Quantum process tomography of two-qubit controlled-Z and controlled-NOT gates using superconducting phase qubits

Abstract

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 |2⟩ 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.

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arXiv:1006.5084v1 [cond-mat.mes-hall] 25 Jun 2010
Quantum process tomography of two-qubit controlled-Z and controlled-NOT gates
using superconducting phase qubits
T. Yamamoto,1,2M. Neeley,1E. Lucero,1R. C. Bialczak,1J. Kelly,1M. Lenander,1Matteo Mariantoni,1A. D.
O’Connell,1D. Sank,1H. Wang,1M. Weides,1J. Wenner,1Y. Yin,1A. N. Cleland,1, ∗and John M. Martinis1, †
1Department of Physics, University of California, Santa Barbara, California 93106, USA
2Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan
(Dated: June 29, 2010)
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 |2? 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.
Quantum computation and quantum communication
rely on excellent control of the underlying quantum sys-
tem [1]. Reasonable control has been achieved with a va-
riety 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
qubits [3] and the entanglement of three qubits [4, 5]. A
key element in these experiments is a two-qubit entan-
gling gate, such as the
Z (CZ) gates [3, 5]. Because the CZ gate is simple to
implement, has high fidelity, and can readily generate
controlled-NOT (CNOT) logic [6], it likely will be an im-
portant component in more complex algorithms such as
quantum error correction. At present, however, the CZ
gate functionality has only been directly tested for a sub-
set of the possible input states.
√iSWAP [4] and the controlled-
In this Letter, we demonstrate the operation of a CZ
gate in a superconducting phase qubit, and fully charac-
terize this gate as well as a CNOT gate using quantum
process tomography (QPT). 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 correspond-
ing 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
gate with the phase qubit [15].
√iSWAP
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 su-
perconducting loop interrupted by a capacitively-shunted
Josephson junction. When biased close to the critical
current, the junction and its parallel loop inductance pro-
duce a non-linear potential as a function of the phase
difference across the junction. Combined with the ki-
netic energy originating from the shunting capacitance,
unequally spaced quantized energy levels appear in the
cubic potential. The two lowest levels are used for the
qubit states |0? and |1?, with a transition frequency fA
(fB
10) that can be controlled by an external magnetic flux
ΦA
ex) applied to the loop. The third energy level |2?
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 (Z-pulse)
or a Gaussian-shaped microwave pulse (X-, Y-pulse) to
its bias coil. For an X- or Y-pulse, we simultaneously
apply the derivative of the pulse to the quadrature (90◦
phase shifted) drive to reduce both unwanted excitation
of the |2? state and phase error due to AC Stark ef-
fect [17]; the derivative scaling factor is determined from
the nonlinearity of each qubit [18]. This procedure en-
ables us to use a Gaussian pulse with a full-width at half
maximum (FWHM) of 10 ns, while maintaining accurate
qubit control [19] in spite of a rather weak qubit nonlin-
earity (∼ 100 MHz). Each qubit state is read out indi-
vidually in a single-shot manner by injecting a large mag-
nitude Z-pulse and then measuring the qubit flux with a
superconducting quantum interference device (SQUID).
The device was fabricated using a photolithographic
process with Al films, AlOxtunnel junctions, and a-Si:H
10
ex(ΦB
SQUID A
Bias
coil A
Cc
CA
CB
LA
LB
I0A
I0B
?exA
?exB
SQUID B
Bias
coil B
qubit Aqubit B
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 parameters of the circuit are IA
CA = CB = 1 pF, LA = LB = 720 pH, and Cc = 2 fF.
0 = IB
0 = 2 µA,

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2
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.
In the present experiment, the two qubits were biased
so that fA
pulse was applied. The relaxation times (T1) were mea-
sured to be 510 ns and 500 ns for qubit A and B, respec-
tively. The dephasing times determined from a Ramsey
interference experiment (TRamsey
2
sian decay proportional to exp[−(t/TRamsey
1/f flux noise [20], were 200 and 230 ns, respectively.
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 µs long Z-pulse (hor-
izontal axis) and the frequency of a microwave X-pulse
(vertical axis) of the same length. Both pulses were ap-
plied simultaneously to qubit B, followed by the Z-pulse
for the readout. In this way, we can probe the change
in the resonance frequency fB
∆i. In addition to the main resonance line corresponding
to fB
10, somewhat broadened because of the large ampli-
tude of the microwave pulse (∆ΦB
line is observed on the low-frequency side of the main
resonance; this corresponds to the two-photon excitation
from the |0? to the |2? state [21]. The vertical distance
between the main and two-photon lines is 1/2 the qubit
nonlinearity ∆f = f10−f21, yielding ∆f = 114 MHz for
qubit A (data not shown) and 87 MHz for qubit B.
We observe an avoided level crossing in the main
resonance at ∆i ≃ 0.13 when the two qubit frequen-
cies overlap fB
10.Here, the degeneracy of the
|AB? = |10? and |01? states produces a splitting with size
14.2±0.2 MHz, determined from a fit to the data, con-
sistent 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√2/2 times as large as
the main resonance, as expected from a |11? and |02? in-
teraction. The slope of the resonance between the two
crossings is 1/2 that of fB
10, as expected for the |11? state.
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 Hamiltonian [22, 23] the energies for the
coupled eigenstates. The energy bands, normalized to
fA
01, are plotted versus the flux bias for qubit B in the up-
per 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 [24]. The (red) thick
lines correspond to the |01? and |10? states, whereas the
(blue) thin lines represent half of the excitation energy
of the |11? and |02? states. The overall structure agrees
well with the experimental data.
As proposed theoretically by Strauch et al. [25], the
10= 7.16 GHz and fB
10= 7.36 GHz when no Z-
), which showed Gaus-
2
)2] due to
10as a function of detuning
ex∼ 10 µΦ0), a sharper
10= fA
7.45
7.35
7.25
7.15
Frequency (GHz)
0.150.10 0.05
∆i (a.u.)
0.00-0.05
0.4
0.2
0.0
Probability
i0
t
1.02
1.00
Frequency
0.900
B/Φ0
0.895
Φex
|01›
|02›
|11›
FIG. 2.
B. The escape probability (gray scale) is plotted versus mi-
crowave frequency and the Z-pulse amplitude ∆i. The single
photon |0? → |1? and two-photon |0? → |2? transitions are
visible, along with two avoided-level crossings.
inset shows the calculated states and eigenenergies, with the
thick (thin) lines representing single (two) photon excitations.
The lower inset illustrates the Z-pulse amplitude i0 for the
CZ = −(SWAP)2operation.
(Color online) High-power spectroscopy for qubit
The upper
avoided crossing due to the degeneracy of the |11? and
|02? states can be used to construct a CZ gate, whose ac-
tion produces no change in state except for |11? → −|11?.
By applying a non-adiabatic Z-pulse, the |11? and |02?
states become degenerate (see lower inset of Fig. 2).
Initially in the |11? state, the system evolves as an
iSWAP interaction, giving |Ψ(t)? = cos(γ∆t/¯ h)|11? +
isin(γ∆t/¯ h)|02?, where 2γ is the splitting energy of the
avoided crossing and ∆t the duration of the Z-pulse. Af-
ter twice the iSWAP time ∆t = h/2γ, the system re-
turns to the initial state |11?, but with a minus sign. If
the system starts in |00?, |01?, or |10?, the state does
not change since it is off-resonance with both avoided
level crossings. A similar scheme using an adiabatic Z
pulse has been used to successfully demonstrate a quan-
tum algorithm [3], and the same (non-adiabatic) 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 non-adiabatic Z-pulse, we directly mea-
sured the coherent oscillation between the |11? and |02?
states. This (iSWAP)2operation sequence is shown in
Fig. 3(a): We first prepare the |11? state with a π pulse
to both qubits, and then apply a Z-pulse with ampli-
tude ∆i and length ∆t to qubit B. Here only qubit B
is probed and we adjust the measurement pulse ampli-

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3
0.08
0.06
0.04
∆i (a.u.)
200150100
∆t (ns)
500
0.0
0.4
0.8
(b)
t0
i0
P|2>
1.0
0.5
0.0
Probability
-0.04 0.00
icmp (a.u.)
π pulse off
π pulse on
B
(d)
qubit A
qubit B
π
meas.
π
∆i
∆t
(a)
qubit A
qubit B
-SWAP2
π/2
π/2
meas.
π
i0
tcmp
t0
icmp
icmp
A
B
(c)
FIG. 3.
state evolution between |11? and |02? states. (b) Plot of |2?
state probability of qubit B versus Z-pulse time ∆t and Z-
pulse amplitude ∆i. The dashed lines correspond to the op-
timal setting for the CZ gate. (c) Operation sequence for (d),
demonstration of the CZ gate. (d) Plot of |1? state probabil-
ity of qubit B as a function of iB
The (red) solid and (blue) dashed curves are for qubit A ini-
tialized to the |0? and |1? states, respectively. The vertical
dot-dashed line indicates the value of iB
(Color online) (a) Operation sequence for (b), the
cmp(iA
cmpis fixed as 3 ×10−4.).
cmpfor the CZ gate.
tude so that the qubit is detected only when in the |2?
(or higher) state [26]. In Fig. 3(b), we plot the tunneling
probability P|2?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 i0de-
termined in Fig. 2(a). The oscillation period t0= 51.8ns
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 discern-
able increase in P|2?is observed (< 1%) at this operation
point (∆t,∆i) = (t0,i0), confirming that we return to
the |11? state after the CZ operation.
Because the qubits themselves also accumulate phase φ
during the CZ pulse, the general unitary evolution from
the gate is given by
U =




1
0 eiφA
0
0
00
0
0
0
00
0
eiφB
0−ei(φA+φB)



.(1)
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 compen-
sation pulses are shown in Fig. 3(c), which consist of a
fixed 10 ns pulse of variable amplitude icmpafter the CZ
pulse. In Fig. 3(d), we plot the tunneling probability of
qubit B as a function of iB
qubit B is measured through a Ramsey fringe experiment.
The (red) solid and (blue) dashed curves correspond to
qubit A being in the |0? or |1? state. They both show a
sinusoidal dependence on iB
cmp, but are shifted by π from
cmpfor fixed iA
cmp. The phase of
(a) CZ
(b) CNOT
Exp.
Exp.
Sim.
Sim.
Re(χ)
Re(χ)
Re(χ)
Re(χ)
FIG. 4.
(a) Left panel: The real part of the experimentally obtained
χ matrix (χp) for the CZ gate, with Fp = 0.70. Right panel:
The real part of the simulated χ matrix for the CZ gate, with
Fp = 0.67. (b) Left panel: The real part of χp for the CNOT
gate, with Fp = 0.56.Right panel: The real part of the
simulated χ matrix for the CNOT gate, with Fp = 0.52. The
open boxes in the figure represent the ideal χ matrix.
(Color online) χ matrices of CZ and CNOT gates.
each other, confirming the correct operation of the CZ
gate. A similar experiment was done for qubit A (data
not shown).
The phases for the CZ gate are set by taking the values
of icmpthat give maximum probability when the control
qubit is in the |0? state, as indicated by the vertical dash-
dotted line in Fig. 3(d). Controlled-NOT (CNOT) gates
are constructed by combining the CZ gate with single
qubit rotations UCNOT = (I ⊗ Rπ/2
where Rθ
yrepresents the rotation of a single qubit state
by an angle θ about the y axis, and I is the identity
operator.
We evaluate the performance of these gates with QPT.
For QPT, we prepare 16 input states in total, chosen from
the set {|0?,|1?,|0?+|1?,|0?+i|1?} for each qubit. After
preparing these input states, we determine the density
matrix of the output state with quantum state tomog-
raphy [16], in which we measure each qubit along the
six directions ±x, ±y and ±z of the Bloch sphere [27].
For each combination of QPT and QST pulses, we repeat
the sequence 1800 times to obtain the joint qubit proba-
bilities PAB= P00,P10,P01and P11. After correcting for
small measurement errors [15], we reconstruct the 16×16
experimental χematrix from the resulting 16 density ma-
trices [28]. With experimental noise, the χematrix found
in this way is not necessarily physical, i.e. completely
positive and trace-preserving. We thus use convex op-
y
) CZ (I ⊗ R−π/2
y
),

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4
TABLE I. Summary of performance for Deutsch-Jozsa algo-
rithm. Deutsch-Jozsa functions are defined as f0(x) = 0,
f1(x) = 1, f2(x) = x, and f3(x) = 1 − x.
Element Deutsch-Jozsa function
Constant
f0
f1
00
measured 0.29 0.28
ideal1
measured 0.71 0.72
Balanced
f2
1
0.76
0
0.24
f3
1
0.74
0
0.26
?00|ρ|00? + ?01|ρ|01?ideal
?10|ρ|10? + ?11|ρ|11?1
timization to obtain the physical matrix χp that best
approximates χe, as used in Ref. [15].
We plot the real part of χpfor the CZ and CNOT gates
in the left panel of Figs. 4(a) and (b). The results are
displayed in the basis formed by the Kroneckerproduct of
Pauli operators {I,σx,−iσy,σz} for each qubit [6]. The
open boxes represent the ideal χ matrix. The imaginary
parts of χp have very small magnitude (< 0.04 for CZ
and < 0.03 for CNOT), and are shown in Ref. [24]. 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(χiχp), where χi represents an ideal χ 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, respectively [24].
To understand the loss of process fidelity, we performed
numerical simulations by solving the master equation
using the experimental parameters [24]. The entire se-
quence, including the QPT and QST pulses, was simu-
lated to construct χsim. The real part of χsimis shown
in Fig. 4, and reproduces reasonably well the reduction
of the expected elements and the appearance of small
unwanted elements. These imperfections are removed as
we increase the single-qubit coherence time in the sim-
ulation, which suggests that loss of Fpin our system is
mostly dominated by single qubit decoherence. We note
that it is possible to obtain more information on the deco-
herence mechanisms by analyzing the magnitude of par-
ticular elements in the χ matrix [29].
By using these conditional gates, we can perform the
Deutsch-Jozsa algorithm [6] using the pulse sequence de-
scribed in Ref. [3]. The experimental probability to ob-
tain 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 calibra-
tion for the measurement error is applied here. The full
density matrix of the final state is given in Ref. [24].
In conclusion, we have demonstrated CZ and CNOT
gates in capacitively-coupled phase qubits using the
higher-energy |2? state. Quantum process tomography
measures a χ matrix that is in good accord with predic-
tions, which is a definitive test of proper gate operation
for any input state.
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. M. M. acknowledges support from an El-
ings fellowship. Semidefinite programming convex opti-
mization was carried out using the open-source MAT-
LAB packages YALMIP and SeDuMi. This work was
supported by IARPA under ARO award W911NF-04-1-
0204.
∗anc@physics.ucsb.edu
†martinis@physics.ucsb.edu
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arXiv:1006.5084v1 [cond-mat.mes-hall] 25 Jun 2010
Quantum process tomography of two-qubit controlled-Z and controlled-NOT gates
using superconducting phase qubits: Supplementary information
T. Yamamoto,1,2M. Neeley,1E. Lucero,1R. C. Bialczak,1J. Kelly,1M. Lenander,1Matteo Mariantoni,1A. D.
O’Connell,1D. Sank,1H. Wang,1M. Weides,1J. Wenner,1Y. Yin,1A. N. Cleland,1and John M. Martinis1
1Department of Physics, University of California, Santa Barbara, California 93106, USA
2Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan
(Dated: June 29, 2010)
I. CALCULATION OF THE ENERGY BANDS AND TRANSITION MATRIX ELEMENTS
We calculated the energy band of the capacitively-coupled flux-biased phase qubits by diagonalizing the following
9 × 9 Hamiltonian [1, 2],
H ≃


0
hf(A)
10
hf(A)
10+ hf(A)
21

⊗I2+I1⊗


0
hf(B)
10
hf(B)
10+ hf(B)
21

−g


0 −1
1
0
0
0−√2
0
√2

⊗


0 −1
1
0
0
0 −√2
0
√2

,
(S1)
where f(A)
coupling energy between the qubits. The last term in the Hamiltonian is based on σyσy-type coupling of the two qubits.
To calculate the transition matrix from the ground state |g? to the excited state |e? in the spectroscopy experiment, we
calculated the transition matrix element of |?e|a†+a|g?|2for one photon excitation and
for two-photon excitation [3], where a (a†) is an annihilation (creation) operator for the harmonic oscillator, Ei is
the energy gap of the state |i? from the ground state, and fdis the frequency of the µ-wave drive, which is set to be
Ee/2h in the calculation.
i,j(f(B)
i,j) is the flux-dependent transition frequency between ith and jth state of the qubit A (B), and g is the
???
?
i
?e|a†+ a|i??i|a†+ a|g?
Ee− Ei− hfd
???
2
II.
χ MATRIX FOR ALL GATES
In Fig. S1, the physical χ matrix χpis plotted for all the CZ and CNOT gates.
III.THE DIFFERENCE BETWEEN χp AND χe
We checked the difference between χe(the experimental χ matirix) and χp(the physical χ matirix) by histogram-
ming the differences in the peak height ∆ = χe− χpof each of the 256 matrix elements in the real part [4]. We fit
it by Gaussian aexp(−∆2/σ2) as shown in Fig. S2. The obtained σ are 0.0020 for CP11and 0.0017 for CNOT gate,
which implies χeand χpare close.
IV.SIMULATION OF QPT
To simulate QPT, we solved the standard master equation, ˙ ρ = −(i/¯ h)[H,ρ]+L[ρ], where H is a 9 by 9 Hamiltonian
for capacitively coupled phase qubits under rotating wave approximation, and L[ρ] =
1
2ρLi †
A, respectively [5]. 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 T2that depends on the length
of the control sequence for a particular experiment tseq. In particular, we used T2= TRamsey
in order for both the Gaussian decay and exponential decay to give the same decay factor at tseq. The actual tseqis
101.8 ns in QPT for CZ gates and 141.8 ns for CNOT gate. In Fig. S3, we show the simulated χ matrix of all CZ and
CNOT gates.
?
i=A,B
?
j=1,2
Li
jρLi †
j−1
2Li †
jLi
jρ−
jLi
j. Here, for example, LA
1= aA/?TA
1and LA
2= a†
AaA
?2/TA
2describe the relaxation and dephasing for qubit
2
2/tseqin the simulation

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2
(a) CZ00
Real
(b) CZ01
(c) CZ10
(d) CZ11
(e) CNOT
Imaginary
Im(χ)
Re(χ)
Im(χ)
Re(χ)
Im(χ)
Re(χ)
Im(χ)
Re(χ)
Im(χ)
Re(χ)
Real
Imaginary
Real
Imaginary
Real
Imaginary
Real
Imaginary
FIG. S1: χp of (a) CZ00 (b) CZ01 (c) CZ10 (d) CZ = CZ11 and (e) CNOT. Process fidelity Fp are 0.68, 0.69, 0.70, 0.70 and
0.56, respectively.

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3
60
40
20
0
# of counts
-10 -505
Diff. in peak height (10-3)
CNOT
σ = 0.0017
(b)
60
50
40
30
20
10
0
# of counts
-404
Diff. in peak height (10-3)
CZ11
σ = 0.0020
(a)
FIG. S2: Histogram of the differences in the peak height of each of the 256 matrix elements in the real part of χ matrix. Data
is for (a) CZ and (b) CNOT. The solid curves are a Gaussian fit to the data.
V.EXPERIMENT ON DEUTSCH-JOZSA ALGORITHM
Figure S4(a) 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
U0= I ⊗ I ,
U1= I ⊗ Rπ
U2= (I ⊗ Rπ/2
U3= (I ⊗ R−π/2
x,
y
Rπ
x)CZ00(I ⊗ Rπ/2
Rπ
y
) ,
y
x)CZ11(I ⊗ R−π/2
y
) .
(S2)
The sequence is same as that used in Ref. [6] except that Rπ/2
This makes the final state a superposition state. Also, in order to shorten the total sequence time, the last Rπ/2
qubit A was applied before the Uipart finishes. The real part of the final density matrices are plotted in Figs S4(b)-(e).
No calibration for the measurement error is applied here.
y
pulse was not applied to qubit B before the tomography.
y
on
[1] M. Steffen, J. M. Martinis, and I. L. Chuang, Phys. Rev. B, 68, 224518 (2003).
[2] A. G. Kofman, Q. Zhang, J. M. Martinis, and A. N. Korotkov, Phys. Rev. B, 75, 014524 (2007).
[3] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, “Atom-photon interaction,” (Wiley, New York, 1992) Chap. IIIC.
[4] J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, Phys. Rev. Lett.,
93, 080502 (2004).
[5] D. F. Walls and G. J. Milburn, Phys. Rev. A, 31, 2403 (1985).
[6] L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Shuster, J. Majer, A. Blais, L. Frunzio, S. M.
Girvin, and R. J. Schoelkopf, Nature, 460, 240 (2009).

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4
(a) CZ00
Real
(b) CZ01
(c) CZ10
(d) CZ11
(e) CNOT
Imaginary
Im(χ)
Re(χ)
Im(χ)
Re(χ)
Im(χ)
Re(χ)
Im(χ)
Re(χ)
Im(χ)
Re(χ)
RealImaginary
RealImaginary
RealImaginary
RealImaginary
FIG. S3:
0.67 and 0.52, respectively.
Simulated χ matrix of (a) CZ00 (b) CZ01 (c) CZ10 (d) CZ and (e) CNOT. Process fidelity Fp are 0.67, 0.67, 0.66,

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5
(b) f0(x) = 0(c) f1(x) = 1
(d) f2(x) = x
0.5
(e) f3(x) = 1-x
0.5
0.5
0.0
-0.5
0.5
0.0
-0.5
0.0
-0.5
0.0
-0.5
qubit A
qubit B
tomo.
tomo.
(a)
FIG. S4: (a) Pulse sequence for the Deutsch-Jozsa algorithm. (b)-(d) Real part of the density matrix of the final state for four
Deutsch-Jozsa functions. Open boxes represent the ideal density matrix.