Download Quantum process tomography of two-qubit controlled-Z

PHYSICAL REVIEW B 82, 184515 共2010兲
Quantum process tomography of two-qubit controlled-Z and controlled-NOT gates
using superconducting phase qubits
T. Yamamoto,1,2 M. Neeley,1 E. Lucero,1 R. C. Bialczak,1 J. Kelly,1 M. Lenander,1 Matteo Mariantoni,1 A. D. O’Connell,1
D. Sank,1 H. Wang,1 M. Weides,1 J. Wenner,1 Y. Yin,1 A. N. Cleland,1,* and John M. Martinis1,†
1
Department of Physics, University of California, Santa Barbara, California 93106, USA
Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan
共Received 19 October 2010; published 9 November 2010兲
2Green
We experimentally demonstrate quantum process tomography of controlled-Z and controlled-NOT gates
using capacitively coupled superconducting phase qubits. These gates are realized by using the 兩2典 state of the
phase qubit. We obtain a process fidelity of 0.70 for the controlled phase and 0.56 for the controlled-NOT gate,
with the loss of fidelity mostly due to single-qubit decoherence. The controlled-Z gate is also used to demonstrate a two-qubit Deutsch-Jozsa algorithm with a single function query.
DOI: 10.1103/PhysRevB.82.184515
PACS number共s兲: 03.67.Lx, 03.67.Ac, 85.25.Cp
I. INTRODUCTION
Quantum computation and quantum communication rely
on excellent control of the underlying quantum system.1 Reasonable control has been achieved with a variety of quantum
systems with superconducting qubits emerging as one of the
most promising candidates.2 Recent experiments using superconducting architectures include demonstrations of quantum algorithms using two qubits3 and the entanglement of
three qubits.4,5 A key element in these experiments is a twoqubit entangling gate, such as the 冑iSWAP 共Ref. 4兲 and the
controlled-Z 共CZ兲 gates.3,5 Because the CZ gate is simple to
implement, has high fidelity and can readily generate
controlled-NOT 共CNOT兲 logic,6 it likely will be an important
component in more complex algorithms such as quantum
error correction. At present, however, the CZ gate functionality has only been directly tested for a subset of the possible
input states.
In this paper, we demonstrate the operation of a CZ gate
in superconducting phase qubits and fully characterize this
gate as well as a CNOT gate using quantum process tomography 共QPT兲. We additionally use the CZ gate to perform the
Deutsch-Jozsa algorithm,3 here with a single-shot evaluation
of the function. The use of QPT provides a more complete
gate evaluation than, for example, measuring the truth table
for the corresponding CNOT gate,7,8 as it verifies that the
gate will properly transform any possible input state. QPT
for two- or three-qubit gates has been reported in NMR,9
optics,10–12 and in ion traps.13,14 In solid-state systems, QPT
has been implemented for the 冑iSWAP gate with the phase
qubit.15
the shunting capacitance, unequally spaced quantized energy
levels appear in the cubic potential. The two lowest levels
are used for the qubit states 兩0典 and 兩1典 with a transition
A B
frequency f 10
共f 10兲 that can be controlled by an external magA
B
共⌽ex
兲 applied to the loop. The third energy
netic flux ⌽ex
level 兩2典 is used as an auxiliary state to realize the CZ gate,
as discussed below.
The operation of a similar device has been reported
previously.15,16 The state of each qubit is controlled by applying a rectangular-shaped current pulse 共Z pulse兲 or a
Gaussian-shaped microwave pulse 共X, Y pulse兲 to its bias
coil. For an X or Y pulse, we simultaneously apply the derivative of the pulse to the quadrature 共90° phase shifted兲
drive to reduce both unwanted excitation of the 兩2典 state and
phase error due to ac Stark effect;17 the derivative scaling
factor is determined from the nonlinearity of each qubit.18
This procedure enables us to use a Gaussian pulse with a full
width at half maximum of 10 ns while maintaining accurate
qubit control19 in spite of a rather weak qubit nonlinearity
共⬃100 MHz兲. Each qubit state is read out individually in a
single-shot manner by injecting a large magnitude Z pulse
and then measuring the qubit flux with a superconducting
quantum interference device 共SQUID兲.
The device was fabricated using a photolithographic process with Al films, AlOx tunnel junctions, and a-Si:H dielectric for the shunt capacitors and wiring crossovers, all on a
sapphire substrate. The device was mounted in a superconducting aluminum sample holder and cooled in a dilution
refrigerator to ⬃25 mK.
SQUID A
Cc
qubit A
qubit B
CA
II. EXPERIMENT
The electrical circuit for the device is shown in Fig. 1,
comprising two superconducting phase qubits A and B,
coupled by a fixed capacitance Cc. Each qubit is a superconducting loop interrupted by a capacitively shunted Josephson
junction. When biased close to the critical current, the junction, and its parallel loop inductance produce a nonlinear
potential as a function of the phase difference across the
junction. Combined with the kinetic energy originating from
1098-0121/2010/82共18兲/184515共8兲
Bias
coil A
LA
Bias
coil B
SQUID B
CB
I0A
ΦexA
I0B
ΦexB
LB
FIG. 1. Circuit diagram for the experimental device, showing
two flux-biased phase qubits coupled by a fixed capacitance Cc. A
bias coil and readout SQUID are coupled to each qubit. The design
B
parameters of the circuit are IA
0 = I0 = 2 ␮A, CA = CB = 1 pF,
LA = LB = 720 pH, and Cc = 2 fF.
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©2010 The American Physical Society
PHYSICAL REVIEW B 82, 184515 共2010兲
YAMAMOTO et al.
|01›
1.00
0.895
0.900
B
Φex /Φ0
t
i0
0.0
∆i (arb.units)
0.2
Probability
0.4
7.25
t0
(b)
7.15
-0.05
0.00
0.05
0.10
∆i (arb.units)
0.15
FIG. 2. 共Color online兲 High-power spectroscopy for qubit B.
The escape probability 共gray scale兲 is plotted versus microwave
frequency and the Z-pulse amplitude ⌬i. The single-photon
兩0典 → 兩1典 and two-photon 兩0典 → 兩2典 transitions are visible, along with
two avoided-level crossings. The upper inset shows the calculated
states and eigenenergies with the thick 共thin兲 lines representing
single 共two兲 photon excitations. The lower inset illustrates the
Z-pulse amplitude i0 for the CZ= −共SWAP兲2 operation.
In the present experiment, the two qubits were biased so
A
B
= 7.16 GHz and f 10
= 7.36 GHz when no Z pulse was
that f 10
applied. The relaxation times 共T1兲 were measured to be 510
ns and 500 ns for qubit A and B, respectively. The dephasing
times determined from a Ramsey interference experiment
兲, which showed Gaussian decay proportional to
共TRamsey
2
兲2兴 due to 1 / f flux noise,20 were 200 ns and
exp关−共t / TRamsey
2
230 ns, respectively.
III. RESULTS AND DISCUSSION
A. Spectroscopy
Figure 2 shows the high-power spectroscopy for qubit B,
which is used to guide formation of the CZ gate. We plot the
escape probability of qubit B in gray scale as a function of
the amplitude ⌬i of a 2 ␮s long Z pulse 共horizontal axis兲
and the frequency of a microwave X pulse 共vertical axis兲 of
the same length. Both pulses were applied simultaneously to
qubit B, followed by the Z pulse for the readout. In this
way, we can probe the change in the resonance frequency
B
as a function of detuning ⌬i. In addition to the main
f 10
B
, somewhat broadened
resonance line corresponding to f 10
because of the large amplitude of the microwave pulse
B
⬃ 10␮⌽0兲, a sharper line is observed on the low共⌬⌽ex
frequency side of the main resonance; this corresponds
to the two-photon excitation from the 兩0典 to the 兩2典 state.21
The vertical distance between the main and two-photon
lines is 1/2 the qubit nonlinearity ⌬f = f 10 − f 21, yielding
⌬f = 114 MHz for qubit A 共data not shown兲 and 87 MHz for
qubit B.
We observe an avoided-level crossing in the main resonance at ⌬i ⯝ 0.13 when the two-qubit frequencies overlap
B
A
= f 10
. Here, the degeneracy of the 兩AB典 = 兩10典 and 兩01典
f 10
states produces a splitting with size 14.2⫾ 0.2 MHz, deter-
tcmp
i0
0.4
0.06
0.0
0.04
100
∆t (ns)
2
i0
P|2>
(d)1.0
0.8
0.08
0
-SWAP
qubit B
meas.
A
icmp
π/2
t0
π/2
∆i
qubit B
|02›
qubit A
∆t
π
|11›
π
(c)
qubit A
200
Probability
7.35
π
(a)
1.02
Frequency
Frequency (GHz)
7.45
B
icmp
meas.
π pulse off
π pulse on
0.5
0.0
-0.04
0.00
B
icmp (arb.units)
FIG. 3. 共Color online兲 共a兲 Operation sequence for 共b兲, the state
evolution between 兩11典 and 兩02典 states. 共b兲 Plot of 兩2典 state probability of qubit B versus Z-pulse time ⌬t and Z-pulse amplitude ⌬i.
The dashed lines correspond to the optimal setting for the CZ gate.
共c兲 Operation sequence for 共d兲, demonstration of the CZ gate. 共d兲
Plot of 兩1典 state probability of qubit B as a function of iBcmp 共iA
cmp is
fixed as 3 ⫻ 10−4兲. The 共red兲 solid and 共blue兲 dashed curves are for
qubit A initialized to the 兩0典 and 兩1典 states, respectively. The vertical
dotted-dashed line indicates the value of iBcmp for the CZ gate.
mined from a fit to the data, consistent with the designed
capacitance Cc. The avoided crossing for the two-photon line
at ⌬i ⯝ 0.066 gives a splitting of 9.7⫾ 0.2 MHz, about 冑2 / 2
times as large as the main resonance, as expected from a 兩11典
and 兩02典 interaction. The slope of the resonance between the
B
, as expected for the 兩11典 state.
two crossings is 1/2 that of f 10
Our interpretation of the spectroscopy is validated by a
numerical calculation. Using three states for each qubit and
the qubit design parameters, we calculate from the resulting
9 ⫻ 9 Hamiltonian22,23 the energies for the coupled eigenA
, are plotted verstates. The energy bands, normalized to f 01
sus the flux bias for qubit B in the upper inset of Fig. 2. Here,
a band is plotted only when its transition matrix element
from the ground state is above a threshold, to simulate the
appearance of the transition in the spectroscopic measurement. The details of the calculation are described in Appendix A. The 共red兲 thick lines correspond to the 兩01典 and 兩10典
states, whereas the 共blue兲 thin lines represent half of the excitation energy of the 兩11典 and 兩02典 states. The overall structure agrees well with the experimental data.
B. Bring up of controlled-Z gate
As proposed theoretically by Strauch et al.,24 the avoided
crossing due to the degeneracy of the 兩11典 and 兩02典 states can
be used to construct a CZ gate, whose action produces no
change in state except for 兩11典 → −兩11典. By applying a
nonadiabatic Z pulse, the 兩11典 and 兩02典 states become
degenerate 共see lower inset of Fig. 2兲. Initially in the 兩11典
state, the system evolves as an iSWAP interaction, giving
兩⌿共t兲典 = cos共␥⌬t / ប兲兩11典 + i sin共␥⌬t / ប兲兩02典, where 2␥ is the
splitting energy of the avoided crossing and ⌬t the duration
of the Z pulse. After twice the iSWAP time ⌬t = h / 2␥, the
system returns to the initial state 兩11典 but with a minus sign.
If the system starts in 兩00典, 兩01典, or 兩10典, the state does not
change since it is off-resonance with both avoided-level
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PHYSICAL REVIEW B 82, 184515 共2010兲
QUANTUM PROCESS TOMOGRAPHY OF TWO-QUBIT…
Exp.
FIG. 4. 共Color online兲 ␹ matrices of CZ and CNOT gates. 共a兲
Left panel: the real part of the experimentally obtained ␹ matrix
共␹p兲 for the CZ gate with Fp
= 0.70. Right panel: the real part
of the simulated ␹ matrix for the
CZ gate with Fp = 0.67. 共b兲 Left
panel: the real part of ␹p for the
CNOT gate with Fp = 0.56. Right
panel: the real part of the simulated ␹ matrix for the CNOT gate
with Fp = 0.52. The open boxes in
the figure represent the ideal ␹
matrix.
Sim.
Exp.
Re(χ)
Re(χ)
(b) CNOT
Sim.
Re(χ)
Re(χ)
(a) CZ
crossings. A similar scheme using an adiabatic Z pulse has
been used to successfully demonstrate a quantum algorithm3
and the same 共nonadiabatic兲 scheme has recently been used
to create a three-qubit entangled state in transmon qubits.5
To experimentally determine the amplitude and length of
the required nonadiabatic Z pulse, we directly measured the
coherent oscillation between the 兩11典 and 兩02典 states. This
共iSWAP兲2 operation sequence is shown in Fig. 3共a兲: we first
prepare the 兩11典 state with a ␲ pulse to both qubits and then
apply a Z pulse with amplitude ⌬i and length ⌬t to qubit B.
Here only qubit B is probed and we adjust the measurement
pulse amplitude so that the qubit is detected only when in the
兩2典 共or higher兲 state.25 In Fig. 3共b兲, we plot the tunneling
probability P兩2典 as a function of ⌬t and ⌬i, which shows the
expected chevron pattern. The minimum oscillation frequency occurs at a value of ⌬i that agrees with i0 determined
in Fig. 2共a兲. The oscillation period t0 = 51.8 ns is also
consistent with the splitting size of the avoided crossing. At
the intersection of these two dashed lines, the time evolution
of the state produces a minus sign, as required for the
CZ gate. We stress that no discernable increase in P兩2典 is
observed 共⬍1%兲 at this operation point 共⌬t , ⌬i兲 = 共t0 , i0兲,
confirming that we return to the 兩11典 state after the CZ
operation.
Because the qubits themselves also accumulate phase ␾
during the CZ pulse, the general unitary evolution from the
gate is given by
U=
冢
1
0
0
0
0 e
i␾A
0
0
0
0
i␾B
0
0
0
e
0
− ei共␾A+␾B兲
冣
.
共1兲
By adding additional Z pulses to both qubits, we can compensate these phases and even place the minus sign at any
diagonal position in the matrix.3 The compensation pulses
are shown in Fig. 3共c兲, which consist of a fixed 10 ns pulse
of variable amplitude icmp after the CZ pulse. In Fig. 3共d兲, we
B
plot the tunneling probability of qubit B as a function of icmp
A
for fixed icmp. The phase of qubit B is measured through a
Ramsey fringe experiment. The 共red兲 solid and 共blue兲 dashed
curves correspond to qubit A being in the 兩0典 or 兩1典 state.
B
, but are
They both show a sinusoidal dependence on icmp
shifted by ␲ from each other, confirming the correct operation of the CZ gate. A similar experiment was done for qubit
A 共data not shown兲.
The phases for the CZ gate are set by taking the values of
icmp that give maximum probability when the control qubit is
in the 兩0典 state, as indicated by the vertical dashed-dotted line
in Fig. 3共d兲. Controlled-NOT 共CNOT兲 gates are constructed
by combining the CZ gate with single-qubit rotations
UCNOT = 共I 丢 R␲y /2兲CZ共I 丢 R−y ␲/2兲, where R␪y represents the rotation of a single-qubit state by an angle ␪ about the y axis
and I is the identity operator.
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PHYSICAL REVIEW B 82, 184515 共2010兲
YAMAMOTO et al.
FIG. 5. 共Color online兲 共a兲 Pulse sequence for the Deutsch-Jozsa
algorithm. 关共b兲–共d兲兴 Real part of the density matrix of the final state
for four Deutsch-Jozsa functions. Open boxes represent the ideal
density matrix.
C. Quantum process tomography
We evaluate the performance of these gates with QPT in
which we determine the ␹ matrix with the elements defined
as6
16
E共␳兲 = 兺 Em␳E†n␹mn .
共2兲
m,n
Here, E共␳兲 is the density matrix obtained by applying the
gate to ␳ and Em’s are the operator bases formed by the
Kronecker product of Pauli operators 兵I , ␴x , −i␴y , ␴z其 for
each qubit. For QPT, we prepare 16 input states in total,
chosen from the set 兵兩0典 , 兩1典 , 兩0典 + 兩1典 , 兩0典 + i兩1典其 for each qubit. After preparing these input states, we determine the density matrix of the output state with quantum state
tomography16 共QST兲 in which we measure each qubit along
the six directions ⫾x, ⫾y, and ⫾z of the Bloch sphere.26 For
each combination of QPT and QST pulses, we repeat the
sequence 1800 times to obtain the joint qubit probabilities
PAB = P00, P10, P01, and P11. After correcting for small measurement errors,15 we reconstruct the 16⫻ 16 experimental
␹e matrix from the resulting 16 density matrices.6 We assumed that input states are ideally prepared.27 With experimental noise, the ␹e matrix found in this way is not necessarily physical, i.e., completely positive and trace preserving.
We thus use convex optimization to obtain the physical matrix ␹p that best approximates ␹e, as used in Ref. 15. The
difference between ␹e and ␹p is small as shown in Appendix
B.
We plot the real part of ␹p for the CZ and CNOT gates in
the left panel of Figs. 4共a兲 and 4共b兲. The open boxes represent the ideal ␹ matrix. The imaginary parts of ␹p have very
small magnitude 共⬍0.04 for CZ and ⬍0.03 for CNOT兲 and
are shown in Appendix C. For both gates, we observe elements with large amplitudes at the proper positions. More
quantitative evaluation is obtained by calculating the process
fidelity Fp, defined by Fp = Tr共␹i␹p兲, where ␹i represents an
ideal ␹ matrix. We obtain Fp = 0.70 for the experimentally
measured CZ gate and 0.56 for the CNOT gate. For CZ gates
with a minus sign at other positions on the diagonal, the
measured Fp’s are 0.68, 0.69, and 0.70 for CZ00, CZ01, and
CZ10 共see Appendix C兲.
To understand the loss of process fidelity, we performed
numerical simulations. We solved the standard master
equation, ␳˙ = −共i / ប兲关H , ␳兴 + L关␳兴, where H is a 9 ⫻ 9
Hamiltonian for capacitively coupled phase qubits
under
rotating
wave
approximation
and
L关␳兴
1 i† i
1
i† i
= 兺i=A,B兺 j=1,2Lij␳Li†
−
L
L
␳
−
␳
L
L
.
Here,
for
example,
j
j
j
j
2 j
2
A
†
冑 A
冑 A
LA
1 = aA / T1 and L2 = aAaA 2 / T2 describe the relaxation
and dephasing for qubit A, respectively.28 The entire sequence, including the QPT and QST pulses, was simulated to
construct ␹sim. Experimental Ramsey interference shows
Gaussian decay, which is not reproduced by the above master
equation. Thus, in order to approximate this situation, we
used an effective T2 that depends on the length of the control
sequence for a particular experiment tseq. In particular, we
2
/ tseq in the simulation in order for both the
used T2 = TRamsey
2
Gaussian decay and exponential decay to give the same decay factor at tseq. The real part of ␹sim is shown in Fig. 4. The
actual tseq is 101.8 ns in QPT for CZ gates and 141.8 ns for
CNOT gate. The simulation reproduces reasonably well the
reduction in the expected elements and the appearance of
TABLE I. Summary of performance for Deutsch-Jozsa algorithm. Deutsch-Jozsa functions are defined as
f 0共x兲 = 0, f 1共x兲 = 1, f 2共x兲 = x, and f 3共x兲 = 1 − x.
Deutsch-Jozsa function
Constant
Element
具00兩␳兩00典 + 具01兩␳兩01典
具10兩␳兩10典 + 具11兩␳兩11典
Ideal
Measured
Ideal
Measured
Balanced
f0
f1
f2
f3
0
0.29
1
0.71
0
0.28
1
0.72
1
0.76
0
0.24
1
0.74
0
0.26
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QUANTUM PROCESS TOMOGRAPHY OF TWO-QUBIT…
(a) 60
(b)
CZ11
50 σ = 0.0020
40
60
# of counts
# of counts
small unwanted elements. These imperfections are removed
as we increase the single-qubit coherence time in the simulation, which suggests that loss of Fp in our system is mostly
dominated by single-qubit decoherence.29 We note that it is
possible to obtain more information on the decoherence
mechanisms by analyzing the magnitude of particular elements in the ␹ matrix.30 The simulated ␹ matrix of all CZ
and CNOT gates including the imaginary parts are shown in
Appendix D.
30
20
By using these conditional gates, we can perform the
Deutsch-Jozsa algorithm6 using the pulse sequence described
in Ref. 3. Figure 5共a兲 shows the pulse sequence for the
Deutsch-Jozsa algorithm. The four different two-qubit gates
Ui correspond to the four Deutsch-Jozsa functions, which we
want to determine by a single-quantum evaluation of the
function. They are given by
U1 = I 丢 Rx␲ ,
0
0
-10 -5 0
5 -3
Diff. in peak height (10 )
FIG. 6. 共Color online兲 Histogram of the differences in the peak
height of each of the 256 matrix elements in the real part of ␹
matrix. Data is for 共a兲 CZ and 共b兲 CNOT. The solid curves are a
Gaussian fit to the data.
APPENDIX A: CALCULATION OF THE ENERGY BANDS
AND TRANSITION MATRIX ELEMENTS
U2 = 共I 丢 R␲y /2Rx␲兲CZ00共I 丢 R␲y /2兲,
U3 = 共I
20
We calculated the energy band of the capacitively coupled
flux-biased phase qubits by diagonalizing the following 9
⫻ 9 Hamiltonian:22,23
U0 = I 丢 I,
−␲/2 ␲
丢 R y Rx 兲CZ11共I 丢
40
10
-4
0
4 -3
Diff. in peak height (10 )
D. Deutsch-Jozsa algorithm
CNOT
σ = 0.0017
R−y ␲/2兲.
H⯝
共3兲
The sequence is same as that used in Ref. 3 except that R␲y /2
pulse was not applied to qubit B before the tomography. This
makes the final state a superposition state. Also, in order to
shorten the total sequence time, the last R␲y /2 on qubit A was
applied before the Ui part finishes. The real part of the final
density matrices are plotted in Figs. 5共b兲–5共e兲. The experimental probability to obtain the correct answer is summarized in Table I. Because our phase qubit has single-shot
readout, we can obtain the correct answer to a single function
query more than 70% of the time, greater than the 50% probability for a classical query and guess. We stress that no
calibration for the measurement error is applied here.
IV. CONCLUSIONS
In conclusion, we have demonstrated CZ and CNOT gates
in capacitively coupled phase qubits using the higher energy
兩2典 state. Quantum process tomography measures a ␹ matrix
that is in good accord with predictions, which is a definitive
test of proper gate operation for any input state.
冢
0
丢
丢
共A兲
hf 10
共A兲
共A兲
hf 10
+ hf 21
冢
冢
冣
0
共B兲
hf 10
1
0
0
冑2
冣冢
0 −1
−g 1
共B兲
共B兲
hf 10
+ hf 21
0 −1
丢 I2 + I1
0
冣
− 冑2 ,
0
0
0
冑2
0
− 冑2
0
冣
共A1兲
共B兲
where f 共A兲
i,j 共f i,j 兲 is the flux-dependent transition frequency
between ith and jth state of the qubit A 共B兲 and g is the
coupling energy between the qubits. The last term in the
Hamiltonian is based on ␴y␴y-type coupling of the two qubits. To calculate the transition matrix from the ground state
兩g典 to the excited state 兩e典 in the spectroscopy experiment, we
calculated the transition matrix element of 兩具e兩a† + a兩g典兩2 for
具e兩a†+a兩i典具i兩a†+a兩g典
one-photon excitation and 兩兺i Ee−Ei−hf d 兩2 for two-photon
excitation,31 where a共a†兲 is an annihilation 共creation兲 operator for the harmonic oscillator, Ei is the energy gap of the
state 兩i典 from the ground state, and f d is the frequency of the
␮-wave drive, which is set to be Ee / 2h in the calculation.
APPENDIX B: DIFFERENCE BETWEEN ␹p AND ␹e
ACKNOWLEDGMENTS
The authors would like to thank F. K. Wilhelm and A. N.
Korotkov for valuable discussion. They would also like to
thank Y. Nakamura for useful comments on the manuscript.
Semidefinite programming convex optimization was carried
out using the open-source MATLAB packages YALMIP and
SeDuMi. This work was supported by IARPA under ARO,
Award No. W911NF-04-1-0204.
We checked the difference between ␹e 共the experimental
␹ matrix兲 and ␹p 共the physical ␹ matrix兲 by histogramming
the differences in the peak height ⌬ = ␹e − ␹p of each of the
256 matrix elements in the real part.10 We fit it by Gaussian
a exp共−⌬2 / ␴2兲 as shown in Fig. 6. The obtained ␴ are
0.0020 for CP11 and 0.0017 for CNOT gate, which implies
␹e and ␹p are close.
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FIG. 7. 共Color online兲 ␹p of 共a兲
CZ00, 共b兲 CZ01, 共c兲 CZ10, 共d兲 CZ
= CZ11, and 共e兲 CNOT. Process fidelity Fp are 0.68, 0.69, 0.70,
0.70, and 0.56, respectively.
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QUANTUM PROCESS TOMOGRAPHY OF TWO-QUBIT…
FIG. 8. 共Color online兲 Simulated ␹ matrix of 共a兲 CZ00, 共b兲
CZ01, 共c兲 CZ10, 共d兲 CZ, and 共e兲
CNOT. Process fidelity Fp are
0.67, 0.67, 0.66, 0.67, and 0.52,
respectively.
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APPENDIX C: ␹ MATRIX FOR ALL GATES
APPENDIX D: SIMULATED ␹ MATRIX FOR ALL GATES
In Fig. 7, the physical ␹ matrix ␹p is plotted for all the CZ
and CNOT gates.
In Fig. 8, the simulated ␹ matrix is plotted for all CZ and
CNOT gates.
16
*[email protected]
†[email protected]
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