Experimental Realization of non-Adiabatic Shortcut to non-Abelian Geometric Gates

# Experimental Realization of non-Adiabatic Shortcut to non-Abelian Geometric Gates

## Abstract

12

Introduction. In quantum information processing, desired operations are achieved by actively manipulating phase evolution of a quantum system under control. Such phase evolution may have a dynamical or geometric nature. In the latter case, as the system Hamiltonian undergoes a cyclic evolution in a parameter space, a pure geometric phase independent of the dynamical evolution accumulates Berry1984 (); Wilczek1984PRL (); Zarnadi1999PRL (); Aharonov1987PRL (); Anandan1988PRA (). Such geometric phase has been shown to be intrinsically resilient to certain types of noise Johansson2012PRA (); Berger2013PRA (); Yale2016Nature (), therefore it may be used to implement fault tolerant geometric quantum computation, also known as GQC Falci2000Nature (); Jones2000Nat (); Duan2001Science (); Wang2001PRL (); Zhu2003PRL (); Sjoqvist2012NJP (). In particular, the matrix form of non-Abelian geometric phase Wilczek1984PRL (); Zarnadi1999PRL () naturally leads to non-commutativity, making it suitable for implementing universal HQC Abdumalikov2013Nature (). Originally, geometric quantum gates were constructed through adiabatic evolutions Falci2000Nature (); Duan2001Science (), which require a long runtime to avoid transitions between the eigenstates. To overcome such a problem, non-adiabatic HQC (NHQC) schemes was proposed Sjoqvist2012NJP (), but it is sensitive to the systematic errors of the control Hamiltonian Shi2016PRA ().

On the other hand, the technique of “shortcut to adiabaticity” (STA) Berry2009JPA (); Chen2010PRL (); Campo2013PRL (); Giannelli2014RPA () represents an alternative non-adiabatic approach to avoid transitions, which is realized by including an auxiliary term to the target Hamiltonian. The auxiliary term allows evolution of the modified system to reach the same state as the adiabatic evolution of the target Hamiltonian, which can be achieved without transitions among the instantaneous eigenstates of the target Hamiltonian. The working principle of the STA technique has been widely demonstrated in various experiments Du2016NatC (); Bason2012NatP (); Zhang2013PRL (); Shuoming2016NatC (); Zhou2016np (); Zhang2017PRA (); Vepsalainenarxiv2017 (); Huarxiv2018 (), including fast quantum state transfer with cold atoms Du2016NatC (), acceleration of Bose-Einstein condensate in an optical lattice Bason2012NatP (), fast control of electron spins in NV centers Zhang2013PRL (), and displacement of trapped ions with minimum excitations Shuoming2016NatC (). In addition, experiments have also confirmed the robustness of STA against dissipation and errors Du2016NatC (); Zhou2016np (); Zhang2017PRA ().

Recently, STA-based techniques have been proposed for realizing of robust geometric quantum gates Zhang2015Srep (); Liang2016PRA (); Song2016NJP (); Liu2017PRA (). However, the existing STA proposals involving non-Abelian geometric phases require, at least, four energy levels, making it technologically challenging for an experimental realization.

Here we propose an alternative way to achieve STA holonomic quantum computation (STAHQC) with only three energy levels instead of four, where any arbitrary geometric single-qubit operation can be realized in a single cycle of a non-adiabatic and non-Abelian operation. For the purpose of demonstration, we also report an experimental realization of this method using a Xmon superconducting qutrit, which is in a ladder structure (Figure 1a, b). In this method, the pulse shape for STAHQC can be designed without obeying the constraint imposed in NHQC. In this way, we not only can have a better noise robustness against control errors, but also have more flexible choices for performing pulse-shape optimization.

In the experiment, we constructed non-commuting holonomic gates from the STAHQC method, varying three independent parameters of the general SU(2) transformation group. Furthermore, taking into account of both control and environmental noises, the experimental results are in good agreement with our numerical simulations. Before optimization, the performance of NHQC and STAHQC are about the same; this is consistent when comparing ComNat () with the results of a recent experimental demonstration Abdumalikov2013 () of NHQC with superconducting qubits. However the approach of Ref. Abdumalikov2013 () is not universal; for many gates, it requires at least two cycles to construct, which takes a much longer time and makes the system more susceptible to noises. Additionally, the noise robustness of STAHQC can be significantly enhanced after optimization; the advantage of STAHQC over NHQC becomes more distinctive when the control error is increased.

Setting the stage.— Consider a three-level system, where the ground state and the second excited state are chosen as logic basis of a qubit, and , and the first excited state as an auxiliary state. The system is driven by a pair of microwave pulses whose frequencies are detuned from or by , and have time-dependent amplitudes and , and phases and (see Fig.1b). When the two-photon resonant condition is satisfied Chen2012PRA (); Giannelli2014RPA (); Kumar2016nc (); Xu2016nc (), under the rotating-wave approximation, the system Hamiltonian can be written as (with ): . Let us define a bright state, , where and . We shall keep and , hence to be time independent. The above Hamiltonian can then be expressed as:

 H0(t)=12(Ω(t)eiϕ—1(t)|b⟩⟨e|+h.c.)+Δ(t)|e⟩⟨e| (1)

where is the Rabi frequency of . The instantaneous eigenstates of are , , and , where is defined by . Note that while the two microwave pulses used for control can be fully specified by their amplitudes () and phases (), the equivalent set of control parameters, namely (), would be more convenient for our discussion later.

STA-based holonomic gates.— The central idea of STA Berry2009JPA () is to include an auxiliary term to the Hamiltonian, , such that the time dynamics is equivalent to that of adiabatic evolution, i.e., for each eigenstate of , one has , where for a cyclic evolution, . Here is the dynamic phase, and is the geometric phase. For our case, the following auxiliary Hamiltonian, , first obtained by Ref. Chen2010PRL (), can be applied to construct a 3-level STAHQC.

Now, let us consider a cyclic evolution of from to . During this interval, the eigenstates are varied in a cyclic fashion, which requires that . Additionally, we impose another constraint at the middle, namely , but it can be varied arbitrarily at other times. In this way, the eigenstate evolves from to , and back to . The corresponding evolution of is illustrated in Fig. 1c. On the other hand, the phase is varied in the following way: for , and for , where and are different constants.

As a result, the geometric phase, , resulted from such a cyclic evolution is given by, . Note that a dynamic phase also accumulates during the evolution, but it can be eliminated with a spin-echo pulse, i.e, a -phase shift of the microwave applied halfway () of the control sequence. Furthemore, the dark state is always decoupled from the system, as .

Consequently, in the subspace spanned by the two states of , and , the holonomy matrix associated with the above cyclic evolution operator is given by , which is non-diagonal in the computational basis ,

 Usub(θ,ϕ,γ)=eiγ2(cγ/2−isγ/2cθ−isγ/2sθeiϕ−isγ/2sθe−iϕcγ/2+isγ/2cθ) , (2)

where and . Alternatively, with , we can also write , which describes a rotation around the n axis by a angle, up to a global phase of . Since n and can be set to any desired values, the geometric gate can be utilized to construct an arbitrary single-qubit geometric gate (see Fig 1.d).

Experimental results and analysis. The three energy levels of our Xmon qutrit are characterized by, GHz, and GHz. The relaxation and dephasing times of the first and second excited states are s, s, s, and s, respectively. Level spacing of the qutrit can be fine tuned by a current on the Z control line. The control microwave pulses are applied to the qutrit through the XY control line. The qutrit is capacitively coupled to a resonator ( GHz) with a coupling strength of MHz, which is in turn coupled to a transmission line. In the dispersive readout scheme Jeffrey2014PRL (), the state of the qutrit can be deduced by measuring the transmission coefficient of the transmission line. More details about the sample can be found in Ref. CSong2017a ().

We first consider a set of gate operations with the following Rabi frequency and detuning: (i) for , we set and ; (ii) for , we set and . Note that the dynamic phase accumulated during the cyclic evolution is automatically canceled. In addition, we can further adjust the control parameters (,,) to realize arbitrary single-qubit gate operations.

In the first part, we experimentally verify the behaviors of the dark and bright states. The qubit is initialized in the ground state . For the realization of the Z gate, where we set , the ground state is the dark state, i.e., . Since the dark state is always decoupled, it should remain unchanged during the entire cyclic evolution (see Fig. 2a). Alternatively, if we set, and , it also yields a geometric Z gate. However, the initial state becomes the bright state instead, i.e., . It follows the evolution of without transition to other states (see Fig.2 b).

To demonstrate that an arbitrary SU(2) transformation can be achieved using our STA protocol, we experimentally verify that all the three parameters, , and , can be varied continuously and independently. We first apply the following gate, , to the initial state of , and investigate the final state as a function of , i.e., . This gate operation corresponds to a rotation along the axis by an angle of (see Fig. 3a). Then, we apply the following gate, , to the initial state , which corresponds to a rotation along the axis by an angle of ; at the end, the system returns to the initial state, as shown in Fig. 3b. Finally, we demonstrate the geometric phase is also continuously tunable by applying a gate to the initial state of . This gate operation is essentially a rotation along the -axis by an angle of , as plotted in Fig. 3c.

Overall, the fidelity of the quantum gate is characterized by quantum process tomography (QPT). Since our STA scheme can generate arbitrary single-qubit gates, we may use them throughout the complete process tomography, including (i) initial-state preparation, (ii) quantum-gate implementation, as well as (iii) final-state rotation for state tomography. For example, Fig. 3d shows the sequence of quantum process tomography using the following set of gates, , , , and , to generate four initial states, , and , and investigate fidelity of the gates , , and . The experimental results are shown in Fig. 4. The process fidelities for , and are , , and , respectively (see Fig 4. a-c). The numerical simulation by master equation, taking into account of dissipation, gives fidelities of , and , which are in good agreement with the experimental results. The major source of errors come from decoherence and the decay processes along with dynamical control errors supp ().

In summary, we have experimentally demonstrated single-looped single-qubit holonomic gates based on the technique of shortcut to adiabaticity, which is robust against control errors and environmental noise Chen2012PRA (); Du2016NatC (); Zhou2016np (). Moreover, the STAHQC approach demonstrated here can be combined with optimal control DaemsPRL2013 (), to further enhance gate fidelity. It is also possible to extend it to construct two-qubit holonomic gate for achieving universal holonomic computation, by driving assisted coherent resonant coupling between interacting resonator and transmons Hong2017arX (); Egger2018arXiv (). Finally, we remark that, currently, the further applicability of the 3-level STAHQC method to superconducting qubits is limited by the spontaneous emission rate of the excited state of the superconducting qubit. However, this method should be of interest to other platforms such as nitrogen-vacancy centers, trapped ions, quantum dots, and nuclear magnetic resonance, etc.

This work was supported by Natural Science Foundation of Guangdong Province (2017B030308003) and the Guangdong Inno- vative and Entrepreneurial Research Team Program (No.2016ZT06D348), and the Science Technology and Innovation Commission of Shenzhen Municipality (ZDSYS20170303165926217, JCYJ20170412152620376). We particularly thank Prof. Haohua Wang at Zhejiang University, where all the experimental data were taken, for providing access to the experimental facilities, as well as the valuable discussions and comments on the manuscript.

## Appendix A Supplementary Material

### a.1 A. Dynamical contributions to the non-Abelian STA holonomies

The transmon(Xmon) Koch2007PRA (); Abdumalikov2013Nature () can be regarded as a slightly anharmonic oscillator, which leads predominantly to allowed microwave transitions between neighboring energy levels. In other words, when a drive applied to a specific transition we interested, it also induced other unwanted transitions with quantum numbers differing by one. This unwanted additional transitions dynamical may contribute to the evolution and degrade the STA holonomic gates fidelity. Our analysis about the dynamical contribution is very similar as the work on non-adiabatic non-Abelian protocol in ref Abdumalikov2013Nature (). The complete hamiltonian including unwanted terms is Koch2007PRA ()

 ˜H(t) = N−1∑i=0ℏωi|i⟩⟨i| (3) + ∑j=0,1N−2∑i=0ℏΩj(t)2(|i⟩⟨i+1|e−iωd,j(t)t+h.c)

where is the number of Xmon levels with energies to be included in the model. denote the amplitude of the time dependent drive at frequency , which is slightly detuning with the transition. Ignoring all rapidly oscillating terms in the interaction picture, we obtain the hamiltonian in Eq. (1) of the main text.

We take the state transferring from state to state as example to show the influence of dynamical terms for our STA holonomic process, and the state density dependence on X gate evolution time is shown in Fig.1, N = 5 transmon levels are taken into account. If dynamical terms isn’t taken into consideration, the system Hamiltonian is as Eq. (1) in the main text, and the qubit state will completely transfer to at the end of the X gate evolution with the auxiliary sate remains unpopulated and the off-diagonal elements of state density being 0(dash lines in left and right figure). However, in the realistic situation, the dynamical terms will also contribute to the process evoltion, more specification, dynamical terms shift the state evolution trace with high frequency oscillations( solidlines in Fig.5 at here and Fig.2 of main text ). The auxiliary state and the higher level of transmon may not totally come back to unpopulated state, hence the final state will not be exact as ideal final state. the process fidelity for the state transferring from state to state by X gate is degraded to due to the dynamical contribution. Similar numerical simulation can be applied to initial states in with X gate, and the process fidelitites are and respectively. Hence the X gate process Fidelity degrades to 99.1% due to the dynamical contribution. Similar considerations can also be applied to analyse any interested gate operation, such as for gate H and , the gate fidelities are 99.0% and 98.2% respectively, which means the gates infidelity induced from the dynamical contribution is dependence on the gates parameters .

### a.2 B. Effect of decoherence

Dissipative processes including decay and decoherence always exist in any realistic quantum system, and these effects can spoil the novel properties of quantum system and degrade the process fidelity. To examine the impact of dissipation on our holonomies gates behavior, we apply the Lindblad Master equation to our system Hamiltonian. The Master equation is expressed as

 ˙ρ(t) = −iℏ[H(t),ρ(t)] (4) + ∑n12[2Cnρ(t)C+n−ρ(t)C+nCn−C+nCnρ(t)]

where is the density matrix of the considered system and is interaction Hamiltonian with or without dynamical terms. are collapse operators, and are the operators through which the environment couples to the system in , and are the corresponding rates. In our Xmon ladder qutrit system, collapse operators including for exited states decay while for exited states decoherence. The decay time of first excited state is and the second excited state is , and dephasing times are and ,respectively. The gate fidelities for and all degrade to about 99.3% due to the dissipative processes by the master equation numerical simulation without the dynamical contribution.

When both the dynamical terms and dissipative effect are taken into account, the X gate fidelity further degrades to 98.4% , gate degrades to 97.8% and H gate degrades to 98.4%. The result is consistent because the process infidelity results from both the dynamical terms and dissipative effect. Our error analysis suggests two approaches to improve the holomonic process fidelity, one way is to improve dissipative behavior of qubit, another one is to minimize the dynamical contribution. The latter one can be realized by utilizing higher nonlinear qubit structure such as flux qubit.

## References

### Footnotes

1. thanks: T.-X. Y. and B.-J. L. contributed equally to this work.
2. thanks: T.-X. Y. and B.-J. L. contributed equally to this work.

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