Alแn Aspuru-Guzik,Anthony Dutoi, Peter Love, and Martin Head-Gordon report on their workin the 9 September issue of the journal Science. Head-Gordon is a staffscientist in Berkeley Lab's Chemical Sciences Division and a professorof chemistry at UC Berkeley; Aspuru-Guzik is a postdoctoral fellow andDutoi a graduate student in the Head-Gordon group. Love is a seniorapplications scientist on the staff of D-Wave Systems, Inc. inVancouver, B.C.

The researchers developed a quantum-computationalalgorithm and ran it on a classical computer to demonstrate thatquantum computers comprised of only tens or a few hundreds of quantumbits (qubits) could calculate significant information about realmolecular systems to high accuracy. Thus a relatively small quantumcomputer could surpass the most powerful quantum-chemistry calculationspossible with today's classical supercomputers.

"What we havedone is demonstrate — by using a quantum algorithm to determine thestates of minimum energy for two real molecules — that quantumcomputing can deliver on the promise of giving highly accuratepractical solutions to interesting chemical problems," saysAspuru-Guzik.

**Confronting virtually unsolvable problems**

The Head-Gordon group concentrates on calculating the electronicstructure of molecules from first principles — that is, from aquantum-mechanical description of the states of all the particles inthe system. Electronic structure calculations allow scientists topredict how molecules react with other molecules and are key tounderstanding and controlling their physical and chemical properties.

Thepractical challenge of such calculations was famously expressed by PaulDirac in 1929, who remarked of quantum mechanics that "The underlyingphysical laws necessary for the mathematical theory of a large part ofphysics and the whole of chemistry are thus completely known, and thedifficulty is only that the exact application of these laws leads toequations much too complicated to be soluble."

Indeed, exactsolutions of the Schr๖dinger equation, the fundamental expression ofquantum mechanics, are so complicated that classical computers are onlyable to exactly solve very small molecules, about the size of water,because the time needed for computation increases exponentially withsize. Practical calculations on real molecules are performed usingapproximations such as density functional theories. These are usefuland usually accurate, but nonetheless are still approximations, whichcan sometimes fail. As long ago as 1982 Richard Feynman suggested thatan easier way to calculate a quantum system might be by using quantumcomputers.

Unlike classical computing, where each bit representseither a 0 or a 1 but not both at once, a quantum bit simultaneouslysuperposes 0 and 1 and only resolves (or "collapses") to a single valuewhen measured. While a classical computer operates serially,essentially dealing with one bit after another, a quantum computer'squbits interact to form very large computational spaces that, whenmeasured, quickly deliver the solution to a complex problem.

Variousphysical systems have been used to perform quantum computations, but noone has yet built a quantum computer large enough to compete withclassical computers. Hardware is only part of the challenge. Another isdevising practical algorithms that can run on quantum computers; inprinciple these can be run — if much more slowly — on classicalsimulations of quantum computers, provided only a few qubits areinvolved.

Aspuru-Guzik calls this the Russian doll approach: "Youbegin with the physical system you want to describe — that's thebiggest doll, with the most information. Inside that is the basicequation that describes the system. Inside that is an 'emulation' ofthe system using a quantum computer. And inside that is a simulation ofthe quantum computer on a classical computer."

**The limits to computation**

Classical supercomputers are hardly tiny; rather they are limited bythe number of operations they can manage within a reasonable time. Onlyvery small molecular systems have been solved exactly from firstprinciples, because the orbital states of each particle in the systemmust be represented in what's known as a basis set, which in a moleculewith many electrons is very large indeed. As the size of the systemincreases, the number of calculations — and thus the time needed tosolve the problem — increases exponentially (the larger the numbergets, the faster it grows).

Using quantum algorithms on a quantumcomputer, however, the number of calculations (and thus the time) growsonly polynomially — faster than linearly, but still "efficiently" — asthe size of the basis set grows.

Aspuru-Guzik says, "We chose tocalculate water and lithium hydride because three-atom water is agood-sized molecule with a small basis set, while two-atom lithiumhydride is a small molecule but has a comparatively large basis set."

Twofactors were key to the group's success. One was finding an efficientway to achieve the essential starting point of any calculation, anapproximation of the ground-state energy sufficiently close to theactual state — the process of "preparing the state that you willsimulate," as Aspuru-Guzik puts it. The researchers showed that byusing a method called adiabatic state preparation (ASP), even arelatively crude initial estimate was practical.

"'Adiabatic' inthis context means reiterating approximations of the state slowly," hesays. "How fast you can prepare the state is determined by the gapbetween the ground state of the molecule and its lowest excited state.We found a way to keep this gap large." The researchers confirmed theaccuracy of the ASP method by calculating the ground state of thetwo-electron hydrogen molecule (H 2).

Even more important wastheir adaptation of a quantum algorithm called a phase estimationalgorithm (PEA), proposed by Daniel Abrams and Seth Lloyd six yearsago. The original version required a read-out register of about 20qubits — prohibitively large for early quantum computers. By modifyingPEA so that it performed recursively, approaching greater accuracy witheach repeated calculation, the researchers reduced the size of theread-out register to a manageable four qubits.

**Don't try this at home**

Applying these and other measures, the researchers were able tosimulate a quantum computer's calculation of the ground states of waterand lithium hydride with accuracy to six decimal places. A real quantumcomputer could have performed the same calculations almost instantly.Its classical simulacrum, however, was an order of magnitude lessefficient than the best conventional methods now available, leading theauthors to emphasize that "while possible as experiments, suchsimulations are not competitive as an alternative" to what peoplealready do on classical computers.

"In other words, we're sayingdon't try this at home," says Aspuru-Guzik. "What we've done isillustrate the truth of the conjecture that to exceed the limits ofclassical computing, quantum algorithms running with at least 40 to 100qubits are needed."

The members of Head-Gordon's group atBerkeley Lab and UC Berkeley continue to explore new theoreticalapproaches and define more specifics for the design of practicalalgorithms to enable quantum chemistry on quantum computers. Making thevast power and speed of quantum computers available to industrialcustomers is the goal of Vancouver's D-Wave Systems, Inc., theHead-Gordon group's research partner and sponsor in the work.

"Simulatedquantum computation of molecular energies," by Alแn Aspuru-Guzik,Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon, appears in the9 September 2005 edition of Science.

Berkeley Lab is a U.S.Department of Energy national laboratory located in Berkeley,California. It conducts unclassified scientific research and is managedby the University of California. Visit our website at http://www.lbl.gov.

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