Optimized dynamical control of state transfer through noisy spin chains

We propose a method of optimally controlling the tradeoff of speed and fidelity of state transfer through a noisy quantum channel (spin-chain). This process is treated as qubit state-transfer through a fermionic bath. We show that dynamical modulation of the boundary-qubits levels can ensure state transfer with the best tradeoff of speed and fidelity. This is achievable by dynamically optimizing the transmission spectrum of the channel. The resulting optimal control is robust against both static and fluctuating noise in the channel's spin-spin couplings. It may also facilitate transfer in the presence of diagonal disorder (on site energy noise) in the channel.

The distribution of coupling strengths between the spins that form the quantum channel, determines the state transfer-fidelities [3,[26][27][28][29][30]. Perfect state-transfer (PST) channels can be obtained by precisely engineering each of those couplings [27][28][29][30][31][32][33][34]. Such engineering is however highly challenging at present, being an unfeasible task for long channels that possess a large number of control parameters and are increasingly sensitive to imperfections as the number of spins grows [34][35][36][37]. A much simpler control may involve only the boundary (source and target) qubits that are connected via the channel. Recently, it has been shown that if the boundary qubits are weakly-coupled to a uniform (homogeneous) channel (i.e., one with identical couplings), quantum states can be transmitted with arbitrarily high fidelity at the expense of increasing the transfer time [36,[38][39][40][41][42][43][44]]. Yet such slowdown of the transfer may be detrimental because of omnipresent decoherence.
To overcome this problem, we here propose a hitherto unexplored approach for optimizing the tradeoff between fidelity and speed of state-transfer in quantum channels. This approach employs temporal modulation of the couplings between the boundary qubits and the rest of the channel. This kind of control has been considered before for a different purpose, namely to implement an effective optimal encoding of the state to be transferred [45]. Instead, we treat this modulation as dynamical control of the boundary system which is coupled to a fermionic bath that is treated as a source of noise. The goal of our modulation is to realize an optimal spectral filter [46][47][48][49][50][51][52][53][54] that blocks transfer via those channel eigenmodes that are responsible for noiseinduced leakage of the QI [55]. We show that under optimal modulation, the fidelity and the speed of transfer can be improved by several orders of magnitude, and the fastest possible transfer is achievable (for a given fidelity).
Our approach allows to reduce the complexity of a large system to that of a simple and small open system where it is possible to apply well developed tools of quantum control to optimize state transfer with few universal control requirements on the source and target qubits. In this picture, the complexity of the channel is simply embodied by correlation functions in such a way that we obtain a universal, simple, analytical expression for the optimal modulation. While in this article we optimize the tradeoff between speed and fidelity so as to avoid decoherence as much as possible, this description [46][47][48][49][50][51][52][53][54][55] allows one to actively suppress decoherence and dissipation in a simple manner, since it may be viewed as a generalization of dynamical decoupling protocols [56][57][58][59]. In what follows, we explicitly deal with a spin-chain quantum channel, but point out that our control may be applicable to a broad variety of other quantum channels. 1. Quantum channel and state transfer fidelity

Hamiltonian and boundary control
We consider a chain of + N 2 spin-1 2 particles with XX interactions between nearest neighbors, which is a candidate for a variety of state-transfer protocols . The Hamiltonian is given by where H 0 and H bc stand for the chain and boundary-coupling Hamiltonians, respectively, σ ( ) i x y are the appropriate Pauli matrices and J i are the corresponding exchange-interaction couplings.

Mapping to a few-body open-quantum system
The magnetization-conserving Hamiltonian H can be mapped onto a non-interacting fermionic Hamiltonian [60] that has the particle-conserving form populates a single-particle fermionic eigenstate ω k of energy ω k , and = j 0 .. 01 0 .. 0 j denote the single-excitation subspace. Under the assumption of mirror symmetry of the couplings with respect to the source and target qubits = − J J i N i , the energies ω k are not degenerate, ω ω < + k k 1 , and the eigenvectors have a definite parity that alternates as ω k increases [28]. This property implies that and allows us to rewrite the boundary-coupling Hamiltonian as For an odd N, there exists a single non-degenerate, zero-energy fermionic mode in the quantum channel, labelled by = = + k z N 1 2 [25,39,44]. As a consequence, the two boundary qubits (0 and + N 1) are resonantly coupled to this mode. Therefore, we consider these three resonant fermionic modes as the 'system' S and reinterpret the other fermionic modes as a 'bath' B. In this picture, the system-bath SB interaction is off-resonant. Then, we rewrite the total Hamiltonian as  (5) is amenable to the application of optimal dynamical control of the multipartite system [46,47,[61][62][63][64]: such control would be a generalization of the single-qubit dynamical control by modulation of the qubit levels [48][49][50][51][52][53][54]. To this end, we rewrite equation (7) in the interaction picture as a sum of tensor products between system S j and bath B j operators (see appendix A) From this form one can derive the system density matrix of the system, ρ ( ) t S , in the interaction picture, under the assumption of weak system-bath interaction, to second order in H SB , as [46,48] denoting the correlation functions of bath operators and being a rotation-matrix in a chosen basis of operators ν i used to represent the evolving system operators, (appendix A). The solution (9) will be used to calculate and optimize the state-transfer fidelity in what follows.

Fidelity derivation
We are interested in transferring a qubit state ψ 0 initially stored on the 0 qubit to the + N 1 qubit . Here ψ 0 is an arbitrary normalized superposition of the spin-down 0 0 and spin-up 1 0 (single-spin) states. To assess the state transfer over time T, we calculate the averaged fidelity [3], which is the state-transfer fidelity averaged over all possible input states ψ 0 . In the interaction picture, and ψ ψ ⊗ S B is the initial state of + S B. In the ideal regime of an isolated three-level system, PST occurs when the accumulated phase due to the modulation control Obviously, this condition does not strictly hold when the system-bath interaction is accounted for, yet it is still adequate within the second-order approximation in H SB used in equation (9). In this approximation, T t 0 0 Here, t are the corresponding dynamical-control functions (appendices A and B). In the calculations we considered ψ = 0 B B . However, in the weak-coupling regime the transfer fidelity remains the same for a completely unpolarized state [44,65] or any other initial state [25] of the bath.

Optimization method
To ensure the best possible state-transfer fidelity, we use modulation as a tool to minimize the infidelity ζ ( ) T in (13)- (14) by rendering the overlap between the interacting bath-and filterspectrum functions as small as possible [46,47].

Optimizing the modulation control for non-Markovian baths
The minimization of ζ ( ) T in (13) can be done for a specific bath-correlation function of a given channel which represents a non-Markovian bath. The Euler-Lagrange (E-L) equation for where λ is the Lagrange multiplier and ϕ α =J z . The optimal modulation can be obtained by solving the integro-differential equation The solution of equation (18) should satisfy the boundary conditions ϕ = ( ) to ensure the required state transfer.
In general the bath-correlations have recurrences and time fluctuations due to mesoscopic revivals in finite-length channels. Therefore, it is not trivial to solve equations (18)- (19) analytically and they need to be solved numerically for each specific channel. We however are interested in obtaining universal analytical solutions for state-transfer in the presence of non-Markovian noise sources. To this end, we here discuss suitable criteria for optimizing the state transfer in such cases.
We require the channel to be symmetric with respect to the source and target qubits and the number of eigenvalues to be odd. These requirements allow for a central eigenvalue that is invariant under noise on the couplings. This holds provided a gap exists between the central eigenvalue and the adjacent ones, i.e. they are not strongly blurred (mixed) by the noise, so as not to make them overlap. At the same time, we assume that the discreteness of the bath spectrum of the quantum channel is smoothed out by the noise, since it tends to affect more strongly the higher frequencies [34,36,37]. Then, if we consider the central eigenvalue as part of the system, a common characteristic of ω ± ( ) G is to have a central gap (as exemplified in figure 1 Therefore, in order to minimize the overlap between ω for general gapped baths, and thereby the transfer infidelity in (14), we will design a narrow bandpass filter centered on the gap.
We present a universal approach that allows us to obtain analytical solutions for a narrow bandpass filter around ω z . Since ω − ( ) G has a narrower gap than ω We seek a narrow bandpass filter, whose form on time-domain via Fourier-transform decays as slowly as possible, so as to filter out the higher frequencies. This amounts to maximizing T T T , , i 0 subject to the variational E-L equation (17), upon replacing ζ by − F T , . Since there is no explicit dependence on ϕ, the second term therein is null, where λ E is the Lagrange multiplier and λ ϕ is an integration constant chosen to satisfy the boundary conditions obeyed by the accumulated phase (11). Analytical solutions of (21) are obtainable for small τ, corresponding to the differential equation The unknown parameters are then optimized under chosen constraints, e.g. on the boundary coupling, the transfer time, the energy, etc.
The frequencies ω v that give a low and flat filter ω the contribution of larger frequencies will be suppressed, while the filter-overlap with the central energy level will be maximized; for larger n, the central peak splits and additional peaks appear at larger frequencies.
Therefore, the analytical expressions for the optimal solutions satisfying ϕ = ( ) , the corresponding filter is the narrowest around 0, but it has many wiggles on the filter tails ( figure 1(b)) which overlap with bath-energies that hamper the transfer. In contrast, the p = 1, 2 bandpass filters are wider (for the same T) and require more energy, = π E E filters out a similar spectral range). The shorter T, the lower is p that yields the highest fidelity, because the central peak of the filter that produces the dominant overlap with the bath spectrum is then the narrowest. However, as T increases, larger p will give rise to higher fidelity, because now the tails of the filter make the dominant contribution to the overlap. As shown in figure 2, the filter for p = 1, 2 can improve the transfer fidelity by orders of magnitude in a noisy gapped bath bounded by the Wignersemicircle, which is representative of fully randomized channels [66] (appendix C).

Optimizing the modulation control for a markovian bath
We next consider the worst-case scenario of a Markovian bath, where the bath-correlation functions Φ τ ± ( ) vanish for τ > 0. This is the case when the gap is closed by a noise causing the bath energy levels to fluctuate faster than the system dynamics. We note that, finding optimal solutions for the noise spectrum of a Markovian bath is important for the case where the gap is reduced or even lost in static cases. The E-L equation under energy constraint (17), is now This equation has a non-trivial analytical solution and the modulation that minimizes ζ ( ) T is given by the following transcendental equation The infidelity for this optimal modulation almost coincides with the one obtained for static

The infidelity function (13) that must be minimized when the correlation time
M q assuming no constraints (λ = 0). An example of the performance of this solution is discussed below and shown in figure 5.

Optimal control of transfer in a homogeneous spin-chain channel
Consider a uniform (homogeneous) spin-chain channel, i.e. ≡ J J i in equation (1), whose energy eigenvalues are ω = π + ( ) J 2 cos k k N 1 [38]. In figure 3, we show the performance of the general optimal solutions (24) for this specific channel as a function of α M and T.
The approach based on equation (13) strictly holds in the weak-coupling regime α ≪ ( ) 1 M [46-50, 53, 54]. In this regime (marked with arrows in figure 3(b), we found that the transfer , and the infidelity decreases by reducing α M according to a power law, aside from the oscillations due to the discrete nature of the bath-spectrum (see appendix C). The filter tails are sinc-like functions, so that when a zero of the filter matches a bath-energy eigenvalue, the infidelity exhibits a dip. Aside from oscillations, the best tradeoff between speed and fidelity within this regime is given by the optimal modulation with p = 2 (for the system described in figure 3(a)).
However, this approach can also be extended to strong-couplings α M , since it becomes compatible with the weak-coupling regime under the optimal filtering process that increases the state fidelity in the interaction picture [51,62,63]. The bandpass filter width increases as T decreases; consequently, in the strong-coupling regime α ∼ bath energies closest to ω z , but still block the higher bath energies, which are the most detrimental for the state transfer [34,36,37]. Then, the participation of the closest bath energies yields a transfer time ≈ T c p p N J 2 . There is a clear minimal infidelity value at the point that we denote as α M opt p which depends on p ( figure 3(b)); thus extending the previous static-control (p = 0) results, where an optimal α M opt 0 was found [26,36,67,68]. The infidelity dip corresponds to a better filtering-out (suppression) of the higher energies, retaining only those that correspond to an almost equidistant spectrum of ω k around ω z , which allow for coherent transfer [36,37]. . If the constraint on α M can be relaxed, i.e. more energy can be used, the advantages of dynamical control can be even more appreciated for both infidelity decrease and transfer-time reduction by orders of magnitude, as shown in figure 3(a). Hence, our main result is that the speed-fidelity tradeoff can be drastically improved under optimal dynamical control.

Robustness against different noises
We now explicitly consider the effects of optimal control on noise affecting the coupling strengths, also called off-diagonal noise, causing: Here ε > 0 J characterizes the noise or disorder strength. When Δ i is time-independent, it is called static noise, as was considered in other state-transfer protocols [34,36,69,70]. When Δ ( ) t i is time-dependent, we call it fluctuating noise [71]. These kinds of noises will affect the bath energy levels, while the central energy ω z remains invariant [34,37]. In the following we analyse the performance of the control solutions obtained in section 2 for these types of noise and later on, in section 4.4 we discuss briefly the effects of other sources of noise.

Static noise
Static control on the boundary-couplings can suppress static noise [34,36] but here we show that dynamical boundary-control makes the channel even more robust, because it filters out the bath-energies that damage the transfer. To illustrate this point, we compare the effect of modulations α ( )

Markovian noise
The worst scenario for quantum state transfer is the absence of an energy gap around ω z . This case corresponds to Markovian noise characterized by , where the brackets denote the noise ensemble average, or equivalent to a bath correlation-function that vanishes at τ > 0. In this case there is an analytical solution for the optimal modulation given by equation (28), although the infidelity achieved by it almost coincides with the one obtained by the static (p = 0) optimal control (figure 5). Counterintuitively, arbitrarily high fidelities can be achieved for such noise by decreasing and thereby slowing down the transfer. This comes about because in a Markovian bath, the very fast coupling fluctuations suppress the disorder-localization effects that hamper the transfer fidelity as we show below for a typical case.

Non-Markovian noise
We now consider a non-Markovian noise of the form Δ + ( ) We observe a convergence of the transfer fidelity to its value without noise as the noise correlation time τ c decreases ( figure 4(b)). Consequently the fidelity can be substantially improved by reducing α M . The effective noise strength scales down as τ c 1 2 ( figure 4(b), inset). By contrast to the Markovian limit τ → 0 c , dynamical control can strongly reduce the infidelity in the non-Markovian regime that lies between the static and Markovian limits and whose bath-spectrum is gapped.

Other sources of noise
Timing errors: in addition to resilience to noise affecting the spin-spin couplings, there is another important characteristic of the transfer robustness, namely, the length of the time window in which high fidelity is obtained. The fidelity F(t) under optimal dynamical control = ( ) p 1, 2 , yields a wider time-window around T where the fidelity remains high compared with its static = ( ) p 0 counterpart. This allows more time for determining the transferred state or using it for further processing. Consequently, the robustness against timing imperfections [29,34] is increased under optimal dynamical control.
On-site energy noise: this kind of noise, alias diagonal-noise, can be either static or fluctuating. The static one can give rise to the emergence of quasi-degenerate central states. Then, the dynamical control approach introduced in this work is still capable of isolating the 'system' defined here (section 1) from the remaining 'bath' levels. It may happen that the spin network is not symmetric with respect to the source and target spins, and then the effective couplings of the source and target qubits with the central level will not be symmetric. This asymmetry can be effectively eliminated by boundary control. On the other hand, a fluctuating diagonal-noise that may produce a fluctuation of the central energy level is here fought by optimizing the tradeoff between speed and fidelity as detailed above. Additional dynamical control of only the source and target spins can be applied to avoid these decoherence effects, by the mapping to an effective three-level system, as a variant of dynamical decoupling [56][57][58][59].

Conclusions
We have proposed a general, optimal dynamical control of the tradeoff between the speed and fidelity of qubit-state transfer through the central-energy global mode of a quantum channel in the presence of either static or fluctuating noise. Dynamical boundary-control has been used to design an optimal spectral filter realizable by universal, simple, modulation shapes. The resulting transfer infidelity and/or transfer time can be reduced by orders of magnitude, while their robustness against noise on the spin-spin couplings is maintained or even improved. Transfer-speed maximization is particularly important in our strive to reduce the random phase accumulated during the transfer when energy fluctuations (diagonal noise) affect the spins [76]. We have shown that, counterintuitively, static noise is more detrimental than fluctuating noise on the spin-spin couplings. This general approach is applicable to quantum channels that can be mapped to Hamiltonians quadratic in bosonic or fermionic operators [12,15,16,44,77]. We note that our control is complementary to the recently suggested control aimed at balancing possible asymmetric detunings of the boundary qubits from the channel resonance [76,77].
1 3 . In the interaction picture     . The latter correlation function represents the limiting case of an infinite channel and it gives a continuous bath-spectrum that becomes a semicircle. In the case of a finite channel, ω ( ) G will be discrete but modulated by the semicircle with a central gap. If disorder is considered, the position of the spectrum lines fluctuates from channel to channel but they are essentially modulated by the semicircle with a central gap as was considered in the figure 1(b)