum channels and entanglementtransformation. Consider quantum error-correction forreliable quantum computation and quantum informa-tion processing. Quantum error-correction is employedin quantum computation and quantum communica-tion to protect quantum information from loss dueto quantum noise and other errors like faulty quan-tum gates. Classical information uses redundancy toachieve error-correction, but the no-cloning theorempresents an obstacle to copying quantum informationand formulating a theory of quantum error-correctionbased on simple redundancy. Again quantum entan-glement comes to the rescue. It is forbidden to copyqubits but we can spread the information of one qubitonto a highly entangled state of several qubits. Shor(1995) first discovered the method of formulating aquantum error-correction code by storing the informa-tion of one qubit onto a highly entangled state of ninequbits. Over time several quantum error-correctioncodes are proposed (Calderbank and Shor, 1996; Coryet al., 1998; Steane, 1996a, 1996b). These quantumerror-correction codes can protect quantum informa-tion against quantum noise, and thus quantum noiselikely poses no fundamental barrier to the performanceof large-scale quantum computing and quantum infor-mation processing.

Here is how quantum error-correction codes work.We consider the single qubit case. First assume that aqubit α0|0〉 + α1|1〉 is passed through a bit flip channelwhich flips the state of a qubit from |0〉 to |1〉 and from|1〉 to |0〉, each with probability p, and leaves each ofstates |0〉 and |1〉 untouched with probability 1−p. Wedescribe a bit flip code that protects the qubit againstquantum noise from the bit flip channel.

We encode states |0〉 and |1〉 in three qubits, with |0〉encoded as |000〉 and |1〉 as |111〉. Thus the qubit stateα0|0〉 + α1|1〉 is encoded in three qubits as α0|000〉 +α1|111〉. We pass each of the three qubits through anindependent copy of the bit flip channel, and assumethat at most one qubit is flipped. The following sim-ple two-step error-correction procedure can be used torecover the correct quantum state.

Step 1. Perform a measurement on a specially con-structed observable and call the measurement result

an error syndrome. The error syndrome can inform uswhat error, if any, occurred on the quantum state. Theobservable has eigenvalues 0, 1, 2 and 3, with corre-sponding projection operators,

Q0 = |000〉〈000| + |111〉〈111| no error,

Q1 = |100〉〈100| + |011〉〈011|bit flip on the first qubit,

Q2 = |010〉〈010| + |101〉〈101|bit flip on the second qubit,

Q3 = |001〉〈001| + |110〉〈110|bit flip on the third qubit.

If one of three qubits has one or no bit flip, the errorsyndrome will be one of 0, 1, 2 and 3, with 0 cor-responding to no flip, and 1, 2 and 3 to a bit flip onthe first, second and third qubit, respectively. For ex-ample, if the first qubit is flipped, the corrupted stateis |ψ〉 = α0|100〉 + α1|011〉. Since 〈ψ |Q1|ψ〉 = 1 and〈ψ |Qj |ψ〉 = 0 for j �= 1, in this case the error syn-drome is 1. Although performing measurements usu-ally causes change to the quantum state, the specialityof the constructed observable is that syndrome mea-surement does not perturb the quantum state: it is easyto check that the state is |ψ〉 both before and after thesyndrome measurement. While the syndrome providesinformation about what flip error has occurred, it doesnot contain any information about the state being pro-tected, that is, it does not allow us to deduce anythingabout the amplitudes α0 and α1. Such a special prop-erty is the generic feature of syndrome measurement.

Step 2. The error type supplied by the error syn-drome can inform us what procedure to use to recoverthe original state. For example, error syndrome 1 in-dicates a bit flip on the first qubit, and a flip on thefirst qubit again will perfectly recover the original stateα0|000〉 + α1|111〉. The error syndrome 0 implies noerror and doing nothing, and error syndromes 1, 2 and3 correspond to a bit flip again on the first, second,third qubit, respectively. The procedure will recover theoriginal state with perfect accuracy, if there is at mostone bit flip in the encoded three qubits. The probabilitythat more than one bit flipped is p3 + 3p2(1 − p) =3p2 − 2p3, which is much smaller than the error prob-ability p of making no-correction for the typical bitflip channel. Thus the encoding and decoding schememakes the storage and transmission of the qubit morereliable.

Next we consider a more interesting noisy quantumchannel: a phase flip channel which, with probability


p, changes a qubit state α0|0〉+α1|1〉 to α0|0〉−α1|1〉,and with probability 1 − p, leaves alone the qubit.The following scheme is to turn the phase flip chan-nel into a bit flip channel. Let |+〉 = (|0〉 + |1〉)/√2and |−〉 = (|0〉 − |1〉)/√2 be a qubit basis. The phaseflip channel leaves alone states |+〉 and |−〉 with prob-ability 1 − p and changes |+〉 to |−〉 and vice versawith probability p. In other words, the phase flip chan-nel with respect to the basis |+〉 and |−〉 acts just likea bit flip channel with respect to the basis |0〉 and |1〉.Thus we encode |0〉 as |+ + +〉 and |1〉 as |− − −〉 forprotection against phase flip errors. The operations forencoding, error-detection and recovery are the same asfor the bit flip channel but with respect to the |+〉 and|−〉 basis instead of the |0〉 and |1〉 basis.

Last we describe Shor error-correction code. It isa combination of the three-qubit phase flip and bitflip codes. First use the phase flip code to encodestates |0〉 and |1〉 in three qubits, with |0〉 encoded as|+ + +〉 and |1〉 as |− − −〉; next, use the three-qubitbit flip code to encode each of these qubits, with |+〉encoded as (|000〉 + |111〉)/√2 and |−〉 encoded as(|000〉 − |111〉)/√2. The resulted nine-qubit code hascodeworks as follows:

|0〉 → |000〉 + |111〉√2

|000〉 + |111〉√2

|000〉 + |111〉√2

,

|1〉 → |000〉 − |111〉√2

|000〉 − |111〉√2

|000〉 − |111〉√2

.

With the mixture of both phase flip and bit flip codes,the Shor error-correction code can protect against bitflip errors, phase flip errors, as well as a combined bitand phase flip errors on any single qubit. In fact it hasbeen shown that this simple quantum error-correctioncode can protect against the effects of any completelyarbitrary errors on a single qubit (Shor, 1995).

7. CONCLUDING REMARKS

Quantum information science gains enormous atten-tion in computer science, mathematics, physical sci-ences and engineering, and several interdisciplinarysubfields are developing under the umbrella of quan-tum information. This paper reviews quantum compu-tation and quantum information from a statistical per-spective. We introduce concepts like qubits, quantumgates and quantum circuits in quantum computationand discuss quantum entanglement, quantum paral-lelism and quantum error-correction in quantum com-putation and quantum information. We present ma-jor quantum algorithms and show their advantages

over the available classical algorithms. We illustratequantum simulation procedure and Monte Carlo meth-ods in quantum simulation. As classical computationand simulation are ubiquitous nowadays in statistics,we expect quantum computation and quantum sim-ulation will have a paramount role to play in mod-ern statistics. This paper exposes the topics to statis-ticians and encourages more statisticians to work inthe fields. There are many statistical issues in theoret-ical research as well as experimental work in quan-tum computation, quantum simulation and quantuminformation. For example, as measurement data col-lected in quantum experiments require more and moresophisticated statistical methods for better estimation,simulation and understanding, it is imperative to de-velop good quantum statistics methods and quantumsimulation procedures and study interrelationship andmutual impact between quantum estimation and quan-tum simulation. Since quantum computation is intrinsi-cally random, and quantum simulation employs MonteCarlo techniques, as we point out in Section 4.3 andWang (2011), it is important to provide sound statis-tical methods for analyzing quantum algorithms andquantum simulation in general and study high-orderapproximations to exponentiate Hamiltonians and theefficiency of the resulted quantum simulation proce-dures in particular. On the other hand, quantum com-putation and quantum simulation have great potentialto revolutionize computational statistics. Below are afew cases in point.

1. The “random numbers” generated by classical com-puters are pseudo-random numbers in the sensethat they are produced by deterministic proceduresand can be exactly repeated and perfectly predictedgiven the deterministic schemes and the initialseeds. On the contrary, superposition states enablequantum computers to produce genuine randomnumbers. For example, measuring (|0〉 + |1〉)/√2yields 0 and 1 with equal probability. In generalwe generate b-bit binary random numbers x =x1 · · ·xb, xj = 0,1 as follows. Apply b Hadamardgates to b qubits of |0 · · ·0〉 to obtain

|0〉 + |1〉√2

· · · |0〉 + |1〉√2

= 1√2b

∑x

|x〉,

x = x1 · · ·xb, xj = 0,1,

where the sum is over all possible 2b values of x,and then measure the obtained qubits and yield b-bit binary random numbers x = x1 · · ·xb with equalprobability. Quantum theory guarantees that such

392 Y. WANG

random numbers are genuinely random. Thus quan-tum computers are able to generate genuine randomnumbers and perform true Monte Carlo simulation.It is exciting to design general quantum randomnumber generator and study quantum Monte Carlosimulation. Perhaps we may need to re-examineMonte Carlo simulation studies conducted by clas-sical computers.

2. It is interesting to investigate the potential of quan-tum computation and quantum simulation forcomputational statistics. We expect that quantumcomputers may be much faster than classical com-puters for computing some statistical problems.Moreover, quantum computers may be able to carryout some computational statistical tasks that areprohibitive by classical computers. Specific exam-ples are as follows: (a) We may use the basic ideasof Grover’s search algorithm to develop fast quan-tum algorithms for implementing some statisticalprocedures. For example, finding the median of ahuge data set is to search for a numerical value thatseparates the top and bottom halves of the data, andquantum algorithms can offer quadratical speedupfor calculating median and trimmed mean. (b) Withgenuine random number generator and faster meanevaluation, quantum computers may offer signifi-cant advantages over classical computers for MonteCarlo integration. For example, Monte Carlo inte-gration in high dimensions may be exponentiallyor quadratically faster on quantum computers thanon classical computers. (c) It might be possible forquantum computers to carry out some prohibitingstatistical computing tasks like the Bayesian com-putation discussed in Section 3. Some preliminaryresearch along these lines may be found in Nayakand Wu (1999) and Heinrich (2003).

3. As quantum computation and quantum simulationare ideal for simulating interacting particle systemslike the Ising model, it is fascinating to explore theinterplay between quantum simulation and Markovchain Monte Carlo methodology and the quantumpotential to speed up Markov chain based algo-rithms. In fact, it has been shown that quantum walkbased algorithms can offer quadratical speedup forcertain Markov chain based algorithms (Magniez etal., 2011; Richter, 2007; Szegedy, 2004, and Wocjanand Abeyesinghe, 2008).

Finally we point out that quantum computers arewonderful but it is difficult to build quantum computerswith present technology. To build a quantum computer

the physical apparatus must satisfy requirements thatthe quantum system realized qubits needs to be wellisolated in order to retain its quantum properties andat the same time the quantum system has to be acces-sible so that the qubits can be manipulated to performcomputations and measure output results. The two op-posing requirements are determined by the strength ofcoupling of the quantum system to the external entities.The coupling causes quantum decoherence. Decoher-ence refers to the loss of coherence between the com-ponents of a quantum system or quantum superposi-tion from the interaction of the quantum system with itsenvironment. It is very crucial but challenging to con-trol a quantum system of qubits and correct the effectsof decoherence in quantum computation and quantuminformation. Quantum computing has witnessed greatadvances in recent years, and quantum computers of ahandful of qubits and basic quantum communicationdevices have been built in research laboratories (seeBarz et al., 2012; Clarke and Wilhelm, 2008; DiCarloet al., 2009; Johnson et al., 2011; Lee et al., 2011;Neumann et al., 2008), but there are technological hur-dles in the development of a quantum computer of largecapacity. History shows that scientific innovations andtechnological surprises are a never-ending saga. It isanticipated that quantum computers with a few dozenof qubits will be built in near future. As we have dis-cussed in Section 3.1, such a quantum computer hascapacity of a classical supercomputer. We are very op-timistic that someday quantum computers will be avail-able for statisticians to crunch numbers. For the timebeing, instead of waiting in the sidelines for that tohappen, statisticians should get into the field of play.It is time for us to dive into this frontier research andwork with scientists and engineers to speed up the ar-rival of practical quantum computers. As a last note,in 2011 a Canadian company called D-Wave has soldthe claimed first commercial quantum computer of 128qubits to the Lockheed-Martin corporation, despite theD-Wave’s quantum system being criticized as a blackbox. Large scale quantum computers may be yearsaway, but quantum computing is already here as a sci-entific endeavor to provoke deep thoughts and integrateprofound questions in physics and computer science.

ACKNOWLEDGMENTS

Wang’s research was supported in part by NSF GrantDMS-10-05635. He thanks editor David Madigan andtwo anonymous referees for helpful comments andsuggestions which led to significant improvements inboth substance and the presentation of the paper.


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