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XRDS S P R I N G 2 0 1 2 V O L . 1 8 N O . 332

Why Now is the

Right Time to StudyQuantum Computing

In previous centuries, our determinis-tic view o the world led us to imagineit as a giant clockwork mechanism.But as computers have become ubiq-uitous, they have infltrated our meta-

phors and changed the way we thinkabout science, math, and even society.Not only do we use computers to solveproblems, we also think in ways thatare inormed by building, program-ming, and using computers.

For example, phenomena as diverseas DNA, language, and cognition arenow thought o as mechanisms orinormation transmission that haveevolved to optimize or both com-pression and error correction. Game

theory and economics have begun toincorporate notions o computationalefciency, realizing, as Kamil Jain a-mously said o Nash equilibria, I yourlaptop cant fnd it, then neither canthe market. Computer science hasalso reshaped the goals o these felds.Mathematics increasingly concerns it-sel with matters o efciency, and hasbeen rapidly growing in the computer-related felds o inormation theory,graph theory, and statistics. The P

versus NP question is the newest o

the Clay Millennium problems, and i

resolved would shed light on the oldestpuzzle in mathematics: What makes ithard to fnd proos?

In hindsight, the computational viewseems natural. But when computers

frst appeared, even those ew who pre-dicted their commercial success couldnot have oreseen the intellectual revo-lution that would ollow. For example,the inormation-theoretic notion o en-tropy (which is central to compressionand error correction) could have eas-ily been invented in the time o Gauss,or even by medieval Arabs or ancientGreeks, but the impetus came in the20th century when Bell Labs employedClaude Shannon under a wartime con-

tract to study cryptography. This is notunique to computer science. Einsteinswork as a patent clerk helped him de-vise relativity; clock synchronizationon railway networks was an importantengineering problem, and metaphorso clocks on trains provided grist or hisamous thought experiments. In gen-eral, science oten ollows technologybecause inventions give us new waysto think about the world and new phe-nomena in need o explanation.

The story o quantum computing

is similar. Quantum mechanics was

invented in the frst ew decades o the20th century, and its modern orm wasmostly known by 1930. But the ideathat quantum mechanics might givecomputational advantages did not

arise until physicists tried to simulatequantum mechanics on computers.Doing so, they quickly ran into a practi-cal problem: While a single system (e.g.,the polarization o a photon) might bedescribable by only two complex num-bers (e.g., the amplitudes o the hori-zontal and vertical components o thepolarization), a collection o n suchsystems requires not2n but2n complexnumbers to ully describe even thoughmeasurement can extract only n bits.

Physicists coped by developing closed-orm solutions and physically motivat-ed approximations that could handlean increasingly large number o caseso interest (more on this later).

The exponentially large state spaceso quantum mechanics should have beena clue that nature contains vastly morecomputational resources than we everimagined. Instead, it and other strangeeatures o quantum mechanics wereseen mostly as limitations and quirkso the theory. For example, Heisenbergs

uncertainty principle was oten thought

Quantum computing is a good way to justiy difcult physics experiments. But howcrucial is quantum computing i we are still waiting or quantum computers to bebuilt? And more importantly, do computer scientists really need to know anythingabout quantum inormation? In this article, I describe where quantum computingcame rom, what it is, and what we, as computer scientists, can learn rom it.

Quantum computing is not merely a recipe for new computing devices, but anew way of looking at the world.By Aram HarrowDOI: 10.1145/2090276.2090288


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XRDS S P R I N G 2 0 1 2 V O L . 1 8 N O . 3 33

o as a restriction on measurements.Phenomena such as entanglement wereconsidered part o quantum ounda-tions or the philosophy o quantummechanics, but were not considered op-erationally relevant until quantum com-puting and quantum cryptography were

independently developed in the 1980sand 1970s, respectively.Quantum computing (or more pre-

cisely, the idea o quantum advantagein computing) did not arise until 1982,

when Richard Feynman suggestedthat since conventional computersappeared to require exponential over-head to simulate quantum mechan-ics, perhaps a quantum-mechanicalcomputer could perorm the task moreefciently. The model was ormalizedin 1985 by David Deutsch, who also

showed, surprisingly, that a quantum-mechanical computer could be asterthan a conventional computer or aproblem that on its ace had nothingto do with quantum mechanics (com-puting the XOR o two bits). A series ostronger speedups ollowed, albeit orcontrived problems, until Peter Shors1994 quantum algorithm or actoringintegers in polynomial time.

Much earlier (1970), then grad stu-dent Stephen Wiesner proposed usingHeisenbergs restrictions on measure-

ments to prevent an adversary romlearning a secret message. Thus quan-tum cryptography was born, although

Wiesners paper was almost universal-ly rejected rom journals, and its ideas

were largely unknown until CharlesBennett and Gilles Brassard publisheda scheme or quantum key distributionin 1984. Even the Bennett-Brassardproposal wasnt taken seriously untilthey implemented it in 1991 (a muchsimpler task than building a general-

purpose quantum computer).The key conceptual shit enablingboth quantum computing and quan-tum cryptography was to start think-ing about the eects o quantum me-chanics on inormation in operationalterms instead o as a source o limita-tions, curiosities, and paradoxes. Oncethis shit was made, the technical as-pects o quantum inormation weremuch simpler than earlier develop-ments in quantum mechanics (e.g., theunifcation in the 1950s o quantum

mechanics with special relativity).

COMPUTING WITH QUBITS

One o the basic contributions o thecomputational approach to quantummechanics is the idea o a qubit (pro-nounced q-bit). In general, the state oa quantum system with dperectly dis-tinguishable states can be describedas a unit vector in (where is the seto all complex numbers). The simplestinteresting case is when d=2, and theresulting vector is called a qubit:

x=

xo

x1

!

"#

$

%&

This ormalism includes the case o aclassical system, or which statejis rep-resented by the vector e

jwith a 1 in po-

sition jand zeroes elsewhere. Since anystate can be written as a linear combi-nation (also called a superposition)o dierent e

j, it is tempting to imagine

that the classical states are somehowundamental. But mathematically, onebasis is as good as any other.

The vector x describes the binaryquantum system in the sense that mea-

suring its state by means o an experi-

ment yields outcomejwith probability

|xj|2 orj=0,1 (or 0,1,d-1 in general);

hence x is indeed a unit vector. Oneconsequence o quantum measure-ment is that upon outcomej, the stateo the system collapses to e

j. Since any

external record o the quantum statecauses the same eect as a measure-ment, this explains why most systemslook classical even though the underly-ing physics is always quantum.

The beauty o qubits is that theyare completely independent o the

underlying system, be it a photonspolarization, the energy level o anelectron bound to an atom, the spin oa nucleus, or the direction o a loop osuperconducting current. In this way,a qubit is a device-independent wayo describing quantum inormation,

just as bits enable us to reason aboutclassical inormation without need-ing to know whether it is encoded inRAM, a hard drive, or an abacus. Fur-thermore, its also possible to describelinear dynamics o the system, as long

as they preserve the norm o the qu-

Figure 1. This molecule was used in a 5-qubit nuclear magnetic resonance (NMR)

quantum computer. While NMR was useul in the early QC implementations, it has

limitations that make it unlikely to lead to a large-scale implementation.

Courtesy

ofIsaac

Chung


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bit: Evolution o the quantum state isgiven by mappingx to Ux, where Uis aunitary matrix, meaning that it alwayspreserves length (mathematically,

, where ).Single qubits are interesting or

physics experiments, but or compu-

tational purposes, we are more inter-ested in what happens when we haven qubits. In this case, the state x is aunit vector in , with entries labeledby n-bit strings. Most n-qubit statesare entangled, meaning that their am-plitudes are in some way correlatedacross the n bits. Dynamics are nowgiven by2nx2nunitary matrices.

While this seems both abstract,and potentially complicated (thereare a lot o2nx2n unitary matrices!),a similar ormalism can be used to

describe classical deterministic andrandomized computations. An n-bit string can be represented, albeitsomewhat wasteully, as a vector olength two inches with a single entryequal to one and the rest equal to zero.Dynamics can be represented by map-ping x to Mx where M is a 0/1 matrix

with a single 1 in each column. Ran-domized computation is similar. Thistime the state is a vector o nonnega-tive real numbers summing to one,and x! Mx is a valid transition or

anyMwith nonnegative entrieswhose

columns each sum to one.In each case, this picture neglects

the act that some transormations on bits are easier than others. The trans-ormations that are eectively comput-able are typically comprised o a rea-sonable number o basic operations o

the orm replace bit x3 with the NANDo bits x2

and x7or set x

1to a uniorm-

ly random bit. The basic transorma-tions correspond to easible22 ma-trices, and so by taking all the productso, say, 109 o them, we obtain all thetransormations that can be achieved

with 109 basic gates. The easible22unitaries can similarly be obtained bymultiplying together short strings oa set o basic quantum operations; ormore details about how this works, seethe reerences at the end o this article.

Thus the key dierences betweenclassical (randomized) computationand quantum computation appear tobe merely the shit rom nonnega-tive real to complex numbers (most othis dierence is captured by allowingnegative reals), and the shit rom l

1to

l2

norm or the state (that is, quantumstates have the squared absolute valueso their amplitudes summing to one,

while probabilities sum to one withoutsquaring). However, a crucial act isthat dierent branches o a computa-

tion can have dierent phases, mean-

ing that when they recombine, theiramplitudes can either add up (calledconstructive intererence), or can can-cel each other (called destructive inter-erence). On the one hand, this is justa ancy way o saying that multiplyingunitary matrices by vectors involves

adding up terms that can have dier-ent signs (or more generally, dierentphases); on the other, it allows a totallynew orm o computing.

For example, i 0 and 1 are repre-sented by the vectors

eo =1

0

!

"#

$

%&, e1=

0

1

!

"#

$

%&

respectively, then the NOT operationcan be expressed (quantumly or classi-cally) as

0 1

1 0

!

"#

$

%&

Geometrically, this is a rotation by/2. However, only a quantum comput-er can perorm the ollowing squareroot o NOT, which corresponds to arotation by/4.

NOT =

1 / 2 !1 / 2

1 / 2 1 / 2

"

#

$$

%

&

''

I we start with the 0 state and ap-ply NOT , then we obtain the state

max Ai, j,k

i, j,k

! xiyjzk Were we to measure, the outcomes

zero or one would each occur with prob-ability .5. However, i we apply NOT asecond time beore measuring, then wealways obtain the outcome one. (To seethat this is indeed perorming a NOT check the result o applying it twice toe

0.) This demonstrates a key dierence

between quantum superpositions and

random mixtures; placing a state intoa superposition can be done withoutany irreversible loss o inormation.

FAMOUS, CUTTING-EDGE QUANTUM

ALGORITHMS

When we have n qubits, superpositionand intererence enable us to achievegreater computational advantages.One amous example is Grovers algo-rithm, which, given a binary unction

f on n bits, allows us to search or aninput x! 0,1{ }

n

with f(x)=1 using 2n/2

evaluations of; a square-root advan-

Figure 2. When light, or any other wave, is split into t wo beams and recombined,

the relative phase o the two beams determines whether they interere construc-

tively or destructively. Since the phase o light depends on how ar its traveled,

this is a ver y sensitive way to measure distance. Similarly, quantum computers

can split computations into multiple branches and recombine them to obtain

either constructive or destructive intererence.


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tage. What makes Grovers algorithmpossible is the act that probabilitiesare the squares o amplitudes. Thus,a uniorm superposition over 2n statesassigns amplitude 1 2n to each one.Moreover, one can show that with e-ort comparable to that required to

calculate f, it is possible to increasethe amplitude o each target x (i.e. withf(x)=1) by roughly 1 2n . This breaksdown only when their total amplitudeis large. Thus, the total eort is on theorder o2n/2.

A more dramatic speedup over clas-sical computing occurs with Shors al-gorithm or actoring integers. Shorsalgorithm has a substantial classicalcomponent, which reduces the prob-lem o actoring to the more abstractproblem o period fnding. This prob-

lem takes as input a unction on theintegers {0, 1, , 2n-1} with the proper-ty thatf(x)=f(y)i and only ix-y is divis-ible byror some hidden period r. Thegoal is to fnd r. Since rcan be exponen-tially large in n, classical computers re-quire exponential time to fnd it i theyare required to treat f as a black box.(This assumption is important. Clas-sical computers can actor n-bit num-bers more quickly than).

However, quantum computers canfnd r much more quickly. The frst

step is to prepare the uniorm super-position

1

2n

ex

x=0

2n!1

" Next, the quantum computer calcu-latesf(x) and then measures the result.

Whatever the outcome is, the residualstate will be a superposition over thosexs compatible with the measurementoutcome, which takes the orm z, z+r,

z+2r, or some z between 0 and r1.This quantum version o Bayes rule

leaves the state r2

n e

z+e

z+r+e

z+2r+( )

or a random z {0,,r1}. The nextstep is uniquely quantum, and in-

volves applying a unitary matrix calleda quantum Fourier transorm (QFT).This matrix hasy,z entry equal to

e2!iyz/2

n

2n

Perorming it efciently is possibleby adapting the classical ast Fouriertransorm (FFT) to the quantum set-

ting, but its action is dramatically di-

erent, since it transorms the ampli-tudes o a quantum state rather than

transorming a list o numbers as theclassical FFT does. Applying the QFTin this case maps our superposition toa state where the amplitude oy is

!

r

2n

1+ e2!i

yr

2n

+

"

#$ e

2!i2yr

2n

+ e2!i

3yr

2n

+

%

&''

Iyr is (approximately) divisible by2nthen this sum will be large, and iyris ar rom divisible by2n then it willinvolve many complex numbers withdierent phases that tend to cancel

out (as can be verifed by a quick calcu-lation). Thus, measuringy will returnan answer that is close to a multipleo2n/r. From this, some more classicalnumber theory can recover r.

Shors algorithm and its variantscan be used to break almost everypublic-key cryptosystem currently inuse. But it has another deeper implica-tion about the complexity o quantummechanics. Beore Shors algorithm,it was easier to imagine that quantummechanics might not really be as com-

plicated as the equations seem, and

that there might be a shortcut to simu-late it without needing exponentially

large vectors. But now we know anyalternate theory o quantum mechan-ics recreating the same physics can-not be much simpler than quantummechanics itsel (or at least no easierto simulate) unless actoring andperiod-fnding can be perormed muchmore quickly on classical computers.This situation is like the developmento NP-completeness, which showedthat eorts in disparate areas to solveNP-complete problems were in a sense

equivalent.Shors algorithm and Grovers algo-rithm are the most amous quantumalgorithms, but are not the only two.One recent algorithm solves large lin-ear systems o equations [1]; that is,given a matrixA and a vector b, it fndsx such thatAx=b. However, unlike clas-sical algorithms or the problem, in thequantum version x and b are quantumstates (and thus can be2n-dimensional

vectors using onlyn qubits). Further,Aneeds to be sparse and given by an im-

plicit representation so that given any

Figure 3. Any unction can be writ ten as a sum o waves o dierent requencies.

The coefcients o this sum are called Fourier coefcients. The quantum Fourier

transorm (QFT) transorms a quantum state into one whose amplitudes are the

Fourier coefcients o the original state. Measuring this state can be an incredibly

efcient way to detect periodic structure.


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row index i it is possible to efcientlyfnd all the nonzero A

ij. These restric-

tions make it possible in principle tosolve exponentially large systems oequations in polynomial time. How-ever, there is an important caveat: Theruntime o the algorithm also scales

with the condition number oA, a pa-rameter which is related to the numeri-cal instability o solving the system oequations classically. Finding a natu-ral scenario in which this algorithmcan be used remains a compellingopen problem.

Another recent algorithmic devel-opment is a quantum analogue o theMetropolis sampling algorithm [2].Classically, the Metropolis method isa technique or sampling rom hard-to-analyze distributions over exponen-

tially large state spaces (indeed, expo-nential speedup via use o randomnessis an old example o how powerul achange o computational model canbe). Its amazing range o applicationsincludes statistical inerence and ap-proximation algorithms or the per-manent o a nonnegative matrix, butit was originally developed to samplerom the thermal distribution o aphysical system. I state x has energy

E(x), then the thermal distributionat temperature T assigns x probabil-

ity proportional to e-E(x)/T, so that lowertemperatures push the system harderinto low-energy confgurations.

The quantum Metropolis algo-rithm, by analogy, produces quantumstates in thermal distributions. Likeits classical cousin, the quantum Me-tropolis algorithm takes longer to pro-duce lower-temperature states, as thiscorresponds to solving harder optimi-zation problems, but proving rigorousbounds on its runtime is difcult in all

but a handul o cases. Without ormalproos, we can nevertheless run theclassical Metropolis algorithm and ob-serve empirically that it works quicklyin many cases o interest. An empiricalevaluation o the quantum Metropolis

will o course have to wait or the arriv-al o a large-scale quantum computer.But we may already have enough tools,i we can just fgure out how to com-bine them to develop new quantumalgorithms that use quantum Metrop-olis as a subroutine, just as the classi-

cal algorithm or the permanent used

classical Metropolis.New proposed uses or quantum

computers continue to arise. One ex-citing development is the increasingnumber o applications being ound

or quantum computers with 10 or so

qubits; small enough to be easily simu-latable classically, while large enoughto interact in ways not previously dem-onstrated. Such quantum computingdevices could be used to improve pre-

cision quantum measurements (e.g.

Some physical excitations, likephotons or the vibrations o

a solid, can have arbitrarily

low energy as their requency

gets lower and lower. Others, like extra

electrons moving through semiconduc-

tors, only exist above some threshold (e.g.

,in Silicon, 1.1 electron-Volt). The least

amount o energy added by an excita-

tion corresponds to the gap between the

lowest energy level o a system and the

second-lowest level.

Physicists care about energy

gaps because o the belie that smallgaps are associated wit h long-range

correlations, roughly speaking, because

it takes little energy to create an

excitation that w ill be highly mobile.

However, the reality turns out to be

more subtle. For any single observable,

say the strengt h o the magnetic eld at

a particular point, correlations indeed

decay rapidly when the gap is large. But

the state o the system as a whole may

still exhibit long-range correlations.

A good analogy or the situation arises

rom considering xing onethousand

random one-to-one unctions on some

large space, such as the set o bit strings

o length 106. I the unction is chosen

randomly, then the pair (x, f(x)) will have

a nearly maximal amount o mutual

inormation, meaning that learning xwill

narrow down the number o possibilities

or f(x) rom 21,000,000 to 1000. On the

other hand, any individual bit in xwill be

nearly uncorrelated with any particular

bit in f(x). This argument also works i

the random unctions are replaced by an

expander graph with suitable parameters,

meaning that this behavior can also

be achieved constructively. (See Dana

Moshkovitzs article on page 23 by or

another application o expander graphs.)

Guided by this intuition, researchers

have developed quantum analogueso expander graphs that demonstrate

systems with a large gap and rapidly-

decaying correlations o any individual

observable, but nevertheless a large

amount o overall mutual inormation

between distant pieces o the system.

Why should we care about these

dierent kinds o correlations, apart

rom the desire to use theory to predict

observable quantities in the real world?

One exciting application is the old

problem o simulating quantum systems

on classical computers. I a system on qubits had no entanglement, then

the number o parameters required to

describe it would not be 2n, but merely 2n.

I the qubits are arranged on a line, with

correlations roughly bounded in range to

some short distance k, then the number

o parameters scales as nexp k( ) . Thus,controlling correlations in quantum

systems also helps us simulate them

efciently.

This line o research can be seen as

part o a larger project to divide quantum

systems into those that are efciently

simulatable classically (say because they

have limited entanglement), and those

that are capable o universal quantum

computing. On the other hand, a handul

o systems o apparently intermediate

complexity have been ound, which

are neither known to be classically

simulatable, nor universal or quantum

computing. These include systems o

non-interacting photons, as well as the

case when the noise rate is too high or

known quantum error-correcting codes

to unction, but too low to rule out the

possibility o large-scale entanglement.

Resolving the complexity o these

boundary cases is a ascinating source o

open problems.

Physics, Algorithms andComplexity: An Example


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in atomic clocks, or to detect gravitywaves), as quantum repeaters in anetwork to distribute entanglementor use in cryptographic protocols oreven to construct arrays o telescopesthat could synthesize apertures o un-limited size. Just as the usage model o

classical computers has gone beyondthe Turing machine model, quantumcomputing devices are likely to bemore exible than we can currentlyimagine.

SCIENCE THROUGH THE (QUANTUM)

ALGORITHMIC LENS

Building large-scale quantum com-puters will undoubtedly cause largeand unpredictable changes in how wethink about science. How long this

will take depends less on physics than

on engineering, error-correction, andeconomics. But even without a physi-cal quantum computer our theoreticalprogress so ar has led to many concep-tual advances.

One major advance is the idea oseparating the inormational contento quantum mechanics rom the phys-ics. Teaching these together, as is donenow, mixes counterintuitive eaturessuch as measurement and entangle-ment with the mathematically com-plicated picture o the Schrdinger

equation as a PDE (partial dierentialequation). It is as though probabilitytheory were only taught in the contexto statistical mechanics, and the frstdistributions that students saw werethe thermal distributions o idealgases. Instead, I believe quantum me-chanics would make more sense iits implications or inormation weretaught frst, and then this ramework

were used to explain atoms, photons,and other physical phenomena. This

would also make this feld more acces-sible to computer science majors.Students are not the only ones who

can beneft rom the quantum inor-mational perspective. Many phenom-ena involving quantum mechanicsare related to issues such as entropy,entanglement, or correlation that havephysical relevance but are best de-scribed in terms o inormation. Oneearly success story concerns the hard-ness o fnding the lowest-energy stateo a quantum system (o n qubits,

with pairwise interactions). For clas-

sical systems such problems are NP-complete, except in special cases, suchas when the systems are arranged in aline or on a tree. For quantum systems,the energy-minimization problemturned out to be QMA-complete. QMAstands or Quantum Merlin-Arthur

games, and is believed to be (rough-ly) as ar out o the reach o quantumcomputers as NP is out o reach oclassical computers. This gives sometheoretical justifcation to the empiri-cally observed phenomenon that manyphysical systems, such as glasses, havetrouble fnding their ground states.Surprisingly, energy minimization isQMA-complete even or systems on aline with identical nearest-neighborinteractions, contrar y to the previouslyheld intuition o physicists that the 1-D

case should be easy.In other cases, the quantum inorma-

tion perspective yields positive results,and even new classical algorithms. Seethe sidebar, Physics, Algorithms andComplexity: An example, or an ex-ample relating physics, algorithms, andcomplexity.

The scientifc benefts o the quan-tum inormation perspective are notrestricted to quantum mechanics.Some important problems, with no ap-parent relation to quantum mechan-

ics, relate to perorming linear algebraon multidimensional arrays. For exam-ple, given a 3-D collection o numbers

Aijk

, with i,j,k ranging over {1,..,n}, howhard is it to compute the ollowing ana-logue o the largest singular value:

max Ai, j,ki, j,k

! xiyjzk

over all unit vectors x,y,z? While com-puting this to accuracy 1/poly(n) canbe readily shown to be NP-hard, or the

oten more realistic case o requiringconstant accuracy, the only hardnessresult known involves quantum tech-niques, as does the most promisingclassical algorithm. One possible rea-son or the eectiveness o the quan-tum inormation perspective here isthat multidimensional arrays corre-spond naturally to entangled states,and our increasingly quantitative un-derstanding o entanglement oten

yields results about linear algebra thatare more widely useul. Linear algebra

appears to be gaining in importance

within theoretical computer science.Examples include using Fourier analy-sis on the Boolean cube and takingadvantage o the way that graphs andmatrices can be viewed interchange-ably, thereby mixing combinatorialand algebraic pictures. In the uture,

I expect that our view o linear algebraand probability will become increas-ingly shaped by tools rom quantuminormation.

Finally, most o the contributionso quantum inormation to computerscience are at this stage theoretical,since large-scale quantum computershave yet to be built. But once they are,

we can expect them to be used in waysthat theorists will struggle to explain,

just as we see classically with success-ul heuristics such as the simplex algo-

rithm. For example, periodic structurewas exploited by Shors algorithm, andcan be applied to obtain several otherexponential speedups. In the uture,

we might use tools such as period-fnding or exploratory data analysis,much as linear regression is used to-day. Many o these new tools will ini-tially be deeply counter-intuitive, butas we master them, they promise radi-cally new ways o looking at the worldscientifcally.

Further Reading

M. A. Nielsen and I. L. Chuang. Quantum Computation and

Quantum Information. Cambridge University Press, 2001.

J. Preskill. Lecture notes or Ph219. Caliornia Institute o

Technology; http://theory.caltech.edu/~preskill/ph229/

D. Mermin. Lecture notes or CS483. Cornell University;

http://people.ccmr.cornell.edu/~mermin/qcomp/CS483.

html

Biography

Aram Harro w hasnt been able to decide whether he is a

physicist or a computer scientist, with degrees in physics

and math (undergrad) and physics (grad) rom MIT,

ollowed by aculty jobs at the University o Bristol (mathand computer science) and the University o Washington

(computer science). He has, however, always remained a

believer in quantum computing.

Reerences

[1] Harrow, A.W., Hassidim, A., Lloyd, S. Quantum

algorithms or linear systems o equations. Physical

Review Letters103, 15 (2009).

[2] Temme, K. et al. Quantum Metropolis sampling. Nature

471, 7336 (2011), 8790.

2012 ACM 1528-4972/12/03 $10.00

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