Fast non-Abelian geometric gates via transitionless quantum driving

A practical quantum computer must be capable of performing high fidelity quantum gates on a set of quantum bits (qubits). In the presence of noise, the realization of such gates poses daunting challenges. Geometric phases, which possess intrinsic noise-tolerant features, hold the promise for performing robust quantum computation. In particular, quantum holonomies, i.e., non-Abelian geometric phases, naturally lead to universal quantum computation due to their non-commutativity. Although quantum gates based on adiabatic holonomies have already been proposed, the slow evolution eventually compromises qubit coherence and computational power. Here, we propose a general approach to speed up an implementation of adiabatic holonomic gates by using transitionless driving techniques and show how such a universal set of fast geometric quantum gates in a superconducting circuit architecture can be obtained in an all-geometric approach. Compared with standard non-adiabatic holonomic quantum computation, the holonomies obtained in our approach tends asymptotically to those of the adiabatic approach in the long run-time limit and thus might open up a new horizon for realizing a practical quantum computer.

effectively reduces the complexity of the control parameters, which in turn simplifies the physical implementation of the TQDA-gates. Finally, we show how to realize the effective Hamiltonians in superconducting tunable coupling transmons.

Results
Degenerate TQDA. Let us consider a quantum-mechanical system characterized by a Hamiltonian λ ( ) H 0 that depends on a set of control parameters λ . For an eigenvalue (λ) E n of the Hamiltonian, we assume there exists a set of m n eigenstates ϕ | (λ)〉( = , , ) k m { 1 } k n n that span the proper subspace  n of Hilbert space . When the parameter λ is varied from an initial value λ i to a final value λ f , the adiabatic theorem entails that  n does not mix with other subspaces in  when the corresponding run time tends to infinity. The states ϕ  k n satisfying the Schrödinger equation along the path in parameter space are related to ϕ k n as ϕ ϕ = ∑  C k n l l n lk n with C n being unitary matrices. By substituting this relation into the Schrödinger equation, we arrive at  coincide, i.e., the dynamical phases acquired are the same, although the eigenstates ϕ k n of H 0 are not eigenstates of the transitionless Hamiltonian H. In cyclic evolution, a non-Abelian geometric phase, induced by the matrix-valued connection appearing in the additional transition-suppressing Hamiltonian, is picked up by each subspace. In the following, we demonstrate how to arrive at desired quantum gates from non-Abelian geometric phases under the degenerate TQDA.
Fast holonomic single-qubit gates. To implement our proposed holonomic gates, we consider a generic system consisting of four bare energy eigenstates , , a 0 1 , and e , coupled by external oscillating fields in a tripod configuration 7 . Here, , 0 1 are qubit states; e and a are excited and auxiliary states, respectively. By assuming that all fields are resonant with the transitions, the target Hamiltonian of such a system is written as where each f kl serves as a time-dependent control parameter for the ↔ l k transition in a frame that rotates with the fields. H 0 has two dark and two bright eigenstates, and the former eigenstates provide a proper subspace to encode a qubit. To obtain our desired holonomic gates, we evolve the subspace along some loops in parameter space. In the standard adiabatic scheme 7 , this task is accomplished by changing the control parameters slowly so that transitions between the dark and bright subspaces become negligible. In the following, we show explicitly how to speed up this process with the help of degenerate TQDA. Since the qubit information is stored in the dark subspace, there is no dynamical contribution. Therefore, the desired quantum holonomy is acquired at the end of the evolution in the same manner as in ref. 7.
We first construct a phase-shift gate = γ U e , with S being the surface enclosed by the path shown in Fig. 1(a). Thus, γ 1 is minus half the solid angle enclosed on the parameter sphere with polar angles θ and φ.
The shortcut to this adiabatic process is realized by adding an extra Hamiltonian term H P 1 to the target Hamiltonian H P 0 . It is evident that H P 1 relies both on the structure and rate of change of H P 0 . In order to simplify H P 1 , we propose evolving the system in three steps forming an "orange slice" path on the parameter sphere along which the connection A vanishes (see Fig. 1(a,b) and Supplementary information for details). First, ϕ = 0 while changing θ from 0 to π. During this step, terms containing ϕ  and ϕ sin vanish, and = + H H H P P P 0 1 becomes a Δ -like Hamiltonian, in which the Rabi frequency between 1 and a is θ/  2. Therefore, a ↔ a 1 transition is added to the system caused by the temporal change in θ. In the second step, we keep θ π = , but vary ϕ from 0 to ϕ 1 . Here, the total Hamiltonian is just a transition between e and 1 with detuning ϕ  . Thus, the detuning is caused by the temporal change in the parameter ϕ. Finally, we keep ϕ ϕ = 1 , while θ is decreased to 0. During this final step, H P is again a Δ -like Hamiltonian. When θ = 0, H P has returned to its initial form. As shown in Fig. 1(a), the acquired geometric phase γ 1 is equal to minus the azimuthal angle ϕ 1 , i.e., minus half the solid angle enclosed by the orange slice shaped loop. We note that both the detuning in the second step and the ↔ a 1 transition in the first and third steps are caused by the fast parameter change. Thus, H P 0 is recovered in the adiabatic limit when the rate of the parameter change tends to zero. The steps of the control Hamiltonian needed to implement U P are shown in Table 1.
Similarly, an adiabatic gate , together with two non-degenerate bright eigenstates. Similar to the adiabatic phase-shift gate, our qubit is encoded within the dark subspace. The desired holonomic gate is obtained after traversing a loop on the parameter sphere with polar angles θ and ϕ, resulting in γ 2 as the solid angle swept 7 .
Shortcut to U B is realized by choosing a geodesic triangle path on the corresponding parameter sphere, again in order to force the connection A to vanish (see Fig 1(c,d) and Supplementary information for details). The purpose of this choice is to simplify the implementation of the gate, just as the choice of an "orange slice" path simplified 1 exhibits a Δ -like structure involving the states , e 0 , and a . Thereafter, we keep θ unchanged but rotate ϕ to ϕ 2 . H B has again a Δ -like structure, but now involving , e 0 , and 1 . Finally, θ is decreased to zero along the geodesic curve and then ϕ is tuned to zero. The level structure for the last step is depicted in Fig. 1(d). Unlike the "orange slice" scheme, no detuning emerges in the "geodesic triangle" scheme. Therefore, all the photon-assisted transitions are resonant. The steps of the control Hamiltonian needed to implement U B are shown in Table. 1. We may note that all transitions between the four levels , , a 0 1 , and e are needed to realize an arbitrary single-qubit gate by combining U P and U B .
Fast holonomic two-qubit gate. Since the combination of the gates U P and U B allows the realization of an arbitrary single-qubit gate, it remains to construct an entangling two-qubit holonomic gate to realize universal quantum computation 28,29  By using the same "orange slice" scheme as in the realization of U P , a geometric phase is acquired by the state 10 while the other states remain unchanged. This amounts to applying an entangling gate = γ U e i 2 10 10 3 on the two-qubit space, where γ 3 is the geometric phase obtained after the cyclic evolution.
The single-and two-qubit gates in superconducting circuits. To realize our non-adiabatic schemes experimentally, we consider a superconducting tunable coupling transmon (TCT) 30 (see Fig. 2(a)), in which we assume that the transition frequency of the system and system-resonator coupling strength can be tuned independently. In the symmetric configuration, where a a a , = ± a , the system Hamiltonian is described by 30 , the energy level splitting is observed with increase in E I as shown in Fig. 3(a), giving rise to the four energy levels , , a 0 1 and e needed to implement our proposal.
When a time-dependent microwave field Φ( ) t impinges onto our four-level TCT (see with Φ ( ) 0 the microwave amplitude, ω kl a carrier frequency of the incoming microwave pulse that connects two desired eigenstates k and l , and . In Fig. 3(b) is a complex Rabi frequency. When we invoke the "orange slice" and "geodesic triangle" schemes, H TCT gives rise to H P and H B , respectively. We note that since our schemes relax the adiabatic constraint, the holonomic gates can be obtained at any speed. However, from a practical perspective, the desired gate time depends on details of the specific experimental setup such as the energy gaps and how fast we can vary the input microwave pulses.
To implement our two-qubit gate, we consider a superconducting architecture with two TCTs placed inside a coplanar resonator (see Fig. 2(b,c)). The direct coupling between the two transmons (labeled as l and r) is mediated by a cavity in the resonant regime 32 . With the built-in tunability of the transmons, we realize the Hamiltonian H 2 as follow. We assume that the transmon levels are anharmonic and each level transition can be tuned at will by

Step1
Step2 Step3 Table 1. Transitions needed in the "orange slice" scheme and the "geodesic triangle" scheme. The "orange slice" scheme and the "geodesic triangle" scheme are proposed to achieve the single qubit gates U P and U B , respectively. Each scheme contains three steps and needed transitions for each step are listed. The symbol ↔ i j denotes the transition between the states i and j .
matching with the resonator frequency via ac-Stark-shift fields 33

Discussion and Conclusions
We would like to put forward some remarks on the holonomic gates based on TQDA proposed here and the nonadiabatic holonomic gates proposed in ref. 10. On one hand, we notice that both schemes are genuinely non-adiabatic in that they can be performed at any desired rate. On the other hand, the holonomies obtained in the proposed TQDA approach tends asymptotically to those of the adiabatic approach in the long run-time limit 2,7 ; a feature which is not shared by the scheme in ref. 10. As a consequence, the TQDA-based gates have an immediate geometric meaning in terms of solid angles enclosed in parameter space of the underlying target Hamiltonian, while no such explicit geometric meaning is present in the other scheme. This also makes our scheme explicitly robust to parameter fluctuations that preserve the solid angle. In addition, our scheme has the advantage over standard adiabatic scheme in that non-adiabatic transitions, which inevitably degrades the control in any realistic finite-time implementation of the adiabatic gates 19 , are suppressed using TQDA.
In conclusion, we have proposed a non-Abelian generalization of the TQDA and show how it can be used for holonomic quantum computation. The key feature of this algorithm is to realize quantum holonomies that can be performed at arbitrary rate by applying an additional transition-suppressing Hamiltonian so that the path of the original eigensubspaces as well as the purely geometric nature of the resulting matrix-valued phases are preserved. Non-adiabatic TQDA-based single-and two-qubit gates are proposed in a four-level transmon and two four-level transmons coupled with a cavity, respectively. With high controllability 35,36 and scalability 37,38 of the superconducting circuit, our scheme can also be incorporated within a more general landscape where the transmons are  . embedded in a resonator lattice [39][40][41][42][43] . This scheme therefore opens up new experimental horizons towards a robust all-geometric high-speed large-scale quantum computation architecture.

Methods
Superconducting tunable coupling transmon. The four level system can be provided by a superconducting tunable coupling transmon (TCT) 30 (see Fig. 2(a)). To obtain the effective Hamiltonian H 0 TCT , we define