Fast topological pumping for the generation of large-scale Greenberger−Horne−Zeilinger states in a superconducting circuit

doi:10.1007/s11467-022-1193-y

Front. Phys.

2022, Vol. 17

Issue (6) : 62504 https://doi.org/10.1007/s11467-022-1193-y

RESEARCH ARTICLE

Fast topological pumping for the generation of large-scale Greenberger−Horne−Zeilinger states in a superconducting circuit

Jin-Xuan Han¹, Jin-Lei Wu²(

), Zhong-Hui Yuan¹, Yan Xia³, Yong-Yuan Jiang^1,^4,^5,⁶, Jie Song^1,^4,^5,⁶(

)

¹. School of Physics, Harbin Institute of Technology, Harbin 150001, China
². Department of Optoelectronics Science, Harbin Institute of Technology, Weihai 264209, China
³. Department of Physics, Fuzhou University, Fuzhou 350002, China
⁴. Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
⁵. Key Laboratory of Micro-Nano Optoelectronic Information System, Ministry of Industry and Information Technology, Harbin 150001, China
⁶. Key Laboratory of Micro-Optics and Photonic Technology of Heilongjiang Province, Harbin Institute of Technology, Harbin 150001, China

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Abstract

Topological pumping of edge states in the finite lattice with nontrivial topological phases provides a powerful means for robust excitation transfer, requiring extremely slow evolution to follow an adiabatic transfer. Here, we propose fast topological pumping via edge channels to generate large-scale Greenberger−Horne−Zeilinger (GHZ) states in a topological superconducting circuit with a sped-up evolution process. The scheme indicates a conceptual way of designing fast topological pumping related to the instantaneous energy spectrum characteristics rather than relying on the shortcuts to adiabaticity. Based on fast topological pumping, large-scale GHZ states show greater robustness against on-site potential defects, the fluctuation of couplings and losses of the system in comparison with the conventional adiabatic topological pumping. With experimentally feasible qutrit-resonator coupling strengths and moderate decay rates of qutrits and resonators, fast topological pumping drastically improves the scalability of GHZ states with a high fidelity. Our work opens up prospects for the realization of large-scale GHZ states based on fast topological pumping in the superconducting quantum circuit system, which provides potential applications of topological matters in quantum information processing.

Keywords topological pumping superconducting ciruit large-scale Greenberger−Horne−Zeilinger states

Corresponding Author(s): Jin-Lei Wu,Jie Song

About author:

Tongcan Cui and Yizhe Hou contributed equally to this work.

Issue Date: 20 September 2022

Cite this article:

Jin-Xuan Han,Jin-Lei Wu,Zhong-Hui Yuan, et al. Fast topological pumping for the generation of large-scale Greenberger−Horne−Zeilinger states in a superconducting circuit[J]. Front. Phys. , 2022, 17(6): 62504.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-022-1193-y
https://academic.hep.com.cn/fop/EN/Y2022/V17/I6/62504

Fig.1 (a) The diagrammatic sketch of a topological superconducting qutrit-resonator chain with the size of

2 N

. The chain belongs to an SSH model whose

n

-th unit cell contains one flux qutrit

A n

and one single-mode resonator

B n

. The intra-cell and inter-cell coupling strengths are

J 1

and

J 2

, respectively. (b) Schematics of energy level transitions for qutrits

A 1

A n

(

1 < n < N

), and

A N

. The energy level structure of a flux qutrit holds two ground states (

| L ?

and

| R ?

) and one excited state (

| e ?

). The coupling strengths between the resonator

B n

and the qutrit

A n

(

A n + 1

) is

J L = J 1

(

J R = J 2

) when

n

is odd, or

J L = J 2

(

J R = J 1

) when

n

is even.

Fig.2 Two types of trajectories of the vector

g (k)

with different topologies when

k

runs across the Brillouin zone: (a)

J 1 / J 2 < 1

and (b)

J 1 / J 2 > 1

. The spectrum of the SSH model versus

J 1 / J 2

for the even size

L = 2 N = 62

of chain in (c) and the odd size

L = 2 N − 1 = 61

in (d).

Fig.3 Functions

J 1, 2

and the corresponding instantaneous energy spectrum as a function of time by setting different parameters

α = 1

in (a) and (b),

α = 6

in (c) and (d), and

α = 11

in (e) and (f). We choose the total evolution time to be unity and the size of chain is

L = 2 N − 1 = 61

Fig.4 The evolution process of zero-energy mode with different values of the parameters

α = 1

in (a),

α = 6

in (b) and

α = 11

in (c). (d) Numerical scatters of the minimum energy gap versus values of

α

with the size of chain

L = 2 N − 1 = 61

Fig.5 (a) Population evolution of states

| r ? 61

| l ? 61

| G ? 61

| Ψ ideal ?

and

| Ψ initial ?

based on fast topological pumping with exponential couplings and the conventional adiabatic topological pumping with Gauss couplings. (b) Final fidelity of 31-body GHZ state

| Ψ ideal ?

as a function of the total evolution time

T

based on fast topological pumping with exponential couplings and the conventional adiabatic topological pumping with Gauss couplings. We choose the size of chain as

L = 2 N − 1 = 61

and set the free parameter

α = 6

Fig.6 (a−c) Final fidelity of

N

-body GHZ state as a function of the total evolution time with different values of the parameter

α

by increasing the number of qutrits

N = 5

in (a),

N = 18

in (b) and

N = 31

in (c). (d−f) The corresponding instantaneous energy

E N + 1

as a function of time with different values of the parameter

α

by increasing the number of qutrits

N = 5

in (d),

N = 18

in (e) and

N = 31

in (f).

Fig.7 The fidelity of N-body GHZ state versus the varying

α

and the total evolution time

T

for (a)

N = 5

, (b)

N = 18

, (c)

N = 31

and (d)

N = 44

. The red and blue solid lines represent 0.995 and 0.999 fidelity contour lines of

N

-body GHZ state, respectively.

Fig.8 (a) Fitting functions and numerical scatters between the number of qutrits and total evolution time with

99.9 %

fidelity for the conventional adiabatic topological pumping with Gauss couplings and the fast topological pumping with exponential couplings. (b) Time evolution of

log 10 ? (1 − F)

for generating

N

-body GHZ states with

N

ranging from 10 to 30 at intervals of 5 based on the conventional adiabatic pumping with Gauss couplings and the fast topological pumping with exponential couplings, respectively.

Fig.9 Final fidelity of

31

-body GHZ state against the unexpected coupling strength and the on-site potential defect for all

A

-type lattice sites and

B

-type lattice sites with disorder

δ

Fig.10 The fidelity of the

31

-body GHZ state versus the varying

δ

and the total evolution time

T

for unexpected couplings with exponential couplings in (a) and with Gauss couplings in (b) and for on-site potential defect with

A

-type lattice sites in (c) and with

B

-type lattice sites in (d). The red (purple) and blue solid lines represent 0.995 (0.997) and 0.999 fidelity contour lines of

N

-body GHZ state, respectively.

Fig.11 Effects of losses on the final fidelity of 31-body GHZ state for the conventional adiabatic topological pumping with Gauss couplings and the fast pumping with exponential couplings. We choose

J 0 / (2 π) = 10

MHz, the corresponding total evolution time

T Gauss N = 31 = 106.42

μ

s and

T Exp N = 31 = 5.04

μ

s with a

99.9 %

fidelity for generating a 31-body GHZ state.

Fig.12 Final fidelities of the GHZ state with the scalability of entanglement

N

under the coupling strengths (e.g.,

J 0 / (2 π) = 1

MHz, 10 MHz and 50 MHz) and decay rates of qutrits and resonators (e.g.,

γ q / (2 π) = κ / (2 π) = 1

kHz) for the conventional adiabatic topological pumping with Gauss couplings and the fast topological pumping with exponential couplings.

Fig.13 The diagrammatic sketch of two-dimensional square lattice with the size of

L × L

in superconducting qutrit-resonator system.

N

-body GHZ state is generated by the

i

th line and the

(2 N − 1)

th column of SSH chain, including three steps: (i) Initialize from the qutrit

A 1, N

A N, N

in the

(2 N − 1)

th column of SSH chain; (ii) The fast topological pumping from the qutrit

A i, 1

A i, N

in the

i

th line of SSH chain; (iii) The fast topological pumping from the qutrit

A i, N

A N, N

in the

(2 N − 1)

th column of SSH chain.

Fig.14 Equivalent circuit of one unit cell in superconducting qutrit-resonator chain. Circuit elements are used to model the three-junction flux qutrit

A n

the

L C

resonator

B n

and the coupler with the additional Josephson junction and the coupler capacitor mounted in a dilution refrigerator (with a temperature

T ∼

mK). The probe microwave signal is sent from a network analyzer and attenuated in the signal input line before arriving at the sample, which is placed in a magnetic shield. The transmitted signal from the sample is amplified by cryogenic LNA and measured by the network analyzer. The resonator

B n

is an

L C

circuit composed of a spiral inductor

L B

and a capacitor

C B

. A flux qutrit

A n

consists of a superconducting loop interrupted by three Josephson junctions. The flux qutrit

A n

and the

L C

resonator

B n

are coupled to the coupler by the capacitor

C 1, 2

and

C 3, 4

, respectively. The coupling strength can be adjusted independently via changing the magnetic flux

Δ Ψ

threading on the loop of coupler, which can add the flux basis line (FBL) to connect with an AWG by adopting controlled voltage pulses.

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