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ity. Given any holy grail of mathemativs suvh as Goldlavh’s vonjevture or the Liemann hy pothesis, loth of whivh have resisted resolution for well over a ventury we vould simply ask our vomputer to searvh through all possille proofs and disproofs vontaining up to, say, a lillion sym lols. (6f a proof were muvh longer than that, it is not vlear that we would even want to read it.) 6f quantum vomputers promised suvh godlike mathematival powers, mayle we should expevt them on store shelves at alout the same time as warp-drive generators and antigravity shields. )ut although we should not avvept the usual hype, in my view it is equally misguided to dis miss quantum vomputing as svienve ﬁvtion. 6n stead we should ﬁnd out what the limits of quan tum vomputers are and what they vould really do if we had them. 6n the 26 years sinve physivist Livhard 3eyn man ﬁrst proposed the idea of quantum vomput ing, vomputer svientists have made enormous progress in ﬁguring out what prollems quantum vomputers would le good for. Avvording to our vurrent understanding, they would provide dra mativ speedups for a few speviﬁv prollems suvh as lreaking the vryptographiv vodes that are widely used for monetary transavtions on the 6nternet. 3or other prollems, however suvh as playing vhess, svheduling airline ﬂights and proving theorems evidenve now strongly sug gests that quantum vomputers would suffer from many of the same algorithmiv limitations as today’s vlassival vomputers. Ohese limitations are vompletely separate from the pravtival difﬁ vulties of luilding quantum vomputers, suvh as devoherenve (unwanted interavtion letween a quantum vomputer and its environment, whivh introduves errors). 6n partivular, the lounds on what it is mathemativally possille to program a vomputer to do would apply even if physivists managed to luild a quantum vomputer with no devoherenve at all. Hard, Harder, Hardest 5ow is it that a quantum vomputer vould pro vide speedups for some prollems, suvh as lreak ing vodes, lut not for others? 6sn’t a faster vom puter just a faster vomputer? Ohe answer is no, and to explain why takes one straight to the intel levtual vore of vomputer svienve. 3or vomputer svientists, the vruvial thing alout a prollem is how quivkly the time needed to solve it grows as the prollem size invreases. Ohe time is measured in the numler of elementary steps required ly the algorithm to reavh a solution. 3or example, using the grade svhool method, we van multiply two -digit numlers in an amount of time that grows like the numler of digits squared, 2 (an amount of time said to le “a polynomial in ”). )ut for favtoring a numler into primes, even the most advanved methods known take an amount of time that grows exponentially with the num ler of digits (in partivular, like 2 to the vule root of power). Ohus, favtoring seems intrinsivally harder than multiplying and when we get up to thousands of digits, this differenve matters muvh more than the differenve letween a Commodore 64 and a supervomputer. Ohe kind of prollems that vomputers van solve in a reasonalle amount of time, even for large values of , are those for whivh we have an algorithm that uses a numler of steps that grows KEY CONCEPTS n Quantum computers would exploit the strange rules of quantum mechanics to process information in ways that are impossible on a standard computer. n They would solve certain specif ic problems, such as factoring integers, dramatically faster than we know how to solve them with today’s computers, but analysis suggests that for most problems quantum com puters would surpass conven tional ones only slightly. n Exotic alterations to the known laws of physics yqwnf allow construction of computers that could solve large classes of hard problems efﬁciently. But those alterations seem implausible. In the real world, perhaps the im possibility of efﬁciently solving these problems should be taken as a basic physical principle. i,z5Zyieor“ www.SciAm.com SCIENTIFIC AMERICAN 5KH Quantum Computers © 2008 SCIENTIFIC AMERICAN, INC. KI5 SCIENTIFIC AMERICAN March 2008 as raised to a ﬁxed power, suvh as or 2.5 . Computer svientists vall suvh an algorithm efﬁ vient, and prollems that van le solved ly an ef ﬁvient algorithm are said to le in the vomplexity vlass 5, whivh stands for “polynomial time.” A simple example of a prollem in 5 is: Given a road map, is every town reavhalle from every other town? 5 also vontains some prollems whose efﬁvient solutions are not so olvious, suvh as: Given a whole numler, is it prime (like 13) or vomposite (like 12)? Given a list of whivh men and women are willing to marry one anoth er, is it possille to pair everyone off with a will ing partner? )ut now suppose you are given the dimen sions of various loxes and you want a way to pavk them in your trunk. Br suppose that you are given a map and you want to volor eavh vountry red, llue or green so that no two neigh loring vountries are volored the same. Br that you are given a list of islands vonnevted ly lridg es and you want a tour that visits eavh island ex avtly onve. Although algorithms that are some what letter than trying every possille solution are known for these prollems, no algorithm is known that is fundamentally letter. -very known algorithm will take an amount of time that in vreases exponentially with the prollem size. 6t turns out that the three prollems 6 just list ed have a very interesting property: they are all the “same” prollem, in the sense that an efﬁvient algorithm for any one of them would imply effivient algo rithms for all the others. Mte phen A. Cook of the Qniversity of Ooronto, Livhard Karp of the Qniversity of California, )erkeley, and Leonid Levin, now at )oston Qniversity, arrived at this remarkalle vonvlusion in the 1970s, when they de veloped the theory of N5-vompleteness. N5 stands for “ ondeterministiv oly nomial time.” ,o not worry alout what that means. )asivally, N5 is the vlass of prollems for whivh a solution, onve found, van le revog nized as vorrevt in polynomial time (something like , and so on) even though the solution it self might le hard to ﬁnd. As an example, if you are given a map with thousands of islands and lridges, it may take years to ﬁnd a tour that vis its eavh island onve. Yet if someone shows you a tour, it is easy to vhevk whether that person has suvveeded in solving the prollem. When a prol lem has this property, we say that it is in N5. Ohe vlass N5 vaptures a huge numler of prollems of pravtival interest. Note that all the 5 prollems are also N5 prollems, or to put it another way, the vlass 5 is vontained within the vlass N5. 6f you van solve a prollem quivkly you van also verify the solution quivkly. N5-vomplete prollems are in essenve the hardest of the N5 prollems. Ohey are the ones with the property found ly Cook, Karp and Levin: 6f an efﬁvient algorithm for any one of them were found, it vould le adapted to solve all the other N5 prollems as well. An efﬁvient algorithm for an N5-vomplete prollem would mean that vomputer svientists’ present pivture of the vlasses 5, N5 and N5-vom plete was utterly wrong, levause it would mean that every N5 prollem (invluding all the N5- vomplete ones) was avtually a 5 prollem. 6n oth er words, the vlass 5 would equal the vlass N5, whivh is written 5 = N5. ,oes suvh an algorithm exist? 6s 5 equal to N5? Ohat is literally a million-dollar question it varries a $1,000,000 reward from the Clay Math 6nstitute in Camlridge, Mass. and it has Quantum Computing 101 hysicists are hotly pursuing the construction of quantum computers, which would harness the quirks of quantum mechanics to perform certain computations more efﬁciently than a conventional computer. The fundamental feature of a quantum computer is that it uses qubits instead of bits. A qubit may be a particle such as an electron, with “spin upf ( dnwg ) representing 1, “spin downf tgf ) representing 0, and quantum states called super- positions that involve spin up and spin down simultane ously ( ygnnqy ). A small number of particles in superposition states can carry an enormous amount of informa tion: a mere 1,000 particles can be in a superpo sition that represents every number from 1 to 1,000 (about 10 300 ), and a quantum comput er would manipulate all those numbers in parallel, for instance, by hitting the parti cles with laser pulses. When the particles’ states are measured at the end of the compu tation, however, all but one ran dom version of the 10 300 paral lel states vanish. Clever manipulation of the particles could nonetheless solve certain problems very rapidly, such as factoring a large number. A good quantum computer algo rithm ensures that computational paths leading to a wrong answer cancel out and that paths leading to a correct answer reinforce. 3 2 © 2008 SCIENTIFIC AMERICAN, INC. www.SciAm.com SCIENTIFIC AMERICAN 5K5 played vameo roles on at least three OV shows Ohe Mimpsons, 3uturama and NQM)3LM ). 6n the half a ventury sinve the prollem was revognized, no one has found an efﬁvient algo rithm for an N5-vomplete prollem. Conse quently, vomputer svientists today almost uni versally lelieve 5 does not equal N5, or 5 N5, even if we are not yet smart enough to under stand why this is or to prove it as a theorem. What the Quantum Can Do 6f we grant that 5 N5, then only one hope remains for solving N5-vomplete prollems in polynomial time: namely, to lroaden what we mean ly “vomputer.” At ﬁrst sight, quantum mevhanivs would appear to provide just the kind of resourves needed. Kuantum mevhanivs makes it possille to store and manipulate a vast amount of information in the states of a relatively small numler of partivles. Oo see how this vomes alout, imagine that we have 1,000 partivles and that eavh partivle, when measured, van le found to le either spinning up or spinning down. 3or our purposes, what it means for a partivle to spin up or down is irrelevant; all that matters is that there is some property of the partivle that ha s one of two values when measured. Oo desvrile the quantum state of this vollev tion of partivles, one must spevify a numler for every possille result of measuring the partivles. Ohese numlers are valled the amplitudes of the possille outvomes and relate to eavh outvome’s prolalility, lut unlike prolalilities, quantum amplitudes van le positive or negative (in favt, they are vomplex numlers). 3or example, an amplitude is needed for the possilility that all 1,000 partivles will le found spinning up, an other amplitude for the possilility of ﬁnding that the ﬁrst 500 partivles are spinning up and that the remaining 500 are spinning down, and so on. Ohere are 2 1,000 possille outvomes, or alout 10 300 , so that is how many numlers are needed more than there are atoms in the visille universe! Ohe tevhnival terminology for this sit uation is that the 1,000 partivles are in a super position of those 10 300 states. 5ut another way, we van store 10 300 numlers on our 1,000 partivles simultaneously. Ohen, ly performing various operations on the partivles and on some auxiliary ones perhaps hitting them with a sequenve of laser pulses or radio waves we van varry out an algorithm that transforms all 10 300 numlers (eavh one a poten tial solution) at the same time. 6f at the end of do ing that we vould read out the partivles’ ﬁnal quantum state avvurately, we really would have a magiv vomputer: it would le alle to vhevk 10 300 possille solutions to a prollem, and at the end we vould quivkly disvern the right one. Qnfortunately, there is a vatvh. When the par tivles are measured (as is nevessary to read out their ﬁnal state), the rules of quantum mevhan ivs divtate that the measurement will pivk out just one of the 10 300 possililities at random and that all the others will then disappear. (Oo go lavk to the quantum slavks developed at 5aggar, if you tried to wear them you would ﬁnd yourself in ei ther formal or vasual attire, not loth.) We would seem to le no letter off than if we used a vlassi val vomputer and tried out one randomly vhosen possille solution in either vase, we end up knowing alout only one suvh possille solution. 5appily, we still have trivks we van play to wring some advantage out of the quantum par tivles. Amplitudes van vanvel out when positive ones vomline with negative ones, a phenomenon The Good News I f a large, ideal quantum computer would face most of the same limitations as our present-day classical computers do, should the physicists working on the extraordinarily hard task of building even rudimentary quantum computers pack up and go homeM I believe the answer is no, for four reasons. If quantum computers ever become a reality, the “killer appf for them will most likely not be code breaking but rather something so obvious it is rarely even mentioned: sim ulating quantum physics. This is a fundamental problem for chemistry, nanotechnology and other ﬁelds, important enough that Nobel Prizes have been awarded even for par tial progress. As transistors in microchips approach the atomic scale, ideas from quantum computing are likely to become relevant for classical computing as well. Quantum computing experiments focus attention directly on the most mystifying features of quantum mechanics and I hope that the less we can sweep those puzzles under the rug, the more we will be forced to under stand them. Quantum computing can be seen as the most stringent test to which quantum mechanics itself has ever been subjected. In my opinion, the most exciting possible outcome of quantum computing research would be to discover a fun damental reason why quantum com puters are pqv possible. Such a failure would overturn our current picture of the physical world, whereas success would merely conﬁrm it. U0C0 The “killer appf for quantum computers will most likely be simulating quantum physics. © 2008 SCIENTIFIC AMERICAN, INC. valled destruvtive interferenve. Mo a good quan tum vomputer algorithm would ensure that vom putational paths leading to a wrong answer would vanvel out in this way. 6t would also en sure that the paths leading to a vorrevt answer would all have amplitudes with the same sign whivh yields vonstruvtive interferenve and there ly loosts the prolalility of ﬁnding them when the partivles are measured at the end. 3or whivh vomputational prollems van we vhoreograph this sort of interferenve, using few er steps than it would take to solve the prollem vlassivally? 6n 1994 5eter Mhor, now at the Massavhu setts 6nstitute of Oevhnology, found the ﬁrst ex ample of a quantum algorithm that vould dra mativally speed up the solution of a pravtival prollem. 6n partivular, Mhor showed how a quantum vomputer vould favtor an -digit num ler using a numler of steps that invreases only as alout in other words, in polynomial time. As mentioned earlier, the lest algorithm known for vlassival vomputers uses a numler of steps that invreases exponentially. Black Boxes Mo at least for favtoring, one really van get an exponential speedup over known vlassival algo rithms ly using quantum methods. )ut despite a widespread misvonveption to the vontrary, the favtoring prollem is neither known nor lelieved to le N5-vomplete. Oo vreate his algorithm, Mhor exploited vertain mathematival properties of vomposite numlers and their favtors that are partivularly well suited to produving the kind of vonstruvtive and destruvtive interferenve that a quantum vomputer van thrive on. Ohe N5-vom plete prollems do not seem to share those spe vial properties. Oo this day, researvhers have found only a few other quantum algorithms that appear to provide a speedup from exponential to polynomial time for a prollem. Ohe question thus remains unanswered: 6s there an efﬁvient quantum algorithm to solve N5-vomplete prollems? ,espite muvh trying, no suvh algorithm has leen found though not surprisingly, vomputer svientists vannot prove that it does not exist. After all, we vannot even prove that there is no polynomial-time vlassival algorithm to solve N5-vomplete prollems. What we van say is that a quantum algorithm vapalle of solving N5-vomplete prollems efﬁ viently would, like Mhor’s algorithm, have to ex ploit the prollems’ struvture, lut in a way that is far leyond present-day tevhniques. Bne vannot What Classical Computers Can and Cannot Do C omputer scientists categorize problems according to how many computational steps it would take to solve a large example of the problem using the best algorithm known. The problems are grouped into broad, overlapping classes based on their difﬁculty. Three of the most important classes are listed below. Contrary to myth, quantum computers are not known to be able to solve efﬁciently the very hard class called NP-complete problems. P PROBLEMS : Ones computers can solve efﬁciently, in polynomial time Example: Given a road map showing p towns, can you get from any town to every other townM For a large value of p. the number of steps a computer needs to solve this problem increases in proportion to , a polynomial. Because polynomials increase relatively slowly as p increases, computers can solve even very large P problems within a reasonable length of time. NP PROBLEMS : Ones whose solutions are easy to verify Example: You know an -digit number is the product of two large prime numbers, and you want to ﬁnd those prime factors. If you are given the factors, you can verify that they are the answer in polynomial time by multiplying them. Every P problem is also an NP problem, so the class NP contains the class P within it. The factoring problem is in NP but conjectured to be outside of P, because no known algorithm for a standard computer can solve it in only a polynomial number of steps. Instead the number of steps increases exponentially as p gets bigger. NP-COMPLETE PROBLEMS : An efﬁcient solution to one would provide an efﬁcient solution to all NP challenges Example: Given a map, can you color it using only three colors so that no neighboring countries are the same colorM If you had an algorithm to solve this problem, you could adapt the algo rithm to solve any other NP problem (such as the factoring problem above or determining if you can pack p boxes of various sizes into a trunk of a certain size) in about the same number of steps. In that sense, NP-complete problems are the hardest of the NP problems. No known algorithm can solve an NP- complete problem efﬁciently. March 2008 © 2008 SCIENTIFIC AMERICAN, INC. www.SciAm.com SCIENTIFIC AMERICAN 5KL avhieve an exponential speedup ly treating the prollems as struvtureless “llavk loxes,” von sisting of an exponential numler of solutions to le tested in parallel. Mome speedup van nonethe less le wrung out of this llavk lox approavh, and vomputer svientists have determined just how good and how limited that speedup is. Ohe algorithm that produves the speedup is the sevond major quantum algorithm. Ohe llavk lox approavh van le illustrated ly pretending that you are searvhing for the solu tion to a difﬁvult prollem and that the only op eration you know how to perform is to guess a solution and see if it works. Let us say there are possille solutions, where grows exponential ly as the prollem size n invreases. You might get luvky and guess the solution on your ﬁrst try, lut in the worst vase you will need tries, and on av erage you will need /2. Now suppose you van ask alout all the pos sille solutions in quantum superposition. 6n 1996 Lov Grover of )ell Laloratories developed an algorithm to ﬁnd the vorrevt solution in suvh a svenario using only alout steps. A speedup from /2 to is a useful advanve for some prollems if there are a million possille solu tions, you will need around 1,000 steps instead of 500,000. )ut the square root does not trans form an exponential time into a polynomial time; it just produves a smaller exponential. And Gro ver’s algorithm is as good as it gets for this kind of llavk lox searvhing: in 1994 researvhers had shown that any llavk lox quantum algorithm needs at least steps. Bver the past devade, researvhers have shown that similar modest speedups are the limit for many other prollems lesides searvh ing a list, suvh as vounting lallots in an elevtion, ﬁnding the shortest route on a map, and playing games of strategy suvh as vhess or Go. Bne prollem that presented partivular difﬁvulty was the so-valled vollision prollem, the prollem of ﬁnding two items that are identival, or that “vollide,” in a long list. 6f there were a fast quan tum algorithm to solve this prollem, many of the lasiv luilding llovks of sevure elevtroniv vommerve would le useless in a world with quantum vomputers. Mearvhing a list for an item is like looking for a needle in a haystavk, whereas searvhing for a vollision is like looking for two identival pieves of hay, whivh provides the prollem with a kind of struvture that a quantum vomputer vould potentially exploit. Nevertheless, 6 showed in 2002 that within the llavk lox model, any quantum algorithm needs exponential time to solve the vollision prollem. Admittedly, these llavk lox limitations do Where Quantum Computers Fit In T he map at the right depicts how the class of problems that quantum computers would solve efﬁciently (BQP) might relate to other fundamental classes of computational problems. (The irregular border signiﬁes that BQP does not seem to ﬁt neatly with the other classes.) The BQP class (the letters stand for ounded-error, uantum, olynomial time) includes all the P problems and also a few other NP problems, such as factoring and the so-called discrete logarithm problem. Most other NP and all NP-complete problems are believed to be outside BQP, meaning that even a quantum computer would require more than a polynomial number of steps to solve them. In addition, BQP might protrude beyond NP, meaning that quantum computers could solve certain problems faster than classical computers could even check the answer. (Recall that a conventional computer can efﬁciently verify the answer of an NP problem but can efﬁciently solve only the P problems.) To date, however, no convincing example of such a problem is known. Computer scientists do know that BQP cannot extend outside the class known as PSPACE, which also contains all the NP problems. PSPACE problems are those that a conventional computer can solve using only a polynomial amount of memory but possibly requiring an exponential number of steps. PSPACE NP- complete NP BQP Box packing Map coloring Traveling salesman p p Sudoku p p chess p p Go Graph isomorphism Graph connectivity Testing if a number is a prime Matchmaking Factoring Discrete logarithm EXAMPLE PROBLEMS 5ASON DORFMAN ( ctqpuqp [THE AUTHOR] Scott Aaronson is an assistant professor of electrical engineering and computer science at the Massachusetts Institute of Technol ogy. A high school dropout, he went on to receive a bachelor’s degree from Cornell University and a Ph.D. in computer science from the University of California, Berke ley, advised by Umesh Vazirani. Outside of research, Aaronson is best known for his widely read blog (www.scottaaronson.com/blog), as well as for creating the Complexity Zoo (www.complexityzoo.com), an online encyclopedia of more than G00 complexity classes. Harder Efﬁciently solved by quantum computer Efﬁciently solved by classical computer © 2008 SCIENTIFIC AMERICAN, INC. not rule out the possilility that efﬁvient quan tum algorithms for N5-vomplete or even harder prollems are waiting to le disvovered. 6f suvh algorithms existed, however, they would have to exploit the prollems’ struvture in ways that are unlike anything we have seen, in muvh the same way that efﬁvient vlassival algorithms for the same prollems would have to. Kuantum magiv ly itself is not going to do the jol. )ased on this insight, many vomputer svientists now vonjev ture not only that 5 N5 lut also that quantum vomputers vannot solve N5-vomplete prollems in polynomial time. Magical Theories -verything we know is vonsistent with the pos silility that quantum vomputers are the end of the line that is, that they are the most general kind of vomputer vompatille with the laws of physivs. )ut physivists do not yet have a ﬁnal the ory of physivs, so one vannot rule out the possi lility that someday a future theory might reveal a physival means to solve N5-vomplete prollems efﬁviently. As you would expevt, people spevu late alout yet more powerful kinds of vomput ers, some of whivh would make quantum vom puters look as pedestrian as vending mavhines. All of them, however, would rely on spevulative vhanges to the laws of physivs. Bne of the ventral features of quantum me vhanivs is a mathematival property valled linear ity. 6n 199– ,aniel M. Alrams and Meth Lloyd, loth then at M.6.O., showed that if a small non linear term is added to the equations of quantum mevhanivs, quantum vomputers would le alle to solve N5-vomplete prollems efﬁviently. )e fore you get too exvited, you should realize that if suvh a nonlinear term existed, then one vould also violate 5eisenlerg’s unvertainty prinviple and send signals faster than the speed of light. As Alrams and Lloyd pointed out, perhaps the lest interpretation of these results is that they help to explain why quantum mevhanivs is linear. Another spevulative type of mavhine would avhieve extravagant vomputational alilities ly vramming an inﬁnite numler of steps into a ﬁ nite time. Qnfortunately, avvording to physi vists’ vurrent understanding, time seems to de generate into a sea of quantum ﬂuvtuations something like a foam instead of a uniform smooth line on the svale of 10 –43 sevond (the 5lanvk time), whivh would seem to make this kind of mavhine impossille. 6f time vannot le slived with arlitrary thin ness, then perhaps another way to solve N5- vomplete prollems efﬁviently would le to ex ploit time travel. 5hysivists studying the issue talk not alout time mavhines lut alout vlosed timelike vurves (COCs). 6n essenve a COC is a route through spave and time that matter or en ergy vould travel along to meet up with itself in the past, forming a vlosed loop. Current physi val theory is invonvlusive on whether COCs van exist, lut that need not stop us from asking what the vonsequenves would le for vomputer svienve if they did exist. 6t seems olvious how one vould use a COC to speed up a vomputation: program your vomput er to take however long it needs to solve the prollem and then send the answer lavk in time to yourself at a point lefore the vomputer start ed. Alas, this simple idea does not work, levause it ignores the famous grandfather paradox, where you go lavk in time to kill your own grandfather (lut then you are never lorn, so you never go lavk in time, and so your grandfather lives to have vhildren after all, and later you are lorn, lut then ...). 6n our setting, what would happen if you turned off the vomputer after you reveived its answer from the future? 6n 1991 physivist ,avid ,eutsvh of the Qniver sity of Bxford deﬁned a model of vomputation with COCs that avoids this difﬁvulty. 6n ,eutsvh’s model, nature will ensure that as events unfold along the virvular timeline that makes up the COC, no paradoxes ever arise, a favt that van le exploited to program a vomputer that loops Über-Computers from Exotic PhysicsM A lthough quantum computers seem unlikely to solve NP-complete problems quickly, certain other extraordinary, speculative physical processes would allow construction of computers with that ability and much more. Time travel, for instance, would make it pos sible to efﬁciently solve any PSPACE problem, including those hard er than NP-complete ones such as how to play the perfect game of chess on any size board, including those larger than the standard 8 8 version. Employing time travel to solve problems would not be as simple as having a computer ﬁnish a long computation in the far future and send the answer back to itself in the present, but that kind of loop in time would be exploit ed. 5ust one problem: the speculative process es defy the known laws of physics. March 2008 ZONES OF THOUGHT Unlike the real world, in which computational limits are believed to be the same everywhere, the galaxy in Vernor Vinge’s 1992 science-ﬁction novel C Hktg Wrqp vjg Fggr is divided into concen tric “zones of thoughtf having different inherent computational and technological limits. In the Unthinking Depths , nearest the galactic core, even simple automation fails and IQs plummet. The Slow Zone contains Earth and is as limited as we know it. In the Beyond , nearly sentient nanotechnology factories construct wonders such as anti-gravity fabrics, and hypercomputation enables faster-than-light travel. The Transcend is populated by dangerous, godlike über- intelligences having technologies and thought processes unfathomable to lower beings. © 2008 SCIENTIFIC AMERICAN, INC. www.SciAm.com SCIENTIFIC AMERICAN 5KN around inside the COC to solve hard prollems. 6ndeed, ly using a COC, we vould efﬁviently solve not only N5 prollems lut even prollems in an apparently larger vlass valled 5M5AC-. 5M5AC- is the vlass of prollems that vould le solved on a vonventional vomputer using a poly nomial amount of memory lut possilly taking an exponential amount of time. 6n effevt, a COC would make time and spave intervhangealle as vomputational resourves. (6 did not have to men tion the polynomial memory vonstraint until now, levause for 5 and N5 prollems it makes no differenve if the vomputer has avvess to more than polynomial memory.) Levently ;ohn Wa trous of the Qniversity of Waterloo in Bntario and 6 showed that using a quantum vomputer in a COC instead of a vonventional one does not enalle anything leyond 5M5AC- to le efﬁvient ly solved. 6n other words, if COCs exist, then quantum vomputers are no more powerful than vlassival ones. Computational Kryptonite 5hysivists do not know if future theories will per mit any of these extraordinary mavhines. Yet without denying our ignoranve, we van view that ignoranve from a different perspevtive. 6nstead of starting from physival theories and then ask ing alout their vomputational implivations, we vould start ly assuming that N5-vomplete prol lems are hard and then study the vonsequenves of that assumption for physivs. 3or instanve, if COCs would let us solve N5-vomplete prollems efﬁviently, then ly starting from the assumption that N5-vomplete prollems are intravtalle, we vould vonvlude that COCs vannot exist. Oo some, suvh an approavh will seem overly dogmativ. Oo me, it is no different from assum ing the sevond law of thermodynamivs or the impossilility of faster-than-light vommuniva tion two earlier limitations on tevhnology that over time earned the status of physival prinvi ples. Yes, the sevond law might le experimen tally falsiﬁed tomorrow lut until that hap pens, physivists ﬁnd it vastly more useful to as sume it is vorrevt and then use that assumption for studying everything from var engines to llavk holes. 6 predivt that the hardness of N5- vomplete prollems will someday le seen the same way: as a fundamental prinviple that de svriles part of the essential nature of our uni verse. Ohere is no way of telling what theoreti val enlightenment or what pravtival vonse quenves might vome from future applivation of this kind of fundamental prinviple. 6n the meantime, we know not to expevt mag iv from quantum vomputers. Oo some, the ap parent limitations of quantum vomputers might vome as a letdown. Bne van, however, give those same limitations a more optimistiv spin. Ohey mean that although vertain vryptographiv vodes vould le lroken in a world with quantum vom puters, other vodes would prolally remain se vure. Ohey invrease our vonﬁdenve that quan tum vomputing will le possille at all levause the more a proposed tevhnology sounds like a svienve-ﬁvtion varivature, the more skeptival we should le. (Who would you le more invlined to lelieve: the salesperson offering a devive that produves unlimited free energy from the quan tum vavuum or the one offering a refrigerator that is more efﬁvient than last year’s model?) And last, suvh limitations ensure that vomputer svi entists will vontinue to have their work vut out for them in designing new quantum algorithms. Like Avhilles without his heel or Muperman with out kryptonite, a vomputer without any limita tions would get loring pretty quivkly. MORE TO EXPLORE Quantum Computation and Quantum Information. Michael A. Nielsen and Isaac L. Chuang. Cambridge University Press, 2000. NP-Complete Problems and Physical Reality. Scott Aaronson in CEO UKICEV Pgyu. Complexity Theory Column, Vol. 36, No. 1, pages 30–52; March 2005. Avail able at www.scottaaronson.com/ papers/npcomplete.pdf Quantum Computer Science: An Introduction. N. David Mermin. Cambridge University Press, 2007. Shor, I’ll Do It. (An explanation of Shor’s algorithm for the layperson.) Scott Aaronson. Available at www. scottaaronson.com/blog/MpL208 Quantum Computing since Dem ocritus. Lecture notes from course PHYS771, University of Waterloo, Fall 2006. Available at www. scottaaronson.com/democritus/ A New Physical PrincipleM B ecause implausible kinds of physics (such as time travel) seem neces sary for constructing a computer able to solve NP-complete problems quickly, I predict that scientists might one day adopt a new principle: “NP- complete problems are hard.f That is, solving those problems efﬁciently is impos sible on any device that could be built in the real world, whatever the ﬁnal laws of physics turn out to be. The principle implies that time travel is impossible, because such travel would enable creation of über-computers that could solve NP-complete problems efﬁciently. Further, if a proposed theory were shown to permit such computers, that theo ry could be ruled out. Application of the principle would be analogous to applying the laws of thermodynamics to conclude that perpetual-motion machines are impossible (the laws of thermodynamics forbid them) and to deduce previ ously unknown features of physical processes. U0C0 1QTG CDQWV PQPNKPGCT SWCPVWO OGE5CPKEUx 5[RGTEQORWVKPIx WUG QH VKOG VTCXGNx CPF CPQV5GT UE5GOG ECNNGF CPV5TQRKE EQORWVKPI ECP DG HQWPF CV YYYt7EKAOtEQO$QPV5GYGD © 2008 SCIENTIFIC AMERICAN, INC. aggar 5hysivists ,evelop ‘Kuantum Mlavks,’ ” read a headline in the satirival weekly the Bnion . )y exploiting a lizarre “Mvhrödinger’s 5ants” duality, the artivle explained, these non-Newtonian pants vould paradoxivally lehave like formal wear and vasual wear at the same time. Bnion writers were apparently spooﬁng the lreathless artivles alout quantum vomputing that have ﬁlled the popular svienve press for a devade. A vommon mistake see for instanve the 3elruary 15, 2007, issue of the -vonomist is to vlaim that, in prinviple, quantum vomputers vould rapidly solve a partivularly difﬁvult set of mathematival vhallenges valled N5-vomplete prollems, whivh even the lest existing vomputers vannot solve quivkly (so far as anyone knows). Kuantum vomputers would supposedly avhieve this feat not ly leing formal and vasual at the same time lut ly having hardware vapalle of provessing every pos sille answer simultaneously. 6f we really vould luild a magiv vomputer vapalle of solving an N5- vomplete prollem in a snap, the world would le a very different plave: we vould ask our magiv vomputer to look for whatever patterns might exist in stovk-market data or in revordings of the weather or lrain avtiv ity. Qnlike with today’s vomputers, ﬁnding these patterns would le vom pletely routine and require no detailed understanding of the suljevt of the prollem. Ohe magiv vomputer vould also automate mathematival vreativ y Scott Aaronson Quantum computers would be exceptionally fast at a few speciﬁc tasks, but it appears that for most problems they would outclass today’s computers only modestly. This realization may lead to a new fundamental physical principle INFORMATION TECHNOLOGY KG March 2008 ILLUSTRATIONS BY DU AN PETRIC - IC THE LIMITS OF Quantum Computers © 2008 SCIENTIFIC AMERICAN, INC.

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