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DOI: 10.1126/science.1130886

, 1423 (2006);

313

Science

et al.Matthias Steffen,

Superconducting Qubits via State Tomography

Measurement of the Entanglement of Two

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and used models developed in laboratories from

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Supporting Online Material

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Materials and Methods

25 April 2006; accepted 26 July 2006

10.1126/science.1129158

Measurement of the Entanglement of

Two Superconducting Qubits via

State Tomography

Matthias Steffen,* M. Ansmann, Radoslaw C. Bialczak, N. Katz, Erik Lucero, R. McDermott,

Matthew Neeley, E. M. Weig, A. N. Cleland, John M. Martinis†

Demonstration of quantum entanglement, a key resource in quantum computation arising from a

nonclassical correlation of states, requires complete measurement of all states in varying bases. By

using simultaneous measurement and state tomography, we demonstrated entanglement between

two solid-state qubits. Single qubit operations and capacitive coupling between two super-

conducting phase qubits were used to generate a Bell-type state. Full two-qubit tomography yielded a

density matrix showing an entangled state with fidelity up to 87%. Our results demonstrate a high

degree of unitary control of the system, indicating that larger implementations are within reach.

T

with classical computation (1). Because this

power is achieved through the controlled

evolution of entangled quantum states, a clear

demonstration of entanglement represents a

necessary step toward the construction of a

scalable quantum computer (2, 3). However,

direct demonstration of entanglement is chal-

lenging because all of the DiVincenzo criteria

(4) for quantum computation must be met

simultaneously. To date, only subsets of these

key requirements have been demonstrated for

superconducting qubits (5–9). We demonstrated

all of the DiVincenzo criteria simultaneously,

thus placing superconducting qubits on the road

map for scalable quantum computing.

he laws of quantum physics provide in-

triguing possibilities for a tremendous in-

crease in computational power compared

CircuitsmadeofsuperconductorsandJosephson

junctions are promising candidates for scalable

quantum computation because of their compati-

bility with integrated-circuit fabrication technol-

ogy(5–9). TheJosephsonphasequbitstandsapart

fromothersuperconductingqubitsbecauseitdoes

not use an optimal operating point. Coupling of

phase qubits is thus straightforward, allowing for

multiple control methods (10). With recent im-

provements in coherence times and amplitudes

(11), and in particular the ability to measure both

qubit states simultaneously (5), it is possible to

use phase qubits to produce entangled states

and measure them with high fidelity.

In the phase qubit circuit (Fig. 1A), the

Josephson junction (with critical current I0) has

a superconducting phase difference, d, that

serves as the quantum variable. When biased

close to the critical current, the junction and its

loop inductance, L, give a cubic potential that

has qubit states k0À and k1À, with an energy

spacing that corresponds to a transition fre-

quency w10/2p È 5 GHz (Fig. 1B). This fre-

quency can be adjusted by È30% via the bias

current.

Singlequbitlogicoperations,correspondingto

rotations about the x, y, and z axes of the Bloch

sphere, were generated as follows. Rotations

about the z axis were produced from current

pulses on the qubit bias line that adiabatically

change the qubit frequency, leading to phase

accumulation between thek0À andk1À states of

the qubit (11). Rotations about any axis in the xy

plane were produced by microwave pulses reso-

nant with the qubit transition frequency. They

selectively address only the qubit energy levels,

because transitions to higher-lying energy levels

are off-resonance due to the anharmonicity of the

potential and the shaping of the pulses (12). The

phase of the microwave pulses defines the ro-

tationaxisinthexy plane. The pulse duration and

amplitude control the rotation angle.

The qubit state was measured by applying

a strong pulse, Iz, so that only thek1À state

tunnels out of the cubic well (Fig. 1C). Once

tunneled, the state quickly decays into an ex-

ternal ground state that can be easily dis-

tinguished from the untunneledk0À state by an

on-chip superconducting quantum interference

device (SQUID) amplifier.

Two separate phase qubits were coupled

with a fixed capacitor (5) (Fig. 1D). With the

qubits labeled A and B, the coupling Hamilto-

nian is Hint0 (S/2)(k01Àb10kþk10Àb01k), where

k01À 0k0ÀA`k1ÀB. The coupling strength, S 0

(Cx/C)Iw10, isproportionaltothecouplingcapac-

itance Cx, 3 fF, where C , 1.3 pF is the junc-

tion shunting capacitance (13) and I is Planck_s

constant (h) divided by 2p. The two qubits may

easilybebroughtintoresonance,eventhoughthey

arenotidentical,becauseeachcanbetunedovera

large frequency range. On resonance, the interac-

tion produces an oscillation with frequency S/h

between the statesk01À and ik10À; for an interac-

tion time of h/4S, the coupling produces the gate

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

itselfasanavoidedlevelcrossingofstrengthS/h in

the spectroscopy of the individual qubits (15).

iSWAP

gates,isuniversal(14). Thecouplingalsomanifests

p

. This gate, together with single qubit

Department of Physics and California NanoSystems Insti-

tute, University of California, Santa Barbara, CA 93106,

USA.

*Present address: IBM Watson Research Center, Yorktown

Heights, NY 10598, USA.

†To whom correspondence should be addressed. E-mail:

martinis@physics.ucsb.edu

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The performance of each qubit can be de-

termined separately by strongly detuning the two

qubits relative to S/h so that they behave inde-

pendently. A standard set of experiments, includ-

ing Rabi and inversion recovery experiments,

gives an energy relaxation time of T10 130 ns

and a dephasing time of T2* 0 80 ns for each

qubit. These results are consistent with measured

values of an uncoupled sample (11), indicating

no additional loss due to the second qubit. The

measurement fidelities, defined as the probabil-

ities of correctly identifying statesk0À andk1À,

are F00 0.95 and F10 0.85, respectively.

We next tuned both qubits to w10/2p 0 5.1

GHz and determined the splitting S/h 0 10 MHz

by qubit spectroscopy. The time dynamics of the

coupling was verified by initializing the qubits to

the statek00À and applying to qubit B a 180-

rotation about the x axis (180xpulse) of 10-ns

duration. This pulse is sufficiently long to avoid

unwanted transitions to other energy levels but

short on the time scale of the coupling. The

resulting statek01À is not an eigenstate of the

coupling Hamiltonian and thus evolves in

time according to ky(t)À 0 cos(St/2I)k01À –

isin(St/2I)k10À. After a variable free-evolution

time, tfree, we simultaneously measure the state

of the two qubits. Repeating the experiment

about 1000 times, we determine the occupation

probabilities P00, P01, P10, and P11. This sequence

of operations is depicted in Fig. 2A, and the

measured probabilities are plotted in Fig. 2B.

The occupation probabilities P01and P10

oscillate out of phase with a period of 100 ns,

consistent with the spectroscopic measure-

ments. The amplitude and decay of the data

are also compatible with the separately mea-

sured lifetimes and measurement fidelities of

the single qubits. Compared with earlier ex-

periments (5), the amplitude of the measured

oscillations is substantially larger because of

improvements in single qubit fidelities. We

note that the oscillations persist longer than the

dephasing time, T2* 0 80 ns, because the period

of the coupled qubit oscillations (Fig. 2) is, to

first order, insensitive to the detuning of the

qubits. For these states, this represents a de-

generacy point that is also tunable.

Although these data are consistent with the

production of an entangled state at tfree0 25 ns, a

more stringent test includes performing coherent

single qubit operations on this entangled state to

verify the predicted unitary evolution of the

system. After the application of a 180xpulse on

qubit B and a tfreeof 25 ns, the system is in the

entangledstateky1À0ðk01Àjik10ÀÞ=

applying a 90zpulse on qubit B, we create the

Bell state ky2À 0 ðk01À jk10ÀÞ=

ky2À is an eigenstate of the coupling Hamiltonian,

it should not evolve with time. Implementation of

this sequence of operations is complicated by the

coupling interaction that occurs during the single

qubit operations. Compared with the coupling

interaction time, tfree0 25 ns, the duration of the

single qubit gates 180xand 90zare 10 ns and 4 ns,

respectively, and are thus not negligible. The

excess coupled interaction during the single qubit

gates can be significantly compensated by

reducing the free evolution time (16) to tfree0

16 ns, which we checked numerically. Upon

executing this sequence of operations, we verify

ffiffiffi

. Because

2

p

. By then

ffiffiffi

2

p

Fig. 1. Qubitcircuit and

experimental operation.

(A) Circuit schematic for

a single Josephson phase

qubit,wheretheXsymbol

represents the Josephson

junction. The measure-

mentisimplementedwith

a broadband 50-ohm

transmissionlinewithcold

attenuators that is con-

nectedtothefluxbiasline

with a bias tee. (B) Oper-

ation mode of the qubit

showing the potential en-

ergy, U, versus junction

phase, d. The qubit is

formed from the two

lowest eigenstates k0À

and k1À, with a transition

frequency w10(Idc)/2p 0

5.1 GHz that can be

adjusted by varying the

bias, If. (C) Measurement

mode of the qubit. Dur-

ing the measurement

pulse, the energy barrier DU is lowered to increase the tunneling rate, G, and the tunneling probability of k1À.

(D) Circuit diagram of the coupled qubits.The loop inductance, L, is È850 pH, and the junction capacitance, C,

is È1.3 pF. An interdigitated capacitor with CxÈ 3 fF couples the qubits, giving rise to an interaction strength

of magnitude S/h 0 10 MHz.

Fig. 2. Coherent oper-

ations on coupled phase

qubits. (A) Sequence of

operations. A 10-ns-long

180xpulse is applied to

qubit B, populating the

k01À state. After a free

evolution time tfreein

which the qubits inter-

act, the state occupation

probabilities are mea-

sured by using 10-ns

current pulses that in-

duce selective tunneling

of the k1À state. For data

in (C) and (D), a 90zand

180zpulse, respectively,

is applied to qubit B

after 16 ns. (B) Plot of

measurement probabil-

ities of the states k01À,

k10À, and k11À as a func-

tion of tfree. Note that

P000 1 j P01j P10j

P11. The solid lines are

the results of simulations

using known measure-

mentfidelities,relaxation

times, and microwave cross talk. (C) Plot of measurement probabilities for a sequence that creates the

eigenstateky2À 0 (k01Àjk10À)=

2of the coupling Hamiltonian. After the eigenstate is formed by the 90z

pulse, it ceases to evolve with time. (D) As in (C), but with an 180zpulse. Here, the phase of the oscillation

changes by 180 degrees.

ffiffiffip

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that indeed P01and P10no longer oscillate as a

function of tfree(Fig. 2C).

This observed behavior, however, could also

be attributed to the destruction of coherence be-

tween the statesk01À andk10À caused by the ap-

plication of the 90zpulse. To check this

possibility, we applied a 180zpulse on qubit B

when the system is in the stateky1À, creating the

stateky3À 0 ðk01À þ ik10ÀÞ=

equivalent toky1À but delayed by tfree0 50 ns, a

reversal of the oscillations is predicted for this

experiment. This prediction is verified (Fig. 2D)

and provides further evidence of an entangled

state.

A full and unambiguous test of entanglement

comes from state tomography (2, 3, 17), which

involvesthemeasurementofthequantumstatein

allninecombinationsofthreemeasurementbases

(x, jy, and jz) for each qubit. Each measure-

ment gives three unique probabilities (e.g., P01,

P10, and P11) for a total of 27 numbers, which

are used to compute the 15 independent

parameters of the unknown density matrix, r,

via a least squares fit (17). The measurement

basis change from jz to x and from jz to jy

arises from applying a microwave pulse 90y

and 90x, respectively, before measurement (11).

After calibrating the phase of the microwave

pulses for the two qubits (13), we perform state

tomography onky1À as indicated by the sequence

of operations in Fig. 3A. As in the previous ex-

periment, we reduced the duration of the free

evolution to compensate for coupled qubit inter-

action during the initial 180xpulse and the

tomography pulses. After executing all nine

tomography sequencesandmeasuring the resulting

occupation probabilities, we computed the density

matrix, rexp. The real and imaginary parts of the

reconstructed rexpare shown in Fig. 3B. The

imaginary off-diagonal elements k01Àb10k and

ffiffiffi

2

p

. Becauseky3À is

k10Àb01khave nearly the same magnitude as the

real diagonal componentsk01Àb01kandk10Àb10k,

revealing a coherent superposition of the states

k01À andk10À. This measurement unambiguously

verifies that the two qubits are indeed entangled.

Compared to the ideally expected density

matrix, s 0ky1Àby1k, we computed the fidelity

of the reconstructed quantum state and find

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

To identify the sources of fidelity loss, we

first corrected for measurement error. Based on

the measurement fidelities discussed earlier, we

renormalized the measured occupation proba-

bilities and calculated the intrinsic occupation

probabilities (13). From this we computed a

density matrix corrected for measurement, rexp,M

(Fig. 3C), that gives an improved fidelity,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

most of the remaining fidelity loss to single-

qubit decoherence. By modeling decoherence

effects (16) using the measured relaxation

times, we obtained an expected rththat gives

a fidelity Fexp0 tr

s½rths½

Fexp0 tr

s½rexps½

q

0 0:75.

Fexp,M0 tr

s½rexp,Ms½

q

0 0:87. We attribute

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p

0 0:89, which is

close to the normalized measured value (18).

The fact that our error is dominated by deco-

herence indicates good unitary control of our

system and thus suggests that improvements in

coherence times will directly translate to en-

hanced gate fidelities. Dramatic increases in

coherence should be possible on the basis of

straightforward improvements in the dielectric

material of the shunting capacitor (11, 19).

Our experiments on coupled phase qubits

have verified by state tomography the creation of

an entangled Bell state with 87% fidelity. Given

thatmostofthelossinfidelitycanbeattributedto

decoherence, we believe that more complex

implementations are well within reach with only

modest improvements in qubit coherence times.

References and Notes

1. M. A. Nielsen, I. L. Chuang, Quantum Computation

and Quantum Information (Cambridge Univ. Press,

Cambridge, 2000).

2. H. Ha ¨ffner et al., Nature 438, 643 (2005).

3. D. Leibfried et al., Nature 438, 639 (2005).

4. D. P. DiVincenzo, Fortschr. Phys. 48, 771 (2000).

5. R. McDermott et al., Science 307, 1299 (2005).

6. T. Yamamoto et al., Nature 425, 941 (2003).

7. D. Vion et al., Science 296, 886 (2002).

8. I. Chiorescu, Y. Nakamura, C. J. P. M. Harman, J. E. Mooij,

Science 299, 1869 (2003); published online 13 February

2003 (10.1126/science.1081045).

9. A. Wallraff et al., Phys. Rev. Lett. 95, 060501 (2005).

10. M. Geller et al., personal communication.

11. M. Steffen et al., Phys. Rev. Lett. 97, 050502 (2006).

12. M. Steffen, J. M. Martinis, I. L. Chuang, Phys. Rev. B 68,

224518 (2003).

13. Materials and methods are available as supporting

material on Science Online.

14. N. Schuch, J. Siewert, Phys. Rev. A. 67, 032301 (2003).

15. A. J. Berkley et al., Science 300, 1548 (2003); published

online 15 May 2003 (10.1126/science.1084528).

16. L. M. K. Vandersypen et al., Nature 414, 883 (2001).

17. L. M. K.Vandersypen et al., Appl. Phys. Lett. 76, 646 (2000).

18. A more stringent measure that quantifies the amount of

entanglement, even for mixed states, is the entanglement

of formation, E(r) (20). We find E(rexp,M) 0 0.42 compared

with E(rth) 0 0.61.

19. J. M. Martinis et al., Phys. Rev. Lett. 95, 210503 (2005).

20. W. K. Wootters, Quantum Inf. Comput. 1, 27 (2001).

21. We acknowledge S. Waltman and the National Institute of

Standards and Technology for support in building the

microwave electronics. Devices were made at the UCSB and

Cornell Nanofabrication Facilities, a part of the NSF-funded

National Nanotechnology Infrastructure Network. N.K.

acknowledges support of the Rothschild fellowship.This work

was supported by Disruptive Technology Office under grant

W911NF-04-1-0204 and by NSF under grant CCF-0507227.

Supporting Online Material

www.sciencemag.org/cgi/content/full/313/5792/1423/DC1

Materials and Methods

5 June 2006; accepted 14 July 2006

10.1126/science.1130886

Fig. 3. State tomography of entangled qubits. (A) Sequence of

operations. A 180xpulse is first applied to qubit B, followed by a

free evolution period of about 16 ns, creating the entangled

state ky1À 0 (k01Àjik10À)=

2 . State tomography is then

performed by using 4-ns single qubit rotations. (B) Reconstructed

rexp[real (Re) and imaginary (Im) parts] using the directly

measured occupation probabilities. (C) Reconstructed density

matrix rexp,Mafter correcting the state occupation probabilities

based on the single qubit measurement fidelities. Note that

matrix diagonal runs from left to right.

ffiffiffip

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