Excitation dynamics in inductively coupled fluxonium circuits

Excitation dynamics in inductively coupled fluxonium circuits

 A. Barış Özgüler,1, ∗ Vladimir E. Manucharyan,2 and Maxim G. Vavilov3
 1
 Fermi National Accelerator Laboratory, Batavia, Illinois, 60510
 2
 Department of Physics, Joint Quantum Institute,
 and Center for Nanophysics and Advanced Materials,
 University of Maryland, College Park, MD 20742
 3
 Department of Physics, University of Wisconsin–Madison, Madison, WI 53706
 (Dated: April 7, 2021)
 We propose a near-term quantum simulator based on the fluxonium qubits inductively coupled to
 form a chain. This system provides long coherence time, large anharmonicity, and strong coupling,
 making it suitable to study Ising spin models. At the half-flux quantum sweet spot, the system is
 described by the transverse field Ising model (TFIM). We evaluate the propagation of qubit excita-
 tions through the system. As disorder increases, the excitations become localized. A single qubit
arXiv:2104.03300v1 [quant-ph] 7 Apr 2021

 measurement using the circuit QED methods is sufficient to identify localization transition without
 introducing tunable couplers. We argue that inductively coupled fluxoniums provide opportunities
 to study localization and many-body effects in highly coherent quantum systems.

 Introduction – Classical simulators hit their limitations been used to encode optimization problems and is a com-
 of analyzing large systems with many degrees of freedom mon tool for quantum computing [25, 26]. Further im-
 as the Hilbert space grows exponentially with the sys- portant insights into condensed matter applications of
 tem size [1]. Quantum simulators have potential to push the fluxonium chain come from the mapping of the sys-
 the limits [2, 3]. Digital quantum simulators (DQS) use tem onto a chain of fermions using the standard Jordan-
 gates to approximate the unitary evolution, whereas ana- Wigner transformation [24]. Mapping of the Kitaev chain
 log quantum simulators (AQS) mimic the time evolution onto the TFIM may help to study topologically protected
 given by the Hamiltonian of another system by its con- quantum states or Majorana bound states [27]. Even
 trollable and tunable components [4]. AQS can currently though the Ising chain does not have topological protec-
 simulate a small class of Hamiltonian models, but they tion, the possibility to explore Majoranas with the chain
 are not as prone to the Trotterization and gate errors as of fluxonium qubits may shed light on the feasibility of
 DQS, so AQS are practical for the NISQ era [5]. the Majorana-based quantum computing [28].
 Quantum simulators have been performed in many In the chain formed by fluxonium circuits, imperfec-
 platforms such as atomic spins [6], vacancy centers [7], tions of the fabrication and flux-tuning will produce dis-
 Rydberg atoms [8], trapped ions [9], ultracold atoms order. For example, the fabrication of the small Joseph-
 [10], optical lattices [11] and superconducting qubits [12]. son junction of the fluxonium results in a variation of
 Transmons have been one of the most successful and the Josephson energy and, consequently, in the qubit en-
 widely used superconducting qubits due to the ease of ergy splitting. We evaluate conditions for the Joseph-
 their engineering and decent coherence properties [13– son energy fluctuations to be weak enough for the chain
 18]. The effect of charge noise is reduced in transmons excitations to remain delocalized. Then, we investigate
 due to capacitative shunting, but transmons are far from how random flux detuning from the sweet spots brings
 perfect two-level systems due to their small anharmonic- the system to the localized regime. We demonstrate that
 ity. An alternative qubit that is still protected from the random magnetic fluxes through different fluxoniums can
 low-frequency charge noise but has no sacrifice in an- realize quenched disorder, allowing one to experimentally
 harmonicity is the fluxonium qubit [19–22]. In addition study statistical properties of many disorder realizations
 to its long coherence, the fluxonium can be designed to using a single device.
 have multiple strong connections with its neighbors via Fluxonium chain as transverse field Ising model – The
 galvanic coupling [23]. fluxonium qubit consists of a phase-slip Josephson junc-
 In this paper, we demonstrate that a fluxonium chain tion shunted by a large inductor, commonly formed by a
 (Fig. 1(a)) is an efficient tool for simulations of clean and long Josephson junction array with the total inductance
 disordered transverse field Ising model (TFIM) [24]. We L. The Hamiltonian for the fluxonium has the form [19]:
 study the dynamics of excitation propagation through a
 one-dimensional chain of inductively coupled fluxonium EL 2
 qubits. When biased at their half-flux quantum sweet Hl = 4 EC n2l +V (θl ), V (θl ) = θ −EJ,l cos(θl −φl ),
 2 l
 spot, the chain is equivalent to the TFIM. TFIM has (1)
 where nl and θl are the Cooper pair number and phase
 operators, respectively. These operators satisfy the fol-
 lowing commutation relation [θl , nl ] = i. Hamiltonian (1)
 ∗ The initial part of the work was performed at the University of is characterized by three energies: the charging energy
 Wisconsin–Madison. EC = e2 /2C, the Josephson energy of the phase-slip

2

 (a) Hamiltonian of the system becomes
 =1 =3 = −2
 L−1 L−2
 1X X
 ! " # $%# Hchain = − εl σ̂lz + J θ̂l θ̂l+1 , (3)
 2
 l=0 l=1
 =0 =2
 … = −1

 where εl = εl (δφl ) is the flux-dependent level spacing
 between the ground and first excited qubit states, σ̂lα is
 (b) l=0 the α = x, y, z Pauli matrix, and the phase operators are
 1.00 l=4
 represented by the matrices:
 Excitation, pl(t)

 l=8
 0.75 l=12
 l=16
  
 θgg (φl ) θge (φl )
 0.50 θ̂l = . (4)
 θeg (φl ) θee (φl )
 0.25 For fluxonium at the half-flux quantum sweet spot, we
 0.00 have θ̂l (φl = π) = aσ̂z , where a ≈ 2.36 for the cho-
 0 10 20 30 sen fluxonium parameters. Below, for the interaction
 Time t, ns strength between fluxoniums, we take Jl = J = 20MHz.
 We consider the chain dynamics when all fluxoniums
 but one are initialized in their ground states, and the
FIG. 1. (a) Open chain of L fluxonium qubits with near- only fluxonium l = 0 is in its first excited state. This
est neighbor inductive coupling. (b) Time dependence of the
 excitation can move to its neighbor in time ∝ ~/J. This
excitation probability for qubits l = 0, 4, 8, 12, 16 in the clean
system of L = 17 fluxoniums. Initially, all qubits were in
 single excitation of the chain will move to other fluxoni-
their ground states, but l = 0 was excited. The parameters ums until it reaches the opposite end of the chain, then
of fluxoniums and the coupling strength are provided in the gets reflected and moves back. Solving the Schrödinger
text. equation for the state of chain of fluxoniums, we calculate
 the excitation probability of fluxonium l as:

junction EJ , and the inductive energy EL = Φ20 /(4π 2 L), pl (t) = hψ(t)| P̂l |ψ(t)i , (5)
where Φ0 = h/2e is the flux quantum, h = 2π~ is the
Planck constant. In the presence of the external mag- where P̂l = |el i hel | is the projection operator to the ex-
netic flux Φl , the energy of the junction is offset by cited state of qubit l.
φl = 2πΦl /Φ0 . We present the evolution for a chain of L = 17 identi-
 We assume that all fluxoniums are flux biased near cal fluxoniums in Fig. 1(b). We estimate that the time
the half-flux quantum sweet spots, φl = π, where the of excitation propagation through the chain is given by
fluxonium exhibits its longest coherence times [20]. We (M − 1)~/J = 12 ns. We also note that the maxima of
consider a high-frequency fluxonium with EC /h = 1.45 pl (t) are lower for the qubits with a higher index l as the
GHz, EL /h = 4.0 GHz, and EJ /h = 9.0 GHz. The dispersion of the propagating excitation waves leads to
energy splitting for 0-1 transition ∆E0−1 /h ' 2.0 GHz the broadening of pl (t) and lowering its maxima. The
is significantly lower than the splitting between states 1 speed of excitation propagation is consistent with that of
and 2, ∆E1−2 /h ' 10.2 GHz. For further discussion, we the TFIM [24]. The maximal probability increases closer
restrict our analysis of the fluxonium dynamics by taking to the ends of the chain as the reflection causes refocus-
into account only its two lowest energy states. ing of the excitation wave packet there, compare l = 0
 The interaction between fluxoniums is realized via a and l = 16 with other intermediate qubit locations.
few shared junctions of superinductor arrays. The energy Uncorrelated disorder – The disorder in the fluxonium
of interaction is determined by the common phase drop chain could originate from the fluctuations of EJ , mu-
along shared junctions and is proportional to the product tual inductances Ml , and magnetic flux fluctuations φl
of the phase operators for the pair of fluxoniums [23]: through the fluxonium loop. The first two sources of
 the disorder are not easily adjustable for a specific de-
 L−2  2 vice and are expected to be a consequence of fabrication
 X ~ Ml
 Hint = Jl θl θl+1 , Jl = . (2) imperfections. The magnetic field can be used as a syn-
 2e Ll Ll+1 thetic disorder to study localization transition using the
 l=0
 same device by randomly changing fluxes through the
Here, Ll is the effective inductance of fluxonium l and Ml fluxoniums.
is the mutual inductance of neighboring fluxoniums l and We assume that the interaction between qubits is uni-
l + 1. Depending on the fraction of the shared Josephson form throughout the chain, Jl = J, which is determined
junctions, the interaction between fluxoniums can reach by the number of shared Josephson junctions of the su-
relatively large values, J ' EL . perinductor and is not expected to exhibit large fluctu-
 In the eigenstate basis of individual fluxoniums, the ations between fluxoniums of the chain. On the other

3

 Random EJ
 (a) 0 (c) 30
 EJ = 0.1 GHz (c) EJ = 0.3 GHz Random EJ
 (a) 1 Random flux
 3.0

 Mean of ln(Pl)
 20 l=2 2 0.05GHz 25
 l=4 2.5 0.1 GHz
 15 l=6 3 0.2 GHz
 Distribution

 Distribution
 l=8 2.0 0.3 GHz 20
 4
 l = 10 1.5 0.5 GHz
 10 50 1

 Loc. length,
 2 3 4 5 6 7 8 9 10
 1.0 Qubit index, l 15
 5 Random flux
 0.5 (b) 0
 0
 10 10 100 0.0
 10 10 10 100 10

 Mean of ln(Pl)
 2 1 3 2 1
 Excitation, Pl Excitation, Pl 2
 0.02
 0.05
 EJ = 0.2 GHz (d) EJ = 0.5 GHz 4 5
 (b) 6 0.07
 0.1
 1.25 0.12
 5 60 1 2 3 4 5 6 7 8 9 10 0.0 0.5 1.0 1.5
 1.00
 Distribution

 Distribution

 4 Qubit index, l Energy fluctuations, /(Ja2)
 3 0.75
 2 0.50 FIG. 3. Mean value of the excitation probability as a func-
 1 0.25 tion of spin index l = 1, . . . , 9 with (a) fluctuating Josephson
 energy with standard deviation δEJ = 0.05 . . . 0.5 GHz and
 0
 10 2 10 1 100 0.00
 10 3 10 2 10 1 100 zero flux detuning from the sweet spot, and (b) flux disorder
 Excitation, Pl Excitation, Pl with standard deviation δφ = 0.02 . . . 0.12 and fixed EJ /h = 9
 GHz. The thick x-marks show the maximal excitation proba-
 bility for the uniform chain when all qubits are identical and
FIG. 2. The distribution function of the excitation probabil-
 biased at the sweet spot. Panel (c): the localization length,
ity of spins l = 2, 4, 6, 8, 10 in the chain with fluctuating
 obtained from the linear fit of data in panels (a) and (b),
Josephson energy EJ with δEJ = 0.1 GHz (a), 0.2 GHz (b),
 shown as functions of the energy fluctuations δε, Eq. (6).
0.3 GHz (c), and 0.5 GHz (d). At least 1000 disorder realiza-
tions are taken for all distributions in the paper.

 Fig. 2(b)) the distribution broadens and shifts to smaller
hand, the Josephson energy fluctuations EJ is the ma- values. At even stronger disorder, (δEJ = 0.3 GHz),
jor challenge to produce a uniform chain of fluxoniums. Fig. 2(c)) the distribution of Pl shifts to even smaller
Therefore, we consider the effect of Gaussian fluctuations values with more than half of the realizations showing
of EJ on the chain dynamics and establish the allowance Pl < 0.1 for l = 8 and 10. For δEJ = 0.5 GHz, the exci-
for its standard deviation δEJ to have a chain in the de- tation propagates to nearest qubits only and is extremely
localized regime. Thus, we assume that the Josephson unlikely to reach qubits further away, as illustrated by the
energies of individual fluxoniums EJ,l have a Gaussian broad distributions for l ≥ 6 with tails going below 0.01.
distribution with average value ĒJ and the standard de- We notice that the distributions for more distant qubits,
viation δEJ . The main effect of fluctuations of EJ,l is i.e. larger l, is broader and shifted to smaller values of
the random variation of energies εl in Eq. (3) with the Pl .
standard deviation: Localization length – To quantify the suppression of ex-
 q citation propagation through the chain, we evaluate the
 δε = ε2l − ε̄2l . (6) average value of ln Pl for different qubit locations l, see
 Fig. 3(a). We observe that ln Pl ≈ c0 −2l/ξ, where ξ has a
Here the bar denotes averaging over disorder configu- meaning of the localization length that characterizes de-
rations. For small δEJ  EJ , we can use a pertur- cay of wave functions as exp(−x/ξ). To avoid the effect
bation theory to find the relation δε = ηδEJ , where of reflection at the opposite end of the chain, we perform
η = | he| cos θ |ei − hg| cos θ |gi | ≈ 0.31. the linear fit for l ≤ L/2 to obtain values of ξ, as illus-
 To characterize the mobility of excitations, we study trated by dashed lines in Fig. 3(a). We presented ξ for
statistics of the maximal probability of excitation several values of δEJ in the range from 0.05 to 0.5 GHz in
 Fig. 3(c). The horizontal axis shows the value of the qubit
 Pl = max pl (t) . (7) energy fluctuations δε, Eq. (6), calculated for correspond-
 t

4

values ξ ' 1, when only partial excitation propagation (a) 5 (b)
 EJ = 0.1 GHz = 0.05
occurs to nearest neighboring sites. We identify the local- EJ = 0.2 GHz 2.5 = 0.07
 4 EJ = 0.3 GHz = 0.1
ization transition when ξ ' L/2 that occurs at δEJ ' 0.3 2.0 = 0.12

 Distribution

 Distribution
GHz or δε/(Ja2 ) ' 0.5. EJ = 0.5 GHz
 3
 1.5
 This estimate for the onset of the localization is consis-
 2 1.0
tent with the behavior of the distribution functions, pre-
sented in Fig. 2. In weak disorder, the probability density 1 0.5
vanishes for small Pl . 0.1, Fig. 2(a,b). At δEJ = 0.3
 0 0.0
GHz shown in Fig. 2(c), the distribution moves to the left 10 4 10 3 10 2 10 1 100 10 4 10 3 10 2 10 1 100
and spreads over two orders of magnitude, with non-zero Excitation, Pl Excitation, Pl
probability to have 0.01 < Pl < 1. As disorder increases
more, ∆EJ = 0.5 GHz, values of Pl < 0.01 become com- FIG. 4. The distribution function of the logarithm of the ex-
mon, see Fig. 2(d). citation probability Pl=16 of spin l = 16. Panel (a): All flux-
 Excitation at the edges – We discussed the excitation oniums are at the sweet spot and the distribution functions
dynamics of a qubit at an arbitrary location along the represent different strengths of fluctuations of the Josephson
chain. The excitation probability of each qubit can be energy, δEJ = 0.1, 0.2, 0.3, and 0.5 GHz. Panel (b): All flux-
measured. However, such measurements are hard in oniums have the same Josephson energy EJ /h = 9 GHz with
 random flux detunings from the sweet spot with standard de-
near-term quantum simulators. Readout of individual
 viation δφ = 0.05, 0.07, 0.1, and 0.12.
qubits is usually slow, while a typical excitation time of
the qubit is characterized by h/(2Ja2 ) and is well under
10 ns long in our case. Below we focus on the excitation
dynamics of the last qubit, l = 16. An edge qubit is easier ergy fluctuations, δε/(Ja2 ), in Fig. 3(c). We notice that
to be coupled to a readout resonator without disturbing the localization lengths obtained for the chain with either
its neighboring qubits, and its excitation can be locked in fluctuating Josephson energy or flux align well along the
time by applying a fast-flux detuning to its neighbor. In same curve when the localization length is plotted as a
this case, the large energy mismatch for the two qubits function of the dimensionless energy fluctuations. Thus,
forbids the energy exchange between neighboring fluxo- we expect that the dominant cause of single excitation
niums [29]. In such flux configuration, slow readout of localization in the chain is the energy mismatch between
the fast propagating qubit excitation is possible. neighboring qubits and is consistent with the Anderson
 The remote qubit maximal excitation probability P16 localization [30].
characterizes the propagation of excitations through the We also present statistics of P16 for the excitation of
chain. In weakly disordered chain, δEJ = 0.1 GHz, the the last qubit for four values of flux disorder in Fig. 4(b).
maximum of excitation probability P16 is a narrow dis- The behavior of the distribution functions is similar to
tribution over disorder ensemble and is slightly reduced the cases with fluctuations of the Josephson energy. For
 (0)
from P16 ≈ 0.67, when compared to the clean system. weak flux noise, δφ . 0.07, the distributions are nar-
The distribution function vanishes for P16 < 0.1, see row, and the excitation probabilities mostly exceed 0.1.
Fig. 4(a). As the disorder strength increases, e.g. as At the transition, which happens around δφ = 0.1, the
δEJ = 0.2 GHz, and the distribution of P16 shifts to distribution becomes broad. It shifts to much smaller val-
smaller values and broadens. At δEJ = 0.3 GHz which ues of Pl , and observing the remote spin’s excitation is
we identified above as the transition point to the local- very unlikely at stronger flux disorder. We again observe
ized regime, the distribution of P16 become the broadest a qualitative similarity between disorder introduced by
with the range 10−3 . P16 . 1. At larger fluctuations of fluctuations of the Josephson energy and random mag-
the Josephson energy, e.g. δEJ = 0.5 GHz, the distribu- netic flux detunings.
tion narrows and is centered at the smaller values of P16 , Conclusions – The chain of fluxonium qubits with
with the majority of P16 < 0.01. record-high coherence provides means for studying the ef-
 Flux Disorder – Here, we also consider a chain with fects of single- and many-body localization of excitations
random flux detunings from the sweet spot φl − π. We in the coherent setting, as well as multiqubit tunneling
describe the localization onset as the standard deviation and multiqubit gates due to strong inductive coupling.
δφ of the flux detunings increases. We note that the flux We studied fluxonium qubit systems simulating clean and
detunings effectively introduce the longitudinal field, and disordered TFIM. The spin-flip experiment identifies er-
thus the system acquires deviations from the TFIM (see godic and localized regimes in the TFIM that require
the Appendix for more details). measurements of edge qubits. Such measurements can
 First, we evaluate ln Pl for several values of flux disor- be performed for a simple fluxonium chain and do not re-
der strength, characterized by δφ. We observe that ln Pl quire a complicated circuit design. Our results show that
is well-fitted by the linear dependence on the qubit loca- the fluxonium chain mimics TFIM well and is a promi-
tion l and the fitting coefficient provide the localization nent candidate to be a near-term quantum simulator of
length in the chain, see Fig. 3(b). We plot the localiza- strongly correlated spin systems. While we analyzed the
tion length as a function of the dimensionless qubit en- properties of one-dimensional chain in this paper, fluxo-

5

nium devices can also be used to explore two-dimensional
lattices and Cayley trees. (a) (b) 4
 The ideal starting point to study spin systems is to 12.5 3 gg
 ge

 ij
fabricate a uniform chain of fluxonium qubits and gener-
 10.0

 Energy /h, GHz
 ee
 2

 Phase operator,
ate random flux detuning from the sweet spot to study
the effects of disorder in the TFIM. However, the cur- 7.5 1
rent fabrication process often provides about 10% fluc-
tuations of the Josephson energy. These fluctuations are 5.0 0
sufficient to bring the chain to the localized regime at the
coupling strength considered in this paper. The fabrica- 2.5 Exact 1
 Perturbation
tion process has to be improved to reduce fluctuations 0.00.0 0.5 1.0 20.0 0.5 1.0
of the Josephson energy of the fluxonium qubit and to
achieve a delocalized regime. Alternatively, the interac-
 Flux detuning, / Flux detuning, /
tion between qubits can be made stronger, thus bringing FIG. 5. (a) Dependence of the lowest excitation energy
the system to the delocalized regime even if fluctuations of a fluxonium as a function of the magnetic flux detuning
of EJ remain significant. For fluxoniums with param- from the half-integer sweet spot. (b). Matrix elements for
eters analyzed here, the coupling between them has to the phase operator θ̂l (δφ) defined by Eq. (4) are shown as
be about 60 MHz, which is still weak compared to the a function of the magnetic flux detuning δφ. The excitation
achievable galvanic coupling between fluxoniums [23]. matrix element of the phase operator, θge , has maximum at
 Acknowledgments – We thank Mark Dykman, and the sweet spot and then decreases as detuning increases. The
Kostya Nesterov for fruitful discussions. This work was diagonal matrix elements, θgg and θee , vanish at the sweet
supported by the U.S. Department of Energy, Office of spots φ = π and φ = 0 but are finite when qubits are detuned
Science, Office of Basic Energy Sciences, under Award from their sweet spots.
Number DE-SC0019449. The work of A.B.Ö. at Fermilab
was supported by the DOE/HEP QuantISED program
grant Large Scale Simulations of Quantum Systems on states of the fluxonium Hamiltonian. The off-diagonal
HPC with Analytics for HEP Algorithms (0000246788). matrix element, θeg responsible for the XX coupling at
This manuscript has been authored by Fermi Research the sweet spot decreases from its value at the sweet spot,
Alliance, LLC under Contract No. DE-AC02-07CH11359 where θeg (φ = π) = a ≈ 2.36. The off-diagonal matrix
with the U.S. Department of Energy, Office of Science, elements θgg and θee vanish at the sweet spot, but their
Office of High Energy Physics. The simulations were per- magnitude increases fast with detuning, see Fig. 5(b).
formed using QuTiP [31] and the computing resources of To better illustrate the properties of the fluxoniums
the UW-Madison Center For High Throughput Comput- away from the sweet spot, we also analyze the system
ing (CHTC) and resources provided by the Open Science Hamiltonian in the basis of fluxonium eigenstates at the
Grid [32, 33], which is supported by the National Science sweet spot. In this basis, the phase operator θ = aσx
Foundation award 1148698 and the U.S. Department of with a ≈ 2.36 for small flux detuning and the choice of
Energy’s Office of Science. parameters introduced in the main text. The Hamilto-
 nian of the system in this basis and at small detuning
 has the form
 Appendix: Fluxonium chain away from the sweet L−2 L−1
 spot X X
 HTFIM = Ja2 σ̂lx σ̂l+1
 x
 + Hl , (A.1)
 l=0 l=0
 The Hamiltonian of the chain is given by Eqs. (3) and
(4). In this Appendix, we focus on the system of iden- where the Hamiltonian of qubit l in the basis of eigen-
tical fluxoniums but consider small detuning from the states of the fluxonium at the sweet spot is
half-flux quantum sweet spot with a random distribution
of δφl . We analyze how detuning from the half-integer ε0 + δεzl z δεxl x
 Hl = − σ̂l − σ̂ , (A.2)
flux quantum sweet spot changes the energy and phase 2 2 l
matrix elements. We also analyze how the uniform flux
detuning affects the dynamics of excitation propagation where ε0 = (E1 − E0 ) is given by the difference of the
through the chain. We present the plot for the depen- qubit energies at the sweet spot.
dence of the qubit energy splitting as a function of the We now evaluate fields δεz and δεx for small devi-
flux detuning δφl = φl −π in Fig. 5(a). We notice that the ations of the flux δφl . The phase-dependent energy
minimal energy splitting is at the half-integer sweet spot V (θ) acquires the correction δV (θ) = EJ (δφ sin θ −
and then monotonically increases to the maximal split- (δφ2 /2) cos θ). We apply the perturbation theory in the
ting at the integer sweet spot. In Fig. 5(b), we present basis of eigenstates of the fluxonium at the sweet spot.
the dependence of matrix elements of the phase operator We find that flux-dependent energy shift determined by
θ̂ on the flux detuning, written in the basis of the eigen- the diagonal matrix elements of cos θ, and δεx is deter-

6

mined by the off-diagonal matrix element of sin θ: all qubits initialized in their ground states and only qubit
 l = 0 in its first excited state. The dynamics is charac-
 EJ (δφl )2 terized by the excitation probability of different qubits
 δεzl = [he| cos θ |ei − hg| cos θ |gi], (A.3)
 2 as a function of time. We can evaluate the location of
 δεxl = 2EJ δφl hg| sin θ |ei . (A.4) the peaks and their height for different qubits. The peak
 time is consistent with the maximal group velocity of the
This Hamiltonian corresponds to the transverse Ising
 TFIM estimated from the spectrum of the TFIM [24]:
Hamiltonian with random transverse and longitudinal
fields. We note that the contribution of δεzl is quadratic
in the flux detuning and, therefore, the main contribution ∂ωq 1 Ja2 ε0 sin q
 u(q) = = p 2 , (A.5)
of the fluctuating field is equivalent to a random longitu- ∂q ~ ε0 + 2Ja2 ε0 cos q + J 2 a4
dinal field. When Hamiltonian (A.1) is rewritten in the
eigenstate basis of the individual qubits, it acquires the where q is the quasimomentum for excitation with energy
form given by Eq. (3), when the interaction is no longer ~ωq . For q = π/2 and ε0  Ja2 , we have u(π/2) ≈
represented by σ̂lx σ̂l+1
 x
 terms only, at the same time, the Ja2 /~.
coefficient in front of σ̂lx σ̂l+1
 x
 term is reduced as the phase We now explore the excitation propagation when the
matrix element θeg decreases, see Fig. 5(b). flux detuning from the sweet spot is identical for all flux-
 oniums. As we argued above, a weak detuning from the
 1.0 sweet spot is equivalent to the additional longitudinal
 l=4 field applied to the TFIM. As the detuning increases, in-
 l=8
 0.8 l=16 dividual qubits’ level spacing increases but does not affect
 Excitation pl(tmax)

 the resonant transfer of excitations between qubits. The
 0.6 main consequence of the detuning is a modification of
 the interaction term, which in the qubit eigenstate basis
 corresponds to the appearance of σlz σl+1z
 channel, while
 0.4 x x
 the σl σl+1 coupling decreases. This decrease results in
 a reduced velocity of the excitation propagation and also
 0.2 in stronger dispersion. As a result, the propagation of
 the excitation in the system uniformly detuned from the
 0.00 5 10 15 20 25 30
 sweet spot results in the longer propagation of the exci-
 Peak arrival time tmax, ns tation along the chain and lower probability excitation
 of remote qubits due to the broadening of the excitation
FIG. 6. The parametric plot of the peak excitation proba-
 wave packet. We compute the relation between the peak
bility for qubits l = 4, 8, 16 as a function of the peak arrival excitation probability of qubits and the corresponding
time tmax in the homogeneous system of L = 17 fluxoniums peak time. The plot of the maximal excitation probabil-
with flux detuning from the sweet spot. Initially, all qubits ity as a function of its arrival time is shown in Fig. 6.
were in their ground states, but fluxonium l = 0 was excited. The monotonic decrease of the maximal excitation prob-
The parameters of fluxoniums and the coupling strength are ability on the peak arrival time is consistent with a ho-
provided in the text, J = 10 MHz. mogeneous reduction of the group velocity due to weaker
 excitation exchange in the longitudinal field and stronger
 In the main text, we discussed the chain dynamics with dispersion.

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