Simulation of Quantum Adiabatic Search in the Presence of Noise

0.96 MB·English

by  Frank Gaitan

Simulation of Quantum Adiabatic Search in the Presence of Noise
Frank Gaitan
Department of Physics; Southern Illinois University; Carbondale, IL 62901-4401
(Dated: February 1, 2008)
Results are presented of a large-scale simulation of the quantum adiabatic search (QuAdS) al-
gorithm in the presence of noise. The algorithm is applied to the NP–Complete problem N–Bit
Exact Cover 3 (EC3). The noise is assumed to Zeeman-couple to the qubits and its effects on the
algorithm’s performance is studied for various levels of noise power, and for 4 different types of
noise polarization. We examine the scaling relation between the number of bits N (EC3 problem-
size) and the algorithm’s noise-averaged median run-time hT(N)i. Clear evidence is found of the
algorithm’s sensitivity tonoise. Two fitstothesimulation resultsweredone: (1) power-law scaling
hT(N)i = aNb; and (2) exponential scaling hT(N)i = a[exp(bN)−1]. Both types of scaling rela-
6 tions provided excellent fits, although the scaling parameters a and b varied with noise power, and
0 with the type of noise polarization. The sensitivity of the scaling exponent b to noise polarization
0 allowsarelativeassessmentofwhichnoisepolarizationsaremostproblematicforquantumadiabatic
2 search. We demonstrate how the noise leads to decoherence in QuAdS, and estimate the amount
of decoherence present in our simulations. An upper bound is also derived for the noise-averaged
n
QuAdSsuccess probability in the limit of weak noise that is appropriate for our simulations.
a
J
PACSnumbers: 03.67.Lx,03.65.Yz,05.40.Ca
7
1

I. INTRODUCTION provetobe “yes”,itis widelyrecognizedthatthis would
1
haveprofoundconsequencesfortheoreticalcomputersci-
v
ence [4].
6
1 Incomputationalcomplexitytheory[1],computational
In 2001, Farhi et. al. [5] examined the question of
1 problemsareclassifiedaccordingtothe resourcesneeded
whether a quantum computer might be able to solve an
1 to obtain a solution. Often such a problem is reformu-
NP–Complete problem in polynomial-time. They ap-
0 lated as a decision problem whose solution is a “yes” or
6 plied the quantum adiabatic search (QuAdS) algorithm
“no” answer. An algorithm that solves all instances of
0 introduced in Ref. [6] to find solutions of randomly gen-
a decision problem is said to be a polynomial-time algo-
/ erated hard instances of the NP–Complete problem N-
h rithmifthetimeneededtofindasolutiontoanarbitrary
p instance is O(nk), where n is the size of the instance Bit Exact Cover 3 which they believe to be classically
- intractable for sufficiently large inputs. The algorithm
t and k is a fixed positive integer. The computational
n wassimulatednumericallyonanexisting(classical)com-
complexity class P is composed of decision problems for
a puter. Becauseaquantumcomputercannotbesimulated
which polynomial-time algorithms exist. Another im-
u efficiently on a classical computer, the simulations were
q portant computational complexity class is the class NP
restricted to 7 ≤ N ≤ 20. They found that the median
: which is composed of decision problems for which, for
v run-time T(N) for QuAdS to succeed on this class of in-
each such problem, a polynomial-time algorithm exists
i stances could be fit with a quadratic scaling relation in
X to verify the “yes” output of a candidate solution. The
thenumberofbitsN (problem-size): T(N)∼N2. Itwas
r question of whether the classes P and NP are equal is
a oneofthebiggestopenproblemsintheoreticalcomputer pointedoutthatshouldclassicalalgorithmstrulyrequire
exponential time on this class of instances, and should
science. It is widely conjectured that P 6= NP. A deci-
the quadratic scalingbehavior ofQuAdS persist to large
sion problem p is said to be polynomially transformable
N, then QUAdS could outperform classical algorithms
to a decision problem q if: (1) there exists a function f
on randomly generated hard instances, though not nec-
that maps every instance x of p into an instance f(x)
essarilyonworstcaseinstances. Theirworkhasdrawna
of q for which the answer to x is “yes” if and only if
greatdealofinterestandsuggeststwopossibledirections
the answer to f(x) is “yes”; and (2) a polynomial-time
for further research: (1) examining the large N scaling
algorithm exists to compute f(x) for every x in p. A
behaviorofQuAdS;and(2) examiningthe robustnessof
problem is said to be NP–Complete if it belongs to NP
QuAdS to noise. Recent progress on (1) appears in the
and every problem in NP is polynomially transformable
work of Roland and Cerf [7]; Mitchell et. al. [8]; and La-
to it. Thus, if a polynomial-time algorithm is found for
torreandOru´s [9, 10]. Inthis paperwe willfocus onthe
an NP–Complete problem, it follows that P = NP. In
question of robustness of QuAdS to noise.
light of Shor’s polynomial-time quantum factoring algo-
rithm[2],Bennettet.al.[3]askedwhetherapolynomial- The analysis of Ref. [5] assumed that the quantum
time quantum algorithm might exist to solve an NP– computerwasperfectlyisolatedfromnoise. Childset.al.
Complete problem. In other words, they asked, “Can a [11]werethefirsttoconsidertherobustnessofQuAdSto
quantum computer solve an NP–Complete problem in noise. Theypresentednumericalsimulationresultswhich
polynomial-time?” Should the answer to this question suggested QuAdS might have an inherent robustness to
2
two types of noise-induced errors: (1) transitions out of tum adiabatic search algorithm and how it is used to
the instantaneous ground state; and (2) unitary control solve instances of Exact Cover 3 (section IIB); and (iii)
errors. Their simulation did not, however, attempt to describethesimulationprotocolofRef.[5](sectionIIC).
match the scale of the simulation presented in Ref. [5].
Recently,RolandandCerfcarriedoutananalyticalstudy
of QuAdS in the presence of noise using random matrix A. Exact Cover 3
theory [12]. They showed that the error probability of
QuAdS would not increase with problem-size N if the Consider a collection of N binary variables (bits)
noise varies either very slowly or very rapidly with re- z ,··· ,z each of which can take on the value 0 or 1.
1 N
specttothe naturaltime-scaleofthe quantumcomputa- The state of this N-bit system is specified by assigning
tionh¯/E,whereE isthe energyscaleforthe eigenvalues values to each bit in the string z = z ···z . A total of
1 N
of the instantaneous Hamiltonian H(t). Their analysis 2N states z are possible.
assumed weak noise which justified a perturbative anal- AspecificinstanceofExactCover3(EC3)iscomposed
ysis of their model. of L clauses, each of which imposes a constraint on the
As stated above, in this paper we will also examine values of three of the N bits. The number of clauses L
the robustness of QuAdS to noise. Our work comple- will generally vary from one EC3-instance to another. If
ments the analysis of Ref. [12] in that we consider noise the i-th clause involves the bits (a(i), b(i), c(i)), then a
that varies on a time-scale comparable to ¯h/E. We ex- particular bit-string z = z ···z satisfies the clause if
1 N
tendthesimulationprotocolofFarhiet.al.[5]toinclude z +z +z = 1. Otherwise, one says that z does
a(i) b(i) c(i)
noise, and determine the noise-averagedmedian runtime not satisfiedthe clause. A bit-string z is said to solvean
hT(N)iforQuAdStosucceedonrandomlygeneratedin- instance of EC3 if it satisfies all the clauses that make
stancesofN-BitExactCover3whichhaveauniquesolu- up the instance. If one is given an instance of EC3 and
tion. We find clear evidence of the sensitivity of QuAdS a candidate solution string z∗, one can check whether
to noise. Two fits to the simulation results were carried z∗ solves the instance in polynomial-time since checking
out: (1) power-law scaling hT(N)i = aNb; and (2) ex- each of the L clauses simply requires adding three inte-
ponential scaling hT(N)i = a[exp(bN)−1]. Both types gers,andthiscanbedoneinpolynomial-time. ThusEC3
of scaling relations provided excellent fits, although the belongstoNP. ItcanbeshownthatallproblemsinNP
scaling parameters a and b varied with noise power, and are polynomially transformable to EC3 [13] so that EC3
withthenoisepolarization(definedinSectionIIIA). Al- is also NP-Complete.
though we did encounter examples of noise realizations
that reduce the runtime for QuAdS to succeed, at the
largest noise power levels that we examined, the pre- B. QuAdS and EC3
dominant effect of noise was to slowdown QuAdS. The
sensitivityofQuAdSto noisepolarizationallows,forthe By combining the dynamics of the quantum adiabatic
first time, a determination of which noise polarizations theoremwith a cleverchoiceofHamiltonianH(t), quan-
are most problematic for QuAdS. tum adiabatic search (QuAdS) causes the state of a
The structure of this paper is as follows. In section II quantum computer (QC) to home-in on a solution of
webrieflydescribetheNP–CompleteproblemN-BitEx- an instance of an NP-Complete problem [5]. We as-
act Cover 3, and review how QuAdS can be used to find sume the NP-Complete problem is EC3. The quan-
a solution to an instance of this problem. In section III tum adiabatic theorem [14] assures us that if the initial
wedescribeournoisemodelandexplainourextensionof state of a quantum system is the groundstate |E (0)i
g
the simulation protocol of Farhi et. al. to include noise. of the initial Hamiltonian H(0), and H(t) varies suffi-
In section IV we present the results of our simulation, ciently slowly, then the quantum state at time T can
and discuss their significance in section V. Section V be brought arbitrarily close to the groundstate of H(T):
also demonstrates how noise-induced dephasing leads to |ψ(T)i=|E (T)i+|δψi. Theprobabilitynottobeinthe
g
decoherence in QuAdS, estimates the amount of deco- groundstate,ǫ(T)=|hδψ|δψi|2, vanishes as T →∞. For
herence presentin our simulations,andderives anupper a given instance of EC3, QuAdS evolves the state |ψ(t)i
bound for the noise-averagedQuAdS success probability overatimeinterval0≤t≤T,andthefinalHamiltonian
in the weak noise limit that is appropriate for our simu- H(T) = H is constructed so that its groundstate en-
p
lations. Finally, in section VI, we provide a summary of codesasolutionofthegiveninstance. TheinitialHamil-
our work. tonian H(0) = H is chosen so that its groundstate is
i
non-degenerate and can be easily constructed, and H(t)
smoothly morphs H →H :
i p
II. EXACT COVER 3 AND QUANTUM
t t
ADIABATIC SEARCH H(t)= 1− H + H . (1)
i p
T T
(cid:18) (cid:19) (cid:18) (cid:19)
Inthis sectionwe: (i)presentthe NP-Completeprob- We now briefly describe how H and H are constructed
i p
lem Exact Cover 3 (section IIA); (ii) describe the quan- for a given instance of EC3.
3
Suppose that we are given an instance of N-bit EC3 the measurement result z = s. The algorithm’s failure
composed of L clauses C = (a(i), b(i), c(i)) with probability ǫ(T) will be much less than 1 if [14]:
i
i = 1,··· ,L, and that the bit-string s = s ···s is a
1 N ε
solution of this instance. Our QC is assumed to contain T ≫ , (4)
∆2
N qubits and the computational basis states |z ···z i
1 N
are chosen to be eigenstates of σ1⊗···⊗σN. Although where,
z z
theQuAdSalgorithmcanbeformulatedmoreabstractly,
dH
it proves convenient to adopt the language of NMR.
ε = max hE (t)|T |E (t)i ; (5)
1 g
Hi, then, describes the Zeeman coupling of the qubits tǫ[0,T] (cid:12) dt (cid:12)
to an external magnetic field which points along the x- ∆ = min (cid:12)|E (t)−E (t)| ; (cid:12) (6)
(cid:12) 1 g (cid:12)
direction. The strength of the magnetic field at the site tǫ[0,T] (cid:12) (cid:12)
of the i-th qubit is equal to the number of clauses in
and |E (t)i (|E (t)i) is the groundstate (first excited
the EC3instance thatcontainbit i. The non-degenerate g 1
state) of H(t).
groundstate of H thus has all qubit spins aligned along
i
xˆ so that |E (0)i is simply the uniform superposition of
g
all 2N computational basis states |z ···z i. The final
1 N C. Noiseless Simulation Protocol
HamiltonianH isconstructedtobediagonalinthecom-
p
putational basis |z ···z i:
1 N Farhi and co-workers [5, 6] studied the performance
of QuAdS by numerically integrating the Schrodinger
H |z ···z i=h(z ,··· ,z )|z ···z i . (2)
p 1 N 1 N 1 N
equation using H(t) (eqn. (1)) to drive the dynamics.
Because the Hilbert space for N-qubits has dimension
The eigenvalue h(z ,··· ,z ) is the sum of energy func-
1 N 2N, numerical simulation quickly becomes impractical
tions h (z ,z ,z ),
Ci a(i) b(i) c(i) since the number of wavefunctioncomponents thatmust
be tracked by the numerical integration grows exponen-
L
h(z ,··· ,z )= h (z ,z ,z ) , (3) tially with N. This practical difficulty caused the simu-
1 N Ci a(i) b(i) c(i) lations done in Refs. [5, 6] to be restricted to the range
i=1
X 7 ≤ N ≤ 20. Only randomly generated USA instances
where C =(a(i),b(i),c(i)) is the i-thclausein the given were simulated as these authors believed that this set
i
EC3 instance and, would provide hard cases for QuAdS, though not neces-
sarily the worst cases.
0 if z ···z satisfes C AnN-bitUSAinstanceofEC3isgeneratedbythefol-
h (z ,z ,z )= 1 N i
Ci a(i) b(i) c(i) 1 otherwise . lowingprocedure. Thefirstclauseisgeneratedbypicking
(cid:26)
3 distinct integers in the range [1, N] at random (uni-
Thus h(z ,··· ,z ) indicates how many clauses are vio- form deviate). A count of the number of bit-strings of
1 N
latedbythestringz =z ···z . Allstringss=s ···s length N that satisfy the clause is then done. Distinct
1 N 1 N
that solve the given instance correspond to computa- clausescontinuetobegeneratedinthismanner,andwith
tional basis states |s ···s i that have zero energy, and each new clause generated, a new count of the number
1 N
which together span the groundstate eigenspace of H . of bit-strings that satisfy all clauses generated to that
p
EC3 instances which have a unique solution are referred point is done. Eventually, the point is reached where
to as unique satisfying assignment (USA) instances, and enough clauses have been generated so that one of two
are believed to be the most difficult for QuAdS. In situations occurs. (1) Only one bit-string s of length N
essence, it’s harder to find a needle in a haystack that remains that satisfies all the clauses generated. The col-
contains only one needle, than it is to find one in a lection of clauses then corresponds to an EC3-instance
haystack that contains many needles. For further dis- which has a unique solution s, and the procedure has
cussionofinstanceswithmultiplesatisfyingassignments, thus produced a USA instance of EC3. (2) Addition of
see Ref. [15]. the most recently generated clause causes the set of all
Thus,ifourQCisinitiallypreparedinthegroundstate clauses produced to have no satisfying assignment (i. e.
of H , and T is chosen to be sufficiently large that H(t) anEC3-instancehasbeenproducedwithnosolution). In
i
evolves adiabatically, then the final state |ψ(T)i will lie this case, the instance is discarded and the above proce-
inthe groundstateeigenspaceofH withprobability1− dure is repeated until a USA instance is generated. The
p
ǫ(T), where ǫ(T)=|hδψ|δψi|2 ≪1. For a USA instance USAinstanceproducedis thenusedtoconstructH and
i
with solution string s = s ···s , measuring |ψ(T)i in H , andfrom them, H(t). The USA instances generated
1 N p
the computational basis will give the solution string s by this procedure were found to contain approximately
with probability1−ǫ(T). Let the string z ···z be the as many clauses L as number of bits N: L∼N.
1 N
actual measurement result. It can be quickly tested to In the adiabatic limit, T → ∞, the QuAdS success
determine whether it solves the USA instance. If “yes”, probability P → 1. Numerical simulations must nec-
s
the algorithm has succeeded, and z = s. If “no”, then essarily work with finite T so that a protocol is needed
thequantumadiabaticsearchprocedureisrepeateduntil to determine how to pick T. Farhi et. al. carried out
4
N =13and14(seeSectionIV). Ournoiselesssimulation
27
Power-Law Fit willbe usedasabaselineforoursimulationofQuAdSin
Exponential Fit
24 thepresenceofnoise. Sincethesimulationswithnoisedo
not go beyond N = 14, we did not extend the noiseless
21
simulations beyond N =14.
18
N)
T(15
III. NOISY QUANTUM ADIABATIC SEARCH
12
9
In this section we discuss QuAdS in the presence
6 of noise. Section IIIA presents our noise model and
describes how noise realizations are constructed. Sec-
3
6 8 10 12 14 tionIIIBdiscusseshowthe noiselesssimulationprotocol
N
ismodifiedtoallowastudyofthenoise-averagedperfor-
mance of QuAdS.
FIG.1: SimulationresultsgeneratedbyourQuAdSnumeri-
calcodeforthenoiselessmedianruntimeT(N)(dimensionless
units)versusthenumberofbitsN. Thesolid lineisthebest
A. Noise Model
power-law fit to the data and the dash-dot line is the best
exponential fit. Respective values for a and b are given in
the text. The error bars give 95% confidence limits for each We consider noise N (t) that couples to the qubits via
i
median. Each datapoint is the outcome of averaging over 75 the Zeeman interaction:
USAinstances.
N
H =− γ σ ·N (t) . (7)
int i i i
a hunting procedure which searches for a T that causes Xi=1
the simulationtoproducea successprobabilityP inthe
s The qubits are labeled by the index i and the subscript
range [0.12, 0.13]. This corresponds to a success proba-
i on the noise field N (t) and coupling constant γ indi-
bility in the vicinity of P = 0.125 = 1/23. As noted in i i
s cates that they can vary from one qubit site to another.
Ref. [5], this value of P is somewhatarbitrary,although
s We assume the noise is a stationary random process. To
for the values of N used in the simulations, it is much
simplify the presentation of our noise model we initially
largerthan 1/2N which is the probability of the solution
consider a one-component noise field N(t). The presen-
strings in the initial state. For eachvalue ofN, 75USA
tation is easily generalized to three-component spatially
instances are generated and QuAdS is used to find the
varying noise N (t).
solutiontoeachinstance. ThisgeneratesruntimesT (N) i
i The rate at which the noise field N(t) can do work
(i = 1,··· ,75) from which the median runtime T(N) is
(i. e. noise power) is [16],
determined. Farhiet.al.[5,6]foundthattheirsimulation
results for T(N) could be fit with a quadratic scaling re- P =N2(t) ,
lationforthe rangeofN-valuesconsidered: T(N)∼N2.
As a check on the soundness of our own simulation and the energy that can be delivered in a time interval
code,werepeatedthenoiselesscalculationofFarhiet.al. dt is,
for 7 ≤ N ≤ 14. Figure 1 shows our results for T(N)
versus N, together with a power-law fit to the simula- dE =N2(t)dt .
tion data: T(N) = aNb. The best-fit parameters are
a = 0.11966 and b = 2.0034. The χ2 value for the fit is We consider power-type noise for which the time-
0.092, and the probability P(χ2 ≥ 0.092) = 0.9999844. averagednoise power
As noted in Section IV, the closer this probability is to
1, the more consistent the data-set is with the fitting 1 T/2
P = lim N2(t)dt (8)
faugnrecetimonen.tWweitsheeoutrhasitmaulpaotwioenr-rleaswulfitst,parnodvioduerseexxpcoelnleenntt T→∞ T Z−T/2
b=2.0034is consistentwith the quadraticfit ofRef. [5]. is finite. The total noise energy
Figure 1 also includes an exponential fit to the noiseless
∞
data. The best-fit parameters are: a = 3.89566; b =
E = dtN2(t) (9)
0.140235; χ2 = 0.214; and P(χ2 >0.214)=0.9998121.
fit −∞
Z
Weseethatexponentialscalingalsoprovidesanexcellent
fit to the data. We will comment further on this in Sec- diverges for this class of noise. The divergence is due to
tions IV and V. The addition of noise to the simulation theoccurrenceofaninfinitenumberofnoisefluctuations
ofQuAdScausessimulationtobecomeimpracticalwhen in the time interval −∞ < t < ∞. The energy of an
N >∼ 12. In a couple of cases we were able to simulate individual fluctuation is, however,finite.
5
The mean noise power P can be related to the noise Asmentionedabove,xisGaussiandistributedwithmean
correlation function, x = 0 and x2 = σ2. From eqn. (14), ε = 2x2τ, and the
mean energy per fluctuation ε is,
1 T/2
N(t)N(t−s)≡ lim dyN(y)N(y−s) . (10) ε=2x2τ =2σ2τ . (15)
T→∞ T Z−T/2
From Campbell’s theorem [19], the power spectral den-
Comparing eqns. (8) and (10) we see that,
sity for N(t) is
P =N2(t) . (11)
S (f)=n|g(f)|2 , (16)
N
The Weiner-Khintchine theorem [17] shows that the
where g(f) is the Fourier transform of the fluctuation
noisecorrelationfunctionandthe powerspectraldensity
profile F(t). Thus, using eqns. (13), (16), and Paresval’s
S (f) form a Fourier transform pair:
N theorem gives,
∞
N(t)N(t−s)= dfS (f)e−2πifs . (12) ∞
N P =n dtF2(t) . (17)
−∞
Z −∞
Z
Thus, it follows from eqns. (11) and (12) that
Finally, using eqns. (14) and (15) gives,
∞
P = dfS (f) , (13) P =2nσ2τ . (18)
N
−∞
Z
Thusweseethatournoisemodelischaracterizedbyany
whichidentifiesSN(f)asthemeannoisepoweravailable 3 of the parameters P, n, σ2, and τ.
in the frequency interval (f, f +df).
The numerical simulation constructs a realization of
Havingreviewedthesebasicfactsaboutstationaryran-
the noiseasfollows. We sampleapositiveintegerN ac-
f
dom processes, we are now in a position to describe our
cordingto the PoissondistributionwithmeanN =nT,
noise model and to write out the basic relations that are f
where T is the duration of the search. N is the num-
necessary to construct an individual realization of the f
ber of fluctuations present in the noise realization. The
noise.
noise model assumes these fluctuations occur indepen-
ThenoiseN(t)isproducedbyasequenceofrandomly
dently with probability dp = (1/T)dt. We sample N
occurring noise fluctuations F(t). The fluctuations: (1) f f
numberst (i=1,··· ,N )fromthe interval(0, T). The
occur independently of each other at average rate n per i f
t correspond to the temporal centers of the N fluctua-
unittime;and(2)haveapeakvaluexwhichisGaussian i f
tions. For simplicity, we’ve assumed that the fluctuation
distributed with mean x = 0, variance x2 = σ2, and
profileh(t)isasquarepulse. WenextcarryoutN sam-
f
temporal width 2τ, where τ is the thermal relaxation
ples x (i = 1,··· ,N ) of a Gaussian distribution with
i f
time. The bandwidth of F(t) is ∆ω ∼1/2τ. Thus,
mean x = 0 and variance x2 = σ2. Here x is the peak
i i i
value of the i-th fluctuation. These sample results pro-
N(t)= F(t−t ) ,
i duce the noise realization N(t):
i
X
where i labels the noise fluctuations, and ti specifies the Nf sgn(t−til)−sgn(t−tir)
N(t)= x , (19)
centeroftheithfluctuation. Themeannumberoffluctu- i
2
ationsNf occurringinthetimeinterval[0,T]isNf =nT. Xi=1 (cid:20) (cid:21)
It is well-knownthat for noise with these properties, the
wheret =t −τ,andt =t +τ. We shallneedtopro-
il i ir i
actualnumberoffluctuationsn thatoccurinatime T is
duce noise realizations with arbitrary mean noise power
governedby the Poissondistribution [18]:
P. We do this by the followingnormalizationprocedure.
First we calculate the mean noise power P of the noise
(N )n
p(n)= f e−Nf . realization N(t) just produced:
n!
1 T
The energy present in a single fluctuation is: P = dtN2(t) . (20)
T
∞ Z0
ε= F2(t)dt . (14)
Then, if the desired value for the noise power is P,
−∞
Z we rescale N(t) in eqn. (19) so that N(t) → n(t) ≡
Let F(t)=xh(t), where h(t) is any convenient function
P/PN(t). The result is a noise realization n(t) with
of finite support with normalization
qmean noise power P. The simulation takes as inputs
∞
the mean noise power P, the variance x2 = σ2, and the
dth2(t)=2τ . i
thermal relaxation time τ. The fluctuation rate n then
−∞
Z
6
follows from eqn. (18): n = P/(2σ2τ). This procedure hunting procedure that requires, on average, 5 integra-
is used to produce 4 types of noise that are character- tions of the Schrodinger equation. Thus a single scaling
ized by the polarization, or direction along which the curve corresponds to approximately 22,500 (30,000) in-
noise N (t) fluctuates: (i) xˆ; (ii) yˆ; (iii) ˆz; and (iv) all 3 tegrations of the Schrodinger equation. As mentioned
i
directions simultaneously. We shall refer to these noise above, we generate 4 such curves for each noise power
polarization types as x-type; y-type; z-type; and 3-type level, and we present data for 3 power levels. Putting
noise, respectively. The noise N (t) is then introduced all this together, we see that the simulation results we
i
into eqn.(7)andthe full HamiltonianH(t) is the sumof present in this paper are the outcome of approximately
H(t) (eqn. (1)) and H (t) (eqn. (7)): 300,000 integrations of the Schrodinger equation. This
int
work was done, initially on our own 16-node Beowulf
H(t)=H(t)+H (t) . (21) cluster, and later on the TeraGrid cluster which was ac-
int
cessed through the National Center for Supercomputing
H(t) drives the Schrodinger dynamics of QuAdS in the Applications (NCSA) in Urbana, Illinois.
presence of the noise realizationN (t). We now go on to
i
explain how the noiseless simulation protocol of Ref. [5]
is extended to include the effects of noise. IV. SIMULATION RESULTS
In this Section we present our simulation results for
B. Noisy Simulation Protocol noisy QuAdS. The noise model parameters (see Sec-
tion III) were chosen to be σ = 0.2; τ = 1; and
We would like to compare how the 4 different noise P = 0.001, 0.003, 0.005. This produced noise that was
types introduced above impact the performance of strongenoughto affectQuAdSperformance,thoughnot
QuAdS. To that end, we determine the scaling relation so strong as to make large-scale simulation impractical.
for the noise-averagedmedian runtime hT(N)i versus N Both H and H have energy–levelspacing ∆E ∼1. For
i p
for each type of noise. Just as in the noiseless proto- our choice of τ, ∆E ∼2/τ, so that our noise bandwidth
col,foreachvalueofN,wegenerate75USAinstancesof matches the natural resonance energies of the quantum
EC3. QuAdS,inthepresenceofeachofthe4noisetypes, computer. WearethusprobingQuAdSinanoiseregime
is then applied to the same 75 USA instances. This al- that complements that studied in Ref. [12]. Further dis-
lows anapples-to-applescomparisonof howthe different cussionof ourchoice ofnoise parameterswill be givenin
noise types affect QuAdS performance. For each noise Section V. As this was our first large-scalesimulation of
type, and each USA instance, we generate 10 noise re- noisy QuAdS, we decided to simplify the noise model by
alizations and use the Hamiltonian H(t) to find the 10 restricting the noise field and coupling constants to be
corresponding QuAdS runtimes Tj(N). Here j labels the same for each qubit: Ni(t) = N(t) and γi = 1. We
i
the noise realizations (j = 1,··· ,10), and i labels the planto lift this restrictionin our nextset ofsimulations.
USA instances (i = 1,··· ,75). Thus, for each value of We also plan to parallelize the hunting procedure which
N, and each noise type, 750 runtimes are found. The shouldsubstantiallyspeedupthecode,allowingustoex-
ensemble E of 750 runtimes Tj(N) allows us to esti- plorelargervaluesofmeannoisepowerP. Togetasense
N i
mate the statistical effects of each noise type on QuAdS of how long the following simulation took in real time,
performance over the full set of 75 USA instances. The note that producing one scalingcurve at P =0.005took
noise-averagedmedian runtime hT(N)i is then identified approximately 3 weeks on our 16-node Beowulf cluster.
with the median runtime calculated from the noise en- Thesamesimulationtookapproximately72–96hourson
sembleE . We thenfitthesimulationresultsusingboth the TeraGrid cluster at NCSA.
N
power-law scaling hT(N)i = aNb and exponential scal- We nowpresentoursimulationresults. Foreachmean
ing hT(N)i = a[exp(bN)−1], and calculated the χ2 for noise powerP we present4 figuresand 2 tables. Eachof
each type of fit. We determine 2 such fits for each of the thefigurescontainsournumericalresultsforhT(N)ifora
4noisetypes. Foreachfit-type,wecomparedtherespec- particular noise polarization type (x, y, z, 3), along with
tive scaling exponents to get a quantitative measure of two fits to the data: (i) a power-law fit hT(N)i = aNb;
whichofthe4noisetypesmostadverselyaffectsQuAdS. and (ii) an exponential fit hT(N)i = a[exp(bN)−1].
As will be discussed in Section IV, scaling curves will Information about the best fits to the data are collected
be determined for each of the 4 noise types for 3 dif- in the two tables, one table for each of the fitting func-
ferent noise power levels. As pointed out at the end tions. Each table contains: (i) the best fit parameters;
of Section IIC, most simulations will be restricted to (ii) the chi-squared for the fit χ2 ; and (iii) the prob-
fit
7 ≤ N ≤ 12, though for x-type noise, the upper limit ability P(χ2 > χ2 ) that, assuming the fitting function
fit
could be extended to N = 13 and 14. It is important correctlydescribesthescalingofhT(N)iversusN,asam-
to recognize that each point on a given scaling curve is pling of hT(N)i would yield a χ2 >χ2 . The closer this
fit
based on 750 runtimes, and so each scaling curve with probability is to 1, the more consistent the data-set is
7 ≤ N ≤ 12 (7 ≤ N ≤ 14) is distilled from 4500 (6000) with the fitting function. As such, it provides a quan-
runtimes. Each runtime Tj(N) is itself the result of a titative measure of the quality of fit to the data. Note
i
7
that,duetothefinite-sizeofthedata-set,morethanone 30
fitting function canbe consistent with the data. To help 27 Power-Law Fit
Exponential Fit
thereaderlocatespecificresults,weprovidethefollowing
24
roadmap through the figures and tables.
21
N) >18
1. Figures 2, 3, 4, and 5 correspond to x, y, z, and < T(15
3-type noise with P =0.001,respectively. Tables I
12
and II give the specifics of the power-law and ex-
ponentialfits throughthese data-sets,respectively. 9
Table entries are ordered according to increasing 6
scaling-exponent b.
3
6 8 10 12 14
N
2. Figures 6, 7, 8, and 9 correspond to x, y, z, and 3- FIG. 2: Simulation results for QuAdS with noise polarized
along xˆ and mean noise power P = 0.001. Plotted are the
type noise with P =0.003, respectively. Tables III
noise-averaged median runtime hT(N)i (dimensionless units)
and IV give the specifics of the power-law and ex-
versusthenumberofbitsN. Thesolidlineisthebestpower-
ponentialfits throughthese data-sets,respectively.
lawfittothedataandthedash-dotlineisthebestexponential
Table entries are ordered according to increasing
fit to thedata. Theerror bars give 95% confidence limits for
scaling-exponent b. eachmedian. Eachdatapointistheoutcomeofaveragingover
75USAinstancesand10noiserealizationsperUSAinstance.
3. Figures10,11,12,and13correspondtox,y,z,and
24
3-typenoisewithP =0.005,respectively. TablesV
and VI give the specifics of the power-law and ex- Power-Law Fit
21 Exponential Fit
ponentialfits throughthese data-sets,respectively.
Table entries are ordered according to increasing 18
scaling-exponent b.
>15
N)
T(
< 12
Examination of these figures and tables shows that the
power-lawandexponentialfitsboth provideexcellentfits 9
to the numerical results. One sees that the probability
P(χ2 >χ2 ),andhence the quality offit, showsa slight 6
fit
decrease with increasing mean noise power P, and that 3
6 8 10 12
the rate ofdecreaseinthe fitquality is largestfor y-type N
noise. One also notes that the quality of the exponential
fit decreases more slowly than does the quality of the
FIG. 3: Simulation results for QuAdS with noise polarized
power-lawfit. Furtherdiscussionofthese resultsisgiven
along yˆ and mean noise power P = 0.001. Plotted are the
in Section V. noise-averaged median runtime hT(N)i (dimensionless units)
versusthenumberofbitsN. Thesolidlineisthebestpower-
lawfittothedataandthedash-dotlineisthebestexponential
fit to thedata. Theerror bars give 95% confidence limits for
TABLEI:Summaryofbest-fitparametersforpower-lawscal-
eachmedian. Eachdatapointistheoutcomeofaveragingover
ingcurvesforP =0.001. Forcomparison,best-fitparameters 75USAinstancesand10noiserealizationsperUSAinstance.
for noiseless QuAdSare also included.
P noise type a b χ2 P(χ2>χ2 )
fit fit V. DISCUSSION
0.000 — 1.1966×10−1 2.0034 0.092 0.9999844
0.001 x 1.1447×10−1 2.0267 0.083 0.9999886
A. Summary
0.001 3 1.0635×10−1 2.0594 0.064 0.9994915
0.001 z 1.0395×10−1 2.0695 0.063 0.9995085
We examine here what can be learned from the re-
0.001 y 9.6766×10−2 2.1075 0.078 0.9992617
sults presented in Section IV. To begin, we note that
both power-law scaling hT(N)i = aNb and exponential
scaling hT(N)i = a[exp(bN)−1] provide excellent fits
8
TABLE II: Summary of best-fit parameters for exponential
24
scaling curvesfor P =0.001.
Power-Law Fit
21 Exponential Fit
P noise type a b χ2 P(χ2>χ2 )
fit fit
18
0.001 x 3.81403 0.142561 0.216 0.9998065
>15 0.001 3 2.60771 0.172331 0.075 0.9993111
T(N) 0.001 z 2.46414 0.176985 0.059 0.9995798
< 12 0.001 y 2.42677 0.179578 0.085 0.9991179
9
6
30
3 Power-Law Fit
6 8 10 12 Exponential Fit
N 27
24
FIG. 4: Simulation results for QuAdS with noise polarized 21
along ˆz and mean noise power P = 0.001. Plotted are the N) >18
noise-averaged median runtime hT(N)i (dimensionless units) T(
versusthenumberofbitsN. Thesolidlineisthebestpower- < 15
lawfittothedataandthedash-dotlineisthebestexponential
12
fit to thedata. The error bars give95% confidencelimits for
9
eachmedian. Eachdatapointistheoutcomeofaveragingover
75USAinstancesand10noiserealizationsperUSAinstance. 6
3
6 8 10 12 14
N
21 Power-Law Fit
Exponential Fit
FIG. 6: Simulation results for QuAdS with noise polarized
18 along xˆ and mean noise power P = 0.003. Plotted are the
noise-averaged median runtime hT(N)i (dimensionless units)
>15 versusthenumberofbitsN. Thesolidlineisthebestpower-
N) lawfittothedataandthedash-dotlineisthebestexponential
< T(12 fit to thedata. Theerror bars give 95% confidence limits for
eachmedian. Eachdatapointistheoutcomeofaveragingover
9
75USAinstancesand10noiserealizationsperUSAinstance.
6
yˆ; (iii) along zˆ; and (iv) along all 3 directions simul-
3
6 8 10 12
N taneously. At the noise power levels considered, x-type
noise was found to have the smallest impact on QuAdS
performance, while y-type noise was found to cause the
FIG. 5: Simulation results for QuAdS with noise polariza-
largest slowdown. As noted in Section IV, although the
tion along all 3 directions and mean noise power P = 0.001.
quality of both the power-law and exponential fits is ex-
Plotted are the noise-averaged median runtime hT(N)i (di-
mensionless units) versus the number of bits N. The solid cellent in all cases considered, we did see a slight re-
line is the best power-law fit to the data and the dash-dot duction in the quality of fit as noise power increased for
line is the best exponential fit to the data. The error bars all noise-types except x-type noise. We also noted (see
give 95% confidence limits for each median. Each datapoint Tables V and VI especially) that the quality of the ex-
is the outcome of averaging over 75 USA instances and 10 ponential fit decreased at a slightly slower rate than did
noise realizations per USAinstance. the power-law fit. It would clearly be of interest to ex-
tend the simulations to larger noise power levels to ex-
amine: (i) how rapidly the scaling exponent b increases
tooursimulationresults,althoughthefitparametersare with noise power; and (ii) how quickly the quality of fit
noisedependent. TablesI–VIshowthatthescalingexpo- forbothpower-lawandexponentialfitsdeteriorateswith
nentbincreaseswithincreasingmeannoisepowerP,and increasingnoise power. Given that a quantum computer
the rate at which it increases depends on the noise po- will eventually crossover to classical behavior with suffi-
larization, or direction along which the noise field N (t) cient noise power, and should classical algorithms truly
i
fluctuates. Clearly, as the scaling exponent increases, requireexponentialtimeontherandomlygeneratedhard
QuAdS performance decreases non-linearly. We simu- instancesconsideredinRef.[5]andhere,then onewould
latednoisewithnoisepolarization: (i)alongxˆ;(ii)along anticipate a power-lawfitto eventuallybecome inconsis-
9
24
30
Power-Law Fit Power-Law Fit
27 Exponential Fit 21 Exponential Fit
24
18
21
N) >18 N) >15
< T(15 < T(12
12
9
9
6
6
36 8 10 12 36 8 10 12
N N
FIG. 7: Simulation results for QuAdS with noise polarized FIG. 9: Simulation results for QuAdS with noise polariza-
along yˆ and mean noise power P = 0.003. Plotted are the tion along all 3 directions and mean noise power P = 0.003.
noise-averaged median runtime hT(N)i (dimensionless units) Plotted are the noise-averaged median runtime hT(N)i (di-
versusthenumberofbitsN. Thesolidlineisthebestpower- mensionless units) versus the number of bits N. The solid
lawfittothedataandthedash-dotlineisthebestexponential line is the best power-law fit to the data and the dash-dot
fit to thedata. The error bars give95% confidencelimits for line is the best exponential fit to the data. The error bars
eachmedian. Eachdatapointistheoutcomeofaveragingover give 95% confidence limits for each median. Each datapoint
75USAinstancesand10noiserealizationsperUSAinstance. is the outcome of averaging over 75 USA instances and 10
noise realizations per USAinstance.
24 TABLE III: Summary of best-fit parameters for power-law
scalingcurvesforP =0.003. Forcomparison,best-fitparam-
Power-Law Fit
21 Exponential Fit eters for noiseless QuAdSare also included.
18
P noise type a b χ2 P(χ2>χ2 )
fit fit
>15 0.000 — 1.1966×10−1 2.0034 0.092 0.9999844
T(N) 0.003 x 9.3604×10−2 2.1304 0.052 0.9999971
< 12 0.003 z 8.4209×10−2 2.1774 0.050 0.9996975
9 0.003 3 7.8825×10−2 2.2077 0.083 0.9991666
0.003 y 2.9821×10−2 2.6792 0.570 0.9663310
6
3
6 8 10 12
N subsection we show that noise-induced dephasing leads
to decoherence in QuAdS, estimate how much decoher-
ence is present in our simulations, and derive an upper
FIG. 8: Simulation results for QuAdS with noise polarized
along ˆz and mean noise power P = 0.003. Plotted are the boundforthenoise-averagedQuAdSsuccessprobability.
noise-averaged median runtime hT(N)i (dimensionless units)
versusthenumberofbitsN. Thesolidlineisthebestpower-
lawfittothedataandthedash-dotlineisthebestexponential B. Noise-Induced Decoherence in QuAdS
fit to thedata. The error bars give95% confidencelimits for
eachmedian. Eachdatapointistheoutcomeofaveragingover
Although our focus in this paper is on QuAdS, the
75USAinstancesand10noiserealizationsperUSAinstance.
following analysis can be adapted to the more general
situation of noisy quantum adiabatic dynamics. We will
report on this elsewhere.
tent with the simulation data at sufficient noise power The total Hamiltonian for noisy QuAdS is (see
and the exponential fit to provide the better fit. Beyond eq. (21)),
this threshold value of noise power, QuAdS would then
be expected to have performance comparable with clas- H(t)=H(t)+H (t) ,
int
sical algorithms. Observation of such a crossover would
beextremelyimportantasitwouldgiveadirectmeasure with H(t) and H (t) given in eqs. (1) and (7), respec-
int
of how much noise QuAdS can tolerate. In the following tively. IntheSchrodingerpicture,thedynamicsisdriven
10
TABLE IV: Summary of best-fit parameters for exponential
scaling curvesfor P =0.003. 39
Power-Law Fit
36 Exponential Fit
P noise type a b χ2 P(χ2>χ2 ) 33
fit fit
30
0.003 x 3.24309 0.157716 0.169 0.9999056
27
0.003 z 2.10817 0.193125 0.036 0.9998428 N) >24
0.003 3 2.05131 0.195818 0.075 0.9993165 T(21
<
0.003 y 0.847514 0.285049 0.283 0.9908794 18
15
12
9
30 6
3
27 Power-Law Fit 6 8 10 12
Exponential Fit N
24
21 FIG. 11: Simulation results for QuAdS with noise polarized
along yˆ and mean noise power P = 0.005. Plotted are the
N) >18 noise-averaged median runtime hT(N)i (dimensionless units)
< T(15 versusthenumberofbitsN. Thesolidlineisthebestpower-
lawfittothedataandthedash-dotlineisthebestexponential
12
fit to thedata. Theerror bars give 95% confidence limits for
9 eachmedian. Eachdatapointistheoutcomeofaveragingover
75USAinstancesand10noiserealizationsperUSAinstance.
6
3
6 8 10 12 14
N
30
FIG. 10: Simulation results for QuAdS with noise polarized 27 Power-Law Fit
Exponential Fit
along xˆ and mean noise power P = 0.005. Plotted are the
24
noise-averaged median runtime hT(N)i (dimensionless units)
versusthenumberofbitsN. Thesolidlineisthebestpower- 21
lfiatwtofittthoetdhaetdaa.tTahaendertrhoerdbaasrhs-gdiovteli9n5e%isctohnefibdeesntceexpliomnietnstfioarl N) >18
eachmedian. Eachdatapointistheoutcomeofaveragingover < T(15
75USAinstancesand10noiserealizationsperUSAinstance.
12
9
by 6
∂ 3
6 8 10 12
i |ψi=H(t)|ψi . N
∂t
Transforming to an interaction-like picture (the time-
FIG. 12: Simulation results for QuAdS with noise polarized
ordering symbol is suppressed), along ˆz and mean noise power P = 0.005. Plotted are the
noise-averaged median runtime hT(N)i (dimensionless units)
t
versusthenumberofbitsN. Thesolidlineisthebestpower-
|ψ(t)i=exp −i dsH (s) |χ(t)i , (22)
int lawfittothedataandthedash-dotlineisthebestexponential
(cid:20) Z0 (cid:21)
fit to thedata. Theerror bars give 95% confidence limits for
the equation of motion for |χ(t)i is found to be eachmedian. Eachdatapointistheoutcomeofaveragingover
75USAinstancesand10noiserealizationsperUSAinstance.
∂
i |χi=H(t)|χi , (23)
∂t
where with B =H(t) and
H(t)=ei 0tdsHint(s)H(t)e−i 0tdsHint(s) . (24) t
R R ξA = i dsHint(s)
Using the well-known identity Z0
t N
eξABe−ξA =B+ξ[A,B]+ ξ2 [A,[A,B]]+··· (25) = −iγ x(s)ds σxi , (26)
2 (cid:20) Z0 (cid:21)i=1
X