Implementation of a Toffoli Gate with Superconducting Circuits

A. Fedorov,1 L. Steffen,1 M. Baur,1 M.P. da Silva,2, 3 and A. Wallraff1

1Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland2Disruptive Information Processing Technologies Group,

Raytheon BBN Technologies, 10 Moulton Street, Cambridge, MA 02138, USA3Department de Physique, Universite de Sherbrooke, Sherbrooke, Quebec, J1K 2R1, Canada

The Toffoli gate is a three-qubit operation thatinverts the state of a target qubit conditioned onthe state of two control qubits. It makes universalreversible classical computation [1] possible and,together with a Hadamard gate [2], forms a uni-versal set of gates in quantum computation. Itis also a key element in quantum error correc-tion schemes [3–7]. The Toffoli gate has beenimplemented in nuclear magnetic resonance [3],linear optics [8] and ion trap systems [9]. Ex-periments with superconducting qubits have alsoshown significant progress recently: two-qubit al-gorithms [10] and two-qubit process tomogra-phy have been implemented [11], three-qubit en-tangled states have been prepared [12, 13], firststeps towards quantum teleportation have beentaken [14] and work on quantum computing ar-chitecture has been done [16]. Implementation ofthe Toffoli gate with only single- and two-qubitgates requires six controlled-NOT gates and tensingle-qubit operations [15], and has not been re-alized in any system owing to current limits oncoherence. Here we implement a Toffoli gatewith three superconducting transmon qubits cou-pled to a microwave resonator. By exploitingthe third energy level of the transmon qubits, wehave significantly reduced the number of elemen-tary gates needed for the implementation of theToffoli gate, relative to that required in theoreti-cal proposals using only two-level systems. Usingfull process tomography and Monte Carlo pro-cess certification, we completely characterized theToffoli gate acting on three independent qubits,measuring a fidelity of 68.5± 0.5 per cent. A simi-lar approach [16] realizing characteristic featuresof a Toffoli-class gate has been demonstratedwith two qubits and a resonator and achieveda limited characterization considering only thephase fidelity. Our results reinforce the poten-tial of macroscopic superconducting qubits for theimplementation of complex quantum operationswith the possibility of quantum error correctionschemes [17].

We have implemented a Toffoli gate with three trans-mon qubits (A,B and C) dispersively coupled to a mi-crowave transmission-line resonator, in a sample whichis identical to the one used in ref. [14]. The resonator is

5

6

7

8

9

A

C

B

res

I II III

Rπ/2

-y Rπ/2

y

Fre

quen

cy [G

Hz]

=

A

C

B

H

π 3π

Rπ/2

yRπ/2

-y

a b c

=

d

H

2π

2π

3ππ

j11xi!ij20xi

jx11i!-jx11i

ij20xi!j11xi

prep. tomo.

FIG. 1. Circuit diagram of the Toffoli gate. a, A not-operation (⊕) is applied to qubit C if the control qubits (Aand B) are in the ground (◦) and excited state (•) respectively.b, The Toffoli gate can be decomposed into a CCPHASE gatesandwiched between Hadamard gates (H) applied to qubit C.c, The CCPHASE gate is implemented as a sequence of aqubit-qutrit gate, a two-qubit gate and a second qubit-qutritgate. Each of these gates is realized by tuning the |11〉 stateinto resonance with |20〉 for a {π, 2π, 3π} coherent rotationrespectively. For the Toffoli gate, the Hadamard gates are re-placed with ±π/2 rotations about the y axis (represented by

Rπ/2±y ). d, Pulse sequence used for the implementation of the

Toffoli gate. During the preparation (I), resonant microwavepulses are applied to the qubits on the corresponding gatelines. The Toffoli gate (II) is implemented with three fluxpulses and resonant microwave pulses (colour coded as in c).The measurement (III) consists of microwave pulses that turnthe qubit states to the desired measurement axis, and a sub-sequent microwave pulse applied to the resonator is used toperform a joint dispersive read-out.

used for joint three-qubit read-out by measuring its trans-mission [18]. At the same time, it serves as a couplingbus for the qubits [19]. The qubits have a ladder-type en-ergy level structure with sufficient anharmonicity to allowindividual microwave addressing of different transitions.We use the first two energy levels as the computational

arX

iv:1

108.

3966

v3 [

quan

t-ph

] 1

4 Fe

b 20

13

2

qubit states, |0〉 and |1〉, and use the second excited state,|2〉, to perform two-qubit and qubit-qutrit operations (aqutrit is a quantum ternary digit). From spectroscopy,we deduce a bare resonator frequency νr = 8.625 GHzwith a quality factor of 3300; maximum qubit transi-tion frequencies νmax

A = 6.714 GHz, νmaxB = 6.050 GHz

and νmaxC = 4.999 GHz; and respective charging energies

Ec/h = 0.264, 0.296 and 0.307 GHz (h, Planck’s con-stant) and qubit-resonator coupling strengths g/2π =0.36, 0.30 and 0.34 GHz for qubits A, B and C. At themaximum transition frequencies, we find respective qubitenergy relaxation times of T1 = 0.55, 0.70 and 1.10µsand phase coherence times of T ∗

2 = 0.45, 0.6 and 0.65µsfor qubits A, B and C.

In the conventional realization of the Toffoli gate, anot operation is applied to the target qubit (C) if thecontrol qubits (A, B) are in the state |11〉. In our set-upit is more natural to construct a variation of the Toffoligate shown in Fig. 1a in which the state of the targetqubit is inverted if the control qubits are in |01〉. Thisgate can easily be transformed to the conventional Toffoligate by a redefinition of the computational basis statesof qubit A or by applying two π-pulses on qubit A.

The Toffoli gate can be constructed from a ‘controlled-controlled-phase’ (CCPHASE) sandwiched between twoHadamard gates acting on the target qubit as shown inFig. 1b. A CCPHASE gate leads to a phase shift of πfor state |1〉 of the target qubit if and only if the controlqubits are in state |01〉. In other words, this correspondsto a sign change of only one of the eight computationalthree-qubit basis states: |011〉 ↔ −|011〉.

The basic idea of ‘hiding’ states by tranforming theminto non-computational states to simplify the implemen-tation of a Toffoli gate was theoretically proposed inrefs. 20 and 21 and has been experimentally implementedfor linear optics and ion trap systems [8, 9]. The im-plementation of the scheme of ref. [20] in our set-upwould require three controlled-phase (CPHASE) gates,six single-qubit operations and two single-qutrit opera-tions. Instead, we construct the CCPHASE from a sin-gle two-qubit CPHASE gate and two qubit-qutrit gates.The latter gates are called π-SWAP and 3π-SWAP, re-spectively (Fig. 1c, red frames). The application of a

TABLE I. List of states after each step of the CCPHASEgate. The state |011〉 acquires a phase shift of π during theCPHASE pulse; the states |11x〉 are transferred to i|20x〉,‘hiding’ them from the CPHASE gate; and the initial states|x0y〉 and |010〉 do not change during the sequence.

Initial state After π-SWAP After CPHASE After 3π-SWAP

|011〉 |011〉 −|011〉 −|011〉|11x〉 i|20x〉 i|20x〉 |11x〉|x0y〉 |x0y〉 |x0y〉 |x0y〉|010〉 |010〉 |010〉 |010〉

single CPHASE gate to qubits B and C (Fig. 1c, blueframe) inverts the sign of both |111〉 and |011〉. To cre-ate the CCPHASE operation, the computational basisstate |111〉 is transferred to the non-computational statei|201〉 by the π-SWAP gate, effectively hiding it fromthe CPHASE operation acting on qubits B and C. Afterthe CPHASE operation, |111〉 is recovered from the non-computational level i|201〉 by the 3π-SWAP gate. Al-ternative approaches using optimal control of individualqubits for implementing a Toffoli gate in a single stephave been proposed [22] and recently analyzed in thecontext of the circuit quantum electrodynamics architec-ture [23].

All three-qubit basis states show three distinct evolu-tion paths during our CCPHASE gate (Table 1). Onlyinput state |011〉 is affected by the CPHASE gate actingon qubits B and C, which transfers |011〉 to the desiredstate, −|011〉. The states |11x〉 with x ∈ {0, 1} are trans-ferred by the π-CPHASE gate to the states i|20x〉. Thesubsequent CPHASE gate therefore has no influence onthe state. The last gate (3π-CPHASE) transfers i|20x〉back to |11x〉. Together the two SWAP gates realize arotation by 4π, such that the state |11x〉 does not acquireany extra phase relative to the other states. The statesof the last group (|010〉 and |x0y〉 with y ∈ {0, 1}) do notchange during the CPHASE gate sequence.

The actual experimental implementation of the Tof-foli gate consists of a sequence of microwave and fluxpulses applied to the qubit local control lines (Fig. 1d).The arbitrary rotations about the x and y axis [24] arerealized with resonant microwave pulses applied to theopen transmission line at each qubit. We use 8-ns-long,Gaussian-shaped DRAG-pulses [24, 25] to prevent pop-ulation of the third level and phase errors during thesingle-qubit operations. Few-nanosecond-long currentpulses passing through the transmission lines next to thesuperconducting loops of the respective qubits controlthe qubit transition frequency realizing z-axis rotations.All two-qubit or qubit-qutrit gates are implemented bytuning a qutrit non-adiabatically to the avoided cross-ing between the states |11x〉 and |20x〉 or, respectively,|x11〉 and |x20〉 (refs 12, 26, and 27). During this time,the system oscillates between these pairs of states withrespective frequencies 2JAB

11,20 and 2JBC11,20. With inter-

action times π/(2JAB11,20) = 7 ns, 3π/(2JAB

11,20) = 21 ns

and π/(2JBC11,20) = 23 ns, we realize a π-SWAP and a

3π-SWAP between qubits A and B and a CPHASE gatebetween qubits B and C, respectively. Our use of qubit-qutrit instead of single-qutrit operations allows for a moreefficient construction of the Toffoli gate. Direct realiza-tion of the scheme proposed in ref. 20 in our system wouldrequire eight additional microwave pulses (used to imple-ment six single-qubit and two single-qutrit gates) with atwofold increase in overall duration of the pulse sequencewith respect to our scheme.

3

0.

0.2

0.4

0.6

0.8

1.

001

100

111

0

0

0

0

0

1

0

1

0

0

1

1

1

0

0

1

0

1

1

1

0

1

1

1

0.0

0.5

1.0

popu

latio

n

FIG. 2. Truth table of the Toffoli gate. The state of qubitC is inverted if qubits A and B are in the state |01〉. Thefidelity of the truth table is F = (1/8)Tr [UexpUideal] = 76.0%

We have characterized the performance of this realiza-tion of a Toffoli gate by measuring the truth table, byfull process tomography [28] and by Monte Carlo processcertification [29, 30]. The truth table depicted (Fig. 2)shows the population of all computational basis statesafter applying the Toffoli gate to each of the computa-tional basis states. It reveals the characteristic propertiesof the Toffoli gate, namely that a NOT operation is ap-plied on the target qubit (C) if the control qubits (Aand B) are in the state |01〉. The fidelities of the out-put states show a significant dependence on qubit life-time. In particular, input states with qubit A (with theshortest lifetime) in the excited state generally have theworst fidelity, indicating that the protocol is mainly lim-ited by the qubit lifetime. The fidelity of the measuredtruth table, Uexp, with respect to the ideal one, Uideal,namely F = (1/8)Tr [UexpUideal] = 76.0%, shows the av-erage performance of our gate when acting onto the eightbasis states.

As an essential addition to the classical characteriza-tion of the gate by the truth table, we have performedfull, three-qubit process tomography and reconstructedthe process matrix, χexp, to characterize the quantumfeatures of the Toffoli gate completely, overcoming thelimited characterization provided by measurements of thephase fidelity only [16]. For this purpose, we prepared acomplete set of 64 distinct input states by applying allcombinations of single-qubit operations chosen from theset {id, π/2x, π/2y, πx} for each qubit, and performedstate tomography on the respective output states. Theprocess matrix reconstructed directly from the data hasa fidelity of F = Tr[χexpχideal] = 70 ± 3% (the errorrepresents a 90% confidence interval), where χideal is the

ideal process matrix. Using a maximum-likelihood pro-cedure [31] to correct for unphysical properties of χexp,we find that the obtained process matrix, χML

exp, has a fi-

delity of F = Tr[χMLexpχideal] = 69 % with expected errors

at the level of 3 %. In Fig. 3a, χexp shows the same keyfeatures as χideal (Fig. 3b).

To gain an accurate alternative estimate of the processfidelity without resorting to a maximum-likelihood proce-dure, we implemented Monte Carlo process certificationfollowing the steps described in ref. 29. First we define aPauli observable as Pn =

∏⊗j=1,...,6 pn,j , a product of six

single-qubit operators chosen from the set of the identityand the Pauli operators (pn,j ∈ {1, σx, σy, σz}). Then wedetermine the 232 observables with non-vanishing expec-tation values Pn = Tr[ρT Pn] 6= 0, where ρT is the Choimatrix of the Toffoli process. For each Pn we prepare all(23 = 8) eigenstates of the product of the first three oper-ators comprising Pn, apply the Toffoli operation to thesestates and measure the expectation value of the prod-uct of the last three operators in Pn. Averaging overthe results obtained with all eigenstates provides an esti-mate of Pn. Extracting all 232 expectation values in thisway allows us to estimate the fidelity of the Toffoli gateas 68.5 ± 0.5% using Monte Carlo process certification,which is in good agreement with the fidelity evaluatedusing tomography.

The scheme that we use to implement the Toffoli gateis generic and can readily be applied to other systems be-cause the majority of the quantum systems used as qubitshave additional energy levels at their disposal. Reductionof the total gate time by use of qubit-qutrit gates togetherwith the recent advances in the extension of the coherencetimes of the superconducting circuits [32, 33] indicates apath towards the realization of practical quantum errorcorrection.

We thank Stefan Filipp, Alexandre Blais for usefuldiscussions and Kiryl Pakrouski for his contributionsin early stages of the experimental work. This workwas supported by the Swiss National Science Foundation(SNF), the EU IP SOLID and ETH Zurich.

[1] Toffoli, T. Reversible computing, vol. 85 of Lecture Notesin Computer Science (Springer Berlin / Heidelberg,1980).

[2] Shi, Y. Both Toffoli and Controlled-NOT need little helpto do universal quantum computation. arXiv:0205115(2002).

[3] Cory, D. G. et al. Experimental quantum error correc-tion. Phys. Rev. Lett. 81, 2152–2155 (1998).

[4] Knill, E., Laflamme, R., Martinez, R. & Negrevergne, C.Benchmarking quantum computers: The five-qubit errorcorrecting code. Phys. Rev. Lett. 86, 5811–5814 (2001).

[5] Chiaverini, J. et al. Realization of quantum error correc-tion. Nature 432, 602–605 (2004).

[6] Pittman, T. B., Jacobs, B. C. & Franson, J. D. Demon-

4

III

IXI

IY�

I

IZI

XII

XXI

XY�

I

XZI

Y�

II

Y�

XI

Y�

Y�

IY�

ZI

ZII

ZXI

ZY�

IZZI

III

IXI

IY�

I

IZI

XII

XXI

XY�

I

XZI

Y�

II

Y�

XI

Y�

Y�

I

Y�

ZI

ZII

ZXI

ZY�

I

ZZI

0.0

0.1

0.2

0.3

È ΧmeasML

Èa

III

IXI

IY�

I

IZI

XII

XXI

XY�

I

XZI

Y�

II

Y�

XI

Y�

Y�

IY�

ZI

ZII

ZXI

ZY�

IZZI

III

IXI

IY�

I

IZI

XII

XXI

XY�

I

XZI

Y�

II

Y�

XI

Y�

Y�

I

Y�

ZI

ZII

ZXI

ZY�

I

ZZI

0.0

0.1

0.2

0.3

È ΧidealÈb

0.

0.05

0.1

0.15

0.2

FIG. 3. Process tomography of the Toffoli gate. Bar chart of the absolute value of the measured process matrix χMLexp (a)

and the ideal process matrix χideal (b). The elements are displayed in the operator basis {III, IIX, IIY , . . . ZZZ}, where I is

the identity and {X, Y , Z} are the Pauli operators {σx,−iσy, σz}. The fidelity of the process matrix is F = Tr[χMLexpχideal] = 69%.

The process fidelity estimated using the Monte Carlo certification method is 68.5± 0.5%.

stration of quantum error correction using linear optics.Phys. Rev. A 71, 052332 (2005).

[7] Aoki, T. et al. Quantum error correction beyond qubits.Nat Phys 5, 541–546 (2009).

[8] Lanyon, B. P. et al. Simplifying quantum logic usinghigher-dimensional hilbert spaces. Nat. Phys. 5, 134–140(2009).

[9] Monz, T. et al. Realization of the quantum Toffoli gatewith trapped ions. Phys. Rev. Lett. 102, 040501 (2009).

[10] DiCarlo, L. et al. Demonstration of two-qubit algorithmswith a superconducting quantum processor. Nature 460,240–244 (2009).

[11] Yamamoto, T. et al. Quantum process tomography oftwo-qubit controlled-z and controlled-not gates using su-perconducting phase qubits. Phys. Rev. B 82, 184515(2010).

[12] DiCarlo, L. et al. Preparation and measurement of three-qubit entanglement in a superconducting circuit. Nature467, 574–578 (2010).

[13] Neeley, M. et al. Generation of three-qubit entangledstates using superconducting phase qubits. Nature 467,570–573 (2010).

[14] Baur, M. et al. Benchmarking a teleportation proto-col realized in superconducting circuits. Preprint athttp://arxiv.org/abs/1107.4774 (2011).

[15] Barenco, A. et al. Elementary gates for quantum compu-tation. Phys. Rev. A 52, 3457–3467 (1995).

[16] Mariantoni, M. et al. Implementing the quantum vonneumann architecture with superconducting circuits. Sci-ence 334, 61–65 (2011).

[17] Reed, M. D. et al. Realization of Three-Qubit quantumerror correction with superconducting circuits. Preprintat http://arxiv.org/abs/1109.4948 (2011).

[18] Filipp, S. et al. Two-qubit state tomography using a jointdispersive readout. Phys. Rev. Lett. 102, 200402 (2009).

[19] Majer, J. et al. Coupling superconducting qubits via acavity bus. Nature 449, 443–447 (2007).

[20] Ralph, T. C., Resch, K. J. & Gilchrist, A. Efficient toffoligates using qudits. Phys. Rev. A 75, 022313 (2007).

[21] Borrelli, M., Mazzola, L., Paternostro, M. & Maniscalco,S. Simple trapped-ion architecture for high-fidelity toffoligates. Phys. Rev. A 84, 012314 (2011).

[22] Sporl, A. et al. Optimal control of coupled josephsonqubits. Phys. Rev. A 75, 012302 (2007).

[23] Stojanovic, V. M., Fedorov, A., Bruder, C. & Wallraff, A.Quantum-control approach to realizing a toffoli gate incircuit QED. Preprint at http://arxiv.org/abs/1108.3442(2011).

[24] Gambetta, J. M., Motzoi, F., Merkel, S. T. & Wilhelm,F. K. Analytic control methods for high-fidelity unitaryoperations in a weakly nonlinear oscillator. Phys. Rev. A83, 012308 (2011).

[25] Motzoi, F., Gambetta, J. M.,Rebentrost, P. & Wilhelm,F. K. Simple pulses for elimination of leakage in weaklynonlinear qubits. Phys. Rev. Lett. 103, 110501 (2009).

[26] Strauch, F. W. et al. Quantum logic gates for coupled su-perconducting phase qubits. Phys. Rev. Lett. 91, 167005(2003).

[27] Haack, G., Helmer, F., Mariantoni, M., Marquardt, F. &Solano, E. Resonant quantum gates in circuit quantum

5

electrodynamics. Phys. Rev. B 82, 024514 (2010).[28] Chuang, I. L. & Nielsen, M. A. Prescription for exper-

imental determination of the dynamics of a quantumblack box. J. Mod. Opt. 44, 2455–2467 (1997).

[29] da Silva, M. P., Landon-Cardinal, O. & Poulin, D.Practical characterization of quantum devices withouttomography. Phys. Rev. Lett. (to appear). Preprint athttp://arxiv.org/abs/1104.3835 (2011).

[30] Flammia, S. T. & Liu, Y.-K. Direct fidelity estimationfrom few Pauli measurements. Phys. Rev. Lett. 106,

230501 (2011).[31] Jezek, M., Furasek, J. & Hradil, Z. c. v. Quantum in-

ference of states and processes. Phys. Rev. A 68, 012305(2003).

[32] Bylander, J. et al. Noise spectroscopy through dynami-cal decoupling with a superconducting flux qubit. NaturePhysics 7, 565–570 (2011).

[33] Paik, H. et al. How coherent are Josephson junctions?Preprint at http://arxiv.org/abs/1105.4652 (2011).