An Exact Approach to Elimination of Leakage in a Qubit Embedded in a Three-level System

Leakage errors damage a qubit by coupling it to other levels. Over the years, several theoretical approaches to dealing with such errors have been developed based on perturbation arguments. Here we propose a different strategy: we use a sequence of finite rotation gates to exactly eliminate leakage errors. The strategy is illustrated by the recently proposed charge quadrupole qubit in a triple quantum dot, where there are two logical states to encode the qubit and one leakage state. We found an su(2) subalgebra in the three-level system, and by using the subalgebra we show that ideal Pauli x and z rotations, which are universal for single-qubit gates, can be generated by two or three propagators. In addition, the magnitude of detuning fluctuation can be estimated based on the exact solution.

a clear view of the relation between the fluctuation and the error. In a more practical manner, we apply the approximation in the case when fluctuation is small and evaluate the corresponding error scales, supported by our numerical analysis. Moreover, our exact solution itself also provides an option to reckon noise strength.

Brief introduction of the Model
We start with a model Hamiltonian represented in the basis spanned by two logical states and one leakage state 19 , where we use the same notations as in the ref. 19 . ε q and g are independent control parameters for rotations with respect to the z and x directions. H leak stands for a coupling between the leakage state and one of the logical states, and ζ is the scaled leakage state energy in the absence of coupling 23,24 . A charge quadrupole (CQ) qubit is formed in three adjacent semiconducting quantum dots sharing a single electron and is embedded in the localized charge basis {|100〉, |010〉, |001〉}, where the basis states denote the electron being in the first, second or the third dot, respectively. The system Hamiltonian reads  where U 1,2,3 are the on-site potentials for the three dots. t A,B are tunnel couplings between adjacent dots, and ε d = (U 1 − U 3 )/2 (ε q = U 2 − (U 1 + U 3 )/2) denotes the dipolar (quadrupolar) detuning parameter. A new set of bases consisting of logical qubtis |C〉, |E〉 and a leakage state |L〉 is defined by 19,20 and a schematic diagram is presented in Fig. 1. The Hamiltonian in the new basis is transformed into . In case that t A = t B and ε d = 0 are satisfied, ∼ H CQ supports a decoherence-free subspace against uniform electric field fluctuations 20 .
In the triple quantum dot system, ε d corresponds to an average dipolar detuning control parameter. Although ε d is set to be zero, its fluctuation δε d breaks the DFS and causes leakage. It has been shown that the fluctuation of quadrupolar detuning control parameter is smaller than δε d and is thus neglected. Now we focus on the influence of δε d on the CQ qubit operations. Noise spectrum of δε d is dominated by low-frequency fluctuations which are slow in comparison with gate operations 21,22 . Therefore δε d is assumed to remain constant during a given gate operation 19,23 . As a result, unitary operators for x and z rotations can be given by x d x l eak d z q d z q l eak d q with arbitrary angles θ and ϕ. In the bang-bang limit where the control pulses switch instantaneously between two values, the angles are associated with the corresponding bang-bang gate time intervals t z and t x , which are θ = t x (2g/ℏ), ϕ = t z (ε q /ℏ). As shown by Eq. (5), rotation operators are obviously polluted by δε d . Below, we will explain our exact solution to this problem.

finite Rotations and exact elimination of Leakage
To suppress the fluctuation δε d in U x (g, δε d , θ), we start with the following three matrices It can be shown that their commutation relations satisfy indicating that these operators generate an su (2) algebra. An arbitrarily given finite rotation can be represented in an exponential form 25 where γ 1 , γ 2 , γ 3 are three continuous parameters and a linear combination of M i (i = 1, 2, 3) indicates a specific rotation axis and the corresponding angle. On the other hand, the finite rotation can also be expressed by three Euler's angles φ 1 , φ 2 and φ 3 , The relation between the two sets of parametrizations can be found by setting where the two sets φ i and γ i (i = 1, 2, 3) are in one-to-one correspondence 26  where n is a positive integer, I is the three dimensional identity matrix, and Δ(i) is a matrix whose ith diagonal matrix element is one and all other elements are zeros. Based on the above properties, we can derive an exact matrix equation representing a system of nine nonlinear equations, among which only three equations are independent. The independent equations determine φ 1 = φ 3 and t an 2 , sin 2 sin 2 (11) 1 2 Therefore angles φ 1 and φ 2 can be expressed in terms of α, a and b. Substituting Eq. (8) to U x (g, δε d , θ), we obtain ix 2 x 0 2 1d leak 0 x d 1 d leak 0 and eliminate the leakage H leak . Notice that constrains (13) hold for any magnitude of fluctuation δε d . Therefore, one can expand constrains (13) about any specific δε d . In fact, factor δε d can be completely removed in the Taylor expansion of the above constrains. For example, when δε d is relatively small compared to g, the Taylor expansion of constraints (13) in terms of (δε d /g) can be given by   where F 1 and F 2 denote the coefficients of (δε d /g) 2 in the expansion of β 1 and β 2 respectively. The last step in the above expression is also based on Taylor expansion. Seen from Eq. (17), if the x-rotation is performed using Eq. (16), the terms in relation with computational and leakage error scale as (δε d /g) 2 and (δε d /g) 3 respectively. Therefore, the probability of computational error and leakage error scale as (δε d /g) 4 and (δε d /g) 6 respectively 19 . It is worth mentioning that the implementation of objective rotation angle β′ 2 depends on θ which is the multiplier of g and t x . Then one can enlarge g and shorten t x to further suppress δε d /g for the same given β′ 2 . Besides above example, one can obtain expansions about other points rather than δε d /g = 0 by constraints (13). Due to the mathematical form of the constrains, a singular point exists. Eq. (16) indicate that the value of β ′ 2 can be arbitrarily chosen except π. When β π = ′ 2 , θ/4 = π/2 and tan(θ/4) goes to infinity. This can be solved by applying two sequences when β 2′ = π/2, which does not change the error scales.

numerical Results
We firstly show the parameter dependence of exact solution (13) by the surfaces in Fig. 2. We set g/ℏ = 3.0 GHz. t x varies from 0 to 2.0 ns and δε d /g varies from 0 to 0.6. One of our main concern is the ability of rotating a qubit about x-axis under constrains (13), which is given by the domain of β 2 . Seen from the lower panel of Fig. 2, which is obtained by the second constrain (13), β 2 exactly changes from 0 to 2π when δε d = 0 (left subfigure in the lower panel). For the case when δε d is bigger than 0, such as 0.6g/ℏ (right subfigure in the lower panel), one can only obtain a good approximation of the δε d = 0 case when t x is small. Besides, from the upper panel which is obtained by the first constrain (13), we can see that the control parameter β 1 is also insensitive to noise, except the neighbour area of singular point t x = 1.05 ns. The left (right) subfigure of the upper panel shows a sectional view of the case when δε d /g = 0 (0.6). In general, one can conclude from the above results that the circuit is insensitive to www.nature.com/scientificreports www.nature.com/scientificreports/ noise when control time is short. Also, an x-rotation of the CQ qubit through an arbitrary angle can be performed under constrains (13) with one or two sequences (14).
Next, we show a numerical analysis of the leakage errors of our whole x rotation scheme, with the probability of leakage errors defined by P LC,LE = |〈L|U|C, E〉| 2 19 . We consider the cases of different types of system noise, i.e., different δε d s, and the results are given by Fig. 3. g/ℏ is also set to be 3.0 GHz and β 2 is set to be π/2. The control parameter β 1 and θ of the circuit (14) applied in all the cases are obtained by solving constrains (13) under above conditions together with the assumption that δε d /ℏ is 0.3 GHz, no matter how it is set in the simulation of a specific evolution. The leakage probabilities are obtained from circuit (14) under quasistatic noise assumption, when system δε d /ℏ (the one used for the simulation of the evolution by the circuit) is 0.1, 0.3, 0.5, or randomly varies from 0.1 to 0.5 GHz in a multi-channel case. The results of δε d /ℏ = 0.1, 0.3, 0.5 GHz are represented by dash-dotted, triangle-dotted, and dashed curves respectively. The solid lines display the results of leakage probabilities of the multi-channel case. In such a case, more than one noise channel is calculated. The δε d /ℏ of a certain channel is static, but randomly varies from 0.1 to 0.5 GHz for different channels. The solid curves in Fig. 3 are obtained by averaging over 100 noise channels. The final leakage probabilities P LC (red curves) and P LE (blue curves) are both below 1.43 × 10 −4 . The infidelity 1 − F (F is the average gate fidelity 27,28 ) in 2D logical subspace of above cases are below 9.76 × 10 −6 which gives the order of the computational error. For this part of results, two conclusions can be drawn. One is that the effectiveness of the circuit (14) does not depend upon an exact knowledge of the noise amplitude. This is indicated by the good performance of the circuit (14) whose parameters are obtained by assuming a fixed value of δε d /ℏ (0.3 GHz), rather than directly using the noise information of corresponding numerical simulation of evolution (system δε d /ℏ = 0.1, 0.5 GHz and random). As for other assumptions of the value of δε d /ℏ, one can also adjust the control parameters of circuit (14) properly according to constrains (13) for a required elimination of leakage. The other is that the changing character of the leakage probabilities of the multi-channel case coincides with that of the single channel case whose system δε d /ℏ is set to be the average of the random one, indicating that multi-channel average will not disturb the effectiveness of the whole process. Therefore, one could expect a stability of circuit (14) in the potential applications.  www.nature.com/scientificreports www.nature.com/scientificreports/ A noise-free version of constrains (13) requires Taylor expansion. So we numerically analyze an example of such approximation. We consider circuit (14) under Eq. (16). β′ 2 is set to be π/2. In order to evaluate a case when δε d /g is not relatively big, we set g/ℏ = 0.5 GHz. By replacing the β 1 and β 2 in circuit (14)  These numbers support the effectiveness of the approximation. Also, we consider the case when δε d /ℏ in circuit (14) changes with time frequently. The simulation is performed by decomposing each exponential operator into a sufficient large number of pieces and δε d /ℏ randomly varies from piece to piece in the range of 0.15 to 0.45 GHz.
The final values of average P LC,LE are 7.85 × 10 −4 and 1.05 × 10 −3 and the infidelity is 2.92 × 10 −4 , which also shows a insensitivity to the noise.

estimation of noise Strength
Based on our exact formula, we can estimate the strength of δε d . For a CQ qubit under noise δε d , an ideal x rotation with angle β 2 is generated by experimental parameters β 1 , θ, g and δε d in terms of the constraints (13). In semiconducting quantum dots, gate operations are implemented by microwave pulses so that θ can be modulated by the pulse width, and g is determined by tunnel couplings t A,B . The spectrum of the noise δε d in range of 5 kHz to 1 MHz has been shown by Hahn echo curves 21 . Here our derivation suggests a new perspective to look into the noise δε d . An estimation of δε d can be done by following steps. (i) Prepare an initial state, for example |C〉.
(ii) Perform the three operations on the right side of Eq. (14) with given g, β 1 and θ which has no limitation other than the first Eq. (13). A good approximation can be obtained when g is set to be sufficiently large, such as the first Eq. (16). The resultant operation in the logical subspace is an ideal x rotation. (iii) Measure the output state, and then β 2 can be given. (iv) Substitute g, θ and the measurement result of β 2 to the second Eq. (13), then δε d is estimated. Our analysis in the previous sections indicates that the first Eq. (13) is mainly in relation with leakage error. So the estimation based on the second Eq. (13) in the logical subspace is little affected by the imperfect control of β 1 . In experiments on semiconducting quantum dots, state initialization and readout take about 4 ms to 5 ms, and state manipulation needs about 1 ms 21 . Therefore our estimation is allowed to be performed and repeated for several times and an effective strength curve of δε d in the time domain can be concluded.
www.nature.com/scientificreports www.nature.com/scientificreports/ conclusion We provide an exact solution to elimination of leakage errors in a three-level quantum model using simple circuits of gates. The model comprises of two logical states and a leakage state, which can be used to describe a triple quantum dot system supporting a DFS. Encoding qubits in a DFS is a well-known strategy in error suppression for quantum computation, which attracts significant attentions because of its minimal overhead requirements. The concatenation of DFS and the circuits promises to give this approach a twofold resilience, against decoherence and stochastic leakage errors. In comparison with the previous work 19 , our formalism is based on finite rotations and su (2) subalgebra. Reasonable approximations for application can be obtained from the formula. For example, one will trace back to the results in ref. 19 by assuming the noise is relatively small. We numerically analysis the parameter dependence of the exact solution, especially the affection of the noise. Numerical simulation shows that the performance of our circuits does not rely on an accurate knowledge of the noise, and is excellent in a multi-channel model when the noise strength randomly varies from channel to channel, indicating the stability of these circuits. Furthermore we propose an estimation of dipolar detuning control fluctuation to extract strength information of noise. The feasibility of our approach is ensured by the development of sophisticated experimental techniques 19,20 .

Data Availability
The datasets generated and analysed during the current study are available from the corresponding author on reasonable request.