Optimal diabatic dynamics of Majorana-based quantum gates
OOptimal diabatic dynamics of Majorana-based quantum gates
Armin Rahmani, Babak Seradjeh, and Marcel Franz Department of Physics and Astronomy and Advanced Materials Science and Engineering Center,Western Washington University, Bellingham, Washington 98225, USA Department of Physics, Indiana University, Bloomington, Indiana 47405, USA Department of Physics and Astronomy and Stewart Blusson Quantum Matter Institute,University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4 (Dated: August 24, 2018)In topological quantum computing, unitary operations on qubits are performed by adiabatic braiding of non-Abelian quasiparticles, such as Majorana zero modes, and are protected from local environmental perturbations.In the adiabatic regime, with timescales set by the inverse gap of the system, the errors can be made arbitrarilysmall by performing the process more slowly. To enhance the performance of quantum information processingwith Majorana zero modes, we apply the theory of optimal control to the diabatic dynamics of Majorana-basedqubits. While we sacrifice complete topological protection, we impose constraints on the optimal protocol totake advantage of the nonlocal nature of topological information and increase the robustness of our gates. Byusing the Pontryagin’s maximum principle, we show that robust equivalent gates to perfect adiabatic braidingcan be implemented in finite times through optimal pulses. In our implementation, modifications to the deviceHamiltonian are avoided. Focusing on thermally isolated systems, we study the e ff ects of calibration errors andexternal white and 1 / f (pink) noise on Majorana-based gates. While a noise-induced antiadiabatic behavior,where a slower process creates more diabatic excitations, prohibits indefinite enhancement of the robustness ofthe adiabatic scheme, our fast optimal protocols exhibit remarkable stability to noise and have the potential tosignificantly enhance the practical performance of Majorana-based information processing. PACS numbers: 71.10.Pm, 02.30.Yy, 03.67.Lx, 74.40.Gh
I. INTRODUCTION
Non-Abelian quasiparticles such as Majorana zero modes(MZMs) provide a promising platform for robust quantum in-formation processing . Qubits are encoded in the fermionparities of MZM pairs nonlocally and are protected from lo-cal environmental perturbations. Quantum gates are imple-mented as unitary transformations in the degenerate ground-state manifold via adiabatic braiding of the MZM worldlines.There has been remarkable progress in realizing MZMs re-cently and several experimental groups are working towardtheir braiding .One of the challenges for an adiabatic scheme is the finitetime of operations, causing inaccuracies in the unitary opera-tions due to diabatic excitations . Other sources of errorinclude quasiparticle poisoning , the on / o ff ratio of Coulombcoupling , and the information decay due to time-dependentperturbations . While topological protection can certainlydefend the system against many environmental perturbations,they do not make the system immune to errors like quasiparti-cle poisoning and high-frequency noise. These errors, in turn,limit the coherence time of the system, making it impossibleto completely eliminate the diabatic excitations by sacrificingperformance.A few recent studies have addressed the diabatic exci-tations. One idea is adding counterdiabatic terms tothe Hamiltonian of the system . This scheme requires areengineering of the devices and may pose experimental chal-lenges. Another idea is to minimize the diabatic excitationsby using smoother adiabatic protocols . While improving theaccuracy of the gates, this scheme still requires slower dynam-ics than the speed limit of the device. A third approach is through the optimal control of the quan-tum evolution , in which we relax the requirement of re-maining adiabatic during the evolution. Instead, we optimizethe time dependence of the Hamiltonian parameters so as togenerate the same final state as the perfectly adiabatic dynam-ics. This approach relies only on optimizing pulse shapes andcan be applied to existing experimental setups. It also real-izes the characteristic speed limit of the device, resulting inthe fastest possible information processing. Optimal controlhas been applied to the motion of one MZM along a one-dimensional wire but the full optimal creation of the sameunitary gates as the adiabatic braiding remains an open ques-tion.In this paper, we solve the optimization problem exactlyin the context of a simple e ff ective model of MZM braiding.More generally, we address the following key questions: Whatis the speed limit for generating the same unitary evolution op-erator as the adiabatic braiding for two MZMs in our device?How robust are these operations to calibration errors and noisypulses? By relaxing the constraint of adiabaticity during theentire process, we give up strict topological protection. Indeeda fully unconstrained optimal protocol, which only minimizesthe di ff erence between the evolution operator and a target uni-tary operator (corresponding to adiabatic braiding), would notutilize any of the topological features of the MZMs.In our optimal-control approach, we strike a balance be-tween performance and robustness, by imposing constraintsthat can improve robustness against environmental perturba-tions, which utilize the nonlocal nature of information storedin pairs of MZMs. We then explicitly examine the e ff ects ofvarious errors on our gates and demonstrate remarkable prac-tical advantages. For example, by calibrating our gates within a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug a few percent of the desired values of control parameters, theyoutperform a topologically protected adiabatic gate and arefaster by two orders of magnitude in the operation time. Theshorter times of the gate operations defend our optimal pro-tocols against decoherence sources like white noise and theexperimentally important 1 / f noise, allowing them to gener-ate accurate unitary operations in much shorter times.The remainder of this paper is organized as follows. InSec. II, we review an e ff ective low-energy model for the braid-ing of MZMs. In Sec. III, we first formulate the optimal-control problem and impose a constraint to increase the ro-bustness of the protocol by making use of the nonlocal natureof information stored in pairs of MZMs. We briefly review thePontryagin’s minimum principle and use it to obtain the opti-mal protocol that generates the target adiabatic unitary opera-tor exactly in a finite time. Section IV is devoted to an in-depthstudy of the robustness of our optimal protocol. In Sec. IV A,we first present a general noise model through a Taylor expan-sion of the control parameters “seen” by the system in terms ofthe control parameters imparted to the system experimentally.The leading error for a generic nontopological qubits is addi-tive, while for topologically protected Majorana-based qubitthe leading error is multiplicative. In Sec. IV B, we examinethe e ff ects of systematic calibration error. In terms of a mea-sure of distance between the unitary operators, the errors inoptimal protocol grow linearly from zero. With multiplicativeerrors calibrated within 2%, we find the optimal protocol canoutperform an adiabatic protocol that is two orders of mag-nitude slower. In Secs. IV C and IV D, we consider randomtime-dependent errors, i.e., noise, in the control parameters.The e ff ect of noise on adiabatic and optimal protocols is gen-erally found to be very similar. For white noise, the fast op-timal protocol outperforms all adiabatic protocols considered.For 1 / f (pink) noise, an adiabatic gate that is 10 times slowerstarts to perform better at relatively large noise strength. Wediscuss a technique for correcting the errors caused by the lim-itations of our e ff ective model in Sec. IV E and close the paperin Sec. V with a brief summary. II. EFFECTIVE MODEL OF MAJORANA BRAIDING
We start from a minimal e ff ective model of braiding, whichis relevant to the current experimental e ff orts involving one-dimensional topological superconductors, e.g., in the top-transmon . The Hamiltonian can be written in terms offour Majorana fermions as H ( t ) = i γ (cid:88) j = ∆ j ( t ) γ j , (1)where γ j = γ † j and { γ i , γ j } = δ i j . The coupling constant ∆ j represents the hybridization energy between γ j and γ . Weassume that all ∆ j can be tuned as a function of time within arange 0 (cid:54) ∆ j ( t ) (cid:54) D j . Defining two Dirac fermions c = ( γ + i γ ) / d = ( γ + i γ ) /
2, we can write the Hamiltonian in (c)(b)(a)
FIG. 1. (Color online) Optimal diabatic braiding of Majorana zeromodes: (a) the three-step braiding scheme for exchanging γ and γ ; (b) the optimal diabatic trajectories in the Bloch sphere for stepA (the black star indicates a switching from one axis of precessionto another); and (c) the bang-bang optimal protocol for the entireprocess (with D < D < D ). the basis (cid:16) | (cid:105) , d † c † | (cid:105) , c † | (cid:105) , d † | (cid:105) (cid:17) as a block-diagonal matrix, H e(o) = ∆ σ y ∓ ∆ σ x − ∆ σ z , (2)where σ x , y , z are the Pauli matrices. The upper (lower) block H e(o) has even (odd) fermion parity.The standard adiabatic scheme of braiding a MZM pairproceeds in three steps as depicted in Fig. 1. Starting with ∆ = ∆ = ∆ = D , so that γ and γ are decou-pled, we have two degenerate ground states, namely, | (cid:105) and c † | (cid:105) , with opposite fermion parity. In each step, we adiabat-ically turn on one coupling to its maximum value and turno ff another to zero. At the end of the three steps, we returnto the initial Hamiltonian, generating a unitary transforma-tion U = exp ( πγ γ /
4) in the ground-state manifold. In the( | (cid:105) , c † | (cid:105) ) basis, up to an unimportant overall phase, we canwrite U = diag(1 , i ), hereafter referred to as the target unitary.Our goal is to generate (up to a phase) the target unitary viadiabatic evolution of ∆ j ( t ) in a finite total time τ . The permis-sible diabatic protocols are bounded functions 0 < ∆ j ( t ) < D j over the time interval 0 < t < τ . The shortest time, τ ∗ , forwhich it is possible to generate the target unitary with a per-missible protocol sets the speed limit of the device. III. OPTIMAL CONTROL APPROACH
The most general diabatic protocols allow for the hybridiza-tion of all the MZMs, which destroys the topological pro-tection. As discussed in Refs. , adiabatic braiding is notprotected against perpetual dynamical perturbations speciallyif they have high-frequency components. External noise canalso result in an antiadiabatic behavior for very slow ramps(see also our Fig. 3 and its discussion). Moreover, the longtime scales required to create accurate gates with the adia-batic evolution, under which the operation enjoys topologicalprotection, may overshoot the coherence time of the system,which is limited by, e.g., quasiparticle poisoning. Topologi-cal protection, however, implies robustness to a wide range oflocal perturbations and, in particular, static calibration errors.One approach would be to altogether abandon the benefits ofinformation nonlocality and simply optimize 0 < ∆ j ( t ) < D j to minimize the di ff erence of the evolution operator and theadiabatic transformation U . However, this requires extremefine-tuning and exposes the gate operation to an array of un-wanted perturbations. Instead, we take a balanced approachthat utilizes the nonlocal nature of the qubits while improvingits operation speed by orders of magnitude.We constrain the optimal dynamics to track the same three-step dynamics as in the adiabatic scheme, without requiringadiabaticity during each step. For example, throughout stepA, we keep ∆ = ∆ and ∆ in their permis-sible range. Therefore, γ remains decoupled and the parityof the c fermion cannot be accessed by local environmentalperturbations. As the total parity is conserved, the parity ofthe d fermion is also locally inaccessible despite the gener-ation of diabatic excitations during the evolution. Similarly,in step B (C), we keep ∆ = ∆ =
0) and decouple γ ( γ ). This way, step A is protected from local environmentalperturbations. If we execute step A perfectly, then we have adecoupled MZM at the beginning and during step B, and stepB will be protected as well. The sacrifice to topological pro-tection originates from possible inaccuracies in step A, whichcan propagate to the next steps. By design, at the end of eachstep (but not during), the state of the system is optimized tomimic a fully adiabatic evolution.Focusing on step A with ∆ =
0, we have H e = H o =∆ σ y − ∆ σ z . Let us concentrate on one parity sector. Theinitial state is the ground state for ∆ = ∆ >
0, i.e., theeigenstate of σ z with eigenvalue + | + z (cid:105) . The target state atthe end of step A is the ground state for ∆ > ∆ = | − y (cid:105) . Denoting the total time with τ A , we minimize thefollowing functional of ∆ , ( t ): F A = − |(cid:104)− y | T e − i (cid:82) τ A [ ∆ ( t ) σ y − ∆ ( t ) σ z ] dt | + z (cid:105)| , (3)where T indicates time ordering. For a given τ A , the optimalprotocol yields the smallest possible F A . As we increase τ A ,this minimal F A decreases and eventually vanishes for a criti-cal time τ ∗ A , where the target state is prepared exactly.To compute τ ∗ A and the corresponding optimal protocol, weuse Pontryagin’s maximum principle . The principle statesthat for dynamical variables x = ( x , x , · · · x n ) and controlfunctions ∆ = ( ∆ , ∆ , · · · ∆ m ), evolving with the equationsof motion ddt x = f ( x , ∆ ) from a given initial conditions x (0)to a final set x ( τ ), the optimal controls, ∆ ∗ , which minimize acost function F [ x ( τ )] (any function of the final values of thedynamical variables), satisfy p ∗ · f ( x ∗ , ∆ ∗ ) = min ∆ (cid:2) p ∗ · f ( x ∗ , ∆ ) (cid:3) , (4) where p are conjugate dynamical variables with equations ofmotion, ddt p = − p · ∂∂ x f ( x , ∆ ) , (5)and x ∗ and p ∗ are optimal trajectories corresponding to ∆ ∗ .Furthermore, the boundary condition for p is set by the costfunction as p ( τ ) = ∂∂ x F [ x ( τ )] . As a consequence of Eq. (4), when f [and consequently p ∗ · f ( x ∗ , ∆ )] are linear functions of the controls ∆ , the opti-mal protocols are “bang-bang”: each of the control functions ∆ attain either its minimum or maximum allowed value atany given time (unless the coe ffi cient of a component of ∆ identically vanishes over a finite interval ).In the problem at hand, the real and imaginary parts of thewave function serve as dynamical variables, with equationsof motion given by the Schr¨odinger equation, which is in-deed linear in the controls ∆ j . Also, the cost function Eq. (3)depends only on the final wave function. Therefore, of allthe permissible functions ∆ , , the optimal protocols are dis-continuous functions that either vanish or attain their maxi-mum allowed value D , at any given time. We cannot have ∆ = ∆ = /
2, we can visualize the dynamics on the Blochsphere. If only ∆ ( ∆ ) is turned on, the quantum state pre-cesses around the y ( z ) axis in the Bloch sphere. If both cou-plings are turned on, it precesses around an intermediate axisshown in black in Fig. 1(b).We now identify the minimal path corresponding to thecritical time τ ∗ A . This simultaneously determines the optimalprotocol and the minimum required time for an exact statetransformation. As seen in Fig. 1(b), in the special case with D = D = D , the protocol is extremely simple. We turn onboth couplings to their maximum and a single precession pre-pares the target state exactly in a time τ ∗ A = π/ (cid:16) √ D (cid:17) . In thegeneral case, we only need one switching during the processas shown in Fig. 1(b). The general form of the optimal proto-col in a step that transfers a MZM from leg a to leg b is as fol-lows. If D a (cid:54) D b , we first switch on ∆ a = D a while keeping ∆ b =
0, wait for a time D a cos − ( D a / D b ), and then switchon ∆ b = D b for a time √ ( D a ) + ( D b ) cos − (cid:104) − ( D a / D b ) (cid:105) . For D a (cid:62) D b , due to time-reversal symmetry, the process is thesame in reverse. An example of such optimal protocol, com-bining all three steps, is shown in Fig. 1(c). While in steps Band C, H e (cid:44) H o , it turns out that for both blocks, the initialstate is transformed to the target state by the same protocol.We now explicitly compute the non-Abelian unitary opera-tor generated by the optimal protocols above. Without loss ofgenerality, we consider the case D (cid:54) D (cid:54) D . Using thenotation s i j , d i j ≡ (cid:112) D i ± D j , we can write U e(o) = D D / ( s ± id σ x ) ( s d ± i D σ x + i D σ z ) × (cid:16) s − id σ y (cid:17) (cid:16) s d ± i D σ x − i D σ y (cid:17) × (cid:16) s d − i D σ y + i D σ z (cid:17) (cid:16) s − id σ y (cid:17) . (6)Despite the complexity of the above unitaries, it can be veri-fied that the evolution operator [generated by the optimal pro-tocols as in Fig. 1(c)], projected to the ground state manifold, | (cid:105) and c † | (cid:105) ), i.e., diag( U , U ), equals the target unitary U up to an overall phase. IV. ERRORS AND ROBUSTNESSA. Error model
Since our optimal bang-bang protocols are fine-tuned to theparameters of the device, one should naturally wonder howrobust the process is. We consider two types of errors: (i) cal-ibration errors that arise from the absence of precise knowl-edge about the actual e ff ective Hamiltonian parameters; (ii) random errors due to the imperfect control over the externalknobs, e.g., gate voltages, which make the parameters noisy.The errors of type (i) are systematic and can be minimized bycareful calibration. The errors of type (ii), on the other hand,generate a di ff erent final state every time the experiment isrun. We demonstrate that even in the presence of these errors,our scheme presents advantages over the adiabatic methods.We begin by modeling the errors. Generically, attemptingto tune a coupling to ∆ j imparts to the system an e ff ective ∆ Sj .The error can be expanded (at any point in time) in ∆ j ( t ) / D j < ∆ Sj ≈ ∆ j + D j (cid:20) (cid:15) j + (cid:15) j (cid:16) ∆ j / D j (cid:17) + (cid:15) j (cid:16) ∆ j / D j (cid:17) + . . . (cid:21) . (7)Calibration errors are characterized by time-independent (cid:15) jn , whereas random errors are modeled by noisy (cid:15) jn . Herewe focus on Gaussian white noise with second moment R jn j (cid:48) n (cid:48) white ( t − t (cid:48) ) = (cid:15) jn ( t ) (cid:15) j (cid:48) n (cid:48) ( t (cid:48) ) = W jn δ j j (cid:48) δ nn (cid:48) δ ( t − t (cid:48) ) (8)and noise strength W jn as well as 1 / f (pink) noise, whichis expected to be the dominant source of noise in experi-ments. For white noise, the spectral noise density defined asthe Fourier transform of the correlation function, i.e., S ( ω ) ≡ (cid:82) ∞−∞ R ( t ) e − i ω t dt is a constant, while for pink noise it decays as S jn j (cid:48) n (cid:48) pink ( ω ) ∼ δ j j (cid:48) δ nn (cid:48) ω . (9)In the case of white noise, we compute the noise-averageddensity matrix through a numerically exact solution of aLindblad-type master equation. Due to the correlations in pinknoise, the noise-averaged density matrix evolves with an inte-gral equation that is di ffi cult to solve. We therefore resort to − − − − − − − − − − − FIG. 2. (Color online) The e ff ects of calibration errors. The distance E ( S , U ) of the actual evolution operator S to the target unitary U as afunction of additive (cid:15) (top) and multiplicative (cid:15) (bottom) calibrationerrors for optimal diabatic protocol at τ = τ ∗ as well as the linear andsmooth adiabatic protocol at τ = τ ∗ and τ = τ ∗ . The insetshows E ( S , U ) vs. (cid:15) for the linear protocol for τ = τ ∗ . discrete Langevin-type numerical simulation, where we gen-erate many discrete realizations of noise, evolve the system foreach with the Schr¨odinger equation, and average the densitymatrices at the end.For nontopological qubits, the leading error is the additiveerror (cid:15) j . However, for topological qubits, e.g., in the top-transmon, the coupling is generated by the overlap of Majo-rana wave functions; so in the limit ∆ j → . Thus, the additive error is irrelevantfor Majorana-based topological qubits and the leading erroris the multiplicative (cid:15) j . In the following, we present resultsfor both additive and multiplicative errors. However, only themultiplicative error is relevant to topological qubits. B. Calibration errors
We first discuss calibration errors. Evolving the systemwith a given protocol generates an evolution operator S in theground-state manifold. We quantify the deviation from thetarget unitary U by the distance E ( S , U ) ≡ (cid:112) − | tr( S † U ) / tr( | , (10)which is independent of the initial state. The target unitary U = diag(1 , i ) lives in the ground-state manifold and S is theprojection of the full evolution operator to this manifold. Al-though S may not be unitary after this projection, E ( S , U ) stillprovides a sensible measure of distance.For concreteness, we focus on the case D = D = D =
1, where the optimal protocols are simple. In the adi-abatic schemes, each step is done in a time T = τ/
3. Weconsider two types of adiabatic protocols: linear switches − − − − − − − − − − − − − − FIG. 3. (Color online) The e ff ects of random noise. The trace dis-tance between the final and the target density matrices for an equal-weight initial superposition of the ground states as a function of thenoise strength, W , for optimal diabatic, and linear and smooth adia-batic protocols. ∆ on j ( t ) = t / T and ∆ o ff j ( t ) = − t / T ; and smooth switches ∆ on j ( t ) = sin ( π t / T ) and ∆ o ff j ( t ) = cos ( π t / T ) with van-ishing slopes at the boundaries of the steps. Here 0 < t < T is measured from the beginning of each step. For all of theseprotocols (optimal, linear, and smooth), the evolution of thesystem is governed through ∆ Sj ( t ) as in Eq. (7) to leading order(additive (cid:15) j and multiplicative (cid:15) j , respectively, for genericand topological qubits). For simplicity we take (cid:15) jn = (cid:15) n inde-pendent of j . The optimal protocol for (cid:15) k = U exactly in a time τ ∗ = τ ∗ A = π/ (cid:16) √ (cid:17) . Thelinear and smooth protocols over the same time are completelynonadiabatic (see the inset of Fig. 2). Therefore, instead of acomparison over the same time, we compare the optimal pro-tocol with adiabatic protocols that are at least one order ofmagnitude slower.In Fig. 2, we show the error E ( S , U ) as a function of ad-ditive and multiplicative calibration error. As expected, thereare no advantages for an adiabatic protocol in the nontopo-logical case of additive error. On the other hand, topologicalprotection gives rise to robust adiabatic protocols when errorsare multiplicative. For timescales that are an order of mag-nitude larger, the adiabatic methods are sensitive to the pulseshape and the calibration error (cid:15) . At time scales that are twoorders of magnitude larger, the adiabatic method becomes in-sensitive to (cid:15) and starts to outperform the optimal protocolfor errors larger than 2% (note that (cid:15) is dimensionless). Uponfurther increasing τ , the robust error of the adiabatic methoddecreases further. However, the lower speed of the operationis limited by coherence times and it is impractical to keepslowing down the process. The fast optimal protocol, whichhas a fixed short time τ ∗ , can perform better than any adiabaticgate, upon improved calibration. As seen in Fig. 2, the error E ( S , U ) for the optimal protocol has a linear dependence on (cid:15) . We also note that when the error is multiplicative ratherthan additive the optimal protocol also performs better. C. Random white noise
We now turn to the noisy couplings. While systematic er-rors can be potentially corrected by careful calibration, ran-dom time-dependent errors pose a greater challenge to boththe adiabatic and optimal gates. We start by quantifyingthe errors due to noise. Noise averaging is essential whendealing with random protocols. Direct averaging of the uni-taries, however, creates artificial dephasing due to the unim-portant overall U (1) factors. Thus, we need a di ff erent costfunction. We choose to work with the noise-averaged den-sity matrix, ρ . We start from a particular superposition ofthe ground states as the initial state, | ψ (cid:105) = √ ( | (cid:105) + | (cid:105) ),where | (cid:105) ≡ c † | (cid:105) , yielding the initial density matrix ρ = ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ), which is then evolved andaveraged over noise to obtain ρ ( τ ), by solving the master equa-tion , ∂ t ρ = − i [ H , ρ ] − (cid:88) n , j W jn D − n ) j ∆ nj (cid:104)(cid:104) ρ, i γ γ j (cid:105) , i γ γ j (cid:105) . (11)The target state U | ψ (cid:105) yields the target density matrix σ = ( | (cid:105)(cid:104) | − i | (cid:105)(cid:104) | + i | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ). We then quantify the errorby the trace distance, D [ σ, ρ ( τ )] ≡
12 tr (cid:113)(cid:2) σ − ρ ( τ ) (cid:3) . (12)We consider the leading order with j -independent noise, whereonly W j and W j are nonzero, respectively, for the nontopo-logical and topological qubits.Numerically solving for ρ ( τ ) and computing the trace dis-tance for the optimal as well as linear and smooth adiabaticprotocols up to τ = τ ∗ indicates that the optimal protocolgenerally outperforms the adiabatic protocols for both addi-tive and multiplicative noise. For W =
0, the optimal protocolproduces a vanishing trace distance, which then grows as W ,while remaining much smaller than the trace distance corre-sponding to the adiabatic schemes before reaching saturation.Only for τ = τ ∗ the smooth protocol performs slightly betterthan the optimal protocol for multiplicative noise (as seen in abarely noticeable crossing of the green and blue curves in thebottom panel of Fig. 3). However, this occurs in the regime ofrelatively large D [ σ, ρ ( τ )] > − and large W > − . Inter-estingly, there is a crossing of adiabatic curves with τ = τ ∗ and τ = τ ∗ in Fig. 3 for both additive and multiplicativenoise, beyond which increasing the time scales of the adia-batic protocols reduces their robustness. This antiadabatic behavior appears analogous to the anti-Kibble-Zurek behav-ior .We comment that in real experiments, a weakly coupledbath is always present, which is neglected in our analysis.If the bath decoheres the system, both adiabatic and optimalschemes fail (as quantum coherence is necessary for quantuminformation processing). D. Pink, / f noise White noise allows for numerically exact calculationsthrough the solution of a deterministic di ff erential masterequation [see Eq. (11)] . This limit is relevant under moregeneral conditions than those suggested by its precise mathe-matical definition, e.g., to the ubiquitous Ornstein-Uhlenbeckprocess, where the correlations of noise in the time domaindecay exponentially. Intuitively, exponentially decaying cor-relations can be safely cut o ff after a characteristic correla-tion time, recovering the white-noise predictions upon tempo-ral rescaling . However, we expect the noise spectra in realexperiments to have a 1 / f frequency dependence .Before a quantitative analysis of 1 / f noise, we commentthat qualitative similarities between the e ff ects of white noiseand other types of colored noise are expected. Noise intro-duces a rate for the deposition of excess energy, which canbe understood by viewing it as a sequence of small quantumquenches. Each quench deposits some energy into the systemwithout a strong dependence on the deterministic part of theHamiltonian H ( t ). Whether there are correlations betweenthese quenches (colored noise) or they are completely uncor-related (white noise) should not qualitatively alter this generice ff ect. This does not imply, however, that the spectral den-sity of the noise is unimportant. An extreme case is a noisespectrum localized on certain frequencies, which are eitherresonant or lie outside the bandwidth of the system, respec-tively, enhancing or suppressing the absorption of energy bythe system. Such localized noise spectra are not common inexperiment.Unlike white noise, the temporal correlations of 1 / f noisedo not allow us to compute the noise-averaged density ma-trix by solving a single deterministic di ff erential equation. Wetherefore take a brute-force approach of direct Langevin-typenumerical simulations, where we use the method of Ref. [46]to generate the discrete noise signal. This method applies to1 / f α noise spectrum, with α = α =
1) corresponding towhite (pink) noise. In this section, we only present the resultsof Langevin-type simulations for the 1 / f noise with α = N intervals of duration ∆ t = τ/ N . Weonly consider the multiplicative noise in this section (whichis relevant to topological qubits) and keep the simplifying as-sumption D j =
1. In terms of a correlated discrete signal x m ,the discretized noisy coupling constants then take the form ∆ Sj ( t ) = ∆ j ( t ) (cid:18) + W x m ∆ t / − α/ (cid:19) , ( m − ∆ t < t < m ∆ t , (13)with m = . . . N . The discrete noise signal x m is, in turn, gen-erated from an uncorrelated zero-mean Guassian signal w m with a standard deviation of unity ( w m = w m w n = δ mn )by using an autoregression model of finite order M that relates -3 FIG. 4. (Color online) The e ff ects of pink noise on both the optimalbang-bang and a linear adiabatic protocol that is one or two orders ofmagnitude slower. The white-noise data from Fig. 3 is also replottedfor easy comparison. the two signals through : M (cid:88) m = Γ ( m − α/ m ! Γ ( − α/ x n − m = w n . (14)Here, Γ represents the gamma function.We are interested in the limit of ∆ t → ∆ t and gen-erate enough realizations to achieve convergence (within ac-ceptable error bars) for the final numerically computed error D [ σ, ρ ( τ )]. We then increase N so these realization-averagederrors also converge in ∆ t . Achieving perfect convergencein these calculations is time consuming specially for longertimes and larger strengths of noise. Nevertheless, by analyz-ing 20000 realizations and five di ff erent values of ∆ t = τ ∗ / N for N = , , , ,
150 (for the adiabatic processes withlonger τ , we increased N to have the same values of ∆ t ), wewere able to significantly reduce the error bars.The results are shown in Fig. 4. As expected, both thebang-bang optimal protocol and the linear adiabatic protocolare a ff ected by the 1 / f noise in a qualitatively similar man-ner to white noise. Due to the suppression of high-frequencymodes, the 1 / f noise has a milder e ff ect than white noise onboth of these protocols. These numerical results support ourqualitative picture of the e ff ects of noise. The advantages ofthe optimal protocol survive in the noise regime, where the D [ σ, ρ ( τ )] errors are small. Interestingly, the antiadiabatic be-havior, where a slower adiabatic process, created more dia-batic excitations, also occurs for the 1 / f noise. In particular,an adiabatic process with τ = τ ∗ begins to underperform aprocess with τ = τ ∗ at W ∼ .
04. While this e ff ect mightappear counterintuitive, it naturally results from the accumu-lation of noise-induced excitations over a longer period. Theantiadiabatic behavior further motivates the use of the optimalprotocol. FIG. 5. (Color online) The continuous protocol. The plot showsprotocols for ramping up ∆ in step A with D = D = τ = . τ ∗ , aswell as the corresponding distance F A to the target state, for variousweights λ . E. Correcting the errors due to the limitations of the model
Our results are obtained in the context of the e ff ectivemodel Eq. (1), which is written in terms of low-energy de-grees of freedom and has an infinite gap to higher excitations.The optimal protocol involves sharp sudden quenches, which,in a more realistic model with a finite excitation gap, maycause high-energy excitations. In this section, we fix this is-sue by introducing an alternative cost function (for each ofthe three steps of the protocol) that penalizes sharp transitionsand yields continuous optimal protocols that only take slightlylonger than τ ∗ .We introduce a modified optimal-control problem for eachof the three steps of the dynamics, where, e.g., in step A, weminimize G A ( λ ) = (1 − λ ) F A + λ (cid:90) τ A (cid:32) d ∆ dt (cid:33) + (cid:32) d ∆ dt (cid:33) dt , (15)instead of minimizing, e..g., F A , with the constraints ∆ (0) = ∆ (0) = D , ∆ ( τ A ) =
0, and ∆ ( τ A ) = D (and similarlyfor steps B and C). The second term penalizes large deriva-tives in the protocol, turning the sudden jumps into contin-uous ramps. The weight 0 (cid:54) λ (cid:54) λ = λ =
1, in which case we get a simple linear protocol fromEuler-Lagrange minimization of G A (1).While the Pontryagin’s formalism can also shed light onoptimal-control problems with a trajectory-dependent costfunction as in Eq. (15), an analytical solution of the con-strained problem is challenging. We therefore use direct nu-merical minimization. Approximating a general protocol witha piece-wise constant protocol with N =
100 steps, we per-form Monte Carlo simulations over the shape of the proto-cols to minimize G A ( λ ) for several values of λ over a totaltime τ A = . τ ∗ A . The results for ramping up ∆ are shown inFig. 5, indicating a continuous transformation from the bang-bang protocols corresponding to λ =
0. (The protocols forramping down ∆ in this step are reflected about the centerwith a similar timescale.) For finite λ , the sudden jumps arespread over finite time scales.The overall protocol then looks very similar to the bang-bang protocol of Fig. 1(c) except each sudden jump is spreadover a time window of length τ (cid:48) (cid:28) τ ∗ . We need to increase − FIG. 6. (Color online) The cost function ˜ F A as a function of tunnelingmatrix element δ to a high-energy mode with E =
20, for the bang-bag protocol with time τ ∗ A and the continuous protocol of Fig. 5 withtime t = . τ ∗ A and leakage-free cost function F A = . the total time of the operation by the sum of these ramp timesto to get a small final error. For example, in Fig. 5, when weadd 10% to the time of step A, the protocol with a negligiblysmall F A (for λ = − ) spreads the jump over a time intervalof approximately τ ∗ A / / E , where E is a large but finite energy gap to higherexcitations [neglected in Hamiltonian Eq. (1)], to prohibitleakage to these excitations. We may compare this timescaleto that in a more realistic model. The gap may be associatedwith low energy Andreev bound states in some of the severaljunctions which are part of the circuits. It may also originatefrom finite Josephson ( E J ) and charging energy ( E C ) of themesoscopic superconducting islands, with E ∼ √ E C E J . In practice, this energy scale is much larger than D j . Wecan use a simple toy model with one more mode to comparethe smooth and bang-bang protocols and quantify the advan-tages of using the smooth protocols obtained above. Focusingon step A of the process with H = ∆ σ y − ∆ σ z , we enlarge thetwo-dimensional Hilbert space to a three-dimensional spacewith ˜ H = − ∆ − i ∆ δ i ∆ ∆ δδ δ E , (16)where δ is a tunneling matrix element to the high energymode. To account for leakage, we compute the cost function˜ F A , using the 3 × H above and three-dimensional vectorsfor | + z (cid:105) and | − y (cid:105) with a vanishing third element [see Eq.(3)]. Focusing on the protocol in Fig. 5 with λ = − andsetting E =
20, we computed ˜ F A as a function of δ . As seenin Fig. 6, the results demonstrate the adiabatic suppression ofleakage when using the smooth protocol. Note that for δ = V. CONCLUSIONS
In summary, based on Pontryagin’s theorem of optimal con-trol, we proposed the optimal protocols for generating thesame unitary operator as the one corresponding to fully adia-batic braiding of MZMs. While not providing full topologicalprotection, our constrained optimal-control approach makesuse of the nonlocal nature of the information stored in MZMsto make the system robust against a range of environmentalperturbations. Through tailored diabatic pulse shapes, ourscheme can significantly increase the speed of devices suchas the top-transmon, without the need for any change to theexperimental setup. Such fast accurate operations may defendthe system against decoherence e ff ects such as quasiparticlepoisoning. The advantages of our method survive in the pres-ence of white and 1 / f noise and small calibration errors. Therobustness can be further enhanced by making the pulses con-tinuous without significantly sacrificing the performance ofthe device. Our proposed optimal diabatic gates can foster the development of high-performance quantum information pro-cessing with MZMs. ACKNOWLEDGMENTS A. Kitaev, Ann. Phys. , 2 (2003). C. Nayak, S. H. Simon, A. Stern, M. Freedman, andS. Das Sarma, Rev. Mod. Phys. , 1083 (2008). Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. ,177002 (2010). R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. ,077001 (2010). J. Alicea, Rep. Prog. Phys. , 076501 (2012). C. Beenakker, Annu. Rev. Con. Mat. Phys. , 113 (2013). S. R. Elliott and M. Franz, Rev. Mod. Phys. , 137 (2015). V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. a. M.Bakkers, and L. P. Kouwenhoven, Science , 1003 (2012). A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrik-man, Nat. Phys. , 887 (2012). H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T. Deng,P. Caro ff , H. Q. Xu, and C. M. Marcus, Phys. Rev. B , 241401(2013). L. P. Rokhinson, X. Liu, and J. K. Furdyna, Nat Phys , 795(2012). M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caro ff , andH. Q. Xu, Nano Lett. , 6414 (2012). A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni, K. Jung, andX. Li, Phys. Rev. Lett. , 126406 (2013). S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H.MacDonald, B. A. Bernevig, and A. Yazdani, Science , 602(2014). D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham, J. Danon,M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Marcus, K. Flens-berg, and J. Alicea, arXiv:1511.05153. M. Cheng, V. Galitski, and S. Das Sarma, Phys. Rev. B ,104529 (2011). T. Karzig, G. Refael, and F. von Oppen, Phys. Rev. X , 041017(2013). C. S. Amorim, K. Ebihara, A. Yamakage, Y. Tanaka, and M. Sato,Phys. Rev. B , 174305 (2015). D. Rainis and D. Loss, Phys. Rev. B , 174533 (2012). B. van Heck, A. R. Akhmerov, F. Hassler, M. Burrello, andC. W. J. Beenakker, New J. Phys. , 035019 (2012). G. Goldstein and C. Chamon, Phys. Rev. B , 205109 (2011). M. J. Schmidt, D. Rainis, and D. Loss, Phys. Rev. B , 085414(2012). M. V. Berry, J. Phys. A: Math. Theor. , 365303 (2009). A. del Campo, Phys. Rev. Lett. , 100502 (2013). T. Karzig, F. Pientka, G. Refael, and F. von Oppen, Phys. Rev. B , 201102 (2015). J. Zhang, T. H. Kyaw, D. M. Tong, E. Sj¨oqvist, and L.-C. Kwek,Sci. Rep. , 18414 (2015). C. Knapp, M. Zaletel, D. E. Liu, M. Cheng, P. Bonderson, andC. Nayak, Phys. Rev. X , 041003 (2016). A. P. Peirce, M. A. Dahleh, and H. Rabitz, Phys. Rev. A , 4950(1988). J. P. Palao and R. Koslo ff , Phys. Rev. Lett. , 188301 (2002). P. Kr´al, I. Thanopulos, and M. Shapiro, Rev. Mod. Phys. , 53(2007). T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero,V. Giovannetti, and G. E. Santoro, Phys. Rev. Lett. , 240501(2009). P. Doria, T. Calarco, and S. Montangero, Phys. Rev. Lett. ,190501 (2011). A. Rahmani and C. Chamon, Phys. Rev. Lett. , 016402 (2011). A. Rahmani, Mod. Phys. Lett. B , 1330019 (2013). T. Karzig, A. Rahmani, F. von Oppen, and G. Refael, Phys. Rev.B , 201404 (2015). F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker, New J. Phys. , 095004 (2011). T. Hyart, B. van Heck, I. C. Fulga, M. Burrello, A. R. Akhmerov,and C. W. J. Beenakker, Phys. Rev. B , 035121 (2013). A. Dutta, A. Rahmani, and A. del Campo, Phys. Rev. Lett. ,080402 (2016). L. S. Pontryagin,
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