Universal Quantum Simulators

Universal Quantum Simulators
Author(s): Seth Lloyd
Source: Science, New Series, Vol. 273, No. 5278 (Aug. 23, 1996), pp. 1073-1078
Published by: American Association for the Advancement of Science
Stable URL: http://www.jstor.org/stable/2899535 .
Accessed: 08/10/2014 15:37
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
.
American Association for the Advancement of Science is collaborating with JSTOR to digitize, preserve and
extend access to Science.
http://www.jstor.org
This content downloaded from 128.104.46.206 on Wed, 8 Oct 2014 15:37:19 PM
All use subject to JSTOR Terms and Conditions
RESEARCHARTICLES
33. G. Felsenfeld et al., J. Am. Chem. Soc. 79, 2023
(1957); A. G. Letai et al., Biochemistry 27, 9108
(1988).
34. M. Riley, Microbiol.Rev. 57, 862 (1993).
35. Supported in part by Department of Energy Cooperative Agreements DE-FC02-95ER61962 (J.C.V.)
and DEFC02-95ER61963 (C.R.W. and G.J.O),
NASA grant NAGW2554 (C.R.W.),and a core grant
to TIGRfrom Human Genome Sciences. G.J.O. is
the recipient of the National Science Foundation
Presidential Young Investigator Award (DIR 8957026). M.B. is supported by National Institutes of
Health grant GM00783. We thank M. Heaney, C.
Gnehm, R. Shirley, J. Slagel, and W. Hayes for soft-
ARTICLES
URESEARCH
Universal
. ... .E
..-.a
........i.....E.E.....
Quantum
Simulators
Seth Lloyd
Feynman's 1982 conjecture, that quantum computers can be programmed to simulate
any local quantum system, is shown to be correct.
Over the past half century, the logical
devices by which computersstore and process informationhave shrunkby a factorof
2 every2 years.A quantumcomputeris the
end point of this process of miniaturization-when devices become sufficiently
small, their behavioris governedby quantum mechanics. Informationin conventional digitalcomputersis storedon capacitors.An unchargedcapacitorregistersa 0
and a chargedcapacitorregistersa 1. Informationin a quantumcomputeris storedon
individual spins, photons, or atoms. An
atom can itself be thoughtof as a tiny capacitor.An atom in its groundstate is analogousto an unchargedcapacitorand can be
takento registera 0, whereasan atom in an
excited state is analogousto a chargedcapacitorand can be taken to registera 1.
So far, quantumcomputerssound very
muchlike classicalcomputers;the only use
of quantummechanicshas been to make a
correspondencebetweenthe discretequantum states of spins, photons, or atoms and
the discretelogical states of a digital computer. Quantumsystems,however,exhibit
behavior that has no classical analog. In
particular,unlike classical systems, quantum systemscan exist in superpositionsof
differentdiscretestates.An ordinarycapacitorcan be eitherchargedor uncharged,but
not both:A classicalbit is either 0 or 1. In
contrast,an atom in a quantumsuperposition of its ground and excited state is a
quantum bit that in some sense registers
both 0 and 1 at the same time. As a result,
quantumcomputerscan do thingsthat classical computerscannot.
Classical computerssolve problemsby
using nonlineardevices such as transistors
to performelementarylogicaloperationson
The author is at the D'ArbeloffLaboratoryfor Information
Systems and Technology, Department of Mechanical En7
gineering, Massachusetts Instituteof Technology, Cambridge, MA 02139, USA. E-mail:[email protected]
the bits stored on capacitors. Quantum
computerscan also solve problems in a
similarfashion;nonlinear interactionsbetween quantumvariablescan be exploited
to performelementaryquantumlogical operations.However,in additionto ordinary
classical logical operationssuch as AND,
NOT, and COPY,quantumlogic includes
operationsthat put quantumbits in superpositionsof 0 and 1. Becausequantumcomputerscan performordinarydigital logic as
well as exotic quantumlogic, they are in
principle at least as powerfulas classical
computers.Just what problems quantum
computerscan solve more efficiently than
classicalcomputersis an open question.
Since their introduction in 1980 (1)
quantumcomputershave been investigated
extensively (2-29). A comprehensivereview can be foundin (15). The best known
problem that quantumcomputerscan in
principlesolve moreefficientlythan classical computersis factoring(14). In this article I presentanothertype of problemthat
in principlequantumcomputerscouldsolve
moreefficientlythan a classicalcomputerthat of simulatingotherquantumsystems.In
1982, Feynmanconjecturedthat quantum
computersmight be able to simulateother
quantumsystemsmoreefficientlythan classical computers(2). Quantumsimulationis
thus the first classicallydifficult problem
posedfor quantumcomputers.Here I show
that a quantumcomputercan in fact simulate quantumsystemsefficientlyas long as
they evolve accordingto local interactions.
Feynman noted that simulatingquantum systemson classicalcomputersis hard.
Over the past 50 years, a considerable
amountof efforthas been devoted to such
simulation.Muchinformationabouta quantum system's dynamics can be extracted
from semiclassicalapproximations(when
classicalsolutionsare known), and ground
state propertiesand correlationfunctions
ware and database support; T. Dixon and V. Sapiro
for computer system support; K. Hong and B. Stader for laboratoryassistance; and B. Mukhopadhyay
for helpfuldiscussions. The M.jannaschii source accession number is DSM 2661, and the cells were a
gift from P. Haney (Departmentof Microbiology,University of Illinois).
can be extractedwith MonteCarlomethods
(30-32). Such methods use amounts of
computertime and memoryspacethat grow
as polynomialfunctionsof the size of the
quantumsystem of interest (where size is
measuredby the numberof variables-particles or lattice sites, for example-required
to characterizethe system). Problemsthat
can be solved by methodsthat use polynomialamountsof computationalresourcesare
commonly called tractable;problemsthat
can only be solved by methods that use
exponentialamountsof resourcesare commonly called intractable.Feynmanpointed
out that the problemof simulatingthe full
quantumsystems
time evolutionof arbitrary
on a classicalcomputeris intractable:The
states of a quantumsystemare wave functionsthat lie in a vectorspacewhosedimension growsexponentiallywith the sizeof the
system.As a result, it is an exponentially
difficultproblemmerelyto recordthe state
of a quantumsystem,let alone integrateits
equationsof motion.Forexample,to record
the state of 40 spin-1/2particles in a classical
1012
computer's memory requires 240
numbers, whereas to calculate their time
evolution requires the exponentiation of a
240 X 240 matrixwith 1024 entries.Feynman asked whether it might be possible to
bypass this exponential explosion by having
one quantum system simulate another directly, so that the states of the simulator
obey the same equations of motion as the
states of the simulated system. Feynman
gave simple examples of one quantum system simulating another and conjectured
that there existed a class of universal quantum simulators capable of simulating any
quantum system that evolved according to
local interactions.
The answer to Feynman's question is,
yes. I will show that a variety of quantum
systems, including quantum computers, can
be "programmed"to simulate the behavior
of arbitraryquantum systems whose dynamics are determined by local interactions.
The programming is accomplished by inducing interactions between the variables
of the simulator that imnitatethe interactions between the variables of the system to
be simulated. In effect, the dynamics of the
properly programmedsimulator and the dynamics of the system to be simulated are
one and the same to within any desired
accuracy. So, to simulate the time evolution
of 40 spin-1/2particles over time t requires a
simulator with 40 quantum bits evolving
SCIENCE * VOL.273 * 23 AUGUST 1996
This content downloaded from 128.104.46.206 on Wed, 8 Oct 2014 15:37:19 PM
All use subject to JSTOR Terms and Conditions
1073
over a periodof time proportionalto t (Fig.
1). QuantumcomputersareFeynman'suniversalquantumsimulators.
To factora 100-digitnumberusingShor's
algorithm,a quantumcomputerwouldhave
to performmillionsof quantumlogic operations coherentlyand without errors.As a
result,despitethe recentinventionof quantum error-correctingroutines (29), constructinga quantumcomputerthat can factor largenumbersis likelyto provedifficult
(11, 19, 25-28). With the methodsdetailed
here, in contrast,a quantumcomputerneed
only performa few tens or hundredsof operationsto simulatea quantumsystem,such
as the set of spins describedabove, that
wouldtake a classicalcomputerAvogadro's
numberof operationsto simulate.Nor do the
quantumoperationsneed to be performed
entirelycoherentlyor withouterrors.In fact,
decoherence and thermal effects in the
quantumcomputercan be exploitedto mimic decoherenceand thermaleffects in the
systemto be simulated.Consequently,the
set of quantumeffectsthat mightbe exploited to constructa quantumcomputercapable
of simulatingotherquantumsystemsis larger
than the set appropriatefor performing
Shor'salgorithm:Essentiallyany nonlinear
interactionbetween quantumsystemsthat
an experimentalistcan modulatein a controlledfashion could be used for quantum
simulation.
Simulation
Simulationis a processby which one system
is made to mimic another.To simulatean
engine on a classicalanalog computer,for
example,one patches up an electricalcircuit whose dynamics mimic the engine's
dynamics.To simulateone quantumsystem
using another,one needs methodsfor controllingpreciselythe dynamicsof quantum
systems,so that the dynamicsof the simulatorcan be madeto mimicthe dynamicsof
the systemto be simulated.
Atomic physics,quantumoptics, solidstate physics, and quantum chemistryall
supplymethodsforcontrollingthe behavior
of quantumsystems.Eachoperationthat an
experimentercan performon a quantum
system-for example,the turningon andoff
of a laser pulse or the modulition of a
magnetic field-is in effect a "tool" that
can be applied to the system. Formally,
experimental operations can be divided
into two categories:those that approximatelypreservequantumcoherence,corresponding to Hamiltonian time evolution,
and those that do not, correspondingto
time evolution governedby a superscattering operatoror masterequation.Practically,
of course, no operationentirely preserves
quantum coherence, and operationsvary
considerablyin the amountof decoherence
they cause. Simulationof a closed system
that evolves accordingto the Schrodinger
equation requiresoperationsthat approximatelypreservecoherence.(The moregeneral case of simulatingopen systemswith
and coherenceboth coherence-preserving
destroyingoperationswill be addressedbelow). An experimenterwho applies such
operationscan be thoughtof as turningon
and turning off Hamiltoniansfrom a set
{FHIH2 . . . He. Essentially, each experimental operationmoves the systema controlled distance along one out of a set of
predetermineddirectionsin Hilbert space.
The experimentermoves the system first
one way,then another,like a driverparking
a car. A familiarexampleof this technique
is the use of extendedsequencesof pulsesin
nuclearmagneticresonanceto drivea set of
spins to a desiredstate (33).
All techniquesthat involve the repeated
applicationof coherence-preserving
operations possessa common algebraicstructure
that allows the specificationat an abstract
level of wherethe systemof interestcan be
driven by using such operations.The followingresult,derivedindependentlyin the
context of quantumcontrol theory and of
quantum computation, determines just
what quantumsystemscan be simulatedby
the repeatedapplicationof such operations.
The straightforwardapplication of the
formula shows
Campbell-Baker-Hausdorff
that by judiciouslyturningon and off Hamiltonians, the experimentercan drive the
system along any time evolution corresponding to a unitaryoperatorU = eiAt,
where A lies in the algebrasd generated
fromthe set {H1,... ., H}Jby commutation
(20, 21, 34). The number of operations
requiredto generatean arbitrarym X m U
is on the orderof m2-the numberof parametersrequiredto specify U in the first
place. The numberof operationsthat need
to be appliedto constructa desiredU can
be thoughtof as the quantumcomputational complexityof the construction.
A particularlyuseful case occurs when
the experimentercan makedifferentquantum variablesinteract accordingto a particular Hamiltonian.Such an interaction
realizesa quantumlogicgate (8, 16-21, 23,
24, 26, 35). In this case it can be seen that
almostany nonlinearinteractionallowsthe
constructionof arbitraryunitary transformations on the 2N-dimensional Hilbert
space (20, 21). For example,Kimbleet al.
have suggestedthat nonlinearinteractions
betweenphotonsand atomsin small-cavity
quantumelectrodynamicscould be used to
"dial up" arbitraryunitarytransformations
of the photons (23, 36). The resultsabove
implythat a varietyof quantumsystemsto
which a set of simple experimentaloperations can be applied can simulate other
quantumsystems.For example, the quantum variables in the simulatorcould be
photons, as describedabove, or spins that
are made to interact by means of double
resonancemethods,or quantumdots that
interact by dipole or exchange forces. In
particular,to the extent that they can be
expanded to include more quantumbits,
ion-trapquantumlogic devices of the sort
that are currently being constructed by
Monroeet al. could act as universalquantum simulators(24).
Simulation Efficiency
Fig. 1. Electrons in quantum dots simulating a set of quantum spins such as protons. The upper line
shows the spins. An arrow pointingdown represents a proton spinning clockwise, and an arrow pointing
up represents a proton spinning counterclockwise. The lower line shows the states of electrons in
quantum dots. The longer wavelengths correspond to electrons in the ground, or lowest energy, state,
and shorter wavelengths correspond to higher energy states. The simulationis set up so that spin-down
corresponds to a dot in its ground state, and spin-up corresponds to a dot in its first excited state.
"Spin-sideways, t' a superposition of spin-up and spin-down, corresponds to a superposition of ground
and excited states, a state with no classical analog. By modulating the interactions between the dots,
one can make their dynamics mimic the dynamics of the spin system.
1074
SCIENCE * VOL.273
*
As noted above, the efficiencyof a simulation dependson how hardit is to set up the
simulator-systemcorrespondence,to control the simulatorto performthe simulation, and to extractits results.A simulation
on a classicaldigital computerof an arbitraryunitarytransformationof N quantum
variables (for example, spin-1/2particles)
involves the multiplicationof a 2N-dimensionalstate vectorby a 2N X 2Nmatrixand
requiresmemoryspace and computational
time on the order of 22N* As noted by
Deutsch (6), to pefform the same simula-
23 AUGUST 1996
This content downloaded from 128.104.46.206 on Wed, 8 Oct 2014 15:37:19 PM
All use subject to JSTOR Terms and Conditions
RESEARCH ARTICLES
tion on a quantum computer requires only
N quantum bits. But the results above imply
that 22N applications of the elementary
tools are required to construct an arbitrary
2N X 2N unitary U. That is, although a
quantum computer gives an exponential
compression of the amount of memory
space required to construct an arbitraryU,
the number of elementary logic steps or
applications of quantum logic gates required
to construct U on a quantum computer is of
the same order as the number of elementary
logic steps required to compute the action
of U on a classical digital computer. For the
case of N spin-?/2 particles, both simulations
require the solution of 2N x 2N matrix
equations, a difficult task for N - 40.
The result just presented might seem at
first glance to contradict the claim that
quantum analog computers can simulate
quantum systems much more efficiently
than can classical computers. For arbitrary
unitary operators U, quantum and classical
simulations both require m2 operations simply because m2 numbers are required to
characterize an arbitrary m X m U. The
time evolution of real quantum systems is
not arbitrary, however. Feynman's conjecture was that quantum computers could
provide efficient simulations not of arbitrary quantum systems but of systems that
evolve according to local interactions. Examples of local systems include hard-sphere
and van der Waals gases, Ising and Heisenberg spin systems, strong and weak interactions, and lattice gauge theories. In fact, any
system that is consistent with special and
general relativity evolves according to local
interactions. As will now be demonstrated,
quantum simulation is potentially much
more efficient than classical digital simulation for local quantum systems.
The method for performing the simulation is conceptually straightforward, if
mathematically involved. The goal is to get
the simulator from point A to point B along
a particular route. But the simulator can
only be driven in certain directions-the
operations that can be applied experimentally are limited-so it is usually not possible to go from point A to point B directly.
But by moving the simulator first a little bit
in one direction, then a little bit in another,
then a little bit in another, and so on, it is
possible to move from A to B. A car can
only be driven forward and backward-it
cannot be driven sideways. But it is still
possible to parallel park. The following construction demonstrates a quantum analog of
a familiar classical fact: By going forward
and backing up a sufficiently small distance
a large enough number of times, it is possible to parallel park in a space only ? longer
than the length of the car.
More precisely, consider a quantum sys-
tem composedof N variableswith HamiltonianH =
1 Hi, whereeach Hi actson
a space of dimension ml encompassingat
most k of the variables.H can depend on
time. Any Hamiltoniansystem with local
interactionscan be written in this form.
Many nonlocal systems such as nonlocal
spin glassesalso have Hamiltoniansof this
form and can be efficiently simulated.As
with simulations on classical computers
(30-32), the quantumsimulationworksby
evolving the system forwardlocally over
small, discrete time slices. BecauseeiHt
m = max{m.i.In orderfor the overallerror
in buildingup the productof exponentials
to be less then ?, the actual experimental
errorin implementingeach of the elementaryoperationsmustbe less than e/nem2, so
that the cumulativeexperimentalerror is
less than ?. In conventionalcomputational
complexity,a simulationis said to be efficient if to simulatea systemwithN variables
takescomputertime that is polynomialin N
(that is, if the simulationis a tractableproblem). Here, the quantumsimulationis effi-
such as nearest neighbor or next-nearest
neighbor,e is proportionalto N.
Equation 1 implies that the minimum
numberof steps n requiredto simulatethe
systemto accuracy? over time t is proportional to t2/e. However, the duration of
each coherent operation requiredto produce the local Hamiltonian evolutions
to t/n, that is, to lt. As
eiHjtln is proportional
a result,the total durationof time required
to simulatethe system over time t is just
proportionalto t. In other words, just as
with a classicalanalogcomputer,the quantum simulation takes an amount of time
proportionalto the time over which the
systemto be simulatedevolves.
As an example,considera lattice of N
spin-?/2particleswith pairwiseinteractions
between neighboringparticles.H = JpN=2
Hi, wherep is the numberof neighborsper
particleand each Hi acts on a local twoparticleHilbertspaceof dimension22 = 4.
Determinationof the correctset of operations to apply to simulatethe local Hi requiresthe solutionof a 4 X 4 matrixequation in 16 variables.(A simulation of a
lattice Hubbardmodel,in contrast,requires
two qubitsper site to indicate the number
and spin of fermionspresent;a 16 X 16
matrix equation must be solved to determine the propersequence of operations.)
The total numberof operationsneeded to
simulatethe spinsto an accuracy? requires
-2pN/c steps [if the spins' spatial wave
functionsoverlap,an extrafactorof at most
N is requiredto keep trackof Fermistatistics (38)].
Next consider simulation of the same
system on a classical computer. Monte
Carlo techniques,which are polynomialin
N, can extractsome informationaboutthe
groundstate of such a local systemby applying relaxationaltechniques,or estimate
the time evolution of such a system by
Green'sfunction techniques,but they typicallydo so by dealingwith a restrictedclass
of wave functions (fermionsare also problematic in morethan one dimension)(3032). In general,merelyto specifyan arbitrary wave function for the particles requires2N memorysites on a classicalcomputer, and to compute its time evolution
requiresthe exponentiation of 2N X 2N
matrices.For directly simulatingthe time
evolutionof an arbitrary
state, the quantum
simulationis exponentiallyfasterthan simulation by the classicalcomputer.This result holds in general.Simulationof a quantum system with N variablesthat evolves
according to a local Hamiltonian on a
quantum computer requiresa number of
steps proportionalto N. Simulationof the
same system on a classical computerre-
cient as long as = t(N) is a polynomial
function of N. For typical local interactions
quires a number of steps exponential in N.
The quantum simulation becomes even
(eiHt...
eiHt/n )n, eiHt
can be simulated by
simulatingthe local time evolution operators eiHltln, eiH2tln,and so on, up to eiHetln
and repeatingn times.
To ensure that the simulation takes
place to within some desiredaccuracy,one
needsto regulatethe time-slicingas in (3032)
eiHt = (e iHl/n . . .eiHtln)n
+ I [Hi,Hj]t2/2n+
i>j
> E(k)
(1)
k= 3
wherethe higherordererrortermsE(k) are
bounded by IIE(k)I1sup
' nllHt/nlIsIk/k!
(where IlAilsupis the supremum,or maximum expectationvalue, of the operatorA
over the states of interest), and the total
error in approximating eiHt (eiHlt/n
is less than IIn(eiHt/n - 1 - iHtl
eiHet/n)n
n)1Isup,which can be made as small as desired by taking n sufficiently large. As a
result,for any? > 0, n can alwaysbe picked
sufficientlylargeto ensurethat the simulator alwaystracksthe correcttime evolution
eiHt to within ?.
Once the accuracyto within which the
simulation is to take place is fixed, the
quantumcomputationalcomplexityof performingthe simulationcan be estimated.
As noted above and in (20, 21), because
each HJactson a local Hilbertspaceof only
mj dimensions, the number of operations
needed to simulate
eiHit/n
is
-mj
(37). Be-
cause each local operator is simulatedn
times,the total numberof operationsneeded to simulatethe time evolutioneiHt to an
accuracy ? is rn(ZnWm-=
SCIENCE * VOL.273
)
*
?
nem2
where
23 AUGUST 1996
This content downloaded from 128.104.46.206 on Wed, 8 Oct 2014 15:37:19 PM
All use subject to JSTOR Terms and Conditions
1075
fasterif the intrinsicparallelizability
of the
simulationcan be exploited,as in the quantum cellular automatadescribedin (13).
Termsin the sum 1i Hi that commutecan
be enacted simultaneouslyby applyingoperationsto differentvariablesin the simulatorat once. The termsin H can be divided up into groupssuch that within each
group,all termscommute.All termswithin
a group can be enacted in parallel. But
becauseH is local, the numberof groupsis
independentof N. In this case, the time
requiredto performthe simulationis independentof the sizeof the local systemsimulated. On a parallelclassicalcomputerit
still takes a numberof steps exponentialin
N merely to write out the matrix H, let
alone to exponentiateit.
In summary,a quantumcomputercan
simulateclosedquantumsystemswith local
Hamiltonians efficiently. The simulation
operatesby inducinginteractionsbetween
the quantumvariablesof the simulatorthat
mimic the interactionsbetween the variables of the system to be simulated.The
simulatormust use a numberof quantum
variablessuch as spin-?/2particlesthat is
proportionalto the numberof variablesof
the systemto be simulated.Severalbits in
the simulatormaybe allocatedto simulatea
given local variable;forexample,two quantum bits are requiredto specifythe number
and type of fermionsat a lattice site in the
Hubbardmodel.Althoughcontinuousvariablescomplicatethe computation,they can
still be approximated
discretely:N quantum
bits are requiredto simulatea continuous
local variable such as the position of a
particleor the value of a field in a lattice
gauge theory to N bits of accuracy.The
simulationtakesan amountof time proportional to the time over which the systemis
to be simulated.The numberof basic experimentaloperationsneededis proportional to the numberof variablesof the system.
This scalingholds true for both time-independent and time-dependentHamiltonians. That is, the quantumsimulationtakes
resourcesof quantumcomputertime and
memoryspace directlyproportionalto the
time andspacetakenup by the systemto be
simulated.This contrastswith simulationof
the systemon a classicaldigital computer,
which requires exponential resources in
time and space.
vironmentis effectivelyclassicaland can be
describedby a time-varyingpotential.However, preparinga system in a desiredstate
and making measurementson the system
necessarilysubject the system to external
influencessuch as dissipationand decoherence, effectsthat areconventionallytreated
with masterequationsor superscattering
operators.As will now be demonstrated,
preparationandmeasurement
as well as environmentaleffectssuch as dissipationand decoherencecan be easilyincludedin a quantum
analogsimulation.
Considera quantumsystem interacting
with its environment.If the systemhas HilbertspaceNCsand the environmenthas HilbertspaceNE, then the Hilbertspaceforthe
systemand environmenttogetheris Ns 0
WE. The joint time evolutionfor the system
and environmentis given by a unitaryoperator U(t) acting on S 0 WE. The system
and environmenttogethercan be described
by a joint density matrix PSE(t) =
closed and do not interact with their surroundings, or their interaction with the en-
m2-dimensional simulated system-environment Hilbert space We 03 We and for some
1076
SCIENCE * VOL.273 * 23 AUGUST 1996
initial simulatedenvironmentstate pE(O)
(39). Findingthe m4 - m2parametersthat
define U(t) and the m2 parametersthat
define pE(O)requiresthe solutionof an m2
X m2 matrixequation[0(t) is uniqueonly
on
up to arbitraryunitarytransformations
WpJ.Once the proper0(t) has been determined, the simulationproceedsas in the
Hamiltoniancase above.
Forexample,considera spin-1/2particle
evolving under an applied magnetic field
and interactingwith its environmentaccordingto the Bloch equation.Simulation
of the time evolutionof such a particleon a
quantum analog computer requires two
qubits.Approximately24 - 22 = 12 operationsarerequiredto take the particlefrom
an initialstateto a state to which it evolves
underthe Bloch equation.
System Preparationand
Measurement
U(t)psE(O)Ut(t) (the dagger indicates Her- Consider an open-system operation that
mitian conjugation),and the state of the puts one of the simulatorvariablesin a
systemon its own is given by the reduced known state, such as cooling an ion in an
densitymatrixps(t) = trEpSE(t)obtainedby ion-trapcomputer,or measuringthe polartaking the trace trEover the states of the izationof a photon in quantumopticallogic
environment.If the systemandenvironment device. Such an operationis either dissipaare initiallyuncorrelated,so that PSE(O) =
tive (as in cooling) or decohering (as in
This operationcan be comPS(O) ( PE(O), then the time-evolvedre- measurement).
duceddensitymatrixcan be writtenps(t) = bined with the sort of coherentoperations
where 9(t) is a trace-preserving used above to simulate Hamiltonian sys2(t)[pS(O)]
linearoperatorcalleda superscattering
oper- tems to preparethe remainingvariablesin
atorthat dependsonly on U(t) and PE(O)* any desiredstate. First,the variableis specTo simulate an open system it is not ified in a knownstate, then a unitaryopernecessaryto simulatethe entirebehaviorof ation is effected that "pumps"entropy to
the environment,but only the aspects of the variablefrom the remainingvariables,
the environment'sbehaviorthat affect the and then this procedureis repeated.If the
system. In general,the effect of the envi- variableis a quantumbit, then with each
ronmentcan be mimickedby usingat most cycle of the pump the entropy of the reas many variablesas are requiredfor the maining variablesis decreasedby one bit.
system. For some cases the effect of the Repeatedpumpingreducestheir entropyto
environmentcan be encompassedby using any desiredvalue. By adjustingthe unitary
a single quantumbit. (Below, it will be "pumping"operation,the remainingvarishownthat sometimesnot even a singlebit ablescan be preparedin any desiredstate.
is required;the effect of the environment
A closely analogoustechnique can be
on the system can be mimicked by the used to make arbitrarymeasurementson
effect of the computer'senvironment on the simulator.Consideran operationthat
the computer.)The reasonis as follows:The extractssome informationfromone of the
goal of the simulationis to make the vari- simulator variables (the "read-out variables in the simulatorcorrespondingto the able") while leaving the other variables
system evolve accordingto some desired unaffected.For example, photon polarizasuperscatteringoperator 9(t). 9(t) is a tions can be detected with beam splitters
linearoper- and photodetectors,and the states of ions
strictlypositivetrace-preserving
atoractingon an m X m Hermitiandensity can be read by using fluorescence. The
matrixand so requiresm4 - m2 parameters read-out operation need not be perfectly
Simulating Open Systems
to verifythat accurate, nor need it leave the read-out
to define. It is straightforward
All physicalsystemsinteractwith their envariableundisturbed.
92(t)[p(O)] = trCJ(t)ps(O)0 pt(O)U'(t)
An arbitrarymeasurementon the state
vironmentto a greateror lesserextent.Up to
simulatorusing this read-outoperathis point, the systemsto be simulatedhave
of
the
(2)
been assumedto undergo a Hamiltonian
tion is made by labelingwith binarynumtime evolution. Such systems are either for some unitary operator U1(t) acting on an bers the set of orthogonalstates between
which the measurement is to distinguish. A
unitary transformation on the simulator is
This content downloaded from 128.104.46.206 on Wed, 8 Oct 2014 15:37:19 PM
All use subject to JSTOR Terms and Conditions
i RESEARCH ARTICLES
then effected that correlatesthe states of
the read-outvariablewith the value of the
first bit of the numberlabelingthe states;
next, the read-outoperationis performed
and these steps are repeateduntil the first
bit has been identifiedto the desireddegree
of certainty.At this pointthe sameread-out
processis performedon the secondbit, and
so on, until all bits requiredto identifythe
state have been determined.For example,
the Kimble quantumlogic gate could be
used to correlate arbitrary polarization
statesof a single photon with the left- and
right-circularpolarizationstates of a sequence of read-outphotons. By increasing
the numberof read-outphotons, one can
accuratelymeasurethe polarizationof the
photon despitethe finite efficiencyof photodetectorsand the destructionof the readout photons.This methodprovidesaccurate
nondemolition measurementsof arbitrary
sets of states even when the only available
read-outoperationis noisy, inaccurate,and
destructive.
System preparationand measurement
are quantum computationsin their own
right. The preparationsand measurements
that can practicallybe performedare those
that can be done efficiently,in a relatively
small number of steps. Fortunately,in a
quantumsimulation,many desiredquantities can be read out directly.For example,
measurementof the correlationfunction
(M(O)M(t))in a simulatedspin glass only
requriesmakingrepeatedmeasurementsin
which one has preparedthe simulatorvariable correspondingto a spin in states of
differentinitial magnetizationM(O),simulated the time evolution of the spins over
time t, then measuredthe final magnetization of the spin M(t). P repetitions are
requiredto build up the correlationfunc-
E HE, where each term in the sum acts
on at most kEvariablesof the system and
environment. Because an open system is
beingsimulatedby embeddingit in a closed
system,Eqs. 1 and 2 hold as above for the
combined system-environmentsimulation:
Simulation of the time evolution of the
joined system-environmentto an accuracy
? overtime t then requires
a numberof steps
((m2 + (EmE4)nt2/ , where m and mE are
the maximumdimensionsof the local system-environmentHilbert spaces acted on
by Hi, HEi,as above (40). The time over
which the simulationtakesplace is proportional to t, as in the case of the closed
system.Justas forthe caseof closedsystems,
the quantum simulation of open systems
uses a numberof variablesproportionalto
the numberof system variablesand takes
time proportionalto the time over which
the systemevolves.
For example, consider the problem of
simulatingthe behaviorof a set of spinsthat
interactwith each other and with the environment.Supposethat the interactionof
each spin with its environmentis Blochlike, so that in the absence of spin-spin
interactions,each spin would evolve accordingto the Bloch equation,with longitudinalrelaxationtime T1,transverserelaxationaltime T2,andnaturalfrequencyw. In
this situation, only one bit is requiredto
simulatethe environmentbecausethe same
bit can be used to simulatefirst the environmentof one spin,then the environment
of another,and so on. Although the simulation progressesby small perturbationsas
in Eq. 1, it can still be used to extract
nonperturbative features. For example,
quantumcomputationcouldbe usedto simulate an annealing process to find the
groundstate of the system.
tion to an accuracy
V.
Exploiting the Environment
Efficient Simulation of Local
Open Systems
So far, the quantum simulatorhas been
assumedto be itselfa closedsystem,isolated
For arbitrary
open systemsas for closed from its environment. However, no real
systems,the difficultyof simulationrises quantumcomputeris totally isolated;quanas the sizeof the systemin- tum computersare open systemssubjectto
exponentially
creases.To simulatean arbitrary
timeevo- thermal fluctuationsand decoherence. In
lution of an open system of N spin-1/2 par- addition,the tools usedto controlthe quanticlestakes-24N steps.Butphysicalsystems tumdynamicsmaythemselvesinducedecodo not evolvearbitrarily.
As withHamilto- herence.Forquantumcomputationssuchas
niansystems,
simulation
showsits factoringdecoherenceis a liability,but for
quantum
powerwhenthe opensystemto be simulat- quantumsimulationsof open systemsdecoed evolvesaccordingto local interactions herence can be an asset. Often, the interbetweenits variables
andbetweenits vari- actionof the variablesof the quantumcomables and the environment.Consideras puterwith their environmentcan be used
abovea systemwith N variables,each of to mimic the interactionof the simulated
whichinteractswithat mostk others,with systemvariableswith their environment.
HamiltonianH = Y.= 1H', where the Hi
Forexample,considerthe use of an ionasbefore.Nowthe trap computer(22) to performa quantum
maybe time-dependent
system is allowed to interact with its environment with local interactions HE =
interactionwith its environmentcharacterized by parametersTI, T2, and C,. These
parameters
aredeterminedby characteristics
of the computer'senvironmentsuch as temperature,pressure,density, viscosity, and
electricfield strengthand can be variedby
manipulatingthe environment.By tuning
the computer'sBloch parameters
so that T1
=cxT1, T2 = xT2,X = t-x1w, the time over
which the computerinteractswith its environmentcan be adjustedso that the decoherenceand noise inducedin the qubitsof
the computerby its environmentsimulate
the decoherenceand noise inducedin the
spinsby their environment.
This method of simulatingthe system
environmentby usingthe computer'senvironmentworksas long as the operatorsthat
determinethe couplingof the computational variablesto their environmenthave the
same form as the operatorsthat determine
the couplingof the systemvariablesto their
environment. The temperature,pressure,
and densityof the computer'senvironment
can then be tuned to make the effect of
both environmentsthe same up to a time
scale. In the case of simulatingspins with
ions, ions at room temperaturecould be
used to simulate spins occurring at microkelvin.The use of environmentsto simulate environmentsis more limited than
the methodfor simulatingthe environment
discussedin the previoussection:The coupling of the computerto its environment
must take the same functionalform as the
couplingof the systemto its environment.
Nonetheless, direct environmentalsimulation has the advantageof exploitinginteractionsthat arepresentin anycaseandthat
wouldotherwiseconstituteunwantednoise.
The use of noise and decoherence to
simulatenoise and decoherenceallowssystems that are far too noisy and decoherent
to factor numbersto function as effective
simulators.Semiconductorquantumdevices such as quantumdots and wells typically
have decoherence time scales that are
shorterthan or comparablewith the time
requiredto flip them from one state to
another.As a result,they arenot an appropriate technologyfor performinglong, coherent quantum computations.But they
could prove suitable for simulatingother
noisy and decoherentsystems.
Conclusion
Feynmanwas correct that quantumcomputerscould provideefficientsimulationof
other quantumsystems.A quantumcomputerwith a few tens of quantumbits could
performin a few tens of steps simulations
that would requireAvogadro'snumberof
simulation of the coupled Bloch spins above.
Suppose that each ion also has a Bloch-like
memory sites and operations on a classical
computer. A mere 30 or 40 quantum bits
23 AUGUST 1996
1077
SCIENCE * VOL. 273
*
This content downloaded from 128.104.46.206 on Wed, 8 Oct 2014 15:37:19 PM
All use subject to JSTOR Terms and Conditions
would suffice to performquantumsimulationsof multidimensional
fermionicsystems 13.
such as the Hubbard model that have 14.
provedresistantto conventionalcomputational techniques.Hundredsto thousands
of bitsmaybe requiredto simulateaccurately systemswith continuousvariablessuchas 15.
16.
latticegaugetheoriesor modelsof quantum 17.
gravity.Currentquantumlogic devicescan
performoperationson two quantum bits 18.
(23, 24, 26); however, ion-trapquantum 19.
computerswith a few tens of quantumbits 20.
apparentlyrequire only minor modifica- 21.
tions of currenttechnology(24). Although
a quantum simulatorwith three or four 22.
quantumbits would be too small to solve 23.
classically intractableproblems, it would
still be large enough to test many of the 24.
ideaspresentedhere. As suggestedin (13), 25.
such simulatorswouldalso be able to create 26.
and test the propertiesof exotic quantum
statessuch as Greenberger-Horne-Zeilinger27.
states. Another possibility(13) is that by
modulatingthe interactionsbetweenspins,
atoms,or quantumdots in largearrays(41),
one could performmassivelyparallelquan- 28.
tum simulationsinvolving many quantum
29.
systemsat once. The widevarietyof atomic,
molecular,and semiconductorquantumdevices availablesuggeststhat quantumsim- 30.
ulation may soon be a reality.
31.
REFERENCESAND NOTES
1. P. Benioff, J. Stat. Phys. 22, 563 (1980); Phys. Rev.
Lett. 48,1581 (1982); J. Stat. Phys. 29, 515 (1982);
Ann. N. Y Acad. Sci. 480, 475 (1986).
2. R. P. Feynman, Opt News 11, 11 (1985); Found.
Phys. 16, 507 (1986); Int. J. Theor. Phys. 21, 467
(1982).
3. W. H. Zurek, Phys. Rev. Lett. 53, 391 (1984).
4. A. Peres, Phys. Rev. A 32, 3266 (1985).
5. N. Margolus,Ann. N. Y Acad. Sci. 480, 487 (1986); in
Complexity, Entropy,and the Physics of Information,
vol. 8 of Santa Fe InstituteSeries, W. H. Zurek, Ed.
(Addison-Wesley, Redwood City, CA, 1991), pp.
273-288.
6. D. Deutsch, Proc. R. Soc. London Ser. A 400, 97
(1985).
7. See, however, D. Deutsch and R. Jozsa, ibid. 439,
553 (1992), for an interesting oracle problem that can
be performed more quickly by quantum computers.
8. D. Deutsch, ibid. 425, 73 (1989).
9. Y. Yamamoto, M. Kitegawa, K. Igeta, in Proceedings
of the 3rd Asia-Pacific Physics Conference, Y. W.
Chan, A. F. Lenng, C. N. Yang, K. Young, Eds.
(World Scientific, Singapore, 1988), pp. 779-799.
10. G. J. Milburn,Phys. Rev. Lett. 62, 2124 (1989).
11. R. Landauer, Int. J. Theor. Phys. 21, 283 (1982);
Found. Phys. 16, 551 (1986); Nature 335, 779
(1988); in Nanostructure Physics and Fabrication,M.
A. Reed and W. P. Kirk,Eds. (Academic Press, Boston, 1989), pp. 17-29; Phys. Today 42,119 (October 1989); in Proceedings of the 3rd International
Symposium on Foundations of Quantum Mechanics, Tokyo, 28 to 31 August 1989 (PhysicalSociety of
Japan, Tokyo, 1990), p. 407; Physica A 168, 75
(1990); Phys. Today 44, 23 (May 1991); in Proceedings of the Workshop on Physics of Computation II,
Dallas, 17 to 20 November 1994, D. Matzke, Ed.
(IEEEComputer Society, Los Alamitos, CA, 1994),
pp. 54-58.
12. K. Obermayer, W. G. Teich, G. Mahler,Phys. Rev. B
37, 8096 (1988); W. G. Teich, K. Obermayer, G.
Mahler, ibid., p. 81 11; W. G. Teich and G. Mahler,
1078
Phys. Rev. A 45, 3300 (1992).
S. Lloyd, Science 261, 1569 (1993); ibid. 263, 695
(1994).
P. Shor, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa
Fe, NM, 20 to 22 November 1994, S. Goldwasser,
Ed. (IEEE Computer Society, Los Alamitos, CA,
1994), p. 124.
D. P. DiVincenzo, Science 270, 255 (1995).
, Phys. Rev. A 50,1015 (1995).
A. Barenco, D. Deutsch, A. Ekert, R. Jozsa, Phys.
Rev. Lett. 74, 4083 (1995).
T. Sleator and H. Weinfurter,ibid., p. 4087.
G. M. Palma, K.-A. Suominem, A. Ekert, Proc. R.
Soc. London Ser. A 452, 567 (1996).
S. Lloyd, Phys. Rev. Lett. 75, 346 (1995).
D. Deutsch, A. Barenco, A. Ekert, Proc. R. Soc.
London Ser. A 449, 669 (1995).
J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091
(1995).
Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi,
H. J. Kimble,ibid. 75, 4710 (1995).
C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano,
D. J. Wineland, ibid., p. 4714.
W. G. Unruh,Phys. Rev. A 51, 992 (1995).
M. Brune et al., Phys. Rev. Lett. 72, 3339 (1994); P.
Domokos, J. M. Raimond, M. Brune, S. Haroche,
Phys. Rev. A 52, 3554 (1995).
R. Landauer, Philos. Trans. R. Soc. London Ser. A
353, 367 (1995); in Proceedings of the Drexel-4
Symposium on Quantum Nonintegrability:Quantum
Classical Correspondence, D. H. Feng and B. L. Hu,
Eds. (InternationalPress, Boston, 1996).
I. L. Chuang, R. Laflamme, P. W. Shor, W. H. Zurek,
Science 270,1633 (1995).
P. W. Shor, Phys. Rev. A 52, 2493 (1995); A. R.
Calderbank and P. W. Shor, Phys. Rev. A, in press;
A. Steane, in preparaticin.
M. H. Kalos, Ed., Monte Carlo Methods in Quantum
Problems (Reidel, Dordrecht, Netherlands, 1984).
K. Binder, Ed., The Monte Carlo Method in Condensed Matter Physics (Springer-Verlag,New York,
1995).
32. C. Rebbi, Ed., Lattice Gauge Theories and Monte
Carlo Simulations (World Scientific, Singapore,
1983).
A
33. W. S. Warren, D. P. Weitekamp, A. Pines, J. Chem.
Phys. 73, 2084 (1980).
34. V. Ramakrishna, M. V. Salapaka, M. Dahleh, H.
Rabitz, A. Peirce, Phys. Rev. A 51, 960 (1995).
35. A. Yao, in Proceedings of the 34th Annual Symposium on Foundations of Computer Science, S. Goldwasser, Ed. (IEEEComputer Society, Los Alamitos,
CA, 1995), p. 352.
36. H. Mabuchi and H. J. K(fmble,Opt. Lett. 19, 749
(1993); A. B. Matsko, S. P. Vyatchanin, H. Mabuchi,
H. J. Kimble, Phys. Lett. A 192, 175 (1994); H. J.
Kimbleet al., in Proceedings of the InternationalConference on Atomic Physics; A. S. Parkins,P. Marte,P.
Zoller,H. J. Kimble,Phys. Rev. Lett. 71, 3095 (1993).
37. Forthe simulated trajectoryalways to remainwithin?
of the desired trajectory, the experimenter must be
able to enact not only eiHit, but e-iHit as well. See,
for example, R. W. Brockett, R. S. Millman, H. J.
Sussman, Eds., DifferentialGeometric Control Theory (Birkhauser,Boston, 1983), and Z. Li and J. F.
Canney, Eds., Nonholonomic Motion Planning (Kluwer Academic, Boston, 1993).
38. The extra factor of N is requiredto conform with the
normal rules for phases of fermions on a lattice in
second quantized form. See, for example, A. L. Fetter
and J. D. Walecka, Quantum Theoryof Many-Particle
Systems (McGraw-Hill,New York, 1971), and J. W.
Negele and H. Ormand,QuantumMany-ParticleSystems (Addison-Wesley, Redwood City, CA, 1988).
39. K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory(Springer-Verlag,
Berlin, 1983).
40. There is an extra subtlety in the open system case.
Forthe quantum analog simulation of open systems
to be efficient, the time evolution of the open system
must be a finite-orderquantum Markovprocess, that
is, the environment must have finite time and space
correlation lengths.
41. C. B. Murray,C. R. Kagan, M. G. Bawendi, Science
270,1335(1995).
42. Supported in part by grant N00014-95-1 -0975 from
the Office of Naval Research. This work is part of the
Quantum Information and Computation project
(QUIC).
8 February1996; accepted 10 June 1996
in
Crosslinked Cofactor
Redox
Function
Oxidase:
Acid
Chains
Amino
Side
Lysyl
for
Sophie Xuefei Wang, Minae Mure,*KatalinF. Medzihradszky,
Alma L. Burlingame,Doreen E. Brown, David M. Dooley,
Alan J. Smith, HerbertM. Kagan, Judith P. Klinmant
A previously unknown redox cofactor has been identified in the active site of Iysyloxidase
from the bovine aorta. Edman sequencing, mass spectrometry, ultraviolet-visible spectra, and resonance Raman studies showed that this cofactor is a quinone. Its structure
is derived from the crosslinking of the --amino group of a peptidyl lysine with the modified
side chain of a tyrosyl residue, and it has been designated lysine tyrosylquinone. This
quinone appears to be the only example of a mammalian cofactor formed from the
crosslinking of two amino acid side chains. This discovery expands the range of known
quino-cofactor structures and has implications for the mechanism of their biogenesis.
Lysyioxidase (LO, E.C.
1.4.3.13) is an
extracellular,matrix-embeddedprotein. It
has a centralrole in the biogenesisof connective tissue by way of posttranslational
oxidative modification of the ?-amino
group of lysine side chains in elastin and
collagen to form inter- and intrachain
crosslinks(1, 2). The physiologic importance of LO is well established.Decreased
LO activity is observedin diseasesof im-
SCIENCE * VOL.273 * 23 AUGUST 1996
This content downloaded from 128.104.46.206 on Wed, 8 Oct 2014 15:37:19 PM
All use subject to JSTOR Terms and Conditions