a r X i v : 1 4 0 6 . 2 4 0 4 v 1 [ q u a n t  p h ] 1 0 J u n 2 0 1 4
A LeakageResilient Scheme for the Measurement of Stabilizer Operators inSuperconducting Quantum Circuits
Joydip Ghosh
1,
∗
and Austin G. Fowler
2,3,
†
1
Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta T2N 1N4, Canada
2
Department of Physics, University of California, Santa Barbara, California 93106, USA
3
Centre for Quantum Computation and Communication Technology,School of Physics, The University of Melbourne, Victoria 3010, Australia
(Dated: June 11, 2014)Superconducting qubits, while promising for scalability and long coherence times, contain morethan two energy levels, and therefore are susceptible to errors generated by the
leakage
of population outside of the computational subspace. Such leakage errors are currently considered tobe a prominent roadblock towards faulttolerant quantum computing with superconducting qubits.Faulttolerant quantum computing using topological codes is based on sequential measurements of multiqubit stabilizer operators. Here we propose a leakageresilient scheme to perform repetitivemeasurements of multiqubit stabilizer operators in superconducting circuits. Our protocol is basedon SWAP operations between data and ancilla qubits at the end of every cycle, requiring readoutand reset operations on every physical qubit in the system, and thereby preventing persistent leakageerrors from occurring.
PACS numbers: 03.67.Lx, 03.67.Pp, 85.25.j
Recent years have witnessed remarkable progress inquantum computing with superconducting components,as far as the scalability and coherence times are concerned [1–3]. With major advances in designing scal
able qubits [1, 4, 5] and highﬁdelity quantum gates
[1, 6–9], a signiﬁcant fraction of this research endeavor is
now directed towards FaultTolerant Quantum Computing (FTQC) with superconducting devices [3, 10]. FTQC
via topological errorcorrecting codes, such as the surface code [11], requires sequential measurements of multiqubit stabilizer operators [12]. The ﬂuctuations in the
measurement outcomes of such stabilizer operators generate characteristic signatures for various discrete Paulierrors, thereby rendering the errorcorrecting scheme robust against error models described by a Pauli channel[12].Superconducting qubits comprise more than two energy levels that are often utilized to design optimal twoqubit gates, such as a controlled
σ
z
(CZ) gate [7–9, 13].
Apart from the decoherence, therefore, superconductingqubits also suﬀer from errors due to
leakage
of population outside of the computational subspace, often referred to as leakage errors and currently considered tobe a prominent barrier towards FTQC [1, 14]. While
decoherenceinduced errors can be approximated by aPauli channel [15], leakage errors lack such a description, consequently compromising the faulttolerance forsuperconducting quantum circuits. It has been shownrecently with numerical simulations, how persistent leakage errors in superconducting circuits destroy an ancillaassisted qubitmeasurement scheme producing randomﬂuctuations in the output of the ancilla qubit [14].
In this letter, we propose a multiqubit stabilizermeasurement scheme, which is resilient to such leakage
(a)
D

ψ
/U
cycle
U
cycle
...
A

0
/

0
...
(b)

0
/
×
U
oddcycle
×
U
evencycle
...

ψ
/
× 
0
×
...
FIG. 1. (a) Standard scheme for repetitive measurements of stabilizer operators. At the end of each cycle, the ancillaqubitregister (denoted by ‘A’) gets measured and initialized, whilethe dataqubit register (demoted by ‘D’) never gets measured.
U
cycle
denotes the sequence of gate operations required for thecorresponding stabilizer measurement. (b) Leakageresilientscheme for repetitive measurements of stabilizer operators.SWAP operations are denoted by vertical lines connectingtwo ‘X’ symbols. Since the roles of data and ancillaregisterget swapped after each cycle, the corresponding sequence of gate operations diﬀer for odd and evennumbered cycles, asdenoted by
U
oddcycle
and
U
evencycle
.
errors. A schematic diagram of the standard and ourleakageresilient scheme for repetitive measurements of stabilizer operators is shown in Fig. 1. In the standard
approach, the data qubit register never gets measured,and therefore, any leaked qubit in the data register remains leaked for many cycles until it undergoes relaxation due to decoherence or leaks back to the computational subspace. As shown in Ref. [14], such leaked
data qubits generate random noises in the measurementoutcomes of ancilla qubits, eﬀectively spoiling the entirescheme. In our leakageresilient protocol, we supplementthe standard approach with SWAP operations at the end
2of every measurementcycle as shown in Fig. 1b. TheSWAP gates exchange the roles of the dataregister andthe ancillaregister, requiring us to measure and initialize every qubit in alternate cycles, eﬀectively eliminatingthe possibility of leakage errors persisting without compromising the stabilizermeasurement scheme.
✚✙✛✘✚✙✛✘✚✙✛✘✚✙✛✘
AAD D
❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅
1 234ZZXX
FIG. 2. (Color online) A Schematic diagram of the architecture for measuring twoqubit stabilizer operators. The circlesdenote superconducting qubits (data qubits denoted by ‘D’and ancilla qubits denoted by ‘A’), and lines denote requirednearestneighbor couplings. The stabilizer operators,
ZZ
and
XX
, are measured via qubit3 and qubit4 respectively.1
•
H
•
H ...
2
•
H
•
H ...
3

0
H
• •
H

0
...
4

0
H
• •
H

0
...
FIG. 3. Standard scheme for repetitive measurements of twoqubit stabilizer operators
XX
and
ZZ
.
H
denotes theHadamard gate, the vertical lines connected by ﬁlled circlesdenote CZ gates, and the numbers denote the indices for thequbits in the same order as shown in Fig. 2. The gates inside
the dashed rectangle represent
U
cycle
in Fig. 1a.
Here, we illustrate the advantage of our protocolassuming a model where both the data and ancillaregisters consist of two superconducting qubits, and wemeasure two 2qubit stabilizer operators,
XX
and
ZZ
,repetitively for many cycles. Fig. 2 shows a schematic diagram of our architecture, where four nearestneighborcoupled superconducting qubits form a 1D circular lattice. The standard scheme for stabilizer measurement forthis model is shown in Fig. 3 [15]. As mentioned earlier,
in this protocol the dataqubit register (qubits 1 and 2)remains unmeasured throughout the routine, while theancillaqubit register (qubits 3 and 4) is measured andinitialized every cycle. Note that, if we encode any of the four Bell states in the data register, then under idealgate operations, the ﬁnal states before reinitialization aregiven by,

00
+

11
√
2
⊗
00
−→

00
+

11
√
2
⊗
00

00
−
11
√
2
⊗
00
−→

00
−
11
√
2
⊗
01

01
+

10
√
2
⊗
00
−→

01
+

10
√
2
⊗
10

01
−
10
√
2
⊗
00
−→

01
−
10
√
2
⊗
11
,
(1)which essentially means that under circuit 3, the four dif
ferent Bell states in the data register are stabilized by theoperators
XX
and
ZZ
as their simultaneous eigenstatescorrespondingto the four diﬀerent possible combinations.Without any loss of generality, in this work we assumethe Bell state (

00
+

11
)
/
√
2 as our encoded initial statein the data register and compare our protocol against thestandard scheme simulating the circuits numerically.
1

0
× •
H
•
H
×
H
• •
H . . .
2

0
× •
H
•
H
×
H
• •
H . . .
3
×
H
• •
H

0
× •
H
•
H . . .
4
×
H
• •
H

0
× •
H
•
H . . .
FIG. 4. The SWAPbased leakageresilient scheme for repetitive measurements of twoqubit
XX
and
ZZ
stabilizer operators.
In order to render this repetitive stabilizer measurement scheme leakageresilient, we introduce SWAP operations between the data and ancilla registers as shownin Fig. 4. SWAP operations between two quantum registers essentially mean sequential SWAP gates betweenthe
k
th
qubits in both the registers, for all
k
. In order tonot transfer or propagate the leakage errors, we expressSWAP operations (between

0
and an arbitrary state

ψ
) as,

ψ
× •
H
•
H
≡
0
×
H
•
H
•
(2)We note that substituting SWAP gates with two CZ gatesalong with some singlequbit Hadamard operations, infact, introduces a negligible overhead in circuit depthbecause of internal cancellations, and the reduced circuit for our SWAPbased stabilizer measurement schemeis shown in Fig. 5. First we discuss how to construct the
model for leakage errors for CZ gates in superconducting circuits and then compare our scheme (5) with thestandard scheme (3) via numerical simulations.
3
1

0
H
• •
H
•
H
•
H
2

0
H
•
H
• •
H
•
H
•
H
• • •
H
3
•
H
•
H

0
H
• •
H ...
4
•
H
• • •
H

0
H
•
H
• •
H
•
H ...
FIG. 5. The leakageresilient scheme for stabilizer measurement where SWAP gates are replaced by CZ and Hadamard gatesas shown in Eq.(2). The gates inside the left dashed rectangle are repeated for odd cycles, while the gates inside the rightrectangle are repeated for even cycles.
In order to estimate the dominant contribution for theleakage errors, we model the superconducting qubits asthreelevel systems (or qutrits) and parametrize the corresponding single and twoqutrit quantum gates, as discussed in Ref. [14]. The Hadamard gate for a qutrit (inthe basis
{
0
,

1
,

2
}
) is given by,
H
=
1
√
21
√
2
0
1
√
2
−
1
√
2
00 0 1
,
(3)which is equivalent to the standard Hadamard gate acting on the computational subspace and an Identity acting on the

2
state. Singlequbit gates can be done withﬁdelities an order of magnitude higher than that of twoqubit gates [6, 16], and will therefore be assumed to be
ideal throughout this work.The CZ gate considered here is performed betweentwo superconducting qubits by tuning and detuning thequbitfrequencies so as to mix the population in theavoided crossing of

11
and

20
eigenstates, therebyacquiring the phase (
π
for CZ) across the

11
state[1, 7, 8, 13]. The dominant errormechanism for this
avoidedcrossingbasedCZ gate is generated by the residual nonadiabatic populationtransfer in singleexcitation(
{
01
,

10
}
) and doubleexcitation (
{
02
,

20
,

11
}
)subspaces. As shown in Ref. [14], such a CZ gate can beparametrized with two generators,
S
(generates the idealpart of the CZ gate) and
S
′
(generates the ﬁrstordererror terms), where the parameters are chosen from afullscale Hamiltonian simulation. The generators
S
and
S
′
can be thought of as Hermitian matrices generatingthe nonideal CZ gate, also obtainable from a fullscalesimulation of the timedependent controlHamiltonian,
U
CZ
=
e
i
(
S
+
S
′
)
.
(4)In the twoqutrit tensorproduct basis

00
,

01

02
,

10
,

11
,

12
,

20
,

21
,

22
,
the generators
S
and
S
′
are given by [14],
S
=
0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0
ξ
1
0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0
π
0 0 0 00 0 0 0 0
ξ
2
0 0 00 0 0 0 0 0
π
0 00 0 0 0 0 0 0
ξ
3
00 0 0 0 0 0 0 0
ξ
4
, S
′
=
0 0 0 0 0 0 0 0 00
ζ
1
0
iχ
1
e
iφ
1
0 0 0 0 00 0 0 0
iχ
2
e
iφ
2
0 0 0 00
−
iχ
1
e
−
iφ
1
0
ζ
2
0 0 0 0 00 0
−
iχ
2
e
−
iφ
2
0
ζ
3
0
iχ
3
e
iφ
3
0 00 0 0 0 0 0 0
iχ
4
e
iφ
4
00 0 0 0
−
iχ
3
e
−
iφ
3
0
ζ
4
0 00 0 0 0 0
−
iχ
4
e
−
iφ
4
0 0 00 0 0 0 0 0 0 0 0
.
(5)Since the CZ gate is based on mixing of population inthe avoided levelcrossing between

11
and

20
states,so we have
11

S

11
=
20

S

20
=
π
. The remainingparameters
ξ
1
−
4
are the dynamical phases acquired bythe corresponding basis states and are assumed to berandom numbers between 0 and 2
π
for our simulation.The parameters in
S
′
,
χ
1
−
4
and
ζ
1
−
4
, are assumed to besmall (
∼
10
−
2
), while the angles
φ
1
−
4
take arbitrary values between 0 and 2
π
. The parameters
χ
1
and
χ
4
causepopulationtransfer in the single and tripleexcitationsubspaces respectively, while
χ
2
,
3
are responsible formixing of population in the doubleexcitation subspace.Since the population transfer probabilities in
{
01
,

10
}
(singleexcitation subspace),
{
02
,

11
,

20
}
(doubleexcitation subspace), and
{
12
,

21
}
(tripleexcitationsubspace) subspaces scalewith

χ
i

2
[14], our choiceof parameters bounds the intrinsic gate errors to about 10
−
4
,which is consistent with what has been obtained from a
4fullscale simulation of the controlHamiltonian for current gatedesign schemes [7, 8].
02004006008001000012(a)
R e a d o u t v a l u e
0200400600800100000.51Cycle
P r o b a b i l i t y
(b)
success of predictionleakagequbit 3qubit 4
FIG. 6. (Color online) Results for the standard approachto repetitive measurements of twoqubit stabilizer operators,
XX
and
ZZ
. (a) The readouts of two ancilla qubits (qubits3 and 4) are shown for various consecutive cycles. The tworegions having random rapid ﬂuctuations indicate that eitherof the two data qubits or both is leaked during those measurement cycles. (b) The solid black curve shows the probabilitythat either of the two data qubits is in the

2
state at the endof every measurement cycle. The regions having almost unitleakage probability exactly coincide with the randomly ﬂuctuating ancilla outcomes in (a). The solid gray (green) curveshows the overlap between the state encoded in the data register and the prediction of it from the corresponding ancillaoutputs at the end of each cycle.
02004006008001000012(a)
R e a d o u t v a l u e
0200400600800100001Cycle
P r o b a b i l i t y
(b)
qubit 3qubit 4success of predictionleakage
FIG. 7. (Color online) Results for our SWAPbased approachto repetitive measurements of twoqubit stabilizer operators,
XX
and
ZZ
. (a) The readouts of two ancilla qubits (qubits3 and 4) are shown for various consecutive cycles. Unlike theresults obtained for the standard scheme, no rapidly ﬂuctuating signals are observed in this case. (b) The solid blackcurve shows the probability that either of two data qubitsis in

2
state at the end of every measurement cycle. Nounit probability for leakage is observed for consecutive cycles.Peaks indicate probabilities for isolated leakage errors. Thesolid gray (green) curve shows the overlap between the stateencoded in the data register and the prediction of it from thecorresponding ancilla outputs at the end of each cycle.
Having parametrized the required quantum gates, wenow simulate the quantum circuits shown in Fig. 3 andin Fig. 5. In order to compare our scheme against thestandard approach, we here assume the nodecoherencelimit (i.e.,
T
1
,
2
→ ∞
). We emphasize that the introduction of decoherence in our calculation only amountsto some more randomly occurred ‘steps’ in the readoutvalues that are neither relevant for, nor inﬂuence the conclusions of this work.The simulation of the standard approach (circuitshown in Fig. 3) is shown in Fig. 6. The measurement
outcomes from the ancilla qubits (qubits 3 and 4) areshown in Fig. 6a for many consecutive cycles. In the presence of leakage errors, we observe regions having randomand rapid ﬂuctuations in the ancilla outcomes, a characteristic signature for a leakage error on either of the dataqubits. Such a signature has been observed in Ref. [14] in
the context of a repetitive ancillaassisted measurementof a single data qubit, and with a rigorous analysis, itssrcin has been conclusively associated with the leakageerror in the data qubit. In order to show the connectionbetween this noise and the dataqubit leakage error moreexplicitly, in Fig. 6b we plot the probability (black curve)that either of the data qubits is leaked at the end of every cycle. For cycles with rapid ﬂuctuations in the outcomes of the ancilla qubits, we also observe a nearunitprobability for leakage errors, clearly signifying the destructive consequences of dataqubitleakage for the standard stabilizermeasurement scheme. Using Eq.(1), it is
possible to predict the quantum state of the dataqubitregister, based on the outcomes obtained in the ancillaqubits [17]. In Fig. 6b, we also plot the probability of
success for such predictions (green curve) at the end of each cycle. It is observed that this successprobabilityis enormously compromised for many consecutive cycleswhere the data qubits remain leaked, essentially indicating a catastrophic failure of the standard scheme underleakage errors.We also simulate our SWAPbased scheme (circuitshown in Fig. 5) and the results are shown in Fig. 7.
Fig. 7a shows the outputs of the ancilla qubits for manyconsecutive cycles and, unlike standard protocol, norapid random ﬂuctuations are observed in the ancilla outcomes for this case. The probability of a leakage error inthe data register is shown (black curve) in Fig. 7b. Incontrast with the standard scheme, we only observe isolated peaks, which means even if there is a dataqubitleakageevent in one cycle, it gets completely removed insubsequent measurement cycles, because in this approachwe are measuring and initializing every physical qubit inthe system in alternate cycles. Fig. 7b also shows theprobability of successful prediction (green curve) of thetwoqubit state encoded in the dataqubit register. Notice that the predictions only get compromised wheneverthere is a leakage error either in the dataregister or inthe ancillaregister, and since the leakage errors are isolated, so are the failure probabilities. The discrete wellseparated peaks in the leakage error plot (or the downward peaks in the curve showing the probability of successful predictions) explicitly signify the resilience of ourSWAPbased scheme against leakage errors. The densityof these peaks is determined by the probability of leakage
5errors occurring in the physical qubits, and with betteroptimization techniques it is possible to suppress such errors even further [9]. In the standard scheme, however,
since the dataregister never gets measured, even a singledataqubit leakageeventruins the entire scheme, becausethe leaked qubit remains leaked either until it undergoesrelaxationto the ground state due to decoherence or leaksback to the computational subspace.In summary, we have devised a leakageresilient protocol for repetitive measurements of multiqubit stabilizer operators in superconducting circuits. In the standard approach, the dataqubit register never gets measured, and therefore a leaked data qubit remains leakedfor many consecutive cycles, producing random ﬂuctuations in the measurement of the ancilla register. Ourscheme relies on SWAP operations between the dataand the ancillaregister, thereby changing the role of these registers each cycle. This requires us to performthe readout and reset operations on every qubit in thequantum circuit, which essentially eliminates the possibility of longlived leakage errors occurring. Sequentialmeasurements of such multiqubit stabilizer operatorsarekey steps to performing FTQC using Topological errorcorrecting codes, such as the surface code. Our analysis motivates the use of SWAPbased gate sequences forstabilizerbased errorcorrectingschemes, rendering themfaulttolerant against the leakage errors.We thank Michael Geller for helpful discussions. Thisresearch was funded by the US Oﬃce of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the US ArmyResearch Oﬃce grant No. W911NF1010334. All statements of fact, opinion or conclusions contained herein arethose of the authors and should not be construed as representing the oﬃcial views or policies of IARPA, the ODNI,or the US Government. J.G. gratefully acknowledges theﬁnancial support from NSERC, AITF and University of Calgary’s Eyes High Fellowship Program.
∗
ghoshj@ucalgary.ca
†
austingfowler@gmail.com[1] R. Barends, J. Kelly, A. Megrant, A. Veitia,D. Sank, E. Jeﬀrey, T. C. White, J. Mutus, A. G.Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro,A. Dunsworth, C. Neill, P. O’Malley, P. Roushan,A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, Nature
508
, 500 (2014).[2] J. M. Chow, J. M. Gambetta, A. D. C´orcoles, S. T.Merkel, J. A. Smolin, C. Rigetti, S. Poletto, G. A. Keefe,M. B. Rothwell, J. R. Rozen, M. B. Ketchen, andM. Steﬀen, Phys. Rev. Lett.
109
, 060501 (2012).[3] J. M. Chow, J. M. Gambetta, E. Magesan, S. J. Srinivasan, A. W. Cross, D. W. Abraham, N. A. Masluk, B. R.Johnson, C. A. Ryan, and M. Steﬀen, ArXiv eprints(2013), arXiv:1311.6330 [quantph].[4] R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus,C. Neill, P. O’Malley, P. Roushan, J. Wenner,T. C. White, A. N. Cleland, and J. M. Martinis,Phys. Rev. Lett.
111
, 080502 (2013).[5] Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang,R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro,A. Dunsworth, E. Jeﬀrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank,A. Vainsencher, J. Wenner, T. C. White, M. R. Geller,A. N. Cleland, and J. M. Martinis, ArXiv eprints(2014), 1402.7367.[6] F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K.Wilhelm, Phys. Rev. Lett.
103
, 110501 (2009).[7] J. Ghosh, A. Galiautdinov, Z. Zhou, A. N. Korotkov, J. M. Martinis, and M. R. Geller,Phys. Rev. A
87
, 022309 (2013).[8] J. M. Martinis and M. R. Geller, ArXiv eprints (2014),1402.5467 [quantph].[9] D. J. Egger and F. K. Wilhelm,Superconductor Science and Technology
27
, 014001 (2014).[10] J. Ghosh, A. G. Fowler, and M. R. Geller,Phys. Rev. A
86
, 062318 (2012).[11] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill,Journal of Mathematical Physics
43
, 4452 (2002).[12] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N.Cleland, Phys. Rev. A
86
, 032324 (2012).[13] F. W. Strauch, P. R. Johnson, A. J. Dragt, C. J.Lobb, J. R. Anderson, and F. C. Wellstood,Phys. Rev. Lett.
91
, 167005 (2003).[14] J. Ghosh, A. G. Fowler, J. M. Martinis, and M. R. Geller,Phys. Rev. A
88
, 062329 (2013).[15] M. R. Geller and Z. Zhou,Phys. Rev. A
88
, 012314 (2013).[16] J. Kelly, R. Barends, B. Campbell, Y. Chen, Z. Chen,B. Chiaro, A. Dunsworth, A. G. Fowler, I.C. Hoi, E. Jeffrey, A. Megrant, J. Mutus, C. Neill, P. J. J. O‘Malley,C. Quintana, P. Roushan, D. Sank, A. Vainsencher,J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, ArXiv eprints (2014), arXiv:1403.0035 [quantph].[17] If the ancilla is in

2
state, we need to map it to somecomputational state for the purpose of such a prediction,and we map the

2
state in ancilla qubit to the

1
statein this work. While the choice is arbitrary, it does notchange the conclusions of this work.