Certification of Boson Sampling Devices with Coarse-Grained Measurements

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by  Sheng-Tao Wang

Certification of Boson Sampling Devices with Coarse-Grained Measurements
Sheng-Tao Wang and L.-M. Duan
Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
(Dated: January 13, 2016)
A boson sampling device could efficiently sample from the output probability distribution of
noninteracting bosons undergoing many-body interference. This problem is not only classically
intractable, but its solution is also believed to be classically unverifiable. Hence, a major difficulty
inexperimentistoensureabosonsamplingdeviceperformscorrectly. Wepresentanexperimental
friendly scheme to extract useful and robust information from the quantum boson samplers based
on coarse-grained measurements. The procedure can be applied to certify the equivalence of boson
samplingdeviceswhilerulingoutalternativefraudulentdevices. Weperformnumericalsimulations
todemonstratethefeasibilityofthemethodandconsidertheeffectsofrealisticnoise. Ourapproach
is expected to be generally applicable to other many-body certification tasks beyond the boson
6
sampling problem.
1
0
PACSnumbers: 03.67.Lx,03.67.Ac,05.30.Jp
2

n
a In the last three decades, quantum computation has from a large-scale boson sampler. In other words, would
J stimulated considerable excitement among physicists, any filtered information be able to verify the equivalence

1 computer scientists, and mathematicians, with the gen- of two identical boson sampling devices while excluding
1
eral belief that quantum computers could solve certain possible fraudulent ones? In light of rapid experimental

tasks significantly faster than current electronic comput- advances,thisissuewillbeincreasinglyrelevantwhenthe
]
h ers [1, 2]. Especially after the discovery of Shor’s fac- system scales up: as the probabilities of generic output
p toring algorithm [3], theorists and experimentalists have eventsbecomeexponentiallysmall,samplingnoisedueto
-
t been teaming up to converge on the goal of demonstrat- limited measurement trials may conceal any distinctive
n
ingquantumsupremacy[4]. Recently,animportantstep information.
a
u made by Aaronson and Arkhipov [5] is to formulate the
In this paper, we introduce an experimental friendly
q boson sampling problem, which is intractable for classi-
scheme to extract useful structures from a boson sam-
[ cal computers but remarkably amenable to quantum ex-
pling device based on coarse-grained measurements. Us-
periments. A number of elegant experiments have since
1 ing standard statistical tools, we simulate the experi-
v implemented the problem with linear optics on a small
mental certification process and show that the coarse-
7 scale [6–13]. Pushing the experiments beyond classical
grained information is able to provide a quantitative as-
2 capabilities would constitute a strong demonstration of
6 sessmenttothe degreesofmatchingbetweentwoalleged
the quantum speedup and in addition lead to important
2 boson samplers. This is important when one needs to
implicationsinthefoundationsofcomputerscience[2,5].
0 verify the equivalence of two quantum samples drawn
.
1 from the same boson sampling device or from different
At the core of the hardness-of-simulation property lie
0 devices with identical processes. It will also be crucial
the exponential cost of computing a matrix permanent
6 in situations wherein we can completely trust one device
1 [14]andtheexponentialnumberofpossibleoutputevents
and need to validate another possibly fraudulent device
: inbosonsampling. Theseposemajordifficultiesincerti-
v against the reliable one. Our numerical simulation in
fying the correctness of a boson sampler on a large scale
i
X addition demonstrates our scheme could tolerate a mod-
[11, 12, 15–18]. The credibility of a certification pro-
erate amount of experimental noise [22, 23] while strong
r cess thus relies on gathering convincing circumstantial
a noise invalidates the equivalence due to mismatched in-
evidence while ruling out alternative explanations. Sev-
terference processes. On a broader scale, our method is
eral efficient schemes have been proposed to validate a
not specific to boson sampling, but could be applicable
boson sampler against the uniform sampler [15] making
to other generic many-body certification problems.
use of the information in the unitary process [11, 12, 16]
and against the classical sampler of distinguishable par- Before proceeding to our proposed scheme, we briefly
ticles[6,11,19]exploitingthebosoniccloudingbehavior introduce the boson sampling problem and clarify why
[12]. It is also possible to depart from the computation- large sampling errors are involved without coarse grain-
ally hard space and design an efficient test based on pre- ing. In a typical setup, there are N indistinguishable
dictable forbidden events with special inputs and scat- bosons prepared in M input modes and allowed to co-
tering process [17, 20, 21], assuming the device would herently interfere with one another in a (random) uni-
be equally operational in general. An unsettled prob- tary process. We abstract away from interactions be-
lem especially pertinent to experiments is whether one tween particles, so the resultant many-body interference
would be able to extract useful and robust information is purely due to the bosonic statistics. To compute the
2
(a) State Space +
+ + + + + + bility
+ + B3 a
+ b
B1 + o
r
+ p
B2 + +
(b)2×10-4 Simulated Exp 1 (c)2×10-4 Simulated Exp 1 + + +Unobserved state observed states
Simulated Exp 2 Simulated Exp 2
Theory Theory
y1 y1 Figure 2. Schematic for the coarse-graining procedure. The
abilit0 abilit0 filled symbols are bubble centers. The bubble structure is
b b formed as follows: (1) Pick one set of experimental samples
o o
pr1 pr1 and sort the observed states in descending probability. (2)
Consider only states that are not included in previous bub-
2 2 bles. Select a bubble center as the state with the highest
1.75 1.8 1.85 1.9 1.95 2 7.25 7.3 7.35 7.4 7.45 7.5 probability (if multiple states attain the same highest prob-
state ×105 state ×105
ability, choose any of them as the bubble center). Form a
newbubblebyenclosingstateswithL distancesmallerthan
Figure 1. Original Distributions. (a) Theoretical distribu- 1
a cutoff radius to this bubble center (the cutoff radius may
tion PQ of a boson sampling device with N = 5 particles
be increased for subsequent bubbles to ensure each one has
in M = 40 modes after a random unitary transformation.
comparable sample size). (3) The bubble structure is estab-
(b) and (c) shows the zoom-in distributions of two simu-
lishedwithcorrespondingcentersafterallobservedstatesare
lated experimental samples (PQ and PQ) with sample size
S1 S2 included. (4)Othersamplesandtheoreticaldistributionsare
N =10000 drawn from PQ. The theoretical distribution is
m coarse grained with the same bubble structure.
superimposed onto the simulated samples.
features and may falsely accept some fraudulent devices.
probabilityofanoutputevent,oneneedstocalculatethe Ourproposedschemetakestheabovefactorsintoconsid-
permanentoftheassociatedN×N matrix[6,24], which eration, and coarse grain the states based on the L dis-
1
requiresanexponentialcostforagenericcomplexmatrix. tancemeasure. TheL distancebetweentwooccupation-
The possible number of output events is D =(cid:0)M+N−1(cid:1), number-basis states i1s defined as L = (cid:80)M|ψ − φ |,
N 1 i i i
which grows exponentially with N (for M ≥ N). As where ψ (φ ) is the occupation number in the ith mode
i i
the system scales up, the number of measurement runs forthestate|ψ(cid:105)(|φ(cid:105)). Detailsofthecoarse-grainingpro-
willbeunabletokeeppacewiththeexponentialgrowth. cedure are shown in Fig. 2. We note that this scheme is
Thisgivesrisetolargesamplingerrorswithlimitedsam- not the only way to perform coarse graining. We expect
ple size. Fig. 1 shows an example in which the sample that other procedures meeting the above considerations
size is less than 1% of the Hilbert space dimension. Two may also work. Below, we show that useful and robust
samplesdrawnfromthesamedevicecouldberatherdis- structures can be extracted from the samples using our
similar (with a low average fidelity F ≈ 0.039±0.002, method.
where F = (cid:12)(cid:12)(cid:112)PQ ·(cid:112)PQ(cid:12)(cid:12)). From these distributions, We simulate the experimental certification process
S1 S2
it is not quite possible to assess whether the samples are with two different systems, one with trapped ions and
drawn from the same bona fide boson sampling device. onewithHaar-distributedrandomunitaries. Forallsim-
With a proper coarse-graining procedure, we show, ulations, we choose a fixed sample size of Nm = 10000,
however, that a reliable comparison between two given whichisareasonabledetectioncountinexperiments,but
outputsamplesisachievable. Thereareafewfactorsthat is nevertheless smaller than 1% of the Hilbert space di-
a reasonable coarse-graining procedure should consider. mension in study.
First, it should be constructed from experimental sam- Fortrappedions,thetransverselocalphononsareused
ples. On a large scale, classical simulation is no longer as indistinguishable bosons with the Hamiltonian given
feasible. Experiments may nevertheless pick out the im- by [26–28]
portant output states with higher probabilities. Second,
M M
(cid:88) (cid:88) (cid:16) (cid:17)
the procedure should be scalable: not only the measured H = (cid:126)w a†a + (cid:126)t a†a +a†a , (1)
c x,i i i ij i j j i
events but all possible events should be grouped into
i i<j
some bubbles, where the number of bubbles should not
be subject to the exponential growth. Third, the filtered where wx,i =−(cid:80)Mj(cid:54)=it0/|zi0−zj0|3, tij =t0/|zi0−zj0|3,
information should still carry some knowledge of the full and t = e2/(8π(cid:15) mω ). z denotes the axial equilib-
0 0 x i0
correlationsintheoutputs. Thisisbecausetheessenceof rium position of the ith ion with mass m and charge e.
the many-body interference lies in the many-mode cor- w (w ) is the axial (transverse) trapping frequency. In
z x
relations: previous works [17, 20, 25] have shown that the simulation, we use experimentally relevant parame-
few-particleobservablesmaynotcapturethefullbosonic tersω =2π×0.03MHzandω =2π×4MHzfor171Yb+
z x
3
(a) 0.1 (b) 0.3
Simulated Exp 1 Simulated Exp 1
Simulated Exp 2 0.25 Distinguishable
0.08 Theory Uniform
y y 0.2
bilit0.06 bilit
a a0.15
ob0.04 ob
pr pr 0.1
0.02
0.05
0 0
0 5 10 15 20 25 0 5 10 15 20 25
bubbles bubbles
(c)0.15 (d)0.15
Simulated Exp 1 Simulated Exp 1
Simulated Exp 2 Distinguishable
Theory Uniform
y 0.1 y 0.1
bilit bilit
a a
b b
o o
pr0.05 pr0.05
0 0
0 5 10 15 20 25 0 5 10 15 20 25
bubbles bubbles
Figure 3. Coarse-grained probability distributions. All samples have a sample size N =10000. The simulated experimental
m
samples (cid:0)PQ and PQ(cid:1) are drawn from the boson sampler (PQ); distinguishable (PC) and uniform (cid:0)PU(cid:1) samples are drawn
S1 S2 S S
respectively from the classical (PC) and uniform (PU) samplers. Errors for the probabilities follow the standard deviations
of the multinomial distribution. (a) and (b): probability distributions for the trapped-ion system with intermediate-time
dynamics. The simulated system has N = 12 phonons in a M = 12 ion chain with one phonon on each ion as the input
state. (c) and (d): probability distributions after a Haar-distributed random unitary transformation. The simulated system
has N =5 particles in M =40 modes with |1,1,1,1,1,0,···,0(cid:105) as the input state.
ions and consider N =12 phonons on M =12 ions. The sensitivetodetailsofthecoarse-grainingmethod,suchas
total Hilbert space size is D = 1352078. The evolution the particular sample used to initiate the bubble struc-
time is chosen at some intermediate time (τ = 100µs) ture.
withinterestingmany-bodydynamics(seeSupplemental
To quantify the degrees of matching between coarse-
Material [29] for further results in the long time limit).
graineddistributions, weemploythetwo-sampleχ2 test.
For the random unitary process, we simulate N =5 par-
Underthenullhypothesiswhereintwosamplesaredrawn
ticlesinM =40modes,asettingcomparabletothecur-
from the same distribution, the χ2 statistics follow the
rentexperimentalregimewithlinearoptics[6–13]andin
χ2-distributionwithdegreesoffreedom(df)equaltothe
a limit where M > N2 to suppress collision events. A
number of bubbles (N ) minus one. If the χ2 statistic
Haar-distributedM×M randomunitarymatrixisused, B
is large with a small associated p-value, one nominally
with a Hilbert space dimension of D =1086008.
rejects the null hypothesis. Therefore, during the certi-
fication process, one should ideally accept the null if the
Based on the coarse-graining procedure outlined in
pair of samples come from the same boson sampling de-
Fig. 2, we group the sample events into different bub-
vice and reject it if one is sampled from an alternative
bles according to one set of experimental samples. The
distribution. Twotypesoferrorscanbeincurred, atype
theoreticaldistributionsarealsosubjecttothesamebub-
Ierror(falsepositive)relatedtofalselyrejectingthetrue
ble structure. Fig. 3 shows an example of the coarse-
null hypothesis and a type II error (false negative) asso-
grained distributions. This extracted information is ro-
ciated with the failure to reject a false null hypothesis.
bust and reliable with small sampling errors. By visual
Prior to the test, one sets a significance level α, which
comparison, we can see that the boson sampling data
PQ and PQ match closely with each other, but differ will be the type I error rate if both samples are from the
S1 S2 same distribution.
significantly from samples drawn from alternative distri-
butions, such as the distinguishable (PC) and uniform In simulation, we could generate many sets of samples
S
(PU) samples [29]. Here, it is also possible to compare and repeat the certification process to better gauge the
S
the simulated data with the theoretical distribution PQ, error rates. Each time, we grouped the sample events
which will not be directly obtainable when experiments into bubbles and compared one set of boson sampling
surpass classical simulation capabilities. By repeating dataPQ withvariousothersamples(PQ,PC,andPU).
S1 S2 S S
the procedure, we observe that the comparisons are not A χ2 statistic and a p-value were computed for each χ2
4
Table I. The pass rates R between simulated experimental
sample 1 and various other samples. The two-sample χ2 test
is performed to assess whether they come from the same dis-
tribution. Thesignificancelevelαissetat1%. Foreachpair
ofgeneratedsamples,ifthep-valueisgreaterthanα,thecom-
parisonpassesthetest. ThisisrepeatedforN =10000runs
s
and pass rates are recorded. Noisy samples for the trapped-
ion system include a 1% (3%) timing error, whereas a 1%
(3%)randomerrorisincludedintherandomunitarymatrix.
pass rate (%) compared to experimental sample 1
Trapped Ions Random Unitary
Figure4. Distributionsofthetwo-sampleχ2test-statisticsfor N 24.2 40.5 69.9 25.9 40.8 70.5
B
therandomunitaryprocesswithN ≈40.8. N =10000sets
B s Exp 2 99.1 99.2 99.1 99.1 99.2 99.1
ofsamplesaregeneratedinthesimulationanddistributionsof
theχ2 statisticsbetweenthecorrespondingpairsareplotted. Exp 2 (1% Noise) 98.0 98.2 98.2 98.9 98.9 99.1
Thesolidcurveistheχ2-distributionwithdf=N −1. The Exp 2 (3% Noise) 71.1 77.1 75.9 98.3 98.4 98.6
B
dashed line marks the cutoff χ2 value at α=1% [30]. Distinguishable 0 0 0 6.7 0.3 0.03
Uniform 0 0 0 0 0 0
test. This process was repeated for N = 10000 runs,
s
with the distributions of the χ2 statistics and p-values
in experiments. In the case of trapped ions, we included
recorded. Fig.4presentsthedistributionsofχ2 statistics
a 1% (3%) systematic error in the timing (1µs (3µs)
for the random unitary process with the average number
shift in τ). This is also equivalent to a 1% (3%) error
of bubbles NB ≈ 40.8. It can be seen clearly that the in the hopping amplitude t ∼ω2/ω [28], which trans-
test statistics between PQ and PQ follow the χ2 distri- ij z x
S1 S2 lates to a respective shift in the trapping frequencies in
bution with df = NB −1, whereas those between PSQ1 experiments. For the random unitary process, we added
and PC fall on the far tail of the χ2 distribution, there- 1%(3%)randomnoisetotheunitarymatrix(seeSupple-
S
fore offering a definitive answer regarding whether the mentalMaterialfordetails[29]). AsseeninTableI,with
samples are from the same distribution. More quantita- smallnoise(∼1%)thetypeIerrorratesarekeptincheck
tively,wecalculatethepassrateRatagivensignificance (<∼2%). When the noise becomes substantial, pass rates
level α = 1%. Specifically, if the p-value is greater than may drop sharply if the unitary process changes consid-
α, the comparison passes the test. Out of Ns tests, the erably. This also shows the sensitivity of our method to
passratesinpercentagearereportedinTableI.Between strong noise and it could serve as a stringent certifica-
two sets of boson sampling data PQ and PQ, the pass tion test. The dissimilar sensitivity to noise for the two
S1 S2
rates R ≈ 99% for all cases, with the type I error rates systemsisduetothedifferentwaysnoiseisincludedand
1−R being very close to α as expected. Between PQ natures of the noise (systematic versus random).
S1
and alternative samples (PSC and PSU), the pass rates re- Inconclusion,wehaveshownthatusefulandrobustin-
flect type II error. They can reach a few percent for formationcanbeextractedfromthecoarse-grainedmea-
smallerNB butdropto<1%asNB increases. Thisalso surementsforbosonsampling. Thecoarse-graineddistri-
presents a tradeoff between the information obscured by butionscanbefurtherusedtocertifythebosonsampling
the sampling noise without coarse-graining and the in- device. We expect this method to be handy when exper-
formationonediscardsbyheavycoarse-graining. Ingen- iments progress beyond classical capabilities. It should
eral, to reduce sampling noise, each bubble should have also be a useful tool for other generic many-body certifi-
a minimum number of observed events. For the χ2 test cation problems.
to work reliably, this minimum number is conventionally
ThisworkissupportedbytheARL,theIARPALogiQ
chosen to be 10. For a detection count of N = 10000,
m program, and the AFOSR MURI program.
a range of 20 <∼ NB <∼ 100 works well, with the require-
ment that the smallest bubble includes no less than 10
events (one could group together very small bubbles).
Noticeably, this only depends on the number of detec-
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B
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6
SUPPLEMENTAL MATERIAL: CERTIFICATION OF BOSON SAMPLING DEVICES WITH
COARSE-GRAINED MEASUREMENTS
In this Supplemental Material, we include additional results for the trapped-ion system with long-time
dynamics and for boson sampling data with experimental noise. Further analysis for the two-sample χ2
test is provided.
RESULTS FOR THE LONG-TIME TRAPPED-ION SYSTEM
(a) 0.1 (b)0.15
Simulated Exp 1 Simulated Exp 1
Simulated Exp 2 Distinguishable
0.08 Theory Uniform
y y 0.1
bilit0.06 bilit
a a
ob0.04 ob
pr pr0.05
0.02
0 0
0 5 10 15 20 25 0 5 10 15 20 25
bubbles bubbles
Figure 5. Coarse-grained probability distributions for the trapped-ion system with long-time dynamics. All samples have
a sample size N = 10000. The simulated experimental samples (cid:0)PQ and PQ(cid:1) are drawn from the boson sampler (PQ);
m S1 S2
distinguishable (PC) and uniform (cid:0)PU(cid:1) samples are drawn respectively from the classical (PC) and uniform (PU) samplers.
S S
Errors for the probabilities follow the standard deviations of the multinomial distribution. The simulated system has N =12
phonons in a M =12 ion chain with one phonon on each ion as the input state.
Inthemaintext,weincludedresultsforboththetrapped-ionsystemwithintermediate-timedynamics(τ =100µs)
and for the random unitary process (Fig. 3 and Table I of the main text). In Fig. 5 and Table II here, we add the
coarse-grained distributions and the pass rate results for the long-time trapped-ion system (at τ = 10ms). With
these coarse-grained distributions, we would like to discuss some differences between the trapped-ion system and
the random unitary process. In the prototypical boson sampling problem, the hardness-of-simulation argument is
based on randomly selected unitaries [5]. This is because for some structured unitary process, fast approximation
algorithms may exist. For the trapped-ion system, the phonon normal modes are fixed (fixed time-independent
Hamiltonian in Eq. 1 of the main text), so we may be able to extract some distinctive structures from the output
probability distributions. Nevertheless, we still expect the complex many-body dynamics in trapped ions to be
classically intractable.
From the coarse-grained distributions, we can see that the boson sampling data PQ for intermediate-time trapped
S
ions are conspicuously different from alternative samples, such as the distinguishable sample PC and the uniform
S
sample PU. On the other hand, the boson sampling data bear resemblance to the distinguishable samples PC for
S S
random unitary processes, and the long-time trapped-ion phonon distributions are instead in closer proximity to the
uniform samples PU. The qualitative difference arises from the fact that the random unitary process has a Haar-
S
distributed random unitary matrix whereas the long-time trapped-ion unitary only has random phases (eigenvalues)
Table II. The pass rates R between simulated experimental sample 1 and various other samples for the trapped-ion system
with long-time dynamics. The two-sample χ2 test is performed to assess whether they come from the same distribution. The
significancelevelαissetat1%. Foreachpairofgeneratedsamples,ifthep-valueisgreaterthanα,thecomparisonpassesthe
test. This is repeated for N =10000 runs and pass rates are recorded.
s
pass rate (%) compared to experimental sample 1
Trapped Ions (Long time)
Average No. of bubbles 24.3 39.0 66.5
Experimental Sample 2 98.9 99.0 99.2
Distinguishable Sample 0 0 0
Uniform Sample 5.7 1.3 0.2
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(a) 0.1 (b)0.25
Simulated Exp 1 Simulated Exp 1
Noisy Exp 2 Noisy Exp 2
0.08 Theory 0.2 Theory
y y
bilit0.06 bilit0.15
a a
ob0.04 ob 0.1
pr pr
0.02 0.05
0 0
0 5 10 15 20 25 0 5 10 15 20 25
bubbles bubbles
Figure6. Coarse-grainedprobabilitydistributionsforthenoisysamples. (a)Trapped-ionsystemofintermediate-timedynamics
witha1%timingerrorincludedinthenoisysample. ThesimulatedsystemhasN =12phononsinaM =12ionchainwithone
phonononeachionastheinputstate. (b)Distributionsafterarandomunitarytransformation. A1%randomnoiseisaddedto
theunitaryprocessinthenoisysample. ThesimulatedsystemhasN =5particlesinM =40modeswith|1,1,1,1,1,0,···,0(cid:105)
as the input state.
withafixedcrystalmodestructure. Wehavefurthertestedthisideabysimulatingthelong-timetrapped-iondynamics
with a unitary matrix given by the fixed mode structure and random phases. Certification results do support the
notion that this system produces output probabilities in closer proximity to the uniform samples (for example, the
pass rate R ≈ 3% between the simulated samples and uniform samples and R ≈ 0% between the simulated samples
anddistinguishablesamplesforN ≈39.1). Thisinadditionshowsourmethodissensitivetosomestructureshidden
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in the many-body interference process.
NOISY SAMPLES
Fig.6presentssomecoarse-graineddistributionsincludingthenoisysamples. Visually,the1%noiseintheunitaries
does not lead to substantial changes to the distributions. The noise we include for the intermediate-time trapped-ion
system is a 1% systematic error in the total time (shift from τ = 100µs to τ = 101µs). For the random unitary
process,inordertopreservetheunitarityoftheprocess,wefirstfindtheeffectiveHamiltonianfortheHaar-distributed
randomunitaryprocess,andsubsequentlyadda1%randomnoisetoeachentryofthehermitianHamiltonianmatrix.
Figure 7. Distributions of p-values and the two-sample χ2 test-statistics for the trapped-ion system with intermediate-time
dynamics and N ≈40.5. A χ2 statistic and a p-value are calculated for each pair of samples, with a total N =10000 runs
B s
generated in the simulation. (a) If two samples come from the same distribution, p-values should be uniformly distributed in
[0,1]. (b)Thesolidcurveistheχ2-distributionwithN −1degreesoffreedom. Thedashedlinemarksthecutoffχ2 valueat
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1% significance level.
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TableIII.Meanandstandarddeviationofp-valuesforthetwo-sampleχ2 testbetweenonesimulatedexperimentalsampleand
variousothersamples. Ap-valueiscalculatedforeachpairofgeneratedsamplesandthisisrepeatedforatotalofN =10000
s
sets. Mean and standard deviation (in brackets) of those p-values are tabled below. If two samples come from the same
distribution, p-values should be uniformly distributed in [0,1] with mean 0.5 and standard deviation 0.289. If not, p-values
should be small. Noisy samples for the trapped-ion system include a 1% timing error, whereas a 1% random error is included
intherandomunitarymatrix. Avalueof0inthetableindicatesthatthep-valueisextremelysmall(<10−323 atthemachine
level).
Simulated experimental sample 1 (p-values)
Trapped Ions (Int. time) Trapped Ions (Long time) Random Unitary
Average No. of bubbles 24.2 40.5 69.9 24.3 39.0 66.5 25.9 40.8 70.5
Experimental sample 2 0.50 0.51 0.51 0.50 0.50 0.51 0.50 0.50 0.50
(0.29) (0.29) (0.29) (0.29) (0.29) (0.29) (0.29) (0.29) (0.29)
Noisy experimental sample 2 0.44 0.45 0.44 - - - 0.50 0.49 0.50
(0.29) (0.29) (0.29) - - - (0.29) (0.29) (0.29)
Distinguishable sample 0 0 0 0 0 0 0.0053 2.1×10−4 7.9×10−5
- - - - - - (0.033) (0.0076) (0.0071)
Uniform sample 0 0 0 0.0043 8.4×10−4 1.1×10−4 0 0 0
- - - (0.029) (0.0098) (0.0027) - - -
FURTHER ANALYSIS ON TWO-SAMPLE χ2 TEST
For the two-sample χ2 test, if the pair of samples comes from the same distribution, the χ2 statistics should follow
the χ2-distribution with degrees of freedom being the number of bubbles minus one. Also, if the null hypothesis is
correct, p-values should be uniformly distributed in [0,1]. In the main text, we plotted the distributions of the χ2
statistics between two boson sampling data and between one boson sampling data and a distinguishable sample. One
could clearly see the distinction in comparison. Here, we include in Fig. 7 the plots for the noisy samples too, in the
case of intermediate-time trapped-ion system with N ≈40.5. Without noise in experiment, it is evident that the χ2
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statistics follow the χ2-distribution and the p-values are almost uniformly distributed. With experimental noise, χ2
statisticsshifttolargervalues, withp-valuestiltedtowardssmallervalues. Therefore, typeIerrorisgoingtoincrease
with noise in experiments. Nevertheless, with small noise (1% in this case), one can still definitively and correctly
conclude that these samples are equivalent with >∼98% confidence level (as tabled in the main text).
OtherthanthepassratesRtabledinthemaintext,hereweaddatableforthep-valuestoo,whichoffersadditional
information regarding the error rates without comparing to a specific significance level α. The mean and standard
deviation of p-values are reported in Table III for a set of N = 10000 runs. In the case of two boson sampling
s
data, the mean and standard deviation are very close to the theoretical values 0.5 and 0.289 respectively. On the
other hand, p-values are distinctively smaller against alternative samples. Some far-fetched distributions even have
p-values smaller than the machine level (<10−323) with the boson sampling data, which illustrates the effectiveness
of our certification method. With an appropriate significance level α and suitable number of bubbles N , one could
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minimize both type I and type II errors in the experimental certification process.