Experimental Realization of Non-Abelian Geometric Gates

A. A. Abdumalikov, Jr.,∗ J. M. Fink,† K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. FilippDepartment of Physics, ETH Zurich, CH-8093, Zurich, Switzerland

(Dated: October 29, 2018)

The geometric aspects of quantum mechanicsare underlined most prominently by the conceptof geometric phases, which are acquired whenevera quantum system evolves along a closed pathin Hilbert space. The geometric phase is deter-mined only by the shape of this path1–4 and is– in its simplest form – a real number. How-ever, if the system contains degenerate energylevels, matrix-valued geometric phases, termednon-abelian holonomies, can emerge5. They playan important role for the creation of syntheticgauge fields in cold atomic gases6 and the descrip-tion of non-abelian anyon statistics7. Moreover,it has been proposed to exploit non-abelian holo-nomic gates for robust quantum computation8–10.In contrast to abelian geometric phases11, non-abelian ones have been observed only in nuclearquadrupole resonance experiments with a largenumber of spins and without fully characterizingthe geometric process and its non-commutativenature12,13. Here, we realize non-abelian holo-nomic quantum operations14,15 on a single super-conducting artificial three-level atom16 by apply-ing a well controlled two-tone microwave drive.Using quantum process tomography, we deter-mine fidelities of the resulting non-commutinggates exceeding 95%. We show that a sequence oftwo paths in Hilbert space traversed in differentorder yields inequivalent transformations, whichis an evidence for the non-abelian character of theimplemented holonomic quantum gates. In com-bination with two-qubit operations, they form auniversal set of gates for holonomic quantum com-putation.

A cyclic evolution of a non-degenerate quantum systemis in general accompanied by a phase change of its wavefunction. The acquired abelian phase can be divided intotwo parts: The dynamical phase which is proportional tothe evolution time and the energy of the system, and thegeometric phase which depends only on the path of thesystem in Hilbert space. This characteristic feature leadsto a resilience of the geometric phase to certain fluctua-tions during the evolution17–19, a property which has at-tracted particular attention in the field of quantum infor-mation processing20. However, universal quantum com-putation cannot be based on simple phase gates, whichmodify only the relative phase of a superposition state,unless they act on specific basis states21. Furthermore,geometric operations acting on degenerate subspaces have

∗ abdumalikov@phys.ethz.ch† Now at Institute for Quantum Information and Matter, Califor-nia Institute of Technology, Pasadena, CA 91125, USA

been proposed for holonomic quantum computation fullybased on geometric concepts8. In this scheme, quantumbits are encoded in a doubly degenerate eigenspace of the

system hamiltonian h(~λ). The parameters ~λ are variedto induce a cyclic evolution of the system. When the sys-tem returns back to its initial state, it can acquire notonly a simple geometric phase factor, but also undergoesa path-dependent unitary transformation, a non-abelianholonomy, which causes a transition between the eigen-states in the degenerate subspace. Because these trans-formations depend only on the geometric properties ofthe path, they share the noise resilience of the geometricphase.

In the original proposal8, the parameters ~λ are changedadiabatically in time to guarantee the persistence of thedegeneracy. Adiabatic holonomic gates have been pro-posed for trapped ions9, superconducting qubits22,23 andsemiconductor quantum dots24. However, they are dif-ficult to realize in experiment because of the long evo-lution time needed to fulfill the adiabatic condition. In-stead, Sjoqvist et al.15 have proposed a scheme based onnon-adiabatic non-abelian holonomies14 which combinesuniversality and speed and can thus be implemented inexperiments.

The main idea is to generate a non-adiabatic and cyclicstate evolution in a three-level system which results in apurely geometric operation on the degenerate subspacespanned by the computational basis states |0〉 and |1〉.The third state |e〉 acts as an auxiliary state and remainsunpopulated after the gate operation. This is achieved bydriving the system with two resonant microwave pulses(Fig. 1a) with identical time-dependent envelope Ω(t),but different amplitudes a and b satisfying |a|2 + |b|2 =1 (see Fig. 1c). The hamiltonian of the system in theinteraction picture is

h(t) =~Ω(t)

2(a|e〉〈0|+ b|e〉〈1|+ h.c.), (1)

and it causes the initial basis vectors to evolve to the

states |ψi(t)〉 = T exp

(− i

~

t∫0

h(t′)dt′)|i〉 (i, j = 0, 1),

where T denotes time ordering. In contrast to adiabaticschemes, the |ψi(t)〉 are not instantaneous eigenstates ofh(t). By keeping the complex amplitude ratio a/b of thepulses constant, no transitions between states are inducedand the evolution satisfies the parallel transport condi-tion, 〈ψi(t)|h(t)|ψj(t)〉 = 0. As a consequence, the evolu-tion is purely geometric with no dynamic contributions.

If the pulse length τ is chosen such thatτ∫0

Ω(t)dt = 2π,

the degenerate subspace undergoes a cyclic evolution, andthe final operator which acts on the |0〉 and |1〉 basis states

arX

iv:1

304.

5186

v1 [

quan

t-ph

] 1

8 A

pr 2

013

2

|1〉

|0〉

|e〉

a Ω(t)

b Ω(t)

a

b preparation pulses

tomographyrotation pulses

geometricgate

holonomy

U|ψ〉

U(2) fib

re

bundle

basespace G

C

C

|1〉

|0〉

|e〉 b Ω(t)

a Ω(t)

t=0t=τ

c

I,Rx(π/2),Ry(π/2),Rx(π)

I,Rx(π/2),Ry(π/2),Rx(π) I,

Rx(π/2),Ry(π/2),Rx(π)

I,Rx(π/2),Ry(π/2),Rx(π)

FIG. 1. Geometric gate operation on a three-level sys-tem. a, Two drive tones with amplitudes a and b and pulseenvelope Ω(t) couple the |0〉 and |1〉 to the |e〉 state. To low-est order, the direct transition between |0〉 and |1〉 states isforbidden. b, Pulse sequence for process tomography: Theinput state is prepared by sequentially applying pulses on the|0〉 ↔ |e〉 and |e〉 ↔ |1〉 transitions (see Methods). The holo-nomic gate is formed by the simultaneous application of twopulses with envelope Ω(t) and amplitudes a and b. A set ofpulses on the |0〉 ↔ |e〉 and |e〉 ↔ |1〉 transitions followed bythe measurement completes the process tomography. c, Holo-nomic gate represented on a fibre bundle. A path in the fiberbundle connects initial and final states. The projection of thepath in the fiber bundle onto the base manifold spanned bythe states |ψi〉〈ψi|, forms a closed loop C along which thesystem is parallel transported. The difference between theinitial and final points lying on a single fiber corresponds tothe matrix valued holonomy U(C), an element of the unitarygroup U(N).

is

U(C) =

(cos θ eiφ sin θ

e−iφ sin θ − cos θ

), (2)

where we have parameterized the drive amplitudes by therelation eiφ tan θ/2 = a/b. In a geometric picture, the dy-namics of the system can be visualized on a fibre bundlein which the closed loop C in the base manifold deter-mines the holonomic operation U(C) (Fig. 1c). Differentvalues of θ and φ correspond to different paths C.

In our experiments, we realize the holonomic gates us-ing a three-level superconducting artificial atom of thetransmon type embedded in a 3D cavity25 (Fig. 2a).The cavity is made of aluminium with inner dimensions32 × 15.5 × 5 mm3. The frequency of the fundamentalmode is ωro/2π ' 8.999 GHz as measured by transmis-sion spectroscopy using the circuit shown in Fig. 2b. Thetransmon is made of two 500× 250µm2 aluminium elec-trodes separated by 130µm and connected via a Joseph-son junction (Fig. 2c,d). It is oriented parallel to theelectric field of the fundamental mode pointing along thesmallest dimension of the cavity (Fig. 2a). The measured

a 500 µm

1 µm

ω0e

ωe1

ωro

ωLO

ADC

I

Q

b 300 K 40 mK 1.6 K 300 K

c

d

I

Q

E

FIG. 2. Transmon qubit in a cavity resonator. a, Alu-minium cavity with an embedded sapphire chip containingtwo transmon qubits. The left qubit is not used in the exper-iment. The electric field profile ~E of the fundamental mode issketched in the upper part of the cavity. b, Microwave controlpulses at the transition frequencies ω0e and ωe1 are created bymodulating two continuous signals using in-phase/quadrature(I/Q) mixers. Modulation pulses in the I and Q channels aregenerated using arbitrary waveform generators. The controlpulses and the readout pulse at frequency ωro are combinedand fed into the cavity. The transmitted signal is amplifiedand detected in a heterodyne measurement at room tempera-ture and analyzed on a computer using a digitizer (ADC). c,Optical and d scanning electron micrograph of the employedtransmon device.

coupling strength between the transmon and the cavityfield is g/2π ≈ 110 MHz. To read out the state of thetransmon we measure the state-dependent transmissionthrough the cavity26. The transition frequencies mea-sured by Ramsey spectroscopy are ω0e/2π ≈ 8.086 GHzand ωe1/2π ' 7.776 GHz. The decay times of both ex-cited states are T1 = 7± 0.1µs, and the dephasing timesare T 0e

2 = 8.0± 0.1µs and T e12 = 3.9± 0.1µs.

Different holonomic gates are realized by adjusting theratio of the two drives a/b = eiφ tan θ/2 to values be-

tween 0 and 1/√

2 with φ = 0. In geometric terms, thiscorresponds to a gradual deformation of the loop C in theprojective Hilbert space resulting in a continuous changefrom the phase-flip gate σz to the NOT gate σx. Theenvelopes Ω(t) are truncated Gaussian pulses27 with awidth of σ = 10 ns and a total pulse length of 40 ns.With pulses of this length, off-resonant driving of neigh-boring transitions is negligible. The performance of theholonomic gates is characterized by process tomographyperformed on all three levels (see Methods). The diago-nal elements of the reduced process matrix χ are shownin Fig. 3a as a function of θ with φ = π. The experimen-tally obtained results are in good agreement with theory.For θ = 0 a single drive on the |e〉 ↔ |1〉 transition causesa phase shift corresponding to χZZ = 1 and the opera-tion U(θ = 0) = σz. In the case θ = π/4, the Hadamard

3

b c

1.0

0.0

0.5

0 π2

3π8

π4

π8

χ ii, t

r(~ )

a

tr(~)χ

χZZ

χXXχYY

χII

θ

0.0

0.5

1.0

-0.5

0.0

0.5

1

-1

0

χ

IX

Y~Z I

X

ZY~

IX

Y~Z I

X

ZY~

FIG. 3. Process tomography of holonomic gates. a,Diagonal elements χii of the process matrices for ideal (lines)and experimental (dots) geometric gates as a function of θ.Black circles correspond to the trace of the reduced processmatrix χ. (b) Bar chart of the real part of the reduced mea-sured process matrix χexp of the geometric Hadamard gate Hand (c) the NOT gate . The wire frames show the theoreti-cally expected values. The small (0.3%) imaginary parts of χare not shown.

transformation H = (σz − σx)/√

2 is generated (Fig. 3b).θ = π/2 corresponds to pulses with equal amplitude andgenerates a NOT gate σx (Fig. 3c). Because of dephas-ing and relaxation of both excited states as well as finitefidelities of the microwave pulses, the |e〉 state is slightlypopulated after the gate operation. This leakage is quan-tified by computing the trace tr (χ) ≈ 0.96 of the reducedprocess matrix χ (black dots).

The fidelity of the Hadamard transformation is FH ≈95.4% and the fidelity of the NOT operation is FNOT ≈97.5%. The numerical solution of a master equation in-cluding dissipative processes results in fidelities of F =97.6% for both processes, in good agreement with theexperimental values.

To explicitly show that different loops of the statevector in the Hilbert space result in non-commutingquantum gates, we sequentially apply the geometricHadamard and the NOT gate in alternating order. Thenon-abelian character of the operation yields either theoperation NOT · H = −(iσy + 1)/

√2 (Fig. 4a) or H ·

NOT = (iσy − 1)/√

2 (Fig. 4b). We visualize the op-erations on the Bloch sphere (Fig. 4c-d): NOT · H cor-responds to a π-rotation about the H-axis followed bya π rotation about the X-axis. This is equivalent to arotation about the Y -axis by π/2. On the other hand,H ·NOT corresponds to a rotation in the opposite direc-tion, which is in clear contrast to the former operation.By concatenation of operations other than the Hadamardand the NOT operations, rotations about arbitrary axescorresponding to a representation of the complete SU(2)group can be realized.

a bH NOT NOT H

Z

X

H

Y

Z

X

H

Y

c d

1

2

3

1

2

3

1

-1

0

-0.5

0.0

0.5

IX

Y~Z I

X

ZY~

-0.5

0.0

0.5

IX

Y~Z I

X

ZY~

FIG. 4. Non-commutativity of holonomic gates. Processmatrices for NOT ·H (a) and H ·NOT (b) gates with fidelitiesof 95%. Because of the non-abelian character of the geometricoperations the resulting processes are different. This can bevisualized on the Bloch sphere by two rotations around X andH axes c-d. The initial state 1 of the system is rotated aroundthe red axis first (red dotted traces) toward the intermediatestate 2 and then rotated around the other axes (green axisand dashed lines) to the final state 3. The resulting combinedrotations are shown by thick black lines.

With the explicit demonstration of two non-commutingpurely geometric gates, we have shown the universal char-acter of the realized non-adiabatic non-abelian transfor-mation acting on a three level quantum system. A sim-ilar scheme applied to two coupled three level systemswould complete the universal set of holonomic quantumgates and would allow for performing all-geometric quan-tum algorithms, which are potentially resilient againstnoise when short pulses are used28. Moreover, holo-nomic gates demonstrated for superconducting quantumdevices, could also be applied to other three-level systemswith similar energy level structure.

A. Methods: Process tomography

In order to characterize the gates, we perform fullprocess tomography on the three-level system and re-construct the process matrix χexp using a maximumlikelihood procedure29. Any SU(3) operator acting ona three-level system can be decomposed into nine or-thogonal basis operators. As a full set of basis op-erators we choose I01, σx01, −iσy01, σz01, σx0e, −iσy0e,σx1e, −iσy1e,E, where σij are Pauli operators actingon the levels i and j, I01 = |0〉〈0| + |1〉〈1| andE = |e〉〈e|. The process is fully determined by itsaction on a complete set of nine input states |0〉,

4

|e〉, |1〉, (|0〉+ |e〉)/√

2, (|0〉+ i|e〉)/√

2, (|0〉+ |1〉)/√

2,

(|0〉+ i|1〉)/√

2, (|e〉+ |1〉)/√

2, (|e〉+ i|1〉)/√

2. Thesestates are prepared by sequentially applying identity, π−,and π/2−operations (I, Rx(π), Rx(π/2), Ry(π/2)) on the|0〉 ↔ |e〉 and |e〉 ↔ |1〉 transitions. After applying thegeometric operation to the input states, we perform fullstate tomography on the respective output states30. Thelength of a typical sequence is 208 ns (five 40 ns pulseswith 2 ns separation). We calibrate the π-pulses on the

|0〉 ↔ |e〉 and the |e〉 ↔ |1〉 transitions by measuring Rabioscillations between the corresponding states. From therecorded 92 = 81 measurements, the process matrix χexp

is reconstructed. The process is compared to the ideal one(2) by calculating its fidelity as F = tr (χexpχth). Sincethe |e〉 state serves only as an auxiliary state, we presentonly the reduced density matrix χ, which describes theprocesses involving the |0〉 and |1〉 states and omits anyoperators acting on the |e〉 state. The set of basis opera-

tors is thus given by I,X, Y , Z = I01, σx01,−iσy01, σz01.

[1] Pancharatnam, S. Generalized theory of interference andits applitactions. Proc. Indian Acad. Sci. A44, 247–262(1956).

[2] Mead, C. A. and Truhlar, D. G. On the determinationof Born-Oppenheimer nuclear motion wave functions in-cluding complications due to conical intersections andidentical nuclei. J. Chem. Phys 70, 2284–2296 (1979).

[3] Berry, M. V. Quantal phase-factors accompanying adia-batic changes. Proc. R. Soc. A 392, 45–57 (1984).

[4] Aharonov, Y. and Anandan, J. Phase-change during acyclic quantum evolution. Phys. Rev. Lett. 58, 1593–1596(1987).

[5] Wilczek, F. and Zee, A. Appearance of gauge structurein simple dynamical systems. Phys. Rev. Lett. 52, 2111–2114 (1984).

[6] Dalibard, J., Gerbier, F., Juzeliunas, G., and Ohberg, P.Colloquium: Artificial gauge potentials for neutral atoms.Rev. Mod. Phys. 83, 1523–1543 (2011).

[7] Read, N. Non-abelian adiabatic statistics and hall viscos-ity in quantum hall states and px+ipy paired superfluids.Phys. Rev. B 79, 045308 (2009).

[8] Zanardi, P. and Rasetti, M. Holonomic quantum compu-tation. Phys. Lett. A 264, 94–99 (1999).

[9] Duan, L. M., Cirac, J. I., and Zoller, P. Geometric manip-ulation of trapped ions for quantum computation. Science292, 1695–1697 (2001).

[10] Pachos, J. K. Introduction to Topological Quantum Com-putation. Cambridge Univerity Press, (2012).

[11] Berry, M. Geometric phase memories. Nat. Phys. 6, 148–150 (2010).

[12] Tycko, R. Adiabatic rotational splittings and Berry’sphase in nuclear-quadrupole resonance. Phys. Rev. Lett.58, 2281–2284 (1987).

[13] Zwanziger, J. W., Koenig, M., and Pines, A. Non-abelianeffects in a quadrupole system rotating around two axes.Phys. Rev. A 42, 3107–3110 (1990).

[14] Anandan, J. Non-adiabatic non-abelian geometric phase.Phys. Lett. A 133, 171–175 (1988).

[15] Sjoqvist, E. et al. Non-adiabatic holonomic quantumcomputation. New J. Phys. 14, 103035 (2012).

[16] Koch, J. et al. Charge-insensitive qubit design derivedfrom the Cooper pair box. Phys. Rev. A 76, 042319(2007).

[17] Leek, P. J. et al. Observation of Berry’s phase in a solid-state qubit. Science 318, 1889 (2007).

[18] Filipp, S. et al. Experimental demonstration of thestability of Berry’s phase for a spin-1/2 particle.Phys. Rev. Lett. 102, 030404 (2009).

[19] Wu, H. et al.Geometric phase gates via adiabatic controlusing electron spin resonance. Phys. Rev. A 87, 032326

(2012).[20] Sjoqvist, E. A new phase in quantum computation.

Physics 1, 35 (2008).[21] Zhu, S. L., and Wang, Z. D. Unconventional Geomet-

ric Quantum Computation. Phys. Rev. Lett. 91, 187902(2003).

[22] Faoro, L., Siewert, J., and Fazio, R. Non-abelianholonomies, charge pumping, and quantum computationwith Josephson junctions. Phys. Rev. Lett. 90, 028301(2003).

[23] Kamleitner, I., Solinas, P., Muller, C., Shnirman, A., andMottonen, M. Geometric quantum gates with supercon-ducting qubits. Phys. Rev. B 83, 214518 (2011).

[24] Solinas, P., Zanardi, P., Zanghi, N., and Rossi, F. Holo-nomic quantum gates: A semiconductor-based implemen-tation. Phys. Rev. A 67, 062315 (2003).

[25] Paik, H. et al. Observation of high coherence in Josephsonjunction qubits measured in a three-dimensional circuitQED architecture. Phys. Rev. Lett. 107, 240501 (2011).

[26] Bianchetti, R. et al. Dynamics of dispersive single-qubit readout in circuit quantum electrodynamics.Phys. Rev. A 80, 043840 (2009).

[27] Motzoi, F., Gambetta, J. M., Rebentrost, P., and Wil-helm, F. K. Simple pulses for elimination of leakage inweakly nonlinear qubits. Phys. Rev. Lett. 103, 110501(2009).

[28] Johansson, M. et al. Robustness of non-adiabatic holo-nomic gates. arXiv:1204.5144 (2012).

[29] Jezek, M., Fiurasek, J., and Hradil, Z. Quantum inferenceof states and processes. Phys. Rev. A 68, 012305 (2003).

[30] Bianchetti, R. et al. Control and tomography of a threelevel superconducting artificial atom. Phys. Rev. Lett.105, 223601 (2010).

Acknowledgement

We are grateful to Erik Sjoqvist for fruitful discussions.Supported by the EU project GEOMDISS, the AustrianScience Foundation (S. F.), and the Swiss National Sci-ence Foundation (SNSF).

Author Contributions

A.A.A. & S.F. developed the idea for the experi-ment; A.A.A. performed the measurements and analysedthe data; J.M.F. designed and fabricated the sample,J.M.F., K.J., M.P. & S.B. contributed to the experi-ment; A.A.A. & S.F. wrote the manuscript; A.W. andS.F. planned the project; all authors commented on themanuscript.