High-fidelity quantum gates via optimizing short pulse sequences in three-level systems

We propose a robust and high-fidelity scheme for realizing universal quantum gates by optimizing short pulse sequences in a three-level system. To alleviate the sensitivity to the errors, we recombine all elements of error matrices to construct a cost function with three types of weight factors. The modulation parameters are obtained by searching for the minimum value of this cost function. The purposes of introducing the weight factors are to reduce the detrimental impact of high-order error matrices, suppress population leakage to the third state, correct the operational error in the qubit space, and optimize the total pulse area of short pulse sequences. The results demonstrate that the optimized sequences exhibit strong robustness against errors and effectively reduce the total pulse area. Therefore, this work presents a valuable method for achieving exceptional robustness and high speed in quantum computations.

As a promising tool, the composite pulse (CP) technique [54][55][56][57][58][59][60][61][62][63][64][65] has come into being and provides sufficient support for the realization of reliable quantum computations in three-level systems.The CP sequence, initially introduced in nuclear magnetic resonance [66,67], refers to a train of pulses with precisely designated phases.In general, these sequences are constructed to compensate for deviations in physical parameters through some controllable variables, such as phase, detuning, and duration.Due to the features of robustness against systematic errors and leakage suppression to undesired states in the three-level system, this technique is now widespread in the control of indirectly coupled qubit [68][69][70][71].
Until now, there are still some challenges for the CP design in three-level systems.Previous approach [72] suggests the nullification of error matrices in the propagator to enhance robustness, which can be further promoted by introducing more pulses.Such approach also implies that a longer duration is always required for the sequence when pursuing better robustness against errors, particularly unfavorable for physical systems with short coherence time.Hence, it is necessary to properly optimize the total duration of sequences on the premise of ensuring robustness.To this end, one can increase the number of modulation parameters instead of pulses, as was done in this work.Nevertheless, an analytical solution is scarcely possible for multiple modulation parameters.A feasible strategy is to exploit the optimal control theory [73][74][75][76].This theory aims to identify suitable control protocols for guiding a physical system to achieve the specific goal via minimizing cost functions [77][78][79].Nevertheless, such functions are currently employed for the optimal CP sequence design in two-level systems [80].Especially, it is useless to construct the cost function for three-level systems by simply generalizing from a two-level one, due to the existence of the third state.Thus, it becomes quite urgent to design an appropriate form of the cost function while taking into account population leakage in three-level systems.
In this paper, we propose a general framework for achieving robust quantum gates in three-level systems by optimizing short pulse sequences.This issue of reducing the impact of error matrices on the composite propagator is converted into minimizing a cost function.In particular, different forms of cost functions, reflecting in three types of weight factors, produces different optimal sequences.The uses of weight factors are as follows: (i) mitigate the impact of high-order error matrices to ensure that the optimal solution primarily addresses the reduction of low-order error matrices; (ii) adjust the proportion of different elements in the error matrix of each order, allowing us to suppress the operational error in the qubit space or population leakage to the third state, or both; and (iii) design a penalty coefficient to optimize the total pulse area.The results indicate that the optimal sequence must work excellent in suppressing population leakage to the third state and correcting the operational error in the qubit space in order to implement high-fidelity quantum gates.Moreover, the penalty coefficient in the cost function can effectively reduce the total pulse area while maintaining a similar robustness.Finally, we construct a three-pulse sequence to demonstrate a greater tolerance range in achieving high-fidelity quantum gates.
This paper is organized as follows.In section 2, we introduce the physical model and develop the general design method for high-fidelity universal quantum gates by tailoring different cost functions.In section 3, we take the two-pulse sequence to exemplify the optimization process of short pulses in detail.In section 3.2, we further optimize the three-pulse sequence and compare it with previous sequences.The conclusion is given in section 4.

Preliminaries
In this section, starting with the physical model, we present the construction of the composite propagator and explain how to transform the design of high-fidelity and robust quantum gates into a problem of finding the minimum value of the cost function.In particular, the cost functions with three different types of weight factors are investigated at length.

Physical model
Consider a Λ-type three-level system with two lower states |1⟩ and |2⟩, and an excited state |3⟩, driven by two external fields that simultaneously excite the transitions |1⟩ ↔ |3⟩ and |2⟩ ↔ |3⟩, where the coupling strengths are denoted as Ω and λ, respectively.The phases ϕ and φ are imposed on the corresponding external field, leading to Ω → Ωe iϕ and λ → λe iφ .Meanwhile, the three-level system is supposed to work in the two-photon resonant regime.In the interaction picture, the Hamiltonian of this system under the rotating-wave approximation reads (h = 1) where ∆ is the single-photon detuning.If all parameters are time-independent, the propagator at the pulse duration T reads (up to a global phase) with ) sin 2 θ, ) sin 2θ, ) sin 2θ, where the pulse area A = Ω g T, Ω g = √ Ω 2 0 + ∆ 2 , Ω = Ω 0 sin θ, λ = Ω 0 cos θ, and δ = ∆T/2.We can see from equation ( 2) that all physically adjustable variables are the detuning ∆, the coupling strengths Ω and λ, the phases ϕ and φ, and the pulse duration T. Without loss of generality, we set Ω 0 as units hereafter, and thus only five variables are freely modulated.For simplicity, we label the propagator (2) as U(p) and the modulation parameters as p = {A, ∆, θ, ϕ, φ}, where the pulse duration T is replaced by the pulse area A.
A train of N pulses, each pulse containing five modulation parameters pn = {A n , ∆ n , θ n , ϕ n , φ n }, fabricates the composite propagator Obviously, the composite propagator (3) depends on all modulation parameters {p 1 , • • • , pN }.This allows us to appropriately alter the modulation parameters to achieve the desired quantum gate in a robust way.Note that the focus of this work is to implement high-fidelity quantum gates by optimizing short pulse sequences, and thus the pulse number N is merely chosen two or three here.For a larger N, our optimal design method is also applicable.Assume that the error occurs in the pulse area and is the same for each pulse, i.e.A n → A n (1 + ϵ).This assumption is reasonable, since the pulses produced in experiments are usually imperfect but identical.Then, the propagator U n (p n , ϵ) of the nth pulse can be expanded into a Taylor series where u (n) 0 is the error-free propagator, and u represents the lth-order error matrix, l = 1, 2, • • • .As a result, equation (3) can be simply recombined by a polynomial, where is the accurate composite propagator in the absence of the pulse area error, and m is the mth-order error matrix for the N-pulse sequence, m = 1, 2, • • • .To provide a clear understanding on the origin of error matrices, we present a schematic diagram in figure 1, where different color arrows represent the source of each order error matrix.
For each order error matrix U (N) m , m = 1, 2, • • • , the expression can be further written as Here, we call the elements U m,32 as leakage terms.Physically, the qubit terms refer to the operational error in the qubit space, and the leakage terms indicate the population leakage to the third state.The existence of leakage terms means that the qubit Figure 1.Schematic diagram for the source of each order error matrix in a two-pulse sequence.When each pulse is truncated to the second order, the highest order of error matrices is no more than 4. The green, yellow, blue, red, and violet arrows point out the source of the zero-, first-, second-, third-, and fourth-order error matrices, respectively.space is not completely decoupled from the third state.Because U (N) m,33 refers to the operational error in an irrelevant space, without loss of generality, we omit it in U (N) m hereafter.Suppose that we intend to implement a universal quantum gate in the three-level system, and its form in the basis {|1⟩, |2⟩, |3⟩} can be expressed as where Θ is the rotation angle, γ and η are relative phases, and G is a global phase with respect to the qubit space {|1⟩, |2⟩}.To implement this quantum gate with high-fidelity, the error-free composite propagator (6) must equal to R, i.e.
Then, to reduce the unfavorable impact on the error-free composite propagator (6), we can nullify the error matrix elements U (N) m,ij in equation (7).Namely, the equations to be solved read In this way, the N-pulse sequence with modulation parameters satisfying equations ( 9) and ( 10) can implement the target gate in a high-fidelity and robust way.
Previous schemes [68,[70][71][72] suggested using the phase as the sole modulation parameter to eliminate these error matrices.Nevertheless, this is extremely challenging for short pulse sequences, after considering the fact that the number of modulation parameters usually requires an order of L 2 to nullify all elements of a L-dimensional error matrix.That is, short pulse sequences cannot offer enough adjustable phases to nullify error matrices.For example, in a three-pulse sequence, four adjustable phases can only eliminates either qubit or leakage terms in the first-order error matrix [72], implying that solutions for equations ( 9) and (10) do not exist even if we just intend to completely nullify the first-order error matrix Alternatively, one may increase the types of modulation parameters to overcome this difficulty for short pulse sequences.As we have proven before that a single pulse can provide five free variables {A, ∆, θ, ϕ, φ}.Thus, each order error matrix seems to be fully nullified by only using two pulses.Nevertheless, this is not the case, because not all modulation parameters are useful for solving equation (10).In other words, the effective modulations parameters offered by a short pulse sequence may be insufficient.As a result, multiple types of modulation parameters of the short pulse sequence still fail to be designed by solving equations ( 9) and (10).
From a physical point of view, if all elements in error matrices are small enough, the robustness against errors can be highly improved as well.This means that, instead of strictly satisfying equation (10), we can focus on reducing the value of the left side of equation (10) to a very small amount.In this way, we can find many alternative solutions for the modulation parameters.It is worth stressing that the unfavorable impacts on the gate fidelity are particularly different for different orders of error matrices.This trait forces us to discriminatively treat those error matrices, which can be managed by constructing an appropriate form of the cost function.

Customized cost functions
First, we need to set up an indicator to quantify each order error matrix.We use f m to represent the sum of the square modulus of both qubit and leakage terms in the mth-order error matrix, i.e. with Obviously, f m = 0 if and only if the mth-order error matrix becomes a null one, m = 1, 2, • • • .When f q m is set to a low level, the operational error in qubit space is largely corrected, and the population leakage to the third state is sharply suppressed when f l m is small.Once both f q m and f l m are restricted to tiny values, the error matrix is very close to a null one and hence the unfavorable influence caused by error matrices on the error-free composite propagator can be alleviated.
Note that completely nullifying each f m is also impossible for short pulse sequences.Instead of nullifying them, we incorporate all f m (m = 1, 2, • • • ) to formulate a cost function.Then, by minimizing this cost function, we can obtain relatively small f m (m = 1, 2, • • • ).At the same time, we demand the minimum value of this cost function to be zero when error matrices are null.In this way, the optimal solution is compatible with the traditional one [72].On the other hand, different orders of f m have varying impacts on the fidelity of quantum gates.Specifically, the lower-order f m prefers to produce a greater unfavorable impact on the error-free composite propagator.Therefore, the cost function we build must possess the capability of distinguishing the contribution of different orders of f m in which lower-order ones require higher proportions.In other words, the contribution of low-order f m to the cost function needs to be enlarged, making the optimization process more focused on minimizing low-order error matrices.
To meet the above requirements, we can construct a cost function for the N-pulse sequence as follows where the highest order of the propagator for each pulse is truncated to K, and the weight coefficients a m must be positive.Indeed, the values of a m determine the contribution of each f m to the cost function, and we demand that a m monotonically decreases as the order of f m increases.In this way, the value of the cost function mainly depends on the low-order f m (m = 1, 2, • • • ) and the contribution of high-order ones can be gradually decreasing.From another point of view, the cost function we build in equation ( 13) can be regarded as a 'functional' of f m , and they are linearly dependent.Then, each f m naturally becomes extremely small if the cost function is minimum.Furthermore, we can individually optimize either qubit terms or leakage terms.To this end, we assign another type of weight coefficients to f m , and the form of f m in equation ( 13) is modified as where the expressions for the qubit term f q m and the leakage term f l m are given in equation ( 12), and the angle α is used to adjust the weight between qubit and leakage terms.Through altering this weight coefficient, the optimal sequence is designed for correcting the operational error in the qubit space, suppressing population leakage to the third state, or both.In the following, we briefly explain the physical effects for some special α.
(i) α = 0.Only leakage terms are involved in f m .As a result, the optimal sequence would show great ability of suppressing the population leakage to the third state.Although the system evolution is mainly confined to the qubit space, the gate fidelity may not necessarily improve significantly.This is because the operational error is not fully considered during the design process, resulting in the deviation of the rotation angle Θ in equation ( 8) and thus a degradation in the gate fidelity.(ii) α = π/4.In this case, each f m contains both qubit and leakage terms with equal weight.The optimal sequence simultaneously reduces the operational error in the qubit space and the population leakage to the third state.However, correcting the operational error and suppressing the population leakage cannot perform well in a short pulse sequence, because both qubit and leakage terms may not be particularly small under the limitation of a finite number of pulses.Consequently, the gate fidelity also sharply drops.
(iii) α = π/2.Here, each f m just contains all qubit terms.As a result, the optimal sequence works well in reducing the operational error in the qubit space, while the population leakage to the third state is completely ignored.When the population leakage is severe, the qubit space fails to be isolated so that we cannot robustly achieve high-fidelity quantum gates in the three-level system.Note that the robustness for this optimal sequence behaves well only when the qubit space is largely decoupled from the third state.
The value of α is not limited to the three cases mentioned above.Actually, it can be divided into two main ranges.When 0 < α < π/4, each f m enlarges the leakage terms and minifies the qubit terms.As a result, the optimal sequence has a greater capability to reduce the population leakage to the third state and slightly corrects the operational error in the qubit space.On the other hand, when π/4 < α < π/2, the qubit terms are accompanied by a large weight.This means that the optimal sequence outstands in correcting the operational error but performs scantily in suppressing the population leakage.Through flexibly adjusting α, we can design different optimal sequences to implement high-fidelity and robust quantum gates.How to choose a suitable α will be studied in detail in the next section using numerical calculations.
While the cost function ( 13) is essential for achieving remarkable robustness for the optimal sequence, some other issues need to be addressed either.For example, the total pulse area is not restricted in the optimization process, which may be larger to achieve better robustness.In fact, the optimal sequence should strike a proper balance between the total pulse area and robustness in consideration of a quantum system with short coherence time.To this end, we can introduce a penalty term into the cost function, i.e.
where the total pulse area A tot = ∑ N n=1 A n , and the positive number a ′ is the penalty coefficient.An appropriate value of a ′ in the cost function (15) not only ensures good robustness against errors, but also rightly optimizes the total pulse area.However, too small a ′ would not have much effect on the optimal solution.If a ′ is too large, it reversely penalizes the robustness instead.Consequently, the optimal sequence may have a small total pulse area but fail to provide strong robustness against errors.

Lagrange multiplier method
Next, we demonstrate how to numerically acquire the optimal solutions for the modulation parameters of short pulse sequences.When there are constraints on the arguments during searching the minimum value of the cost function, it is quite convenient to employ the Lagrange multiplier method that is well-known for its high efficiency in solving optimization problems [73,[80][81][82].The basic idea of this method is to transfer a constrained optimal problem into an unconstrained one.The Lagrange function, combining the cost function and the constraints, has the form where c r (r = 1, • • • , R) represent some specific constraints, and µ r is the corresponding Lagrange multiplier.We can easily yield the analytical expressions for the first partial derivatives of the Lagrange function L (N)  with respect to all modulation parameters, because the cost function in equation ( 16) is only composed of elementary functions with finite terms.Then, we vanish its first partial derivatives to obtain the following algebraic equations grad Note that equation ( 17) has at least one solution, since the cost function we construct is continuously smooth but not monotonic so that at least one extreme point exists for such function.
Actually, a solution that satisfies equation ( 17) is not necessarily an extreme point.It is simply a candidate for the minimum point of the cost function.In order to judge whether these candidate points are minima, one also requires to calculate the second-order condition of extrema, i.e. the Hessian matrix.Here, we are not going to deduce the expression for the Hessian matrix due to its complexity.Instead, we individually solve equation ( 17) enough times by using different initial values of the modulation parameters, because different initial values can make the numerical solutions converge to distinct extreme points.Upon a sufficient number of solves, we are able to gather all possible extreme points of the cost function, and then select the minimum one as the optimal solution of the modulation parameters.

Two pulses
We first take two pulses as an example to illustrate how to optimize short pulse sequences.By setting K = 1 in equation ( 4), the composite propagator of the two-pulse sequence reads Since an overall phase shift of all pulses does not change the robust performance, we choose the phases in the first pulse as ϕ 1 = φ 1 = 0 for simplicity.In this way, the modulation parameters become Assume that the objective is to implement a Hadamard gate, i.e.
According to equations ( 13) and ( 18), the cost function for the two-pulse sequence can be written as Without loss of generality, we set a 1 = 1.Given the prerequisite condition in equation ( 9), we can construct the Lagrangian function as follows, where F(ϵ) = |Tr[ P(U (2) (ϵ)H † ) P † ]|/2 is the fidelity of the Hadamard gate in the presence of the error ϵ, and P = ∑ 2 i =1 |i⟩⟨i| represents the projection operator.First, we investigate the influence of different values of α in f m (m = 1, 2). Figure 2 shows the gate fidelity as a function of the pulse area error for some special α, where the modulation parameters are presented in table 1.The results indicate that changing the value of α can generate the optimal sequences possessing different robust characteristics.
As illustrated by the yellow dotted curve in figure 2, the gate fidelity is highly sensitive to the pulse area error, similar to the case of the single pulse.The reason is that the optimal solution cannot make the values of leakage terms in error matrices U (2) m very small when α = π/2.Accordingly, the third state cannot be completely decoupled from the qubit space, causing severe population leakage.Therefore, merely minimizing the qubit terms in the error matrix should be avoided in the design of the optimal two-pulse sequence.
The red dotted curve in figure 2 demonstrates that the robustness has a slight improvement comparing to the single pulse, because the optimal sequence can restrict the qubit and leakage terms in error matrices U to a low level when α = π/4, simultaneously correcting the operational error and suppressing the population leakage.For π/4 < α < π/2, the population leakage cannot be considerably suppressed such that the operational error is always corrected in a non-isolated qubit space.As a result, we are still unable to robustly achieve the high-fidelity Hadamard gate on this occasion even if the operational error is reduced to a great extent.
When α = 0, the population leakage is sharply suppressed and thus the system evolution is almost confined to the qubit space.As shown by the purple solid curve in figure 2, the gate fidelity is indeed insensitive to the pulse area error.This result reveals that it is priority to avoid the transition between the qubit space and the third state when implementing a robust quantum gate in the three-level system.Due to  completely ignoring qubit terms in the case of α = 0, a slight deviation may occur in the rotation angle given by equation (8).Therefore, designing appropriately small values for α as well as a 2 is necessary to obtain a better robustness for the optimal sequence, and we next study this issue in detail.
To quantify the robustness for the optimal sequence, we define the high-fidelity width as follows, where Through minimizing the cost function with different a 2 and α to achieve different optimal solutions, we display the corresponding high-fidelity widths in figure 3. We can see in figure 3(a) that a large a 2 is hardly to realize a large high-fidelity width W. This is because large a 2 enhances the contribution of f 2 to the cost function, preventing from a preferential reduction of f 1 .Consequently, the first-order error matrix would significantly decline the gate fidelity.As a 2 decreases continuously, f 1 gives a more significant contribution to the cost function.The optimal solution tends to reduce the influence of the first-order error matrix, resulting in a large W; see figures 3(b)-(d).Nevertheless, too small a 2 is also not conducive to enlarging the high-fidelity width W, because the cost function almost equals to f 1 when a 2 approaches 0 such that there is no guarantee that f 2 can take a relatively small value during optimization.As a result, the two-pulse sequence may not be optimal in terms of robustness, cf W = 0.1378 (W = 0.1267) at α = π/25 and a 2 = 0.01 (a 2 = 0) in figure 3(b) (figure 3(d)).
On the other hand, it is better to take α a small value than 0, as demonstrated in figures 3(b)-(d).When α is not equal to 0 but much less than π/4, the optimal sequence can effectively suppress the population leakage to the third state and sightly correct the operation error in the qubit space as well.Note that not all α within 0 < α < π/4 is applicable.In figures 3(b)-(d), the high-fidelity width W gradually decreases when α Table 2. Modulation parameters for the f-STIRAP and STA methods to achieve the Hadamard gate, where ∆ = 0 and ξ = π/4.is close to π/4.Under this circumstance, the optimal sequence fails to sharply suppress population leakage, causing a rapid drop in the gate fidelity.Thus, α can be properly chosen from [π/30, π/8] for the optimal two-pulse sequence.For a comparison with the current optimal sequence, two well-known control methods, the fractional simulated Raman adiabatic passage (f-STIRAP) [83] and shortcuts to adiabaticity (STA) [84], are also employed to implement the Hadamard gate, where the expressions for modulation parameters are presented in table 2. In figure 4, we plot the fidelity F as a function of the pulse area error ϵ by these methods.

Ω(t) λ(t)
It is easily found from tables 1 and 2 that the total pulse area of the f-STIRAP method is about three times than that of the optimal two-pulse sequence.Even though the f-STIRAP method is quite robust against the pulse area error (see the red dotted curve in figure 4), the highest fidelity is only on the order of 0.999.This is due to the fact that the validity of this method heavily depends on the adiabatic condition.Ideally, the adiabatic condition is perfectly satisfied only if the pulse area tends to infinity [83].Under realistic conditions with a limited pulse area, regardless of the pulse area error, there are still a few transitions between different eigenstates in the system, and this unfavorable effects can be alleviated by largely prolonging the pulse area.Therefore, we can only obtain a robust Hadamard gate but not very high fidelity by using the f-STIRAP method.
Different from the f-STIRAP method, the STA method can be used to achieve a high-fidelity Hadamard gate, and a significantly smaller total pulse area confirms the acceleration of the adiabatic process.At the same time, this method possesses remarkable robustness against the pulse area error, as shown by the moderate high-fidelity width in figure 4(b).However, the robustness achieved by the STA method is still worse than the optimal two-pulse sequence, since the corresponding high-fidelity widths are W = 0.0456 and W = 0.1378.In addition, the pulse area of the optimal two-pulse sequence is about 24.2% smaller than that of the STA method, and the square waveform in the optimal sequence is also easy to realize.These results indicate that the optimal two-pulse sequence outperforms both the f-STIRAP and STA methods in terms of robustness and pulse area when implementing the Hadamard gate.
Furthermore, the two-pulse sequence with pulse area optimization can be obtained by incorporating a penalty term into the cost function.According to equation (15), the cost function can be designed as follows with the total pulse area A tot = A 1 + A 2 , and the optimal solution is also given in table 1. Hereafter, we refer to this sequence as the area-optimized sequence, and demonstrate the corresponding fidelity performance in figure 4. Compared to the optimal two-pulse sequence designed without penalty, the area-optimized one has almost the same robustness against the pulse area error under a smaller total pulse area.This can be also verified by the high-fidelity width, which are W = 0.1378 and W = 0.1365 for the optimal two-pulse and area-optimized sequences, respectively.Besides, the area-optimized sequence performs well when the pulse area error is positive, indicating a better resistance to the prolonged pulse area.

Three pulses
We next illustrate the construction of the optimal three-pulse sequence and make brief comparisons with other three-pulse sequences [69,72].Suppose that we intend to obtain a X gate with the following form Through setting K = 1, the composite propagator of the three-pulse sequence is truncated to the third order, i.e.
To guarantee that the composite propagator strictly equals to the X gate in the absence of the pulse area error, we demand the error-free fidelity F(0) = 1, where F(ϵ) = |Tr[ P(U (2) (ϵ)X † ) P † ]|/2.As a result, the Lagrange function becomes Figure 5 shows the fidelity F as a function of the pulse area error ϵ, where the modulation parameters are given in table 3.For briefness, the optimal three-pulse sequence is labeled as the X 3 sequence hereafter.
To make a comparison, we also plot in figure 5 the fidelity of the X gate by the three-pulse sequence using strength modulations [69], denoted as the X s 3 sequence.Notice that the gate fidelity experiences a significant drop around ϵ = 0, and the high-fidelity width is quite narrow, i.e.W = 0.0045.This means that the X gate achieved by the X s 3 sequence is not immune to the pulse area error.The reason can be found as follows.The X s 3 sequence [69] was designed for complete population inversion rather than the X gate.As a consequence, it is sufficient to focus on the population of the target state, and thus the phases of the final state are completely  ignored during the pulse design.Although we are able to properly shift an overall phase of the X s 3 sequence to achieve the X gate, the fidelity cannot maintain a high-level in the presence of errors due to unknown phase derivations of the final state.Hence, the X s 3 sequence is unsuitable for implementing robust quantum gates.As for the X 3 sequence, the phases are applied as the modulation parameters to effectively compensate for the unfavorable impact on the fidelity.Therefore, the current X 3 sequence outperforms the X s 3 sequence in terms of implementing the X gate in the three-level system.
Furthermore, figure 5 also shows the gate fidelity for the three-pulse sequence designed by phase modulations [72], denoted as the X p 3 sequence.We can see that the X p 3 sequence has a less robustness against errors than the X 3 sequence, where their high-fidelity widths are 0.1153 and 0.1686, respectively.In terms of the total pulse area, we find that the X 3 sequence is smaller (about 5% reduction) than the X p 3 sequence according to table 3.

Discussions
Indeed, the significant robustness against errors gives the credit to the efficient use of all physical quantities as modulation parameters in short pulse sequences.One vivid example is that the X p 3 sequence [72] does not reach the efficiency limit of robustness, since the modulation parameters except the phase have been predetermined.Intuitively, introducing more types of physical quantities (such as the detunings and coupling strength ratios) means that more modulation parameters can be used for error compensations to achieve better robustness, as adopted in the X 3 sequence.However, not all modulation parameters are effective, and some of them are redundant and have no effect on improving robustness.
Specifically, the pulse areas of all sub-pulses seem to be the same in the optimal two-and three-pulse sequences in tables 1 and 3. To confirm this, we use the two-pulse sequence to numerically investigate the minimum value of the cost function (20) for all region of the pulse areas.We plot in figure 6(a) the minimum value of the cost function (20) versus the pulse areas A 1 and A 2 , where the gray regions represent that the Lagrange function cannot converge to a specific extreme value after solving equation ( 17) for enough times.Namely, we are unable to find a minimum value of the cost function in gray regions.An inspection of figure 6(a) demonstrates that the minimum value of the cost function is roughly on the diagonal, i.e.A 1 = A 2 .To obtain the optimal solution more accurately, we restrict the region to 8π/5 ⩽ A 1 , A 2 ⩽ 2π, and then resolve the minimum value of the cost function in figure 6(b).It is clear that the optimal solution can be found only when the pulse area of two pulses area identical, i.e.A 1 = A 2 , implying that all pulse areas must be equal each other.Therefore, although we treat all pulse areas A n as free variables during the optimization, only one of them is actually valid and the rest have no effect on the improvement of robustness.Similar situation can also be found in the detuning, where they are equal, as shown in table 1.
It is interesting to study the robust performance for the optimal sequence when some types of modulation parameters are not free variables.To illustrate this, we fix one type of the modulation parameters in the cost function (20), and display the high-fidelity width W for different optimal sequences in figure 7. It is easily observed that different types of modulation parameters have different effects on the robustness for the optimal sequence.Specifically, fixing a phase hardly reduces the robust performance, as all W are equal to or almost close to the optimal value 0.1378.The reason is that the fixed phase can be regarded as shifting an overall phase to all pulses such that it does not change the robustness for the sequence.However, when other modulation parameters are fixed, the sequence can provide good robustness only if the fixed parameter happens to approach the optimal solution in table 1.This can be understood as follows.Fixing a specific parameter (i.e.A 1 , ∆ 1 , or θ 1 ) may change the propagator of the first pulse in an undesired manner, and thus the system cannot evolve along the intended path of resistance to errors.At this point, no matter how we optimize the modulation parameters, the leakage to the third state will increase significantly in the presence of errors, leading to a sharp drop in gate fidelity.
We next explain why we cannot directly generalize the cost function used in two-level systems [80].In the previous work [80], the cost function is designed for achieving robust population transfer in the two-level system.Because the system dynamics are confined in a two-dimensional Hilbert space and there are not population leakage during evolution process, only one weight factor is sufficient to characterize the cost function.If we simply employ the form of the cost function used in the two-level system [80], it is impossible to suppress the population leakage in the three-level system.Therefore, at least two different types of weight factors are required for the cost function in the three-level system.In particular, because the objective of this work is to achieve robust quantum gates rather than population transfer, there are many error terms with the same order and we have to classify them first before constructing the cost function, as shown in equation (11).Moreover, to properly optimize the total pulse area for short pulse sequences, we here introduce the third type of weight factor in the three-level system.Therefore, the cost functions designed in the current work are more elaborated and versatile comparing to the ones used in the two-level system [80].
Finally, we briefly discuss the experimental feasibility of this scheme.In the current scheme, the waveforms of all modulation parameters (including pulse areas, detunings and phases) are the square wave, which can be obtained by various technical means in different physical platforms.For example, in an atomic system driven by laser fields, pulse areas (detunings) is easily modulated through changing the intensity (frequency) of a tunable laser beam [85], while it is feasible to use an electro-optic modulator to impose the required phase shift [86].Another typical example can be found in the superconducting circuit [87,88].In each Xmon qutrit [89], coupling strengths (equivalently, pulse areas) and phases are varied over time when using pulses on the capacitively coupled microwave control line [90].Moreover, one can apply a flux-bias current on the inductively coupled direct current line to achieve the realtime control of the detunings [90].

Conclusion
We have proposed a systematic framework for the implementation of robust and high-fidelity quantum gates by optimizing short pulse sequences in three-level systems.The robustness for short pulse sequences results from a reduction in all elements of error matrices on the composite propagator, and the robust problem is transformed into a task of finding the minimum value of cost functions, where all modulation parameters are regarded as free variables.
We design three types of weight factors in the cost function.The first type aims at increasing the proportion of low-order error coefficients in order to mitigate the unfavorable impact of low-order error matrices on gate fidelity.The second type is introduced to adjust the proportion between qubit terms and leakage terms in each order error matrix.Finally, the third type focuses on properly optimizing the total pulse area on the premise of ensuring robustness.The results show that the optimal short pulse sequences perform well in suppressing population leakage, correcting the operational error in qubit space, and shortening the total pulse area, thereby realize high-fidelity quantum gates in a robust way.In particular, the efficiency limit in terms of the robustness can be successfully reached after introducing more adjustable physical quantities.Therefore, the proposed optimization design provides an useful method that strikes a balance between excellent robustness and high speed in achieving high-fidelity quantum computations.

Figure 2 .
Figure 2. (a) Fidelity F and (b) infidelity 1 − F of the Hadamard gate vs the pulse area error ϵ.The optimal modulation parameters in table 1 are determined after independently solving equation(17) with different initial values for 500 times.

Figure 3 .
Figure 3. High-fidelity width W vs different α and a2.Each optimal solution is obtained by solving equation (17) for 500 times.

Figure 4 .
Figure 4. (a) Fidelity F and (b) infidelity 1 − F of the Hadamard gate vs the pulse area error ϵ by different methods.The weight coefficients and the expressions for modulation parameters are given in tables 1 and 2.

Figure 6 .
Figure 6.Minimum value of the cost function C (2) vs pulse areas A1 and A2.Each minimum value is selected from all solutions obtained by solving equation (17) for 1000 times, where a2 = 0.001 and α = π/15.

Figure 7 .
Figure7.High-fidelity width W for the optimal two-pulse sequence against fixing different types of modulation parameters.Each optimal solution is obtained by solving equation (17) for 1000 times.

Table 1 .
Modulation parameters for the single pulse and optimal two-pulse sequences to achieve the Hadamard gate, where ϕ1 = φ1 = 0 and a2 = 0.01.

Table 3 .
Modulation parameters for different three-pulse sequences to achieve the X gate.