Experimental realization of non-adiabatic universal quantum gates using geometric Landau-Zener-Stückelberg interferometry

High fidelity universal gates for quantum bits form an essential ingredient of quantum information processing. In particular, geometric gates have attracted attention because they have a higher intrinsic resistance to certain errors. However, their realization remains a challenge because of the need for complicated quantum control on a multi-level structure as well as meeting the adiabatic condition within a short decoherence time. Here, we demonstrate non-adiabatic quantum operations for a two-level system by applying a well-controlled geometric Landau-Zener-Stückelberg interferometry. By characterizing the gate quality, we also investigate the operation in the presence of realistic dephasing. Furthermore, the result provides an essential model suitable for understanding an interplay of geometric phase and Landau-Zener-Stückelberg process which are well explored separately.


I. CYCLIC EVOLUTION AND GEOMETRIC PHASE
In the method presented in this study, the quantum gates are implemented based on the geometric phase accumulated in the cyclic evolutions. Specifically, a normalized state |Ψ(t)⟩ is cyclic in the interval 0 ≤ t ≤ τ if and only if where ϕ is a real number [1][2][3]. In a cyclic evolution, there are dynamic and geometric contributions for a total phase of ϕ = ϕ d + ϕ g . The non-adiabatic geometric phase ϕ g can be associated with the evolution path in the project Hilbert space, which is known as the Aharonov-Anandan (AA) phase [1].

III. STATE EVOLUTION OF A LANDAU-ZENER-STÜCKELBERG INTERFER-OMETRY
A physical implementation of the general method is designed for a solid state two-level system. The Hamiltonian can be expressed as i.e., Eq. (2) of the main text. The current technique is based on the cyclic evolution of the two orthogonal states |Ψ + (t)⟩ and |Ψ − (t)⟩, which are the instantaneous eigenstates of the system with energy eigenvalues E + (t) and E − (t), respectively.
The manipulation pulse used to achieve the Landau-Zener-Stückelberg (LZS) interferometry for the qubit is illustrated in Figure 2b. In the adiabatic-impulse model [4], the system dynamics can be observed as a sequence of free evolutions and non-adiabatic transitions localized at the anti-crossing point. The calculation is more conveniently performed in the basis of |0⟩ and |1⟩.
The free evolution from time 0 to t 1 (or from t 2 to τ ) can be described by a diagonal matrix with the form as follows: Between t 1 and t 2 the propagator can be written as follows: The Landau-Zener transition at the anti-crossing point can be described by a transfer matrix of the form as follows: where the Landau-Zener transition probability is given by P = exp(−2π∆ 2 /v ); v is the sweep velocity of the driving pulse through the anti-crossing point; ϕ s is the Stokes phase ; and Γ is the Gamma function [4].
Hence, the LZS interferometry can be treated as a successive unitary transformation between the initial state and the final state as follows: i.e., Eq. (3) of the main text, where T denotes a transposition of the matrix.
Then, we can obtain the total transformation matrix as follows: where the elements of the matrix can be calculated as follows: Furthermore, the phase factors acquired during each stage of the process can be denoted as follows:

IV. DESTRUCTIVE INTERFERENCE AND CYCLIC EVOLUTION CONDI-TION
The key idea is to apply quantum gates or unitary transformations based on the cyclic evolutions, which is referred to the process where the state of the system returns to its original state after the gate operations. Typically, not every closed loop in the parameter space can result in a cyclic evolution in the Hilbert space.
In the following, we will illustrate that the LZS interferometry can support a cyclic evolution under the condition of destructive interference. In particular, we consider the Because the time evolution of the state is explicitly provided in Eq. (S5), we find that the initial state can be transformed into the final state as follows: The qubit state energy evolution between the recurrent Landau-Zener transitions provides a phase-space analog to the two arms and the beamsplitters of an optical Mach-Zehnder interferometer. ∆ϕ is the phase difference between the two evolution paths [5,6]: one through the state |Ψ + (t)⟩, and the other through the state |Ψ − (t)⟩, Therefore the condition for the system returning to the initial state |Ψ + (0)⟩ after a period of evolution can be expressed as follows: ∆ϕ = 2nπ, where n is a positive integer. This condition signifies that the two trajectories interfere destructively [5,6] and enhance the probability of returning to |Ψ + (0)⟩ completely.
Consequently, the entire transformation operator reduces to a phase accumulation matrix, and Eq. (S7) can be rewritten as follows: Next, if a geometric process is realized through the destructive interference in a LZS interferometry, one can use the transformation matrix U (τ,0) or U (θ, ϕ) as defined above to describe the operation results. There are three input control parameters: the initial position ε 0 (0), the amplitude A and the width τ of the driven pulse. There are two output result quantities: angles θ and ϕ which are functions of the above control parameters. One can link the input parameters and output quantities as equation (5) in the main text. Thus one can determine the input parameters for the fixed output quantities θ and ϕ to engineer the system evolution such as some desired gate operations. In general, it is a routine but lengthy job to exactly solve the equation (5) because of the nonlinearity and three numbers of input variables in the problem.

V. DYNAMICAL AND GEOMETRIC COMPONENTS IN THE TOTAL PHASE
The total phase ϕ acquired in the evolution contains both dynamic ϕ d and geometric ϕ g components. On one hand, the dynamical phase can be easily calculated as follows [1][2][3]: On the other hand, the geometric phase for a two-level system is most clearly visualized by a closed path in the projected Hilbert space [1][2][3]7]. We define the polarization vector − → n = (sin Θ cos Φ, sin Θ sin Φ, cos Θ) on the Bloch sphere, which corresponds to the actual state evolution with a mapping relation − → n (t) = ⟨Ψ(t)| − → σ |Ψ(t)⟩. Thus, the geometric phase can be related to the solid angle subtended by the evolution curve on the sphere as follows [3]: where C is a closed path on the Bloch sphere.
In particular, we consider the cyclic evolution from the initial state as |Ψ + (0)⟩ = (sin Θ 1 cos Φ 1 , sin Θ 1 sin Φ 1 , cos Θ 1 ). In the second stage, − → n moves along the parallel to the point − → n (t 2 ) = (sin Θ 2 cos Φ 2 , sin Θ 2 sin Φ 2 , cos Θ 2 ), where Θ 2 = Θ 1 . Then, at time t 2 , the second Landau-Zener transition occurs, and the state vector returns to the north pole and remains there until τ . The path followed by − → n in the three-dimensional Bloch sphere is illustrated in Figure 2c of the main text.
When the state vector completes the closed circular path C, the geometric phase acquired by the state is where Θ c is the solid angle of the cone subtended by C at the origin.
In addition, it is interesting to study the effect of noise on the geometric phase and dynamical phase. On one hand, as illustrated in the Fig. S1 in the supplementary materials, the effect of fast noise on the geometric path can be averaged. However, slow noise can change the solid angle subtended by the path in the Hilbert space from one measurement to the next and cause error in the geometric phase. On the other hand, the noise will influence the dynamical phase in a different way as discussed in the main text.

VI. TOMOGRAPHY AND FIDELITY OF THE GATE OPERATIONS
We fully characterize the gate operations using state tomography [8] which has been implemented in various systems: photons [9], superconducting qubits [10][11][12], respectively. In our experiment, a pulse repetition rate of 30 MHz was chosen to ensure that the qubit is relaxed to the initial state and to carry out a sufficient number of projective measurements (10 7 times) for an adequate signal-to-noise ratio. The ensemble averaging of these measurements, in terms of the average charge detector conductance, allows us to directly obtain the probability of the qubit states. These measurements allow for the evaluation of the qubit density matrix as shown in Figure 3 in the main text.
Furthermore, it is beneficial to quantify the performance of the operations and evaluate the impact of noise effects. Thus, we can estimate the fidelity of each gate operation. We can adopt a gate fidelity defined as follows [8,13]: where U I is the ideal unitary transformation matrix of the gate operation; and U + P is the Hermitian conjugate of the physical unitary transformation matrix. The definition can be naturally expressed as follows: (S11) where ρ P (τ ) and ρ I (τ ) are the physical and ideal density matrices after the gate operation, respectively; and Tr represents the trace.

VII. THEORETICAL MODELING AND SOLVING MASTER EQUATION
To construct a realistic model for our experiment, we find two main sources in the noisy environment [14]: one is electron qubit-phonon coupling and its contributions can be introduced by the dissipation rate parameters Γ i (see below). The other is charge noise, well known to cause slow fluctuations of the electrical potential in semiconductor microstructures [15,16]. Here we include the influence of the slow fluctuations by adding a noise term ε 0 (t) + δε 0 to the Hamiltonian H [17][18][19][20]. Usually it leads in ensemble average to an inhomogeneous, Gaussian broadening of width parameter σ ε .
The time evolution of the qubit is given by solving the time-dependent master equations of the density matrix as follows: H is the Hamiltonian introduced in Eq. (S3), where the operators L j are called Lindblad operators [8,21,22] and they describes the dissipative effect of the environment noise on the system in the Born-Markov approximation. For the case of a two-level system, the incoherent processes, include relaxation and pure decoherence processes: Here σ − and σ + are spin ladder operators and σ z is the Pauli z matrix. They correspond to relaxation from the excited state to the ground state with rate Γ − , relaxation from the ground state to the excited state with rate Γ + , and pure decoherence with rate Γ φ .