Measurement Crosstalk Errorsin Cloud-Based QuantumComputing

Seungchan SeoSchool of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 291Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

Joonwoo BaeSchool of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 291Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

Abstract—Quantum technologies available currently contain noise in general, often dubbednoisy intermediate-scale quantum (NISQ) systems. We here present the verification of noise inmeasurement readout errors in cloud-based quantum computing services, IBMQ and Rigetti, bydirectly performing quantum detector tomography, and show that there exist measurementcrosstalk errors. We provide the characterization and the quantification of noise in a quantummeasurement of multiple qubits. We remark that entanglement is found as a source of crosstalkerrors in a measurement of three qubits.

THE COMPUTATION based on the laws ofquantum mechanics makes it possible to achievethe capability beyond the limitations of conven-tional computing [1], [2], [3]. To understand theexisting gap between theory and implementationin the realization of quantum computing, thekey is to find how noise deteriorates the capa-bility of quantum information processing. Noisein a quantum system signifies its transition toa classical one. Once systems are governed byclassical physics, no quantum advantages canbe obtained. An ultimate solution to deal withquantum noise may be obtained by quantumerror-correcting codes that can preserve quantumstates [4]. The quantum technologies available atpresent, dubbed noisy intermediate-scale quantum(NISQ) technologies [5], do not meet the level toimplement quantum error correction.

Although the errors are present, NISQ tech-nologies can be used to show advantages over

classical systems [6]. Much effort has been de-voted to improving the capability of NISQ-basedinformation processing. Quantum algorithms fit-ted to NISQ technologies are devised, e.g., quan-tum approximate optimization [7] or variationalquantum eigensolvers [8]. Strategies to mitigatethe errors in quantum dynamics have been pro-posed while quantum error-correcting codes can-not be realized [9], see also [10], [11], [12]. Infact, industry vendors such as IBMQ, IonQ, orRigetti immediately provide cloud-based quantumcomputing services.

The main focus of the present article is mea-surement readout errors, in particular, crosstalkerrors, appearing in a quantum measurement withNISQ technologies. The significance of measure-ment errors is twofold. Firstly, no method isknown so far to generally clear measurementerrors. Quantum error-correcting codes that aimto deal with noise appearing in quantum dynam-

Published by the IEEE Computer Society © IEEE 1

arX

iv:2

112.

0098

4v1

[qu

ant-

ph]

2 D

ec 2

021

1

0 2 3

4

Individual measurement3

0

2 15

161

{Mi}?Joint measurement

Probability distributions

pi|k

• Find a POVM • For given data

• with known states

{Mi}

pi|k = tr [ |ψk⟩⟨ψk |Mi]

|ψk⟩ k = 1,...,N

QDT

0.2

0

IBMQ Yorktown Rigetti Aspen-8

<latexit sha1_base64="y13ZnqQJopKDfkP7ZZjNYevX9QI=">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</latexit>

H : Hadamard, S =p

Z, and X, Z : Pauli operators

Mutually Unbiased Bases for QDT

Figure 1. QDT is performed on detectors in IBMQ Yorktown and Rigetti Aspen-8. Qubit states are prepared inthe six states and measured. QDT finds measurement operators from the statistics of measurement outcomes.

ics cannot apply to measurement readout errors.Remarkably, readout errors on individual qubitscan be mitigated with the help of quantum de-tector tomography (QDT) [14]. The step of QDTthat costs expensive experimental resources maybe circumvented by depolarizing various typesof quantum noise with the help of single-qubitgates [15]. Secondly, evidences for crosstalk er-rors beyond individual qubits have been reported[16]. For instance, crosstalk among detectors maycause unexpected correlations in the outcomestatistics. This means that one should not considermeasurement errors on individual qubits only.Moreover, it is generally hard to verify the sourcefor measurement crosstalk errors, which may becaused by interactions within systems but anenvironment. All these make it even harder to dealwith measurement errors in general.

In this work, we show that crosstalk errors in ameasurement of two and three qubits in fact existin the cloud-based quantum computing services,IBMQ and Rigetti, by performing QDT directlyon the detectors for multiple qubits. In particular,entanglement is detected in the measurement op-erators describing three-qubit detectors but two-qubit ones. We introduce an operational methodto quantify measurement crosstalk errors andprovide the analysis. Our results can be used to

devise methods to mitigate measurement errors.The paper is organized as follows. We begin

by a brief review of QDT. We then introduce thecharacterization and quantification of crosstalkerrors in a quantum measurement. Measurementerrors in IBMQ Yorktown and Rigetti Aspen-8are analyzed. Finally, we summarize the structureof correlations appearing in a measurement ofmultiple qubits.

Quantum Detector TomographyLet us begin with the measurement postulate

in quantum theory. The quantum measurementdictates the statistics of measurement outcomesin a quantum experiment. A quantum probabilityis given by a product of a state and a mea-surement operator. A quantum measurement isgenerally described by positive-operator-valued-measures (POVMs), denoted by {Mi}i, that sat-isfy

Mi ≥ 0,∑i

Mi = I. (1)

For a qubit state identified as a two-level quantumsystem, a qubit measurement is described by 2×2 POVM elements. For a quantum state ρ, theprobability of obtaining an outcome i is given by

p(i) = tr[Miρ]

2

which is called the Born rule.

Measurement in quantum computingIn quantum computing, a measurement is per-

formed in the computational basis, for which atwo-outcome measurement contains two POVMelements,

M0 = |0〉〈0| and M1 = |1〉〈1|. (2)

A detection event in the arm 0 (1) is describedby M0 (M1). A measurement for n qubits givingoutcomes aj ∈ {0, 1} for j = 1, · · · , n isdescribed by a POVM element in the following,

M~a = Ma1 ⊗Ma2 ⊗ · · · ⊗Man . (3)

Note that for n detectors for n qubits are char-acterized by 2n POVM elements, each of whichcorresponds to a 2n × 2n non-negative matrix.

QDT aims to verify POVM elements for n-qubit detectors. Thus, non-negative 2n × 2n ma-trices are obtained by QDT. In practice, QDT canbe implemented by applying a set of tomograph-ically complete states, repeating a measurement,collecting the outcome statistics and character-izing a non-negative matrix that is the mostconsistent with outcome statistics. An instanceof the set of tomographically complete states ismutually unbiased states,

|0〉, |1〉, |+〉, |−〉, |+ i〉, | − i〉, (4)

where |±〉 = (|0〉 ± |1〉)/√

2 and | ± i〉 =(|0〉 ± i|1〉)/

√2. For n qubits, the number of

tomographically complete states is O(4n). Then,we apply maximum likelihood estimation to re-construct a POVM element consistent to the out-come statistics [18], see also the QDT algorithm.

Quantum measurement in a realistic scenarioIn a realistic scenario with the NISQ systems,

a measurement may not be the form in Equations(2) or (3). First, POVM elements giving outcomes0 and 1 in Equation (2) may not be rank-one. Thisis due to noise on individual detectors, which weidentify as local noise. Then, POVM elements fordetecting multiple qubits may not be factorized ina form in Equation (3), that is,

M~a 6= Ma1 ⊗Ma2 ⊗ · · · ⊗Man . (5)

The type of noise may be due to crosstalk ofdetectors [17], which we call crosstalk errors

Algorithm 1: QDT with maximum like-lihood estimation

Data: Prepare states ρk = |ψk〉〈ψk| inEq. (4) and the empiricalfrequency of obtaining outcome i,{fi,k}.

Result: A POVM {Mi} that bestdescribes given data, i.e., whichmaximizes log-likelihoodfunction,

logL =∑i

∑k

fi,k log tr[Miρk].

Set a termination threshold ε.Start with M0

i = I/D where D denotesa dimension of a system.

do:pi,k ← tr [M t

i ρk]M t and Rt denote M and R at thet-th iteration stepRti ←

∑kfi,kpi,k×(∑

j

∑l,m

fj,lpj,l

fj,mpj,m

ρlMtjρm

)− 12

ρk

M t+1i ← Rt

iMtiR†,ti

while∑

i ‖M ti −M t+1

i ‖1 ≥ ε

in a measurement. In the following, we devisedistance measures to identify crosstalk errors ina quantum measurement.

Detection of Crosstalk Errors inQuantum Measurements

To analyze crosstalk errors in a quantum mea-surement, it is essential to perform QDT. We ex-ploit the trace distance to quantify crosstalk errorsexisting in POVM elements. The trace distancebetween two normalized POVM elements Π1 andΠ2 is denoted by

D(Π1, Π2) =1

2‖Π1 − Π2‖1,

where ‖ · ‖1 denotes the trace norm.

Two-qubit crosstalk errorsSuppose that a POVM element Πxy

ab over twoqubits labelled by x and y giving outcomes a andb, respectively, is obtained from QDT. We now in-troduce a distance measure for the quantification

3

Table 1. Two-qubit crosstalk errors in IBMQ YorktownQubits 00 : (DN , DC , D∗

L) 01 : (DN , DC , D∗L) 10 : (DN , DC , D∗

L) 11 : (DN , DC , D∗L)

(1,2) (0.1158,0.0219,0.1159) (0.1210,0.0155,0.1216) (0.1710,0.0462,0.1708) (0.1732,0.0140,0.1722)(2,3) (0.0866,0.0105,0.0854) (0.0519,0.0112,0.0544) (0.0793,0.0122,0.0774) (0.0553,0.0136,0.0573)(0,3) (0.3312,0.0351,0.3365) (0.3062,0.0449,0.3050) (0.2614,0.0469,0.2629) (0.2331,0.0625,0.2285)(1,4) (0.1879,0.0056,0.1882) (0.1267,0.0105,0.1257) (0.2018,0.0125,0.2017) (0.1452,0.0117,0.1446)(0,1) (0.3295,0.0208,0.3299) (0.3896,0.0307,0.3915) (0.2588,0.1606,0.2513) (0.2259,0.0273,0.2223)(3,4) (0.1996,0.0064,0.1994) (0.1148,0.0142,0.1145) (0.1309,0.0129,0.1320) (0.1099,0.0095,0.1113)

QDT for two-qubit detectors in IBMQ Yorktown is performed on Nov. 18 2020. The distance measures (DN , DC , D∗L) are

computed for the four POVM elements giving outcomes 00, 01, 10 and 11.

of noise in the POVM element. Throughout, weconsider normalized POVM elements. The totalerror is quantified by DN

DN(a, b|x, y) := D(Πxyab , |ab〉〈ab|), (6)

where a, b ∈ {0, 1}. This shows the distance of aPOVM element from the ideal one |a〉〈a|⊗|b〉〈b|.A non-zero distance DN > 0 may be contributedby two sources of noise: one is the errors in indi-vidual detectors due to local noise, and the otheris the errors that cannot be described by noiseon individual detectors. The latter is identified ascrosstalk errors.

The sources of noise may be elucidated byintroducing crosstalk DC and local errors DL,

DC(a, b|x, y) := minΠx

a⊗Πyb

D(Πxyab , Π

xa ⊗ Πy

b ), (7)

D∗L(a, b|x, y) := D(Πxa ⊗ Πy

b , |ab〉〈ab|) (8)

where the normalized POVM element Πxa ⊗ Πy

b

in Equation (8) corresponds to the optimal onein Equation (7). The measure DC finds the mini-mal distance of a POVM element from POVMelements having no crosstalk noise. One canconclude that crosstalk noise exists if and onlyif DC is positive:

crosstalk in detectors ⇐⇒ DC > 0.

Once DC is computed with optimal local POVMelements, the distance between the optimal localPOVM elements and the ideal one is measuredby D∗L. From the property of distance measures,it holds that

DN ≤ DC +D∗L or DC ≥ DN −D∗Lfor a given set {a, b, x, y}.

Multi-qubit crosstalk errorsFor multiple qubits, crosstalk errors can be an-

alyzed by bipartite splittings over multiple qubits.Let us consider a bipartition analysis for threequbits and the analysis can be generalized tomultiple qubits straightforwardly. Let A, B, andC denote three qubits, and ΠABC

abc a normalizedPOVM element giving outcome a, b, and c.

The total error may be estimated as the tracedistance,

DN(ΠABCabc ) = D(ΠABC

abc , |abc〉〈abc|). (9)

The bipartite crosstalk errors may be quantifiedin the bipartite splittings A : BC, B : CA, andC : AB. For instance, the crosstalk error in thebipartition A : BC is quantified by

DC(ΠA:BCabc ) :=

minΠA

a⊗ΠBCbc

D(ΠABCabc , ΠA

a ⊗ ΠBCbc ). (10)

Similarly, the crosstalk error in other bipartitesplittings can be quantified by,

DC(ΠB:CAbca ) :=

minΠB

b ⊗ΠCAca

D(ΠABCabc , ΠB

b ⊗ ΠCAca ). (11)

and DC(ΠC:ABabc ) :=

minΠC

c ⊗ΠABab

D(ΠABCabc , ΠC

c ⊗ ΠABab ). (12)

Given a POVM element above, one can alsodefine local errors as it is shown in Equation (8).

The genuine multi-qubit crosstalk errors canbe quantified as follows,

DC(ΠA:B:Cabc ) :=

minΠA

a⊗ΠBb ⊗ΠC

c

D(ΠABCabc , ΠA

a ⊗ ΠBb ⊗ ΠC

c ).(13)

Note that DC(ΠA:B:Cabc ) = 0 implies that crosstalk

errors in all bipartite splittings in Equations (10),

4

Table 2. Three-qubit measurement crosstalk errors in IBMQ YorktownPartitions 000 100 010 110 001 101 011 1110:1:2 0.0242 0.1399 0.0591 0.0405 0.0292 0.1350 0.0366 0.02610:(2,3) 0.0225 0.1383 0.0376 0.0238 0.0260 0.1325 0.0332 0.02121:(2,0) 0.0189 0.1352 0.0538 0.0366 0.0268 0.1340 0.0315 0.02292:(0,1) 0.0219 0.0260 0.0537 0.0367 0.0273 0.0247 0.0333 0.0164

Detectors of theree qubits labelled by 0, 1, and 2 are analyzed. From POVM elements giving outcomes 100 and 101, itis found that crosstalk errors in bipartite splittings 0 : 1 are dominant. As for 2 : (0, 1), crosstalk errors are about 1%.

(11), and (12) are zero. Thus, it holds that non-zero crosstalk errors in a bipartite splitting implyDC(ΠA:B:C

abc ) > 0.

In what follows, we apply the quantificationof crosstalk errors in a quantum measurement tothe detectors in cloud-based quantum computingservices, IBMQ Yorktown and Rigetti Aspen-8.

Crosstalk Errors in IBMQ YorktownWe have performed QDT for detectors in

IBMQ Yorktown on Nov. 15 2020. IBMQ York-town contains five qubits, in which the connec-tivity is shown in Figure 1. QDT is performed ontwo- and three-qubit detectors.

For two-qubit detectors, the pairs of qubitslabelled (1, 2), (2, 3), (0, 3), (1, 4), (0, 1), and(3, 4) are analyzed. To find crosstalk errors, wehave computed the distance (DN , DC , D

∗L) in

Equation (7) for the POVM elements of the pairs.They are listed in Table 1. It is found that two-qubit errors DN vary from 5% to 39% dependingon the pairs. Errors around 30% are found for thepairs (0, 3) and (0, 1) for which the qubit labelledby 0 is in common. The pairs without qubit 0,such as (2, 3), show relatively lower errors. Thetotal error DN is higher in the pairs (1, 2) and(0, 3). It is found that crosstalk errors are around1-5% whereas local errors are 10-30%.

For three-qubit detectors, the triple of qubitslabelled (0, 1, 2) is analyzed. We have performedQDT on Nov. 11 2020. In Table 2, crosstalk errorsin all partitions are computed. It is shown thatthe genuine crosstalk errors exist and peak at twoPOVM elements giving outcomes 100 and 101.Similar values are observed in the bipartitions 0 :(1, 2) and 1 : (2, 0), whereas crosstalk errors inthe bipartition 2 : (1, 0) are around 0.1%. Thisconcludes that detectors for qubits labelled 0 and1 share larger crosstalk correlations.

Table 3. Two-qubit total errors DN are computed forpairs of qubits.

Qubits 00 01 10 110,1 0.2678 0.3394 0.2492 0.32620,2 0.0992 0.0960 0.0789 0.06030,3 0.1823 0.1512 0.1406 0.08260,15 0.2657 0.0881 0.3058 0.12720,16 0.1831 0.0560 0.2165 0.08691,2 0.1820 0.1961 0.3449 0.37041,3 0.2474 0.3022 0.4028 0.41191,15 0.3052 0.1651 0.4344 0.31021,16 0.3295 0.9706 0.9965 0.34852,3 0.1212 0.2107 0.2207 0.26542,15 0.2804 0.1614 0.2491 0.12382,16 0.1635 0.9910 0.9997 0.06893,15 0.3325 0.1152 0.3598 0.16563,16 0.2375 0.9899 0.9897 0.123715,16 0.2583 0.9946 0.9979 0.0544

Large values are found when pairs contain the qubit labelledby 16. From the analysis of crosstalk errors in Table 4, thelarge values are mainly due to local noise.

Crosstalk Errors in Rigetti Aspen-8We have performed QDT for detectors in

Rigetti Aspen-8 on Jan. 7 2021. The detectorsof the qubits labelled by 0, 1, 2, 3, 15 and 16are considered, see Figure 1. Note that theseare chosen out of 31 qubits contained in RigettiAspen-8. Two qubits labelled 0 and 1 are notdirectly connected and the qubit labelled 2 isdirectly connected to three qubits 1, 3 and 15.

In Table 3, two-qubit total errors are computedfor all of the pairs of qubits. For the qubit 0, thatis not directly connected to the rest of qubits,show total errors 6-32%. It is found that the pair(0, 2) contain the smallest total error. The qubit2, connected to three qubits, shows errors 12-99%. Large values of the errors are found forthe pairs containing the qubit labelled 16. As itis discussed, the total errors can be decomposedinto crosstalk and local errors.

The source of the errors can be noticed fromthe result of computing crosstalk errors, which is

5

Table 4. Two-qubit crosstalk errors DC are computedfor pairs of qubits.

Qubits 00 01 10 110,1 0.0587 0.0485 0.0622 0.04720,2 0.0299 0.0244 0.0339 0.02590,3 0.0293 0.0288 0.0302 0.02640,15 0.0198 0.0316 0.0378 0.03480,16 0.0262 0.0192 0.0183 0.02851,2 0.0484 0.0715 0.0478 0.02651,3 0.0684 0.1050 0.0912 0.05141,15 0.0469 0.0484 0.0448 0.02791,16 0.0642 0.0590 0.0735 0.06332,3 0.0317 0.0655 0.0409 0.05152,15 0.0319 0.0332 0.0235 0.03382,16 0.0301 0.0328 0.0168 0.02033,15 0.0714 0.0584 0.0512 0.07223,16 0.0660 0.0519 0.0498 0.022015,16 0.0278 0.0432 0.0262 0.0248

Crosstalk errors vary from 2 to 10%, which amount to two-qubit gate or measurement errors.

shown in Table 4. We first note that the crosstalkerrors are not peaked at the pairs containing qubit16. This concludes that large values in two-qubittotal errors in Table 3 are due to local noise.The qubit 0, connected to none of the rest ofqubits, and the qubit 2, connected to three qubits,do not show much difference in the crosstalkerrors. Overall, two-qubit detectors contain rel-atively higher values about 10-40% in the totalerrors, in which crosstalk errors are around 2-7%.

Entanglement in MeasurementCrosstalk Errors

Crosstalk errors in a quantum measurementcan be rephrased as correlations shared by detec-tors. The correlations may be investigated at theangle of entanglement theory that characterizesquantum correlations which do not have a classi-cal counterpart. In what follows, we apply entan-glement theory to the analysis of correlations inmulti-qubit POVM elements of IBMQ Yorktownand Rigetti Aspen-8.

Two-qubit POVM elementsTwo-qubit POVM elements correspond to 4×

4 non-negative operators. We call a normalizedtwo-qubit POVM element contains entanglementif it cannot be decomposed into a probabilisticmixture of local POVM elements. Otherwise, thePOVM element is called separable such that itcan be prepared by local operations and classical

communication:

Πab =∑λ

p(λ)Πa(λ)⊗ Πb(λ)

for some λ. For the two-qubit cases, a simplecondition can tell whether entanglement is con-tained or not. A normalized two-qubit POVM isentangled if it is non-positive under the partialtranspose (NPPT), where the partial transposemay be taken either of the qubits [19]. Otherwise,it contains correlations that can be prepared bylocal operations and classical communication.

All two-qubit POVM elements obtained fromIBMQ Yorktown in Table 1 and Rigetti Aspen-8in Table 3 are analyzed. We have found that noneof the pairs is NPPT. This shows that two-qubitcrosstalk errors are from classical correlationsshared by two detectors.

Three-qubit POVM elementsCorrelations shared by three qubits can be

approached by bipartite splittings. It is clear thatentanglement exists if correlations in a bipartitionare NPPT. For instance, for a detector for threequbits ABC, the partial transpose is performedin its normalized POVM element in the bipartitesplitting A : BC. Negative eigenvalues concludeentanglement in the cut. The partial transpose isapplied to other bipartite splittings, B : CA andC : AB.

Contrast to two-qubit cases, we have foundthat entanglement exists in three-qubit POVMelements. In Table 5, NPPT correlations in IBMQYorktown are summarized. In particular, POVMelements giving outcomes 000, 001, and 011 areNPPT in all bipartite splittings. In the case ofRigetti Aspen-8, four triples of qubits (0, 1, 2),(0, 1, 3), (0, 1, 15), and (1, 2, 16) are considered.It is found that all bipartite splittings of all of thetriples contain NPPT correlations.

The structure of correlationsTo summarize, it is found that three-qubit

POVM elements contain entanglement whereastwo-qubit POVM elements are separable. To beprecise, two-qubit POVM elements marginal fromtheir three-qubit extension are separable wherethe three-qubit POVM elements are entangled.

6

Table 5. Entanglement and crosstalk errors in IBMQ YorktownOutcomes 000 100 010 110 001 101 011 111(0,1,2) (N,N,N) (P,P,P) (P,P,N) (P,P,P) (N,N,N) (N,N,P) (N,N,N) (P,N,P)

For three qubits (0, 1, 2), bipartite splittings (ΓA:BC ,ΓB:CA,ΓC:AB) are considered. NPPT correlations are denoted byN, in which entanglement is contained. Otherwise, it is written P.

<latexit sha1_base64="veExDSXP41fXQb05DrhPtOTCUxs=">AAAB/3icbVDLSgMxFM34rPU1KrhxEyyCq5KRoi6LblxWsA9oh5JJM21oHkOSEcpYwV9x40IRt/6GO//GtJ2Fth4IHM65l3tyooQzYxH69paWV1bX1gsbxc2t7Z1df2+/YVSqCa0TxZVuRdhQziStW2Y5bSWaYhFx2oyG1xO/eU+1YUre2VFCQ4H7ksWMYOukrn/YEdgOtMigSi1Rgo7hI0Ko65dQGU0BF0mQkxLIUev6X52eIqmg0hKOjWkHKLFhhrVlhNNxsZMammAyxH3adlRiQU2YTfOP4YlTejBW2j1p4VT9vZFhYcxIRG5yktbMexPxP6+d2vgyzJhMUkslmR2KUw6tgpMyYI9pSiwfOYKJZi4rJAOsMbGusqIrIZj/8iJpnJWD83LltlKqXuV1FMAROAanIAAXoApuQA3UAQEP4Bm8gjfvyXvx3r2P2eiSl+8cgD/wPn8AbHOVtw==</latexit>

outcome 000<latexit sha1_base64="HP+QwpVA6ed3148ZUfBOK89ITJ8=">AAAB/3icbVDLSgMxFM34rPU1KrhxEyyCqzIjRV0W3bisYB/QDiWT3mlDk8mQZIQyVvBX3LhQxK2/4c6/MW1noa0HAodz7uWenDDhTBvP+3aWlldW19YLG8XNre2dXXdvv6FlqijUqeRStUKigbMY6oYZDq1EAREhh2Y4vJ74zXtQmsn4zowSCATpxyxilBgrdd3DjiBmoESGZWqoFDDGj57nd92SV/amwIvEz0kJ5ah13a9OT9JUQGwoJ1q3fS8xQUaUYZTDuNhJNSSEDkkf2pbGRIAOsmn+MT6xSg9HUtkXGzxVf29kRGg9EqGdnKTV895E/M9rpya6DDIWJ6mBmM4ORSnHRuJJGbjHFFDDR5YQqpjNiumAKEKNraxoS/Dnv7xIGmdl/7xcua2Uqld5HQV0hI7RKfLRBaqiG1RDdUTRA3pGr+jNeXJenHfnYza65OQ7B+gPnM8fbfeVuA==</latexit>

outcome 001<latexit sha1_base64="Sh47Wn/nfbnCtXeNAi6Iuk8Jae8=">AAAB/3icbVDLSgMxFM34rPU1KrhxEyyCqzIjRV0W3bisYB/QDiWT3mlDk8mQZIQyVvBX3LhQxK2/4c6/MW1noa0HAodz7uWenDDhTBvP+3aWlldW19YLG8XNre2dXXdvv6FlqijUqeRStUKigbMY6oYZDq1EAREhh2Y4vJ74zXtQmsn4zowSCATpxyxilBgrdd3DjiBmoESGZWqoFDDGj57vdd2SV/amwIvEz0kJ5ah13a9OT9JUQGwoJ1q3fS8xQUaUYZTDuNhJNSSEDkkf2pbGRIAOsmn+MT6xSg9HUtkXGzxVf29kRGg9EqGdnKTV895E/M9rpya6DDIWJ6mBmM4ORSnHRuJJGbjHFFDDR5YQqpjNiumAKEKNraxoS/Dnv7xIGmdl/7xcua2Uqld5HQV0hI7RKfLRBaqiG1RDdUTRA3pGr+jNeXJenHfnYza65OQ7B+gPnM8fbfiVuA==</latexit>

outcome 010<latexit sha1_base64="6bQttsrIrcYJXfEoDSDegaAxAu8=">AAAB/3icbVDLSgMxFM34rPU1KrhxEyyCqzIjRV0W3bisYB/QDiWTZtrQZDIkd4QyVvBX3LhQxK2/4c6/MW1noa0HAodz7uWenDAR3IDnfTtLyyura+uFjeLm1vbOrru33zAq1ZTVqRJKt0JimOAxqwMHwVqJZkSGgjXD4fXEb94zbbiK72CUsECSfswjTglYqesediSBgZYZVilQJdkYP3q+33VLXtmbAi8SPycllKPWdb86PUVTyWKgghjT9r0Egoxo4FSwcbGTGpYQOiR91rY0JpKZIJvmH+MTq/RwpLR9MeCp+nsjI9KYkQzt5CStmfcm4n9eO4XoMsh4nKTAYjo7FKUCg8KTMnCPa0ZBjCwhVHObFdMB0YSCraxoS/Dnv7xIGmdl/7xcua2Uqld5HQV0hI7RKfLRBaqiG1RDdUTRA3pGr+jNeXJenHfnYza65OQ7B+gPnM8fb3yVuQ==</latexit>

outcome 011<latexit sha1_base64="XGWsAz+w7AQKYrkfbWz5WqSz0+4=">AAAB/3icbVDLSgMxFM34rPU1KrhxEyyCqzIjRV0W3bisYB/QDiWT3mlDk8mQZIQyVvBX3LhQxK2/4c6/MW1noa0HAodz7uWenDDhTBvP+3aWlldW19YLG8XNre2dXXdvv6FlqijUqeRStUKigbMY6oYZDq1EAREhh2Y4vJ74zXtQmsn4zowSCATpxyxilBgrdd3DjiBmoESGZWqoFDDGj77ndd2SV/amwIvEz0kJ5ah13a9OT9JUQGwoJ1q3fS8xQUaUYZTDuNhJNSSEDkkf2pbGRIAOsmn+MT6xSg9HUtkXGzxVf29kRGg9EqGdnKTV895E/M9rpya6DDIWJ6mBmM4ORSnHRuJJGbjHFFDDR5YQqpjNiumAKEKNraxoS/Dnv7xIGmdl/7xcua2Uqld5HQV0hI7RKfLRBaqiG1RDdUTRA3pGr+jNeXJenHfnYza65OQ7B+gPnM8fbfmVuA==</latexit>

outcome 100<latexit sha1_base64="U1qNydId6j9dxEVvNxBLQsZ1z/M=">AAAB/3icbVDLSgMxFM34rPU1KrhxEyyCqzIjRV0W3bisYB/QDiWTZtrQZDIkd4QyVvBX3LhQxK2/4c6/MW1noa0HAodz7uWenDAR3IDnfTtLyyura+uFjeLm1vbOrru33zAq1ZTVqRJKt0JimOAxqwMHwVqJZkSGgjXD4fXEb94zbbiK72CUsECSfswjTglYqesediSBgZYZVilQJdkYP/qe33VLXtmbAi8SPycllKPWdb86PUVTyWKgghjT9r0Egoxo4FSwcbGTGpYQOiR91rY0JpKZIJvmH+MTq/RwpLR9MeCp+nsjI9KYkQzt5CStmfcm4n9eO4XoMsh4nKTAYjo7FKUCg8KTMnCPa0ZBjCwhVHObFdMB0YSCraxoS/Dnv7xIGmdl/7xcua2Uqld5HQV0hI7RKfLRBaqiG1RDdUTRA3pGr+jNeXJenHfnYza65OQ7B+gPnM8fb32VuQ==</latexit>

outcome 101<latexit sha1_base64="RB1i/5rPLX/dA5D40Bg0iAlJ+eM=">AAAB/3icbVDLSgMxFM34rPU1KrhxEyyCqzIjRV0W3bisYB/QDiWTZtrQZDIkd4QyVvBX3LhQxK2/4c6/MW1noa0HAodz7uWenDAR3IDnfTtLyyura+uFjeLm1vbOrru33zAq1ZTVqRJKt0JimOAxqwMHwVqJZkSGgjXD4fXEb94zbbiK72CUsECSfswjTglYqesediSBgZYZVilQJdkYP/q+13VLXtmbAi8SPycllKPWdb86PUVTyWKgghjT9r0Egoxo4FSwcbGTGpYQOiR91rY0JpKZIJvmH+MTq/RwpLR9MeCp+nsjI9KYkQzt5CStmfcm4n9eO4XoMsh4nKTAYjo7FKUCg8KTMnCPa0ZBjCwhVHObFdMB0YSCraxoS/Dnv7xIGmdl/7xcua2Uqld5HQV0hI7RKfLRBaqiG1RDdUTRA3pGr+jNeXJenHfnYza65OQ7B+gPnM8fb36VuQ==</latexit>

outcome 110<latexit sha1_base64="NtbtSb0kOD72+5ygoj1pAAVz89k=">AAAB/3icbVDLSgMxFM34rPU1KrhxEyyCqzKRoi6LblxWsA9oh5JJM21oHkOSEcpYwV9x40IRt/6GO//GtJ2Fth4IHM65l3tyooQzY4Pg21taXlldWy9sFDe3tnd2/b39hlGpJrROFFe6FWFDOZO0bpnltJVoikXEaTMaXk/85j3Vhil5Z0cJDQXuSxYzgq2Tuv5hR2A70CKDKrVECTqGjwihrl8KysEUcJGgnJRAjlrX/+r0FEkFlZZwbEwbBYkNM6wtI5yOi53U0ASTIe7TtqMSC2rCbJp/DE+c0oOx0u5JC6fq740MC2NGInKTk7Rm3puI/3nt1MaXYcZkkloqyexQnHJoFZyUAXtMU2L5yBFMNHNZIRlgjYl1lRVdCWj+y4ukcVZG5+XKbaVUvcrrKIAjcAxOAQIXoApuQA3UAQEP4Bm8gjfvyXvx3r2P2eiSl+8cgD/wPn8AcQKVug==</latexit>

outcome 111

Figure 2. For qubits labeled 0, 1, 2, POVM elements for i, j, k ∈ {0, 1} contain entanglement. N stands forNPPT meaning entanglement in the bipartite splitting. Entanglement is detected in all POVM elements exceptthe cases 100 and 110, see also Table 5.

Note also that three-qubit POVM elements con-tain NPPT correlations depending on a biparti-tion. The analysis shows that correlations amongdetectors are highly non-trivial. For instance, itis necessary to perform QDT for more than twoqubits to understand the correlation structure.

Application to Quantum Error MitigationLet MABC

abc denote a POVM element of threequbits labeled ABC giving outcome abc. Theprobability of outcome abc is given by

pexp(abc) = tr[ρA ⊗ ρB ⊗ ρCMABCabc ].

for a three-qubit state ρA⊗ρB⊗ρC . Let p0(abc)denote the probability with a POVM element|a〉〈a|⊗|b〉〈b|⊗|c〉〈c| in a noiseless environment.In error mitigation, it is aimed to transform aprobability given from an experiment, pexp(abc),to a probability as close to the ideal one p0(abc).When noise is present in qubits individually,i.e., with MABC

abc = MAa ⊗ MB

b ⊗ MCc for

some POVM elements MAa , MB

b and MCc , the

probability can be factorized for individual qubits.Thus, errors can be mitigated by taking individualqubits only into account [14], [15]. This strategyno longer works when crosstalk errors exist.Hence, our results show that measurement errorson multiple qubits can be mitigated by consider-ing outcomes of multiple qubits collectively.

ConclusionIn conclusion, we have analyzed measurement

readout errors in cloud-based quantum computingservices. A method of quantifying crosstalk errorsis introduced and applied to analyzing readouterrors in IBMQ Yorktown and Rigetti Aspen-8.

Our results characterize and quantify measure-ment crosstalk errors in quantum computing withNISQ technologies, and pave a way to devisea method of dealing with measurement readouterrors in multiple qubits, such as measurementerror correction or mitigation. Our results alsosuggest that QDT may be performed before qubitallocations so that the crosstalk errors are ana-lyzed and those qubits having a large fractionof crosstalk in a measurement are avoided bythe command of qubit allocations. It would beinteresting to investigate the relations of errormitigation and the properties and the structure ofcrosstalk errors in a quantum measurement.

Finally, we address that raw data from QDTwere generated at the cloud-based quantum com-puting services IBMQ and Rigetti. Derived datasupporting the findings of this study are availablefrom the corresponding author upon request.

AcknowledgementThis work was supported by Samsung Re-

search Funding & Incubation Center of SamsungElectronics (Project No. SRFC-TF2003-01).

REFERENCES1. D. Deutsch and R. Penrose, ”Quantum theory, the

church–turing principle and the universal quantum

computer”, Proceedings of the Royal Society of

London. A. Mathematical and Physical Sciences,

400(1818):97–117, 1985.

2. P. W. Shor, ”Polynomial-time algorithms for prime factor-

ization and discrete logarithms on a quantum computer”,

SIAM Journal on Computing, 26(5):1484–1509, 1997.

3. L. K. Grover, ”Quantum mechanics helps in searching

7

for a needle in a haystack”, Physical Review Letters,

79:325–328, 1997.

4. P. W. Shor, ”Scheme for reducing decoherence in quan-

tum computer memory”. Physical Review A. 52 (4):

R2493–R2496, 1995.

5. J. Preskill, ”Quantum Computing in the NISQ era and

beyond”, Quantum, 2:79, 2018.

6. F. Arute, et. al., ”Quantum supremacy using a pro-

grammable superconducting processor”, Nature 574:

505–510, 2019.

7. E. Farhi, J. Goldstone, and S. Gutmann, ”A quantum

approximate optimization algorithm”, arXiv:1411.4028,

2014.

8. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q.

Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien,

”A variational eigenvalue solver on a photonic quantum

processor”, Nature Communications, 5(1):4213, 2014.

9. S. Endo, Z. Cai, S. C. Benjamin, and X. Yuan, ”Hybrid

Quantum-Classical Algorithms and Quantum Error Miti-

gation”, J. Phys. Soc. Jpn. 90, 032001, 2021.

10. S. Endo, S. C. Benjamin, and Y. Li, ”Practical Quantum

Error Mitigation for Near-Future Applications”, Phys. Rev.

X 8, 031027, 2018.

11. A. Kandala, K. Temme, A. D. Corcoles, A. Mezzacapo,

J. M. Chow and J. M. Gambetta, ”Error mitigation extends

the computational reach of a noisy quantum processor”,

Nature 567, 491–495, 2019.

12. P. Murali, D. C. Mckay, M. Martonosi, A. Javadi-Abhari,

”Software Mitigation of Crosstalk on Noisy Intermediate-

Scale Quantum Computers”, ASPLOS ’20: Proceedings

of the Twenty-Fifth International Conference on Architec-

tural Support for Programming Languages and Operating

Systems,1001–1016, 2020.

13. A. Lowe, M. H. Gordon, P. Czarnik, A. Arrasmith, P.

J. Coles, and L. Cincio, ”Unified approach to data-

driven quantum error mitigation”, Phys. Rev. Research

3, 033098, 2021.

14. F. B. Maciejewski, Z. Zimboras, and M. Oszmaniec,

”Mitigation of readout noise in near-term quantum de-

vices by classical post-processing based on detector

tomography”, Quantum 4, 257, 2020

15. H. Kwon and J. Bae, ”A hybrid quantum-classical

approach to mitigating measurement errors in quan-

tum algorithms”, IEEE Transactions on Computers, doi:

10.1109/TC.2020.3009664.

16. Y. Chen, M. Farahzad, S. Yoo, and T.-C. Wei, ”Detector

Tomography on IBM 5-qubit Quantum Computers and

Mitigation of Imperfect Measurement”, Phys. Rev. A 100,

052315, 2019.

17. M. Sarovar, T. Proctor, K. Rudinger, K. Young, E.

Nielsen, R. Blume-Kohout, ”Detecting crosstalk errors

in quantum information processors”, Quantum 4, 321,

2020.

18. M. Paris and J. Rehacek, Quantum State Estimation,

Springer-Verlag Berlin/Heidelberg, 2004.

19. A. Peres, ”Separability Criterion for Density Matrices”,

Phys. Rev. Lett. 77, 1413–1415, 1996.

8