Experimental Implementation of Noncyclic and Nonadiabatic Geometric Quantum Gates in a Superconducting Circuit Zhuang Ma1Jianwen Xu1Tao Chen2Yu Zhang1Wen Zheng1

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Experimental Implementation of Noncyclic and Nonadiabatic Geometric Quantum

Gates in a Superconducting Circuit

Zhuang Ma,1, ∗Jianwen Xu,1, ∗Tao Chen,2, ∗Yu Zhang,1Wen Zheng,1

Dong Lan,1, 3 Zheng-Yuan Xue,2, †Xinsheng Tan,1, 3, ‡and Yang Yu1, 3

1National Laboratory of Solid State Microstructures,

School of Physics, Nanjing University, Nanjing 210093, China

2Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,

and School of Physics and Telecommunication Engineering,

South China Normal University, Guangzhou 510006, China

3Hefei National Laboratory, Hefei 230088, China

(Dated: October 10, 2022)

Quantum gates based on geometric phases possess intrinsic noise-resilience features and therefore

attract much attention. However, the implementations of previous geometric quantum computation

typically require a long pulse time of gates. As a result, their experimental control inevitably suﬀers

from the cumulative disturbances of systematic errors due to excessive time consumption. Here,

we experimentally implement a set of noncyclic and nonadiabatic geometric quantum gates in a

superconducting circuit, which greatly shortens the gate time. And also, we experimentally verify

that our universal single-qubit geometric gates are more robust to both the Rabi frequency error

and qubit frequency shift-induced error, compared to the conventional dynamical gates, by using the

randomized benchmarking method. Moreover, this scheme can be utilized to construct two-qubit

geometric operations, while the generation of the maximally entangled Bell states is demonstrated.

Therefore, our results provide a promising routine to achieve fast, high-ﬁdelity, and error-resilient

quantum gates in superconducting quantum circuits.

The superconducting quantum circuit is one of the

promising candidates for future large-scale quantum com-

putation [1] due to its high controllability and scalabil-

ity. At this stage, the major obstacle is relatively short

coherence time and experimental perturbations, which

demand speeding up quantum operations and improv-

ing the robustness against errors under the experimental

controls in superconducting quantum circuits. Therefore,

with their intrinsic noise-resilience features, the gates in-

duced by geometric phases [2–4], attainable in supercon-

ducting systems, are highly anticipated.

The geometric phases depend only on the global prop-

erties of their evolution paths, so that they can be applied

to construct the geometric quantum gates against certain

local noises [5]. Adiabatic geometric quantum computa-

tion (AGQC) based on the Berry phase has been pro-

posed [3,6–8] and ﬁrst experimentally demonstrated in

nuclear magnetic resonance (NMR) [4], aiming to realize

high-ﬁdelity and robust quantum gates. However, the

long gate time due to the adiabatic and cyclic evolution

conditions restricts the practical application of AGQC,

especially in quantum systems with limited coherence

time. Some approaches are proposed to overcome this

problem, including the shortcut acceleration to the adi-

abatic evolution [9–12], while these inevitably sacriﬁce

some robustness and generally increase the control com-

plexity. Recently, nonadiabatic geometric quantum com-

putation (NGQC) has been theoretically proposed and

experimentally implemented based on Abelian [13–21]

and non-Abelian geometric phases [22–31] to break the

limitation of the adiabatic condition. However, to strictly

satisfy the cyclic evolution in NGQC, it usually requires

at least π-pulse time consumption to construct a geomet-

ric gate, so there is still no advantage in operation time

compared to conventional dynamical gates. Meanwhile,

the increase in time consumption will also be accompa-

nied by cumulative disturbances from systematic errors,

making the robust advantage of the geometric gate dis-

plays ambiguous in experiments.

To reduce the gate-operation time and release the re-

striction of the cyclicity in the design of geometric gates

[32], some theoretical schemes based on nonadiabatic but

noncyclic geometric evolution have recently been pro-

posed [33–35]. One of them has been experimentally im-

plemented in a single trapped ultracold 40Ca+ion [36], in

which a special single-qubit geometric gate has demon-

strated its error-resilient feature. But the experimental

veriﬁcation of short-time and error-resilient features for

a set of universal geometric gates is still lacking, espe-

cially for the simultaneous suppression of diﬀerent types

of errors.

Here, we experimentally implement the noncyclic and

nonadiabatic (NCNA) geometric quantum computation

in a superconducting quantum circuit. The method we

adopted to construct NCNA geometric gates is reverse

engineering, which purposefully determines the Hamilto-

nian for the system to generate noncyclic geometric evo-

lution paths [37]. In our experiment, a set of universal

and short-time single-qubit NCNA geometric gates in-

cluding [38]π/8 gate (T), Phase gate (S), and Hadamard

gate (H) are realized, and their high ﬁdelities are charac-

terized via randomized benchmarking (RB). Remarkably,

arXiv:2210.03326v1 [quant-ph] 7 Oct 2022

T gate S gate H gate

(a) (b)

(c)

FIG. 1: Single-qubit NCNA geometric gates. (a) The non-

cyclic evolution path of state vector |ψ+(t)iwith Bloch rep-

resentation to realize NCNA geometric gates. (b) Sketch of a

two-qubit system with a coupler. QA(blue) and QB(red) are

directly coupled with an eﬀective coupling strength gAB and

the coupling strength between the coupler C(black) and QA

(QB) is gAC (gBC ). (c) The experimental pulses to realize

NCNA geometric T,S, and Hgates and the corresponding

evolution trajectories with speciﬁc initial states.

we also experimentally demonstrate the strong resistance

of our universal single-qubit NCNA geometric gates to

both the Rabi frequency error and qubit frequency shift-

induced error. Finally, we implement the nontrivial two-

qubit geometric operation using parametric modulation

[39–41] to generate maximally entangled Bell states.

We ﬁrst brieﬂy elucidate the theoretical proposal [37]

of constructing NCNA geometric gates in the supercon-

ducting qubit. With ~= 1, a general Hamiltonian for a

two-level system is

H(t) = 1

2−∆(t) Ω(t)e−iφ(t)

Ω(t)eiφ(t)∆(t),(1)

where Ω(t) and φ(t) are the time-dependent amplitude

and phase of the driving microwave ﬁeld, respectively;

∆(t) = ωq−ωmis the time-dependent detuning between

the qubit transition frequency and the frequency of a mi-

crowave ﬁeld. According to the Lewis-Riesenfeld invari-

ant methods [42–44], we can choose a set of orthogonal

states as |ψ+(t)i=eif+(t)[cos χ(t)

2|0i+ sin χ(t)

2eiξ(t)|1i]

and |ψ−(t)i=eif−(t)[sin χ(t)

2e−iξ(t)|0i − cos χ(t)

2|1i] in

which f+(t) = f−(t) = γis regarded as a global phase,

and χ(t) and ξ(t) represent the polar and azimuthal an-

gles on a Bloch sphere respectively. To realize our NCNA

Gate Fidelity

0.9943

0.9982

0.9995

0.9994

Ref.

Reference

Interleaved

FIG. 2: Cliﬀord-based RB of single-qubit NCNA geometric

gates. The inset is the sequence of gates from the Cliﬀord

group for the reference RB and interleaved RB. Sequence ﬁ-

delities are functions of the number of Cliﬀords and the ex-

ponential decay curves give ﬁdelities of NCNA gates: the 2T,

S, and Hgates.

geometric gate, the entire noncyclic evolution path com-

posed of three path segments needs to be utilized, as

denoted in Fig. 1(a). We here take the evolution details

of state vector |ψ+(t)ias an illustration: ﬁrst, it evolves

along the longitude line from the initial point (χ1, ξ1)

to (χ2, ξ1) at time t=τ1, with null accumulation of the

global phase; next the state evolves along the latitude line

from (χ2, ξ1) to (χ2, ξ2) at time t=τ2; the third path

is similar to the reverse of the ﬁrst path, which is from

(χ2, ξ2) to the ﬁnal point (χ1, ξ2). Among them, after

strictly eliminating the dynamical phase existing in the

middle segment by setting Rτ2

τ1∆(t)dt = (ξ1−ξ2) sin2χ2,

the accumulated geometric phase can be obtained as

γg=−1

2Zτ

ξ(t)[1−cos χ(t)]dt =−1

2(ξ2−ξ1)(1−cos χ2),

(2)

which is exactly half of the solid angle enclosed by the

noncyclic evolution path and its geodesic connecting the

initial point (χ1, ξ1) and ﬁnal point (χ1, ξ2). Based on

these, the corresponding Hamiltonian parameter φ(t) and

the pulse area associated with Ω(t) can then be reverse-

engineered in these three segments t∈[0, τ1], [τ1, τ2] and

[τ2, τ] as

φ(t) = ξ1+π

2,1

2Zτ1

Ω(t)dt =1

2(χ2−χ1),

φ(t) = ξ(t) + π, 1

2Zτ2

τ1

Ω(t)dt =1

4(ξ2−ξ1) sin(2χ2),

φ(t) = ξ2−π

2,1

2Zτ

τ2

Ω(t)dt =1

2(χ2−χ1),(3)

with detuning ∆(t) = 0, −(ξ2−ξ1) sin2χ2/(τ2−τ1), 0,

where ξ(t) = ξ1−Rt

τ1∆ (t0)dt0+ cot χ2Rt

τ1Ω (t0)dt0. In

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ExperimentalImplementationofNoncyclicandNonadiabaticGeometricQuantumGatesinaSuperconductingCircuitZhuangMa,1,JianwenXu,1,TaoChen,2,YuZhang,1WenZheng,1DongLan,1,3Zheng-YuanXue,2,yXinshengTan,1,3,zandYangYu1,31NationalLaboratoryofSolidStateMicrostructures,SchoolofPhysics,NanjingUniversity,Nanjing21...

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Experimental Implementation of Noncyclic and Nonadiabatic Geometric Quantum Gates in a Superconducting Circuit Zhuang Ma1Jianwen Xu1Tao Chen2Yu Zhang1Wen Zheng1

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