Analytical modeling of parametrically-modulated transmon qubits
Nicolas Didier, Eyob A. Sete, Marcus P. da Silva, and Chad Rigetti
Rigetti Computing, 775 Heinz Avenue, Berkeley, CA 94710
(Dated: June 20, 2017)
Scaling up quantum machines requires developing appropriate models to understand and verify
their complex quantum dynamics. We focus on superconducting quantum processors based on
transmons for which full numerical simulations are already challenging at the level of qubytes. It is
thus highly desirable to develop accurate methods of modeling qubit networks that do not rely solely
on numerical computations. Using systematic perturbation theory to large orders in the transmon
regime, we derive precise analytic expressions of the transmon parameters. We apply our results
to the case of parametrically-modulated transmons to study recently-implemented parametricallyactivated entangling gates.
I.
INTRODUCTION
Scaling up quantum machines is a challenging enterprise that requires accurate modeling of complex quantum dynamics. Precise understanding is crucial to design, manipulate, optimize, and verify the machine. In
the field of superconducting quantum computers, transmons [1, 2] are currently widely used as qubits [3–16] or
quantum devices [17–19]. Transmons are weakly nonlinear oscillators based on the Cooper pair box, a Josephson
junction shunted by a capacitance. The transmon regime
corresponds to a large Josephson energy compared to the
charging energy— it is a compromise between a large anharmonicity and a weak sensitivity to charge noise. The
coherence and gate times of transmons in quantum computing experiments have been steadily improving over
the last several years, and transmons are now one of the
leading candidates to an architecture that can meet the
stringent requirements of fault-tolerant quantum computing [20].
Although analytical expressions for the behaviour of
non-interacting transmons are well understood, the accurate description for the behavior of interacting transmons requires the diagonalization of coupled systems
(i.e., the charge basis description of the transmons with
charge dipole interactions). Numerical diagonalization
of these systems quickly becomes intractable because a
large number of basis states are necessary to obtain high
accuracy even for non-interacting transmons. A more efficient approach is to use analytical expressions of transmon energies and states. Exact diagonalization of the
Cooper-pair box Hamiltonian is achieved with Mathieu
functions [21, 22], but manipulating them can be cumbersome. For example, calculating the Fourier transform of
Mathieu functions, necessary to describe capacitive couplings, leads to rather complex expressions. An alternative is to consider controlled approximations, such as the
approximate diagonalization via standard perturbation
theory, which is widely used in quantum mechanics [23].
For transmons, the natural small parameter is the ratio of the charging energy of the Cooper-pair box to the
Josephson energy of the junction, as this parameter is
typically bellow 2 %.
In this paper, we apply systematic perturbation theory to model interacting transmons at sub-kHz accuracy
with respect to numerical diagonalization. The resulting
analytical expressions are particularly useful to estimate
crosstalk in the dispersive regime. We then use these
analytic expressions to model the parametric control of
transmon qubits to realize two-qubit gates, in a manner
similar to other proposals for parametric gates [17, 24–
31]. Our theory has been already used to successfully
predict and simulate iSWAP and controlled-phase gates
on 2-qubit [15] and 8-qubit devices [16].
We start by presenting the perturbation theory for a
single transmon qubit in Sec. II. We then consider the
case of tunable transmons in Sec. III. We treat the capacitive coupling of transmons in Sec. IV. We use the results
to study flux modulation of coupled tunable transmons in
Sec. V. We finally discuss directions to take into account
dissipation in Sec. VI.
II.
FIXED-FREQUENCY TRANSMON QUBITS
The circuit of a fixed-frequency transmon consists of a
Josephson junction shunted by a capacitance, as depicted
in Fig. 1, and is governed by the Hamiltonian,
ĤF = 4EC N̂ 2 − EJ cos ϕ̂.
F
•
ϕF
(1)
•
ϕT
g
T
φext
EJF
ECF
EJT1
EJT2
ECT
FIG. 1. Circuit of a fixed-frequency transmon F and a tunable transmon T that are capacitively coupled (strength g).
Transmons are characterized by their charging energy EC and
Josephson energy EJ . The transmon regime of the Cooper
pair box corresponds to EC /EJ 1. Tunable transmons are
composed of a SQUID and controlled with pulses on the flux
bias line, φext (t).
∂2
−4EC 2 − EJ cos ϕ ψ(ϕ) = Eψ(ϕ),
∂ϕ
(2)
can be solved exactly in terms of Mathieu functions [1,
21, 22],
EJ
En =EC MA µn , −
,
2EC
1
En
EJ ϕ
ψn (ϕ) = √
MC
,−
,
EC
2EC 2
2π
EJ ϕ
En
2n+1
−i
,−
,
,
MS
EC
2EC 2
(3)
p
i
N̂ = √ (↠− â).
2 ξ
ξ(↠+ â),
(5)
The positive real number ξ is related
point
q to the zero
p
√
√
2
2
fluctuations, hϕ̂ i = 2ξ and hN̂ i = 1/ 2ξ. We
then express Hamiltonian Eq. (1) in terms of â, ↠and
diagonalize the quadratic
ppart; it is characterized by the
plasma frequency, ωh = 8EC EJ , and the dimensionless
parameter,
r
ξ=
2EC
.
EJ
(6)
In what follows, ξ will be the small parameter of the
perturbation theory (typically, ξ < 0.2). The transmon Hamiltonian is then expressed as a function of the
bosonic field, normal ordered and written as a Taylor
series in ξ,
ĤF =
∞
X
ξ u Ĥ (u) ,
u=0
c Copyright 2017 Rigetti & Co., Inc.
108
107
106
105
104
103
102
101
100
10−1
10−2
10−3
ω
η
0
2
4
6
8 10 12 14 16 18 20 22 24
perturbation theory order
FIG. 2. Accuracy of the transmon frequency and anharmonicity as a function of perturbation theory order in ξ compared
to numerical simulation with 30 Fock states. Sub-kHz accuracy of the transmon frequency and anharmonicity is found
at large orders, expressions are reported in Appendix A.
(4)
with MA the Mathieu characteristic value, MC the even
Mathieu function, MS the odd Mathieu function, and
µn = (−1)n+1 n + [n mod 2] the indexes for the eigenenergies. The structure of the solution indicates that transmons are described with the dimensionless parameter
EJ /(2EC ) and the characteristic energy EC . The transmon wavefunction is expressed in Eq. (4) in the phase
representation. Going to the charge representation requires Fourier transforming the Mathieu functions, a calculation that turns out to be rather cumbersome.
The commutation relation between ϕ̂ and N̂ allows us
to express these conjugate variables as the two quadratures of a bosonic field â (characterized by [â, ↠] = 1),
ϕ̂ =
accuracy [Hz]
The two conjugate quantum variables here are the
Cooper pair number operator N̂ and the superconducting phase difference ϕ̂ satisfying the commutation rule
[ϕ̂, N̂ ] = i. The Schrödinger equation for the transmon
in the phase representation,
(7)
with Ĥ (0) = ωh ↠â and, for u ≥ 1,
Ĥ (u) = ωh
u
X
(−1)u
2u−v+1 (u − v)!
v=0
×
v+1
X
w=−(v+1)
â†(v+1+w)
â(v+1−w)
. (8)
(v + 1 + w)! (v + 1 − w)!
The eigenenergies En and eigenstates |ψn i are then
obtained using perturbation theory [23] in ξ,
En =
∞
X
ξ p En(p) ,
|ψn i =
p=0
∞
X
ξ p |ψn(p) i,
p=0
(9)
by solving the Schrödinger equation at each order in ξ.
The unperturbed system is a harmonic oscillator of fre(0)
quency ωh , the energies are En = nωh and the cor(0)
responding eigenstates are the Fock states |ψn i = |ni.
p
The eigenenergies and eigenstates at order ξ , for p ≥ 1,
are obtained by recurrence,
En(p) =
p−1
X
q=0
|ψn(p) i
=
hn|Ĥ (p−q) |ψn(q) i,
X
m6=n
+
p−1
X
q=1
(10)
1
hm|Ĥ (p) |ni
(n − m)ωh
hm|Ĥ
(p−q)
−
|mi.
En(p−q) |ψn(q) i
(11)
To compute the eigenenergies and eigenstates of level n
at order p, Fock states |0i → |n+4pi are required because
terms such as â†(2(q+1)) and â(2(q+1)) are involved. In particular at each new order the Hilbert space is extended
by the action of â†4 and â4 . The eigenstates |ψn i derived
by recurrence are not normalized. The normalized
states
p
|Ψn i are easily obtained via |Ψn i = |ψn i/ hψn |ψn i. The
2
The first five orders are identical to the available asymptotic expansion of Mathieu functions for large arguments [32]. Higher order expansions are easily obtained
through these recursive expressions, and as an example,
we report 25th order expansions in Appendix A.
102
ω
η
101
accuracy [Hz]
diagonalization transformation is represented by the operator Ûeigen that transforms the n + 4p Fock states into
the n eigenstates.
At 5th order, the transmon frequency and anharmonicity read
p
ω ' 8EC EJ
21
19
5319
1
− EC 1 + 2 ξ + 7 ξ 2 + 7 ξ 3 + 15 ξ 4 , (12)
2
2
2
2
9
81 2 3645 3 46899 4
η 'EC 1 + 4 ξ + 7 ξ + 12 ξ + 15 ξ . (13)
2
2
2
2
100
10−1
10−2
10−3
−1.0
−0.5
0.0
φext /(2π)
0.5
1.0
FIG. 3. Accuracy of the tunable transmon frequency and
anharmonicity at 25th order in ξ as a function of parking flux
bias compared to numerical simulation with 30 Fock states.
onalization Ûeigen generates nonadiabatic terms,
†
ĤT (t) = Ûeigen Ûφ Ĥ Ûφ† Ûeigen
III.
TUNABLE TRANSMON QUBITS
˙
˙
†
†
+ iÛeigen Ûφ Ûφ† Ûeigen
+ iÛeigen Ûeigen
The transmon frequency can be tuned with an external
magnetic field by replacing the Josephson junction with
a SQUID, see Fig. 1. The tunable-transmon Hamiltonian
is then that of a split Cooper-pair box,
ĤT = 4EC N̂ 2 − EJ1 cos(ϕ̂ − φext ) − EJ2 cos ϕ̂,
(14)
where EJ1 and EJ2 are the Josephson energies of the
SQUID loop. The Hamiltonian can be recast as an effective single-junction transmon,
ĤT = 4EC N̂ 2 − EJeff cos(ϕ̂ − φeff ),
(15)
with the flux-dependent effective Josephson energy and
offset phase,
q
EJeff = EJ21 + EJ22 + 2EJ1 EJ2 cos φext ,
(16)


sin φext
.
φeff = arctan
(17)
E
cos φext + EJJ2
1
The phase is localized around the offset phase φeff ; it can
removed with the displacement operator
Ûφ = eiφeff N̂ ,
(18)
giving the same form as the fixed-frequency Hamiltonian
Eq. (1), ĤT = 4EC N̂ 2 − EJeff cos ϕ̂. The charge operator is invariant under this unitary transformation — Uφ
will not affect, e.g., the capacitive coupling interaction.
The perturbation theory developed for a fixed-frequency
transmon can be applied to p
the effective Hamiltonian
with the small parameter ξ = 2EC /EJeff .
For time-dependent flux biases, φext (t), the unitary
transformation for phase displacement Ûφ (t) and diagc Copyright 2017 Rigetti & Co., Inc.
=
(19)
2
X
˙
√
ξ(t)
φ̇eff (t)
[λ(t)σ̂y + 2Λ(t)ŝy ] −
υ(t)Ŝy ,
En Π̂n − p
ξ(t)
2 ξ(t)
n=0
(20)
where we note Π̂n = |Ψn ihΨn | the projector on qubit
eigenstate |Ψn i. The nonadiabatic Hamiltonian acts as a
drive between the different transmon levels. The ladder
operators between the three transitions are
σ̂+ = |ψ1 ihψ0 |,
ŝ+ = |ψ2 ihψ1 |,
Ŝ+ = |ψ2 ihψ0 |,
†
σ̂− = σ̂+
,
ŝ− =
Ŝ− =
ŝ†+ ,
†
,
Ŝ+
σ̂y = i(σ̂+ − σ̂− ),
ŝy = i(ŝ+ − ŝ− ),
Ŝy = i(Ŝ+ − Ŝ− ).
(21)
(22)
(23)
The parameters λ, Λ are the weights of the charge number operator in the three-level transmon eigenbasis,
λ
Λ
N̂ = √ σ̂y + √ ŝy .
2 ξ
2ξ
(24)
˙
†
and υ comes from Ûeigen Ûeigen
. At 5th order, these parameters are equal to,
1
11
65
4203
ξ − 8 ξ 2 − 11 ξ 3 − 17 ξ 4 ,
23
2
2
2
1
73 2 79 3 113685 4
Λ = 1 − 2ξ − 9ξ − 9ξ −
ξ ,
2
2
2
219
√
1
11 2 321 3 5609 4
υ = − 2 4 ξ + 8 ξ + 13 ξ + 17 ξ .
2
2
2
2
λ=1−
(25)
(26)
(27)
Higher order expressions are given in Appendix A.
IV.
DISPERSIVELY-COUPLED TRANSMONS
The capacitive coupling of two Cooper-pair boxes, as
depicted in Fig. 1, generates a charge-charge interaction through the coupling capacitance, gC N̂1 N̂2 . To
3
accuracy [Hz]
103
transmon basis is then
102
Ĥm (t) =
n=0
101
+ g(t)[λF σ̂yF +
ω
0
5
η
10
coupling g [MHz]
χ
disp
15
g 2 λ21 λ22
,
ω1 − ω2
(28)
2g 2 λ22,1 Λ21,2
.
ω1 − ω2 ∓ η1,2
(29)
The interaction in the dispersive regime is of the form
Ĥdisp = χ|11ih11| with the dispersive shift,
χ = 2g 2
V.
λ21 Λ22
Λ21 λ22
−
ω1 − ω2 + η2
ω1 − ω2 − η1
.
(30)
TRANSMONS UNDER FLUX MODULATION
We consider two capacitively coupled transmon qubits,
the first at fixed frequency and the second tunable.
The flux bias pulse is modulated; it renders the Hamiltonian time dependent via p
EJeff (t) and φeff (t). We
note
the
parameters
ξ
=
2ECF /EJF and ξT (t) =
F
p
2ECT /EJeff (t) . The coupling in the quadrature basis,
p
g(t) = gC /(4 ξF ξT (t)), is expressed in terms of the capacitive coupling gC . The system Hamiltonian in the
c Copyright 2017 Rigetti & Co., Inc.
√
2ΛT (t)ŝyT ]
We specify the modulation of the flux bias pulse,
treat the capacitive coupling in the transmon eigenbasis, we use√Eq. (24) and define the coupling strength
g = gC /(4 ξ1 ξ2 ). For detunings much larger than the
coupling, |ω1 −ω2 | g, the transverse coupling gives rise
to state-dependent frequency shifts. At lowest order in
the small parameter g/|ω1 − ω2 |, the frequencies and anharmonicities are modified as follows, ω12 → ω1,2 + δω1,2
and η12 → η1,2 + δη1,2 with
δη1,2 = 2δω1,2 ∓
2ΛF ŝyF ][λT (t)σ̂yT +
20
FIG. 4. Accuracy of the capacitively-coupled transmon frequencies, anharmonicities and frequency shift at 25th order
in ξ for four transmon eigenstates as a function of coupling
strength compared to numerical simulation with 30 Fock
states for each transmon. The dashed lines correspond to
the dispersive results.
δω1,2 = ±
√
φ̇eff (t)
ξ˙T (t)
φ̇eff (t)
λT (t)σ̂yT − p
ΛT (t)ŝyT −
υ(t)ŜyT .
− p
ξT (t)
2 ξT (t)
2ξT (t)
(31)
100
10−1
2 h
i
X
EFn Π̂Fn + ETn (t)Π̂Tn
φext (t) = φp + φep cos(ωp t + θp ),
(32)
which oscillates around the parking flux φp at the modulation frequency ωp and amplitude φep . The Hamiltonian
is time dependent via ξT (t), therefore via
cos[φext (t)] = cos φp J0 (φep )
∞ n
X
+2
(−1)n cos φp J2n (φep ) cos[2n(ωp t + θp )]
n=1
o
− sin φp J2n−1 (φep ) cos[(2n − 1)(ωp t + θp )] . (33)
We now consider that the tunable transmon is operated
at the maximum of the spectrum, φp = 0, where flux
noise is suppressed at first order, thereby offering optimal coherence times. For modulation amplitudes smaller
than half a flux quantum, φep < π, the Bessel functions
of orders larger than 4 can be neglected. The timedependent parameters of the Hamiltonian can then be
evaluated as follows,
f [cos φext (t)] ' f¯ + f˜ cos[2(ωp t + θp )],
f¯ = f (0) [J0 (φep )] + J2 (φep )f (2) [J0 (φep )]
(34)
2
+ J22 (φep )J4 (φep )f (3) [J0 (φep )] + 41 J42 (φep )f (4) [J0 (φep )] (35)
f˜ = −2J2 (φep ) f (1) [J0 (φep )] + J4 (φep )f (2) [J0 (φep )]
+ 21 J22 (φep )f (3) [J0 (φep )] + 23 J22 (φep )J4 (φep )f (4) [J0 (φep )] ,
(36)
where we take into account the first four derivatives
with respect to cos φext (t). In Eq. (34), f¯ represents
the static shift of the parameters due to the modulation and f˜ is the Fourier coefficient at 2ωp . This upconversion from the flux modulation at ωp to the parameter modulation at 2ωp comes from the shape of the
spectrum around the parking point. At the maximum,
the slope vanishes and periodic excursions on the curvature doubles the frequency. Notice that the expression
of f [cos φext (t)] at the minimum is found by replacing
{J0 (φep ), J2 (φep ), J4 (φep )} → {−J0 (φep ), −J2 (φep ), −J4 (φep )}
in Eq. (34).
To highlight the parametrically-activated coupling, we
4
go into the interaction picture with the unitary,
Ûint = exp
−i
Z
t
dt0
0
2
X
[EFn Π̂Fn + ETn (t)Π̂Tn ] .
n=0
(37)
To proceed, we note the energy difference ωF,Tij =
EF,Tj − EF,Ti and the time-integral $Tij (t) =
Rt 0
dt ωTij (t0 ). The Fourier expansion Eq. (34) is used
0
to express the shift and the modulation amplitude of
$Tij (t),
eTij cos[2(ωp t + θp )],
ωTij (t) = ω Tij + ω
ω
eT
$Tij (t) = ω Tij t + ij {sin[2(ωp t + θp )] − sin 2θp } .
2ωp
(38)
Expressing ωTij as a function of cos φext (t) is done using the expansion
Eq. (12). We then use the identity
P
eiy sin x = n∈Z Jn (y)einx to expand the Hamiltonian in
the interaction picture,
X (n)
(n)
Ĥm =
g11 ei(2nωp −∆)t eiβ01 |10ih01|
n∈Z
(n)
(n)
(n)
(n)
+g21 ei(2nωp −[∆+ηF ])t eiβ01 |20ih11|
+g12 ei(2nωp −[∆−ηT ])t eiβ12 |11ih02|
(n)
(n)
+g22 ei(2nωp −[∆+ηF −ηT ])t eiβ12
(n)
(n)
−g11 ei(2nωp −Σ)t eiβ01
(n)
|21ih12|
|00ih11|
(n)
(n)
(n)
−g22 ei(2nωp −[Σ−ηF −ηT ])t eiβ12 |11ih22|
(n)
(n)
+Ω01 ei((2n+1)ωp −ωT01 )t ei(β01
(n)
+Ω12 ei((2n+1)ωp −[ωT01 −ηT ])t e
+θp )
1 ⊗ |0ih1|
(n)
i(β12 +θp )
(n)
(n)
−Ω02 ei(2nωp −[2ωT01 −ηT ])t eiβ02
1 ⊗ |1ih2|
1 ⊗ |0ih2| + h.c.
(39)
with the notation |F T i. Each line of the Hamiltonian
Eq. (39) is a coupling between a pair of two-qubit states
that can be brought into resonance with the right modulation frequency ωp . The rate of the corresponding Rabi
oscillations is set by the effective coupling strength, and
hence the modulation amplitude. The capacitive coupling thus give access to a large variety of two-qubit gates:
iSWAP with |10ih01|; controlled-phase with |11ih20| and
|11ih02|; Bell-Rabi with |00ih11|. The flux modulation
then completely controls the activation and rate of the
gates. In Eq. (39), we note the frequency difference ∆
and sum Σ,
∆ = ω T01 − ωF01 ,
Σ = ω T01 + ωF01 .
c Copyright 2017 Rigetti & Co., Inc.
(42)
(43)
(44)
where the time-independent the parameters ḡ and g̃ are
obtained from the couplings
√
g11 (t) = g(t)λF λT (t),
g21 (t) = 2g(t)ΛF λT (t), (45)
√
g22 (t) = 2g(t)ΛF ΛT (t), g12 (t) = 2g(t)λF ΛT (t). (46)
ω
e
−g12 ei(2nωp −[Σ−ηT ])t eiβ12 |01ih12|
(n)
(41)
The phases are βijn = 2ωijp sin 2θp + 2nθp + nπ. The effective drives are written in Appendix B. For modulation
frequencies well below qubit frequencies and close to half
the frequency detuning ∆, a rotating wave approximation
allows us to consider only the first three lines of Eq. (39).
When the parking flux is not exactly at the “sweet
spot”, the slope of the spectrum is not zero and the
modulated Hamiltonian parameters pick up a component
at ωp ,
−g21 ei(2nωp −[Σ−ηF ])t eiβ01 |10ih21|
(n)
The effective couplings are,
ω
eT01
(n)
g11 = ḡ11 Jn
2ωp
ω
eT01
ω
eT01
1
− 2 ge11 Jn−1
+ Jn+1
,
2ωp
2ωp
ω
eT01
(n)
g21 = ḡ21 Jn
2ωp
ω
eT01
ω
eT01
1
− 2 ge21 Jn−1
+ Jn+1
,
2ωp
2ωp
ω
eT12
(n)
g12 = ḡ12 Jn
2ωp
ω
eT12
ω
eT12
1
− 2 ge12 Jn−1
+ Jn+1
,
2ωp
2ωp
ω
eT12
(n)
g22 = ḡ22 Jn
2ωp
ω
eT12
ω
eT12
1
− 2 ge22 Jn−1
+ Jn+1
,
2ωp
2ωp
(40)
ωTij (t) = ω Tij + ω̆Tij cos(ωp t + θp ) + ω
eTij cos[2(ωp t + θp )].
(47)
A resonance condition for the modulation at 2ωp then
coincides with the resonance condition for the second
harmonic of the modulation at ωp . To understand the
effect of this superposition on the interaction, we derive
the Hamiltonian for the flux pulse modulation Eq. (47)
and apply the rotating wave approximation. The effect
of the bichromatic modulation is to renormalize the effective coupling strengths,
X
ω̆T01
ω
eT01
(n)
n0
0
0
g̃11 ' g
(−1) Jn−2n
Jn
,
(48)
ωp
2ωp
n0 ∈Z
√ X
ω̆T01
ω
eT01
(n)
n0
0
0
g̃12 ' 2g
(−1) Jn−2n
Jn
, (49)
ωp
2ωp
n0 ∈Z
√ X
ω̆T12
ω
eT12
(n)
n0
0
0
g̃21 ' 2g
(−1) Jn−2n
Jn
, (50)
ωp
2ωp
0
n ∈Z
5
where we have taken the leading term of the parameters λ, Λ. Missing the “sweet spot” will thus degrade
dephasing time but will not affect gate operation apart
from renormalizing the gate time.
VI.
CONCLUSION AND PERSPECTIVES
We have developed an accurate analytical model for
coupled transmons in the presence of flux pulse modulation. Our approach provides efficient and precise simulations of large-scale transmon-based quantum processors,
which would be otherwise intractable with numerical diagonalization. The tools developed in this work are useful for designing large-scale processors, optimizing design
parameters and predicting the performances of quantum
operations. This model has already been successfully
used in recent experiments on parametrically-activated
two-qubit gates [15, 16].
Moreover, our approach can be straightforwardly ex-
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tended to the calculation of more than three transmon
states to have more accurate expressions of frequency
shifts. This is particularly important for finding the optimal regimes of operation of the controlled-phase and
iSWAP gates.
Developing an accurate model for dissipation in superconducting qubits is a natural extension of this work.
The Keldysh formalism of Green’s functions is a powerful framework particularly well suited for nonlinear open
quantum systems [33–35]. The perturbative expansion
of the transmon parameters can then be calculated in
presence of a dissipative bath, allowing to take into account the relaxation and dephasing rates of the quantum
machine in its optimization.
ACKNOWLEDGMENTS
We acknowledge fruitful daily collaborations with the
experimental teams of Rigetti Computing.
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c Copyright 2017 Rigetti & Co., Inc.
Appendix A: Transmon parameters at 25th order in ξ
ω'
p
8EC EJ
21
19
5319
1
− EC 1 + 2 ξ + 7 ξ 2 + 7 ξ 3 + 15 ξ 4
2
2
2
2
6649 5
+ 15 ξ
2
1180581 6
+
ξ
222
446287 7
ξ
+
220
1489138635 8
+
ξ
231
648381403 9
+
ξ
229
614557854099 10
+
ξ
238
75265839129 11
+
ξ
234
637411859250147 12
ξ
+
246
86690561488017 13
+
ξ
242
405768570324517701 14
+
ξ
253
15191635582891041 15
+
ξ
247
2497063196283456607731 16
+
ξ
263
102281923716042917215 17
ξ
+
257
2292687293949773041433127 18
ξ
+
270
25544408245062216574759 19
+
ξ
262
4971071120163260007203175705 20
ξ
+
278
59956026877695226936825271 21
+
ξ
270
6299936888270974385982624367587 22
+
ξ
285
20465345194746565030172477629 23
+
ξ
275
36984324599399309412347250837528543 24
+
ξ
294
128862667153189778842334459173303 25
+
ξ
.
284
(A1)
7
81
9
3645
46899
η ' EC 1 + 4 ξ + 7 ξ 2 + 12 ξ 3 + 15 ξ 4
2
2
2
2
1329129 5
+
ξ
219
20321361 6
+
ξ
222
2648273373 7
+
ξ
228
45579861135 8
+
ξ
231
1647988255539 9
ξ
+
235
31160327412879 10
+
ξ
238
2457206583272505 11
+
ξ
243
50387904068904927 12
+
ξ
246
2145673984043982897 13
+
ξ
250
47368663010124907041 14
ξ
+
253
17329540083222030375645 15
+
ξ
260
410048712835835979799431 16
ξ
+
263
20066784213453521778111375 17
+
ξ
267
507447585299180759749453827 18
+
ξ
270
53019019946496461235728807475 19
+
ξ
275
1429754157181172012054040903645 20
ξ
+
278
79571741391885949104006842758911 21
ξ
+
282
2283773190022904454409743892590327 22
ξ
+
285
540565733415401595950277192471356985 23
+
ξ
291
16479511149218202447739080120870460083 24
+
ξ
294
1034743270413623494225962156243473940687 25
+
ξ
.
298
(A2)
c Copyright 2017 Rigetti & Co., Inc.
11
65
4203
1
ξ + 8 ξ 2 + 11 ξ 3 + 17 ξ 4
23
2
2
2
40721 5
ξ
220
1784885 6
ξ
225
21465147 7
ξ
228
4455462653 8
ξ
235
61698199851 9
ξ
238
3623317643901 10
ξ
243
56143119646191 11
ξ
246
7321743985484303 12
ξ
252
125280019793719221 13
ξ
255
8984438512815167237 14
ξ
260
168544684286400995331 15
ξ
263
105741913308715347076701 16
ξ
271
2164311753394257835891059 17
ξ
274
184798694135089048676718297 18
ξ
279
4109869091672376619457585371 19
ξ
282
761062061371895548979377743237 20
ξ
288
18317012159331390907042783219855 21
ξ
291
1831630981593132690479908285273395 22
ξ
296
47512263370928552970648689915451821 23
ξ
299
20440707519371829420653298425077482201 24
ξ
2106
569157711742925565406447462105395143103 25
ξ
.
2109
(A3)
λ=1−
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
8
√ 1
73 2 79 3 113685 4
11
1
321
5609
ξ
+
ξ
ξ
ξ
ξ + 8 ξ 2 + 13 ξ 3 + 17 ξ 4
+
+
2
υ
=
−
22
29
29
219
24
2
2
2
747533 5
450555 5
ξ
+
ξ
221
223
175422349 6
10164565 6
+
ξ
ξ
28
2
227
698471247 7
507453429 7
+
ξ
ξ
29
2
232
1520876829389 8
13856203441 8
ξ
+
ξ
39
2
236
13668058962903 9
3280643089875 9
ξ
ξ
+
41
2
243
4122722770459287 10
104433423564937 10
+
ξ
ξ
48
2
247
2534488707574995 11
7105628334651135 11
+
ξ
ξ
46
2
252
26543348405245135937 12
256923396012609391 12
+
ξ
ξ
58
2
256
281548290669062665101 13
39309225873672019119 13
ξ
+
ξ
60
2
262
98933257452818263360213 14
1584176336386469903609 14
ξ
ξ
+
67
2
266
561603848629069641896937 15
134062942734813033556893 15
+
ξ
ξ
68
2
271
3372037991404912212166296765 16
5937992825016447235650113 16
ξ
ξ
+
79
2
275
40819563311626093062783992331 17
4393462009358111483920628355 17
+
ξ
ξ
81
2
283
16314102788878455728540034311379 18
211630177923548593260384339985 18
+
ξ
ξ
88
2
287
52535388424912627194648863334467 19
21195084297362748051328855644603 19
ξ
+
ξ
88
2
292
178610931461508948221684711385383067 20
1101441422698682678884159890620131 20
ξ
ξ
+
98
2
296
2444937960639526361173164055382471707 21
237236307127374537401655462955710741 21
ξ
ξ
+
100
2
2102
1103567409503040799217165335410059740779 22
13218681516317907311568006522672236075 22
ξ
ξ
+
107
2
2106
8017554417550804194373089101907638666069 23
1522482900088767896105176250210633085315 23
ξ
+
ξ
108
2
2111
30711842188423912661533983529887505235301321 24
90520992079359034852853176891693012642775 24
ξ
+
ξ
118
2
2115
473069922042437374183190305740304564254754227 25
44409876028541673056803493111783651485068951 25
ξ
.
+
ξ
.
2120
2122
(A4)
(A5)
Λ=1−
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
c Copyright 2017 Rigetti & Co., Inc.
9
Appendix B: Effective drives
The expression of the effectives drives in Eq. (39) is
ω
eT01
ω
eT01
(n)
Ω01 = (ν 01 − 12 νe01 ) Jn
− Jn+1
2ωp
2ωp
ω
eT01
ω
eT01
+ 21 νe01 Jn−1
− Jn+2
,
(B1)
2ωp
2ωp
ω
eT12
ω
eT12
(n)
Ω12 = (ν 12 − 12 νe12 ) Jn
− Jn+1
2ωp
2ωp
ω
eT12
ω
eT12
+ 21 νe12 Jn−1
− Jn+2
,
(B2)
2ωp
2ωp
ω
eT02
ω
eT02
(n)
Ω02 = ν 02 Jn−1
− Jn+1
,
(B3)
2ωp
2ωp
with
#
EJT1
+ cos φext (t) ,
E J T2
(B4)
"
#
ΛT (t) EJT1 EJT2 EJT1
e
ν12 (t) = ωp φp p
+ cos φext (t) ,
2 2ξT (t) EJ2eff (t) EJT2
(B5)
EJT1 EJT2
ν02 (t) = ωp φep J1 (φep )
(t)υ(t).
(B6)
4EJ2eff
λT (t) EJT1 EJT2
ν01 (t) = ωp φep p
4 ξT (t) EJ2eff (t)
c Copyright 2017 Rigetti & Co., Inc.
"
10