, 1423 (2006);
et al.Matthias Steffen,
Superconducting Qubits via State Tomography
Measurement of the Entanglement of Two
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Supporting Online Material
Materials and Methods
25 April 2006; accepted 26 July 2006
Measurement of the Entanglement of
Two Superconducting Qubits via
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.
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
junctions are promising candidates for scalable
quantum computation because of their compati-
bility with integrated-circuit fabrication technol-
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
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
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
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
the spectroscopy of the individual qubits (15).
. This gate, together with single qubit
Department of Physics and California NanoSystems Insti-
tute, University of California, Santa Barbara, CA 93106,
*Present address: IBM Watson Research Center, Yorktown
Heights, NY 10598, USA.
†To whom correspondence should be addressed. E-mail:
www.sciencemag.orgSCIENCEVOL 3138 SEPTEMBER 2006
on December 18, 2006
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
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
. By then
Fig. 1. Qubitcircuit and
(A) Circuit schematic for
a single Josephson phase
represents the Josephson
junction. The measure-
a broadband 50-ohm
attenuators that is con-
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
is applied to qubit B
after 16 ns. (B) Plot of
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-
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.
8 SEPTEMBER 2006VOL 313 SCIENCE www.sciencemag.org
on December 18, 2006
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
A full and unambiguous test of entanglement
comes from state tomography (2, 3, 17), which
(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
. 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
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,
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
0 0:87. We attribute
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
decoherence, we believe that more complex
implementations are well within reach with only
modest improvements in qubit coherence times.
References and Notes
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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
Materials and Methods
5 June 2006; accepted 14 July 2006
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.
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on December 18, 2006