Simulation of non-Abelian braiding in Majorana time crystals
Raditya Weda Bomantara Jiangbin Gong
quant-ph

	
	
	phyrwb@nus.edu.sg
	
		Department of Physics, National University of Singapore, Singapore 117543
	
	
	phygj@nus.edu.sg
	
		Department of Physics, National University of Singapore, Singapore 117543
	
	
	
	
	
	
	

	
	
		Discrete time crystals have attracted considerable theoretical and experimental studies but their potential applications have remained unexplored.  A particular type of discrete time crystals, termed “Majorana time crystals", is found to emerge
in a periodically driven superconducting wire accommodating two different species of topological edge modes. It is further shown that one
can manipulate different Majorana edge modes separated in the time lattice, giving rise to an unforeseen scenario for topologically protected gate operations   mimicking braiding.  The proposed protocol can also generate a magic state that is important for universal quantum computation.  This study thus advances the quantum control in discrete time crystals and  reveals their great potential arising from their time-domain properties.
		
	
	
	
	  Simulation of non-Abelian braiding in Majorana time crystals
    Jiangbin Gong
    December 30, 2023
==============================================================

Introduction. The idea of time crystals was first coined by Frank Wilczek in 2012 <cit.>. Despite the existence of a no-go theorem which prohibits time crystals to arise in the ground state or equilibrium systems <cit.>, time crystals in periodically driven systems, named discrete time crystals (DTCs), have recently attracted considerable interests <cit.>.  Two experimental realizations of DTCs have been reported <cit.>.
	
	Here we explore the potential applications of DTCs as exotic phases of matter <cit.>.  Specifically, DTCs are exploited to perform topologically protected quantum computation <cit.>. To that end, one needs to first find a particular type of DTCs that can simulate non-Abelian, e.g. Ising <cit.> or Fibonacci <cit.> anyons.  Ising anyons can be described in the language of Majorana fermions in one dimensional (1D) superconducting chains <cit.>.
	
DTCs have recently been proposed in a periodically driven Ising spin chain <cit.>. As learned from the mapping between one-dimensional (1D) superconducting chains and static spin systems <cit.>, we expect the emergence of DTCs in a periodically driven Kitaev superconducting chain.  Indeed, there period-doubling DTCs are obtained using the quantum coherence between
two types of topologically protected Floquet Majorana edge modes <cit.>.
Such DTCs are termed `Majorana time crystals' (MTCs) below.  Next, a scheme is proposed to physically simulate the non-Abelian braiding of a pair of Majoranas
<cit.> at two different time lattice sites.    We also elucidate how our scheme can be used to generate a magic state, which is necessary to perform universal quantum computation <cit.>. These findings open up a new concept in   simulating the braiding of Majorana excitations and should stimulate future studies of the applications of DTCs.
	
	Majorana time crystals. Consider a periodically driven system H(t) of period T. For the first half of each period, H(t) is a 1D Kitaev chain with Hamiltonian H_1= ∑_j^N-1( -J_j c_j+1^† c_j + Δ_j c_j+1^† c_j^†+h.c.) +μ_1∑_j^N c_j^† c_j
<cit.>, and for the second half of each period,
H(t)=	H_2 =μ_2∑_j^N c_j^† c_j. Here c_j (c_j^†) is the annihilation (creation) operator at site j, J_j and Δ_j are respectively the hopping and pairing strength between site j and j+1, μ_1 and μ_2 are chemical potential at different time steps. Throughout we work in a unit system with ħ=1 .
Unless otherwise specified later,  we take J_j=J and Δ_j=Δ for all j=1,⋯ N-1 for our general discussions.
For later use, we also define the one-period propagator 𝒰=𝒯exp(-∫_0^Ti H(t') dt'), where 𝒯 is the time ordering operator.
One candidate for H_1 is an ultracold atom system <cit.>, realizable by optically trapping 1D fermions inside a three dimensional (3D) molecular Bose-Einstein condensate (BEC). In such an optical lattice setup, the hopping term is already present due to the two Raman lasers generating the optical lattice, while the pairing term can be induced by introducing a radio frequency (rf) field coupling the fermions with Feshbach molecules from the surrounding BEC reservoir. Realizing the periodic quenching between H_1 and H_2 is also possible <cit.>.

	
Our motivation for considering the above model system depicted by H(t) is as follows. If J=Δ=Δ^* and μ_1=0, then H(t) can be mapped to a periodically driven Ising spin chain <cit.>, which is known to exhibit DTCs <cit.>.   We thus expect H(t) to support DTCs.  That is, there exists some observable such that for a class of initial states, the oscillation in the expectation value of this observable does not share the period of H(t), but exhibits a period of nT, with n>1 being stable against small variations in the system parameters. Furthermore, in the thermodynamic limit,  the oscillation of this observable with period nT persists over an infinitely long time.





 DTCs in our model emerge from the interplay of periodic driving, hopping, and p-wave pairing. In particular,  H(t) yields a number of interesting Floquet topological phases manifested by a varying number of edge modes, with their corresponding eigenphases of 𝒰 being 0 or π.   These  eigenmodes of 𝒰 localized at the system edge are often called Floquet zero <cit.> or π edge modes <cit.>, possessing all the essential features of a Majorana excitation <cit.>.  For example, by taking μ_2 T =JT=Δ T= π and μ_1=0, the eigenphases of 𝒰 can be explicitly solved, which yield both Majorana zero and π modes. Given that the Majorana zero (π) mode develops an additional phase 0 (π) after one driving period T,
a superposition of Majorana zero and π modes will evolve as a superposition, but with their relative phase being π (0) after odd (even) multiples of T.  That is,
the ensuing dynamics yields period-doubling oscillations for a generic observable.  Further, because these edge modes are protected by the underlying topological phase, they do not rely on any fine tuning of the system parameters <cit.>, yielding the necessary robustness for DTCs.









Define two Majorana operators γ_j^A=c_j+c_j^† and γ_j^B=i(c_j-c_j^†) at each chain site j, with γ_j^A=(γ_j^A)^†, (γ_j^A)^2=1 and similar equalities for γ_j^B, as well as commutation relations {γ^A_j,γ^B_l}=2 δ_ABδ_jl.

  In particular, before the periodic driving is turned on, the choice of system parameters above yield a Majorana zero mode Ψ(0)=γ_1^A. Once the driving is turned on, Ψ(0) becomes a linear superposition of Majorana zero and π modes and will then evolve non-trivially in time. At time t, it can be written in general as Ψ(t)=∑_j∑_l=A,B c_j,l(t) γ_j^l, where ∑_j∑_l=A,B |c_j,l(t)|^2=1. To demonstrate how DTCs can be observed in the system, special attention is paid to the quantity Z(t)=|c_1,A(t)|^2-|c_1,B(t)|^2, which measures the difference between the weight of γ_1^A and γ_1^B in Ψ(t). 


Figure <ref>(a)-(c) show Z vs time in several cases, whereas Fig. <ref>(d)-(f) show the associated subharmonic peak in the power spectrum, defined as Z̃(ω)=∑_n Z(t)  exp(i n ω T ). |Z̃(ω)|^2 is seen to
 be pinned at ω=π/T, confirming the emergence of period-doubling DTCs.
     Under the special system parameter values chosen above, Ψ(0) comprises of an equal-weight superposition of Majorana zero and π modes (shown below) and will therefore undergo period-doubling oscillations between two Majorana operators γ_1^A and γ_1^B as time progresses. As Figs. <ref>(b) and (c) show, tuning the values of μ_1, J, Δ, and μ_2 away from these special values still yields the same period-doubling oscillations for a long time scale, accompanied by some beatings in the time dependence which diminishes as the system size increases. These results thus justify the term
 MTC to describe such DTCs.

 

	

	



	
	
	
	
	  Simulation of braiding protocol. Consider now four Majoranas in our model, labeled as γ_L^A, γ_R^A, γ_L^B, and γ_R^B, with γ_L^A (γ_L^B) and γ_R^A (γ_R^B) representing Majorana edge modes localized in space, at the left and right edges respectively, and in time, at any even (odd) integer multiple of period. That is, γ^B_L and γ^B_R are obtained by evolving respectively γ_L^A and γ_R^A over one period.  During our protocol, γ_L^A and γ_L^B will be adiabatically manipulated   to simulate braiding, while γ_R^A and γ_R^B are left intact.  Such a nonconventional operation is schematically described by Fig. <ref>. Physical implementation of the adiabatic manipulation in the above-mentioned optical-lattice context   <cit.> can be done by slowly tuning the strength of the Raman lasers and the rf field.
		



	
   Before presenting our protocol, we will first recast H_1 and H_2 in terms of Majorana operators as (focusing on the first three lattice sites   and taking μ_1=0)
 
    H_1   =   i (Δ_1/2-J_1/2)γ_1^Aγ_2^B+ i(Δ_1/2+J_1/2)γ_1^Bγ^A_2 
       +   i Im(Δ_2/2-J_2/2)γ_2^Aγ_3^A+ i Re(Δ_2/2+J_2/2)γ_2^Bγ^A_3  
       +   i Im(Δ_2/2+J_2/2)γ_2^Bγ_3^B+ i Re(Δ_2/2-J_2/2)γ_2^Aγ^B_3  
       +   ⋯ ,

  and H_2= -(μ_2/2)(1+iγ_1^Aγ^B_1) +⋯, with J_1, Δ_1, J_2, and Δ_2 are subject to adiabatic manipulations, during which J_2 and Δ_2 may be complex, while J_j and Δ_j for j≠ 2 are assumed to be always real. For the sake of analytical solutions
  and better qualitative understandings, we again take μ_2 T=J_j T=Δ_j T=π at the start to illustrate our idea, so that γ_L^A (γ_L^B), initially prepared from the edge mode of H_1, is precisely γ_1^A (γ_1^B). As demonstrated in Fig. <ref>(a)-(c) below, this fine tuning of the system parameters is not needed in the actual implementation.



  In step 1, we exploit the adiabatic deformation of Majorana zero and π modes, denoted 0̂ and π̂, along an adiabatic path
     with J_2=Δ_2 being real.
     To develop insights into this step,
    we parameterize J_1+Δ_1=2π/T, J_1-Δ_1=2πsin(ϕ_1)/T, J_2=Δ_2=2πcos(ϕ_1)/T. As detailed in Supplementary Material <cit.>,
    we find (up to an arbitrary overall constant)
     
    0̂=[cos(ϕ_1)γ_1^A-sin(ϕ_1)γ_3^A]+ [cos(ϕ_1)γ_1^B-sin(ϕ_1)γ_3^B]; 
    π̂=[cos(ϕ_1)γ_1^A-sin(ϕ_1)γ_3^A]- [cos(ϕ_1)γ_1^B-sin(ϕ_1)γ_3^B].


    By tuning ϕ_1 slowly from 0 to π/2, 0̂ will adiabatically
     change from (γ_1^A+γ_1^B) to -(γ_3^A+γ_3^B), whereas π̂ will adiabatically
     change from (γ_1^A-γ_1^B) to (γ_3^B-γ_3^A), i.e., both zero and π modes
     are now shifted to the third site. Due to this adiabatic following, a superposition of  0̂ and π̂ modes remains a superposition, thus preserving the DTC feature were the adiabatic process stopped at any time. The net outcome of this step can thus be described simply as γ_1^A→ -γ_3^A and γ_1^B→ -γ_3^B.



     Step 2 continues to adiabatically deform 0̂ and π̂.  Starting from J_2=Δ_2=0 as a result of step 1, we consider an adiabatic path with J_2=-Δ_2 being purely imaginary values. If we parameterize J_1-Δ_1=2π/T, J_1+Δ_1=2πcos(ϕ_2)/T, J_2=-Δ_2=iπsin(ϕ_2)/T, then one easily finds <cit.>
      
      
    0̂= [sin(ϕ_2)γ_1^B-cos(ϕ_2)γ_3^A]-[sin(ϕ_2)γ_1^A+cos(ϕ_2)γ_3^B]; 
    π̂=[sin(ϕ_2)γ_1^B-cos(ϕ_2)γ_3^A]+[sin(ϕ_2)γ_1^A+cos(ϕ_2)γ_3^B].

    As ϕ_2 adiabatically increases from 0 to π/2,  0̂ and π̂ undergo further adiabatic changes to
     (γ_1^B - γ_1^A) and (γ_1^B +γ_1^A) respectively. The overall transformation of this step is -γ_3^A→γ_1^B and -γ_3^B → -γ_1^A.




    In step 3, we exploit further the coherence between Majorana zero and π modes so as to recover the system's original Hamiltonian, while at the same time preventing γ_L^A and γ_L^B from completely untwisting and returning to their original configuration. As an innovative adiabatic protocol, we adiabatically change the system parameters every other period. This amounts to introduce a characteristic frequency π/T in our adiabatic manipulation,   resulting in the coupling between zero and π quasienergy space. As the system parameters are adiabatically tuned, Majorana zero and π modes will then adiabatically follow the degenerate eigenmodes of 𝒰^2 (i.e.,
    	the two-period propagator) associated with zero eigenphase. This leads to a nontrivial rotation between the two Majorana modes dictated by the non-Abelian Berry phase in this degenerate subspace.  With this insight, one can envision many possible adiabatic paths to induce a desirable rotation between Majorana 0 and π modes. 

    

   After some trial and error attempts, we discover a class of adiabatic paths for step 3 that can yield a rotation of π/4 between γ_L^A and γ_L^B. Specifically,  we fix J_1  and let    Δ_1= 2π f_3,a(t)/T, J_2=√(2)πexp(iπ/4)[1-if_3,b(t)]/T, and Δ_2=√(2)πexp(-iπ/4) [1+if_3,c(t)]/T, where f_3,l(t), with l=a,b,c, are certain (not necessarily the same) functions that slowly increase from -1 to 1 for every other period.  That is, for each new period, f_3,l(t) are alternatively increased or stay at the values of the previous step.
   At the end of the adiabatic manipulation, this step yields the original Hamiltonian, with γ_1^B→ (γ_1^A+γ_1^B)/√(2) and -γ_1^A→ (γ_1^B-γ_1^A)/√(2) to a high fidelity.

   Finally in step 4, we repeat the 3 steps outlined above to obtain the overall transformations γ_L^A →γ_L^B and γ_L^B→ -γ_L^A, which completes the simulated braiding operation to the two different species of Majoranas and at the same time resets the system configuration. As shown in Fig. 2, at the start of the protocol, γ_L^A (γ_L^B) at our MTC appears at even (odd) multiples of T; by constrast, at the end of the protocol, γ_L^A (γ_L^B) appears at odd (even) multiples of T.
  	
 To confirm the above analysis, we calculate the evolution of Majorana correlation functions during the manipulation process. The system is assumed to be in the even parity state such that initially ⟨iγ_L^A γ_R^A⟩ = ⟨iγ_L^B γ_R^B⟩ = 1 and ⟨iγ_L^A γ_R^B⟩ = ⟨iγ_L^B γ_R^A⟩=0, where γ_i^α≡γ^α_i(t=0), α=A,B, and i=L,R. During the manipulation process, γ_A^L(t) and γ_B^L(t) in general become a superposition of γ_L^A and γ_L^B, thus changing  the correlations ⟨iγ^α_L(t) γ^β_R⟩, where α,β=A,B. The success of our protocol is then marked by the final correlation functions ⟨iγ_L^A(t_f) γ_R^A⟩ =⟨iγ_L^B(t_f) γ_R^B⟩=0 and ⟨iγ_L^A(t_f) γ_R^B⟩ =-⟨iγ_L^B(t_f) γ_R^A⟩=1.   In experiment, Majorana correlation functions ⟨iγ_L^A γ_R^A ⟩ and ⟨iγ_L^B γ_R^B ⟩ may be measured via time-of-flight imaging method as proposed in Ref. <cit.> or indirectly by measuring the parity of the wire <cit.> at even and odd integer multiples of T. To measure cross correlation functions such as ⟨iγ_L^B γ_R^A ⟩, one could first turn off the periodic driving on the right half of the wire after the protocol is completed, then wait for one period. Since γ_R^A is a Majorana zero mode in the absence of periodic driving by construction, it will stay invariant in one period, whereas the left Majorana mode will transform into γ_L^B <cit.>. The same readout process can then be carried out to measure their correlation functions.
	
	The full evolution of Majorana correlation functions is depicted in Figs. <ref>(a)-(c) under different system parameter values. In particular, Fig. <ref>(c) assumes also the presence of disorders and a small hopping term in H_2, which may arise due to the presence of the Raman lasers, even after taking low frequency and large detuning values. More precisely, hopping, pairing, and onsite static disorders are considered by taking J_j= J+δ J_j, Δ_j= Δ +δΔ_j, μ_1→μ_1+δμ_1,j, and μ_2→μ_2+δμ_2,j, where δ J_j T, δΔ_j T, δμ_1,j T and δμ_2,j T uniformly take random values between -0.1 and 0.1, while the small hopping term is of the form -∑_j (𝒥+δ𝒥_j) c_j+1^† c_j +h.c., where 𝒥T=0.025 and δ𝒥_j T∈[-0.01,0.01]. The fact that Figs. <ref>(a)-(c) look qualitatively the same demonstrates the robustness of our protocol against such system imperfections. Finally, plotted in Fig. <ref>(d) is the whole eigenphase spectrum of 𝒰^2, which indicates that its zero eigenphases are well separated from the rest of the spectrum, thus confirming the topological protection needed to realize the rotation between γ_L^A and γ_L^B.

	
	
	

	Discussion.    Due to fermion parity conservation, a minimum of four Majoranas is required to harness their non-Abelian features for nontrivial (single-qubit) gate operations. Our work demonstrates, through exploiting the time-domain features, that this can be achieved in a minimal single wire setup, thus avoiding the necessity to design complicated geometries <cit.>. Moreover, as demonstrated in Supplementary Material <cit.>, our setup can be readily extended to an array of wires to simulate more intricate braiding between various pairs of Majoranas at different time and wires. In view of these two aspects, it is expected that certain quantum computational tasks may now be carried out using significantly less number of wires.
	
	
	
	
	
	
	
	
	
	
	
		
		
	
	
	
	
	
	
	
	
	
	
	
	
	
	

	As another feature of the proposed protocol, the end of step 3 has achieved the transformation γ_L^A→1/√(2)(γ_L^A+γ_L^B) and γ_L^B→1/√(2)(γ_L^B-γ_L^B), which can be written as V=exp(-π/8γ_L^Aγ_L^B). In the even parity subspace, a qubit can be encoded in the common eigenstates of the parity operators iγ_L^Aγ_R^A and iγ_L^Bγ_R^B, such that iγ_L^Aγ_R^A  |0⟩ =iγ_L^Bγ_R^B  |0⟩=|0⟩ and iγ_L^Aγ_R^A |1⟩ =iγ_L^Bγ_R^B⟩  |1⟩ =-|1⟩. It can be easily verified that V maps |0⟩ to a magic state cosπ/8|0⟩ -sinπ/8|1⟩. It is known that a combination of Clifford gates and a magic state is required to achieve universal quantum computation <cit.>. While Clifford gates can be realized in a typical Ising anyonic model alone, the creation of a magic state normally requires an additional dynamical process <cit.>.
  The rather straightforward realization of the operation V here is hence remarkable.
	
	  It is also important to ask to what extent the gate operations here share the robustness of the braiding of two Majorana modes.  On the one hand, if J_2, Δ_1, and Δ_2 in step 3 take arbitrary time-dependence then the desired braiding outcome cannot be achieved.
 On the other hand, to physically implement a braiding operation without the use of direct spatial interchange between two Majoranas,
  we expect some necessary control to restrict the time dependence of J_2, Δ_1, and Δ_2 used in step 3 to a certain degree.
  Our further investigations <cit.> indicate that our protocol does enjoy some weak topological protection, in the sense that
  its fidelity is rather stable under considerable time-dependent deformation: J_2→ J_2+δ J_2, Δ_1→Δ_1+δΔ_1, and Δ_2 →Δ_2 +δΔ_2 in step 3, where δ J_2, δΔ_1, and δΔ_2 represent time-dependent perturbations, vanishing at the start and end of step 3, with relative strength of the order 5% <cit.>.
	
	Conclusion. A coherent superposition of Majorana zero and π modes of a periodically driven 1D superconducting wire is shown to yield period-doubling MTCs. By adiabatic manipulation of the Majorana zero and π modes, we have proposed a   relatively robust scheme to   mimic the braiding of two Majorana modes localized at different physical and time lattice sites. Our approach is promising for physical resource saving. As an important side result, we also obtain a magic state crucial for universal quantum computation <cit.>.

	


	Acknowledgements:  J.G. is supported by the Singapore NRF grant No. NRF-NRFI2017-04 (WBS No. R-144-000-378-281) and by the Singapore Ministry of Education Academic Research Fund Tier I (WBS No. R-144-000-353-112).

	
	
	
	
	
	
	
	
	

	
	
 Supplemental Material 

	
	This supplemental material has eight sections. In Sec. A, we elucidate in detail the exact mapping between H(t) defined in the main text and a periodically driven Ising spin chain. This mapping allows us to deduce the parameter regime in which DTCs may emerge.   In Sec. B, we present some comparisons between our model and typical Kitaev Hamiltonian in static systems. In Sec. C, we formulate the definition of Majorana modes and Majorana time crystals in time periodic systems. In Sec. D, the role of Majorana zero and π modes in Majorana time crystals is discussed. In Sec. E, we verify that 0̂ and π̂ presented in the main text during the first two steps in the protocol are indeed Majorana zero and π modes.   In Sec. F, we present some numerical results to demonstrate the robustness of step 3 in the protocol against small deformations in the adiabatic parameters.   In Sec. G, we discuss the feasibility of our protocol in real experiment and possible error correction against decoherence. Finally, we propose the possibility to extend our protocol to an array of superconducting wires in Sec. H.



§ SECTION A: EXACT MAPPING BETWEEN A PERIODICALLY DRIVEN SUPERCONDUCTING WIRE AND ISING SPIN CHAIN


Consider a periodically driven Ising spin chain as follows,


    ℋ(t)=
    -𝒥∑_j^N-1σ_z,j+1σ_z,j   for  MT < t≤ (M+1/2) T 
    
    -f ∑_j^N  σ_x,j   for (M+1/2)T<t<(M+1)T 
     ,


where σ_x(z),j represent Pauli operators acting on site j, J describes the strength of nearest neighbor spin-spin interaction, M is an integer, T is the period of the Hamiltonian, and f corresponds to the strength of the Zeeman term. Following Ref. <cit.>, we define


    a_j   =   ∏_l<jσ_x,lσ_z,j , 
    
    b_j    =   i a_j σ_x,j .


It can be easily verified that a_j=a_j^†, b_j=b_j^†, { a_j,b_j}=0, and { a_j,a_k}={ b_j,b_k}=2δ_j,k. That is, a_j and b_j are Majorana operators. Upon expressing σ_x(z),j in terms of a_j and b_j,


    σ_x,j   =    -i a_j b_j  ,
    σ_z,j+1σ_z,j   =    -i b_j a_j+1 ,


and Eq. (<ref>) becomes


    ℋ(t)=i𝒥∑_j^N-1 b_j a_j+1   for  MT < t≤ (M+1/2) T
    i f ∑_j^N a_j b_j     for (M+1/2)T<t<(M+1)T 
     .


Given the Majorana operators a_j and b_j, we can form complex fermionic operators as


    c_j    =   1/2(a_j -i b_j)  ,
    
    c_j^†   =   1/2(a_j +i b_j)  ,


so that


    i a_j b_j    =    -(c_j^2+c_j^† c_j -c_j c_j^† +c_j^† 2) 
       =    2c_j^† c_j-1  , 
    i b_j a_j+1   =    -(c_jc_j+1+c_jc_j+1^†-c_j^† c_j+1-c_j^† c_j+1^†) 
       =    c_j+1^† c_j -c_j+1^† c_j^† +h.c.


where we have used the fact that c_j^2=c_j^† 2=0 and c_j c_k^†=δ_j,k-c_k^† c_j. Eq. (<ref>) can further be written as


    ℋ(t)=𝒥∑_j^N-1(c_j+1^† c_j -c_j+1^† c_j^†  +h.c.)    for  MT < t≤ (M+1/2) T
    
    f ∑_j^N (2c_j^† c_j -1)     for (M+1/2)T<t<(M+1)T 
     ,


which is equivalent to H(t) in the main text up to an unimportant constant upon identifying J=Δ=Δ^*=-𝒥, μ_2=2f, and μ_1=0.

  

§ SECTION B: COMPARISON WITH STATIC KITAEV HAMILTONIAN


We start by first writing H(t) in momentum space as


    H(t)    =   ∑_k 1/2Ψ_k^† h_1,kΨ_k    for  MT < t≤ (M+1/2) T
    ∑_k 1/2Ψ_k^† h_2,kΨ_k    for (M+1/2)T<t<(M+1)T 
     , 
    
    h_1,k   =   μ_1 τ_z - J cos k τ_z +Δsin k τ_y  , 
    
    h_2,k   =   μ_2 τ_z  ,


where Ψ_k =(c_k, c_-k^†)^T is the Nambu vector, and τ_a=x,y,z are Pauli matrices acting on the Nambu space. In Nambu basis, the one-period propagator can be simply written as the product of two exponentials as (taking T=2 and ħ=1 units)


    U(k) = exp(-i h_1,k) exp(-i h_2,k)  .


In particular, since both h_1,k and h_2,k are both 2× 2 matrices, we may combine the two exponentials to write U(k)=exp(-i h_f(k)), where h_f(k)=θm̂·τ is the Floquet Hamiltonian, where


    θ   =   arccos[ cos (μ_2) cos(h) -sin(μ_2)sin(h) μ_1-Jcos(k)/h] , 
    m̂   =   Δsin(k)sin(h)/hsinθ[cos(μ_2) ŷ-sin(μ_2) x̂] +hsin(μ_2) cos(h)+(μ_1-Jcos(k))cos(μ_2) sin(h)/hsin(θ)ẑ , 
    
    h    =   √([μ_1-Jcos(k)]^2+[Δsin(k)]^2) .


Note that while h_f takes the same structure as a typical Kitaev Hamiltonian in momentum space <cit.>, the effective chemical potential, hopping strength, and (now complex) p-wave pairing, acquire non-trivial k dependence. Moreover, since h_f appears only as a phase, its eigenvalues (quasienergies) can only be defined up to a modulus of 2π, which leads to the existence of two gaps at quasienergy zero and π. By contrast, Kitaev Hamiltonian in static systems has only one gap at energy zero. Despite these differences, one important similarity between h_f and Kitaev Hamiltonian is the presence of particle hole symmetry, which for the former can be written explicitly by the operator 𝒫=τ_x K which satisfies 𝒫 h_f(k) 𝒫^-1= -h_f(-k). As elucidated further in the next section, it is due to this particle hole symmetry and the existence of two gaps which allow the Majorana condition to hold at quasienergy zero and π.







§ SECTION C: MAJORANA MODES IN TIME PERIODIC SYSTEMS AND MAJORANA TIME CRYSTALS


In treating time periodic systems, Floquet formalism <cit.> is usually employed so that essential information about the systems is encoded in the eigenstates of the one period propagator 𝒰 as defined in the main text. The corresponding eigenphases of 𝒰 are also referred to as quasienergies (up to a factor 1/T) as an analogy with eigenenergies in static systems. However, unlike energy, eigenphase of 𝒰  is only defined modulo 2π due to its phase nature. Consequently, eigenphases ε=±π are identified as the same.

In the second quantization language, we may define a fermion mode Ψ_ε associated with each eigenphase ε of 𝒰. By taking |R⟩ as a reference state satisfying 𝒰|R⟩ =|R⟩, we can then construct another eigenstate of 𝒰 with eigenphase ε as |ε⟩ = Ψ_ε^† |R⟩. In superconducting systems, the existence of particle-hole symmetry guarantees that associated with a fermion mode Ψ_ε of eigenphase ε, there exists another fermion mode Ψ_-ε of eigenphase -ε. These two modes are related by


    Ψ_ε = Ψ_-ε^† .


Equation (<ref>) implies that Ψ_0 and Ψ_±π are Hermitian, which are thus termed Floquet Majorana zero and π modes respectively. However, such Hermitian fermion modes must always come in pairs, e.g. Ψ_0^(1) and Ψ_0^(2), in order to form a complex fermion Ψ_0^(c)=Ψ_0^(1)+iΨ_0^(2), since 𝒰 can only admit terms of the form Ψ_0^(c)†Ψ_0^(c). In this sense, Majorana zero and π modes are usually also referred to as half-fermions.

The above idea can be readily generalized to define Majorana modes in DTCs. Since discrete time translational symmetry is spontaneously broken, eigenphase ε is no longer a conserved quantity. However, since generally a time crystal state still exhibits periodicity of period n≠ 1 times of that of the Hamiltonian itself, a new reference state |R'⟩ can be defined, which satisfies 𝒰^n |R'⟩ = |R'⟩. New set of fermion modes Φ_ε_n can then be defined based on the eigenphases ε_n of 𝒰^n, where particle-hole symmetry implies Φ_0 to be Hermitian, which is analogues to Floquet Majorana modes above. DTCs possessing such Majorana modes are what we termed `Majorana time crystals' (MTCs) in the main text.



 

 

 
 
 
 

 




§ SECTION D: THE ROLE OF MAJORANA ZERO AND Π MODES IN MTCS


In order to construct a DTC with period nT (n≠ 1), it is necessary that the associated Floquet operator (one period propagator) 𝒰 possesses at least n eigenphases, each differing from another by an integer multiple of 2π/n. A state exhibiting nT period can then be constructed as a superposition of such n states with different eigenphases <cit.>. However, in general there is no guarantee that this periodicity is robust against a small change in the system parameters. For the system considered in the main text, setting μ_2 T=π with J=Δ=μ_1=0 leads to two highly degenerate eigenphases ±π/2, as depicted in Fig. <ref>(a). However, upon slightly changing the value of μ_2, the two eigenphases move to other values and are no longer separated by π. As a result, a state formed as a superposition of these two Floquet states will not exhibit a robust periodicity. Hence there is no DTC formed.
For nonzero J=Δ<π/T, some degeneracies of the previous type are lifted, and the eigenphase spectrum form two bands of finite bandwidth, as depicted in Fig. <ref>(b). Owing to this finite bandwidth, two eigenphases at ±π/2 remain in the vicinity of μ_2 T=π [compare the two green diamond marks in panel (a) and (b)]. A superposition of two such states may then exhibit rigid periodicity to some extent, leading to the emergence of DTCs. However, such DTCs are formed by two bulk Floquet states, which are not the main focus of the main text.


To construct Majorana time crystals, we require at least two edge states with eigenphase separation of π. Majorana zero and π modes indeed satisfy this requirement. We thus hope that both Majorana zero and π modes exist under the same system parameters. Fortunately, this can be achieved by setting J=Δ≈π/T, as is evident from Figs. <ref>(c) and (d)  (Magenta squares in both panels). Period-doubling state can then be constructed by creating a superposition of these Majorana zero and π modes. Unlike bulk DTCs elucidated above, the rigidity of Majorana time crystals stems from topology, as both Majorana modes are topologically protected and they can only be lifted when the two bulk bands touch each other. Finally, it is worth noting that spontaneous time translational symmetry breaking phenomenon becomes apparent in this context for the following reason. Consider a state initially prepared in a zero edge state at certain system parameter values which only admit Majorana zero modes. Suppose that the parameter values are rapidly changed so that the system now admit both Majorana zero and π modes. The state, localized at the edge to begin with, will now naturally become a coherent superposition of all the Floquet eigenstates of the new Hamiltonian with most population being on zero and π edge states.  As a result, its periodicity changes from T to 2T, while the Hamiltonian remains periodic with period T.






§ SECTION E: ADIABATIC FOLLOWING OF MAJORANA ZERO AND Π MODES IN THE FIRST TWO STEPS OF THE PROTOCOL


In the Heisenberg picture, the evolution of Majoranas over one period is described as γ(t)=𝒰^†γ(0)𝒰. For 0̂ and π̂ to be instantaneous Majorana zero and π modes during the protocol, it suffices to show that 𝒰^†0̂𝒰=0̂ and 𝒰^†π̂𝒰=-π̂ for instantaneous values of the system parameters.  Note that 𝒰 can be written as a product of two exponentials

    𝒰=exp(-i H_2)×exp(-i H_1),

 where T has been set to 2 for brevity, H_1 and H_2 are defined in the main text. To simplify our notation, we will focus only on relevant terms in H_1 and H_2 which contain γ_j^A and γ_j^B with j=1,2,3; we suppress the rest of the terms since they always commute with both 0̂ and π̂.

In step 1, we have (using the same parameterization described in the main text)


    H_1   =   iπ/2(sinϕ_1 γ^B_2 γ^A_1 +cosϕ_1 γ^B_2 γ^A_3 +γ_1^B γ_2^A +γ_2^Bγ_3^A+γ_3^B γ_4^A) , 
     H_2   =   -iπ/4( γ_1^A γ_1^B+γ_2^Aγ_2^B+γ_3^Aγ_3^B) ,


where ϕ_1 is adiabatically increased from 0 to π/2. According to the main text,


    0̂   =   1/√(2){[cos(ϕ_1)γ_1^A-sin(ϕ_1)γ_3^A]+ [cos(ϕ_1)γ_1^B-sin(ϕ_1)γ_3^B]}, 
    π̂   =   1/√(2){[cos(ϕ_1)γ_1^A-sin(ϕ_1)γ_3^A]- [cos(ϕ_1)γ_1^B-sin(ϕ_1)γ_3^B]}.


It can be easily verified that the first (second) bracket in 0̂ and π̂ commute (anti-commute) with H_1. By using the identity


    exp(θγ_j^A γ_k^B)=cos(θ)+sin(θ) γ_j^A γ_k^B  ,


the first exponential in 𝒰 transforms 0̂→π̂ and π̂→0̂. Since H_2 anti-commute with both 0̂ and π̂, it can be shown that applying Eq. (<ref>) on the second exponential in 𝒰 leads to transformation 0̂→ -π̂ and π̂→0̂. Taken together, 𝒰 yields the desired result 0̂→0̂ and π̂→0̂.

In the second step,


    H_1   =   iπ/2( γ^B_2 γ^A_1 +sin(ϕ_2) γ_3^A γ_2^A +cos(ϕ_2)γ_1^B γ_2^A +γ_2^Bγ_3^A+γ_3^B γ_4^A) , 
    
    H_2   =   -iπ/4( γ_1^A γ_1^B+γ_2^Aγ_2^B+γ_3^Aγ_3^B) ,






where ϕ_2 is adiabatically increased from 0 to π/2. According to the main text,


    0̂= [-cos(ϕ_2)γ_3^A+sin(ϕ_2)γ_1^B]-[cos(ϕ_2)γ_3^B+sin(ϕ_2)γ_1^A], 
    π̂=[-cos(ϕ_2)γ_3^A+sin(ϕ_2)γ_1^B]+[cos(ϕ_2)γ_3^B+sin(ϕ_2)γ_1^A].



Using the same approach as before, we consider the action of each exponential in 𝒰 separately on 0̂ and π̂. Since the first (second) bracket of 0̂ and π̂ commute (anti-commute) with H_1, it immediately follows that exp(i H_1)0̂exp(-i H_1)=π̂ and exp(i H_1)π̂exp(-i H_1)=0̂. Meanwhile, H_2 remains anti-commute with both 0̂ and π̂, so that exp(i H_2)0̂exp(-i H_2)=-π̂ and exp(i H_2)π̂exp(-i H_2)=0̂. Taken together, indeed 𝒰^†0̂𝒰=0̂ and 𝒰^†π̂𝒰=-π̂.

















	


§ SECTION F: TOLERANCE OF STEP 3 AGAINST SMALL DEFORMATIONS IN THE ADIABATIC PARAMETERS


As elucidated in the main text, step 3 of the protocol corresponds to a rotation between γ_L^A and γ_L^B according to Δ_1= 2π f_3,a(t)/T, J_2=√(2)πexp(iπ/4)[1-if_3,b(t)]/T, and Δ_2=√(2)πexp(-iπ/4) [1+if_3,c(t)]/T. In the following, we numerically evaluate the evolution of the Majorana correlation functions under two different choice of the functions f_3,l=a,b,c with additional small deformations in the functional forms of the adiabatic parameters as given by J_2→ J_2+δ J_2, Δ_2→Δ_2+δΔ_2, Δ_1→Δ_1+δΔ_1.

To quantify the precision of our result, we define a normalized fidelity as ℱ=⟨ψ_f,num |ψ_f,th⟩ - 1/√(2)/1-1/√(2), where |ψ_f,th⟩ = 1/√(2)(|0⟩ -|1⟩), with |0⟩ and |1⟩ being parity eigenstates as defined in the main text, is the expected final state resulted from a perfect braiding operation, and |ψ_f,num⟩ is the final state obtained numerically from the proposed protocol. Note that ℱ≤⟨ψ_f,num |ψ_f,th⟩ for any real ⟨ψ_f,num |ψ_f,th⟩, with ℱ=1 if |ψ_f,th⟩ = |ψ_f,num⟩, and ℱ=0 if |ψ_f,num⟩ = |ψ_0⟩ = |0⟩. In Fig. <ref>(a), we take f_3,a(t)=f_3,b(t)=f_3,c(t)=cos[π s(t)] (the same as those used in the main text) with s(t) decreasing from 1 to 0. The normalized fidelity is found to be ℱ=0.9956. In Fig. <ref>(b), we take three different functions f_3,a(t)=1-2sin[π s(t)/2], f_3,b=1-2s(t), f_3,c=cos[π s(t)], which leads to the normalized fidelity of ℱ=0.9949. These two fidelities are above the typically required threshold level of ≈ 1% error for fault-tolerant quantum computation <cit.>. The fact that such high fidelities are achieved even in the presence of considerable time-dependent perturbation demonstrates certain weak topological protection of our protocol.






§ SECTION G: EXPERIMENTAL CONSIDERATION OF THE PROTOCOL


To further verify the feasibility of the proposed protocol in real experiment, it is important to compare the coherence time-scale of the Majorana modes in our system and the typical time required to complete the whole experiment. In the cold atom setup proposed in the main text, after taking into account a variety of mechanisms that induce particle loses, the coherence time-scale can be assumed to be extendable to the order of seconds <cit.>. On the other hand, given that the system parameters are typically of the order of tens of kHz <cit.>, a single period can last of the order of 0.1 ms in order to achieve the parameter regime in which Majorana zero and π modes exist. As demonstrated by our numerics, each step in the protocol may require around 200-400 number of periods in order to achieve a good precision. This means that to complete a single braiding process, it may take around 1200-2400 number of periods, which is of the order of 0.1 s. Comparing the two time scales together, it is expected that a number of braiding operations can be applied via our protocol safely before the system loses its effectiveness.

In real experiment, one may also be wary of decoherence. Due to the nonlocal nature of our qubits, local errors induced by decoherence are typically harmless by themselves. However, over time, such errors may accumulate across the whole lattice, leading to a nonlocal error of the form iγ_L^l γ_R^k, with l,k=A,B, which may become dangerous. Therefore, especially if one needs to perform longer time operations, it may be necessary to devise a correction scheme to avoid such an error. Given that Majorana chain is a natural stabilizer code, the most natural error correction scheme will be to measure these (Floquet) stabilizer operators (e.g. iγ_j^Aγ_j^B with j≠ 1,N) stroboscopically at every period, then perform suitable correction by simply applying the error operators one more time, since local errors ℰ_j compatible with parity conservation are typically of the form iγ_j^B γ_j+1^A, which squares to identity.  In the proposed cold atom experiment, measuring these stabilizer operators should be feasible as one essentially needs to only measure the parity of each lattice site, while applying corrections can be done by injecting/removing fermions at the infected sites. Although this error correction scheme is rather straightforward, the time required to perform corrections scales with the number of lattice sites, so it may become less efficient for longer wires. As such, it may also be interesting to explore other more sophisticated error correction schemes, such as that introduced in Ref. <cit.>. However, these are beyond the scope of this work and may be left for potential future studies.





§ SECTION H: EXTENSION OF THE PROTOCOL TO AN ARRAY OF SUPERCONDUCTING WIRES




Since our original setup only allows the implementation of a single qubit in either even or odd parity subspace of the wire,  its manipulation through braiding and its application in quantum computing may at first look quite limited. However, by adding more wires and making some modifications to our protocol, it is possible to encode and manipulate more qubits, thus unleashing the full power of MTCs to perform more complex quantum computation.

To elaborate on this idea, consider an array of superconducting wires, each described by Hamiltonian H^(l)(t) as defined in the main text, where an index l has been introduced to mark different wires in the array. In particular, the index l is given to all system parameters, namely J_j^(l), Δ_j^(l), μ_1^(l), and μ_2^(l) which to some extents allow each wire not to be fully identical. We will focus our attention on two adjacent wires l and l+1 as shown in Fig. <ref>(a). Upon turning on the periodic driving, the Majorana zero modes at each wire acquire period-doubling behaviour, which leads to four pairs of Majoranas in space-time lattice. In particular, we may denote the four Majoranas localized at the left edge as γ_A^(l), γ_A^(l+1), γ_B^(l), and γ_B^(l+1), where the subscript and superscript indices denote the time lattice and wire array respectively.

The modified protocol consists of four steps, which are summarized in the inset of Fig. <ref>(b). In the first step, we modify J_1^(l), J_2^(l), Δ_1^(l) and Δ_2^(l) according to step 1 in our original protocol, while the rest of parameters are kept constant. This results in γ_A^(l) and γ_B^(l) to localize around the third site of the lth-wire. In the second step, we introduce hopping and pairing between the first site of the (l+1)th-wire and the second site of the lth-wire, while at the same time varying J_1^(l) and Δ_1^(l) according to step 2 in our original protocol and keeping the other parameters fixed. This moves γ_A^(l+1) and γ_B^(l+1) to the first site of the lth-wire. In the third step, we reduce the hopping and pairing strength between the first site of the (l+1)th-wire and the second site of the lth-wire adiabatically to zero, while at the same time varying J_2^(l) and Δ_2^(l) according to step 2 in our original protocol and keeping the other parameters constant. This moves γ_A^(l) and γ_B^(l) to the first site of the (l+1)th-wire. Finally, in the last step, we vary J_1^(l), J_2^(l), Δ_1^(l) and Δ_2^(l) according to step 3 and 4 in our original protocol to return the Hamiltonian to its original configuration. The net outcome of the aforementioned protocol simulates a successive braiding between different pairs of Majoranas as γ_A^(l+1)→γ_B^(l+1)→ -γ_B^(l),  γ_B^(l+1)→ -γ_A^(l+1)→γ_A^(l), γ_A^(l)→γ_A^(l+1), and γ_B^(l)→γ_B^(l+1), which is described schematically in Fig. <ref>(b).  This makes it clear that we can scale up the MTCs described in the main text to implement multi-gate operations.

	
	
	
	
	
	
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 This exchange γ_L^A ↔γ_L^B is trivial as compared with the braiding exchange γ_L^A→γ_L^B and γ_L^B→ -γ_L^A (i.e., an extra negative sign is necessary for a braiding process).
	
	
