High-fidelity non-adiabatic cutting and stitching of a spin chain via local control

We propose and analyze, focusing on non-adiabatic effects, a technique of manipulating quantum spin systems based on local 'cutting' and 'stitching' of the Heisenberg exchange coupling between the spins. This first operation is cutting of a bond separating a single spin from a linear chain, or of two neighboring bonds for a ring-shaped array of spins. We show that the disconnected spin can be in the ground state with a high-fidelity even after a non-adiabatic process. Next, we consider inverse operation of stitching these bonds to increase the system size. We show that the optimal control algorithm can be found by using common numerical procedures with a simple two-parametric control function able to produce a high-fidelity cutting and stitching. These results can be applied for manipulating ensembles of quantum dots, considered as prospective elements for quantum information technologies, and for design of machines based on quantum thermodynamics.


Introduction
Theory of quantum control and its applications [1,2] attract considerable attention of researchers. The abilities of producing and transforming quantum states on demand become important also in connection with research on quantum computation and information [3]. The problems in this field are usually related to the studies of the system evolution under time-dependent Hamiltonians and the resulting preparation of the desired final states with the maximum possible fidelity.
In general, these problems can be formulated as follows. Assume that we impose an external control on a given quantum system in such a way that its initial Hamiltonian H i and the final Hamiltonian after time T , H f , are known. We want to achieve the ground state of H f as a result of the unitary evolution governed by a time-dependent Hamiltonian H(t), with H(0) = H i and H(T ) = H f , whose initial state is the ground state of H i . The adiabatic passage is a well-known way [4] to realize this process. However, in order to satisfy the conditions of the adiabatic theorem, one has to implement the evolution of slow-varying Hamiltonian for a very long time, where decoherence may ruin the quantumness of the system.
One of the strategies for quantum control is the "adiabaticity shortcut" [5] to allow the design of the Hamiltonian H(t) in such a way that the system terminates at the ground state of H f at a relatively short T . There are lots of proposals for making such adiabatic-like passages in a finite, short time; for instance, the Berry's proposal of transitionless quantum driving [6], or pulse/noise control [7,8], and invariant-based inverse engineering [5]. However, these proposals require the ability to control the entire quantum system. For example, to cut a spin chain in a short time with "adiabatic shortcut", one has to control and change with time each single spin and each spinspin interaction bond [9] in the chain. Moreover, the adiabaticity implies that for each intermediate time 0 < t < T , the state of the evolving system is an instantaneous ground state of H(t), but this is not necessary for our goal.
In the present paper, we investigate the shortcut via only local control of a complex quantum system described by the Heisenberg model for interacting spins in an external magnetic field. First, we focus on the problem of "cutting" or disentangling of a such system into two parts. Second, we consider inverse problem of stitching of two systems into one-the process which can be characterized as a quantum assembly. In general, the current adiabatic quantum computation or quantum annealing [10] is a multipartite stitching process, as implemented, for example, in D-wave quantum processors. The initial Hamiltonian H i may describe a non-interacting independent spin system, and with time evolution the system is stitched as a correlated entity described by the sitebond spin model H f , such as the Ising model. Our direct numerical simulations show that such kind of control can be achieved with a high fidelity in the non-adiabatic domain.
Such a quantum spin system can be realized in an ensemble of quantum dots, proposed as hardware elements for quantum information processing [11] as well as in other spin-based systems with a similar prospective [12]. The spin manipulation in a quantum-dot system has become a well-controllable procedure [13,14] by now. The spinspin interaction can be controlled electrically by a fast gating of the electron tunneling channels between the quantum dots [15].

Separation of the quantum system
Here we describe the problem in general, without referring to a specific system. Let us assume that some joint quantum system consists of two parts: A and B. Initially, there is an interaction between A and B, so the ground state |ψ 0 of the joint A+B system generally cannot be presented as a direct product of A and B states: |ψ 0 = |ϕ 0A ⊗|ϕ 0B , where |ϕ 0A(B) is the ground state of system A(B). Assume that initial Hamiltonian of this joint system has the form: where V corresponds to the interaction between A and B parts. We can also write the following relation: (H 0 + V ) |ψ 0 = ǫ min |ψ 0 , where ǫ min is the minimal eigenvalue of the Hamiltonian (1). Let us assume the "switch-off" of the V interaction while the unitary evolution U(t) is governed by the time-dependent Hamiltonian H(t) = H 0 + g(t)V , with initial state |ψ(t = 0) = |ψ 0 , g(t ≤ 0) = 1, and g(t ≥ T ) = 0, where T is the time of cutting. The final state can be written as |ψ(T ) = U(T ) |ψ 0 , where with T denoting the time-ordering. Here and below we use the units with ≡ 1. After the interaction V between A and B is switched off, we may consider A and B as two separate systems, which however, can be entangled. The density matrix for the quantum state of each system can be written as a partial trace over the other one: Our goal is to make the switch-off process in such a way that the system A is finally in its ground state (see figure 1). Note, that we are not interested in the final state of the system B in such a setting. The final Hamiltonian can be written as H(t ≥ T ) = H 0 = H 0A + H 0B , where H 0A and H 0B have its own ground states H 0A(B) |ϕ 0A(B) = λ min A(B) |ϕ 0A(B) . Our goal is to find the optimal shape of g(t) to have the fidelity, as close to 1 as possible. Also, we can reformulate the problem and define another fidelity f ′ = | ϕ 0A | ⊗ ϕ 0B | |ψ(T ) | which is equal to 1 if and only if the A and B systems are both in their ground states after switching off the interaction. It is easy to prove the following inequality: f ≥ f ′ , so the fidelity f ′ also can be used to calculate the lower bound of the target fidelity f . We will use f ′ to describe the reverse problem for controllably entangling the initially-separate A and B parts into the joint A+B system.  (3), are equal to each other [16].
First of all, we notice the importance of the commutator [H 0 , V ] for our consideration. It is easy to see from (2) by using Magnus and Zassenhaus [17] formulas that if [H 0 , V ] = 0, the target fidelity f (f ′ ) does not depend on g(t) and T : Such a trivial case is not interesting for our consideration and we will assume that [H 0 , V ] = 0.
It is hard or impossible to find an analytic solution of propagator (2) for our quantum control problems. Consequently, numerical algorithms for designing optimal quantum control have been proposed (see, e.g., the Krotov optimization method [18] which was successfully implemented in the quantum control [19]). Recently, an optimized algorithm has been proposed for the control pulses for practical qubits [20].
In this paper, we present numerical results of the successful optimization of the g(t) function by choosing appropriate set of parameters for its representation. The target fidelity f or f ′ , being a functional of g(t), becomes then a function of these parameters. We will demonstrate how this set can be optimized and study the properties of the optimized solutions.

The model: Heisenberg chain in magnetic field
We consider a chain with N spins (see figure 2) placed in a uniform external magnetic field with the following Hamiltonian where σ n is the Pauli matrix vector, σ z is its z− component, B is the external magnetic field along the z direction, J = ±1 for (anti-) ferromagnetic coupling, and n is the spin number. For the ring-shaped array of spins we add Jσ N σ 1 −term to equation (5) to assure the system periodicity. We consider the whole chain as the A+B system and the first spin as the A system (see figure 2). In such a case, we where the time dependence of the coupling is due to the time dependence of the exchange interaction in the corresponding bonds. It is obvious that for a ferromagnetic coupling with J < 0, the ground state of the A+B system is disentangled |ψ 0 = |ϕ 0A ⊗ |ϕ 0B and therefore the cutting with the best fidelity f = 1 can be made instantaneously. Thus, we concentrate only on the nontrivial antiferromagnetic coupling J = 1.

Parametrization of g(t) and numerical results
We begin with a polynomial representation of g(t) in the form: where a 2 , . . . , a K are free parameters and a 1 = −(1 + a 2 + a 3 + . . . + a K ). Thus, the fidelity is a function of a 2 , . . . , a K as f = f (a 2 , a 3 , . . . , a K , T ). In the adiabatic regime with the simplest linear decrease of the V term, we have Our goal is to find a set of parameters a = {a 2 , a 3 , . . . , a K } which maximizes the fidelity f for a non-adiabatic process with a given finite time T . To find the proper a, we use Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization method [22]. In figure 3 we show the fidelity f for optimized cutting and f 0 for non-optimized cutting as a function of the cutting time T for the ring-shaped and linear chains with N = 6 and 7, J = 1, and B = 2. We use two free parameters a 2 , and a 3 , with K = 3 in equation (6). Numerical simulations have been performed by using a Python SciPy package with the built-in BFGS optimization method. Unitary operator in equation (2) was approximated with 300 time steps, and the error in fidelity f (f ′ ) is less than 0.001. Gradient for BFGS was approximated as a finite difference with a step of 0.1 in the parameter space. Note, that the magnetic field B should not be very high, because in such a case the ground state becomes fully spin polarized and disentangled. In figure  4 we show the optimal shape of the control function g(t) for the ring-shaped array of spins. The open chain has the similar shape for g(t). Obviously, g(t) becomes closer to the simple linear dependence with increasing the cutting time T . Numerical data for this setting is presented in Table 1.
In figure 5 we show the fidelity f as a function of the parameter a 2 (a 3 ) with a fixed at the optimal value a 3 (a 2 ). Here T = 0.6 for the ring-shaped array with N = 6, J = 1, and B = 2, where the optimal values are a 2 = 54.31 and a 3 = −36.33 (the optimal values correspond to the red dashed vertical lines in figure 5 and are shown in the second column in Table 1). We have a non-optimized fidelity f 0 = 0.86, and f = 0.99 with optimization. As can be seen in figure 5 when parameters a 2 and a 3 are allowed to have a small deviation ≈ 2% from their optimal values, our proposal is still effective. The parameter a 3 must be tuned with more accuracy than parameter a 2 .
As we have mentioned above, the cutting process is not necessarily adiabatic also in the sense that the condition, f ′ (t) = | ψ 0 (t)|ψ(t) | → 1 for 0 ≤ t ≤ T , is not important for us. Here |ψ 0 (t) is the instantaneous ground state of H(t), and |ψ(t) = U(t) |ψ(0) . In figure 6, we show the time dependent f ′ (t) and our target fidelity f (t) versus the time for optimized and non-optimized cutting protocols. One can see that in the middle of optimized cutting process, the fidelity f can be smaller than for non-optimized cutting. Nonetheless, at the end of the process, it becomes high. Moreover, as can be seen from the behavior of f ′ (t), optimized process is less adiabatic than non-optimized one during most of the evolution time. However, as mentioned above, we are interested only in the final state.
In addition, we note that even in the case of the energy-levels crossing, which happens, e.g., in a ring-shaped array with N = 7, B = 2, J = 1, the behavior of cutting fidelity f (or f 0 corresponding to the non-optimized linear switch-off) is the same as without eigen-energy crossing. Fidelities f and f 0 increase up to 1 with the evolution time T (see figure 3(b)). However, fidelity f ′ = 0 in this case. This means that A and B parts become disentangled after cutting, with A part being in its ground state, while B part is in a pure state orthogonal to its ground state.

Robustness of the control
Now we consider the influence of the "apparatus" noise [23,24] in the control function g(t) to the effectiveness of cutting. The noise is simulated as a set of rectangular pulses with a fixed length ∆t and the random strength ∆g(1/2−r), where ∆g is a characteristic strength, and r is a random number in the interval [0, 1). This noise is added to the smooth optimal control function g(t) (given by the representation in equation (6)) for   figure 7. Thus, we conclude that our proposal is robust against small fluctuations in the control function. Also, as can be seen in figure 7, a noise with a high characteristic frequency influences the fidelity less than that with a low-frequency.

Another choice of the control function
Since we are not restricted in choosing the parametrization of the control function g(t), here we compare the results of its different representations. According to the boundary conditions, we can write g(t) as follows: where a 1 , . . . , a K are free parameters. To compare the parametrizations in equations (6) and (8), we chose K = 2 in equation (8) and repeat the optimization for the cutting of a ring-shaped array. The results are presented in figure 8. As can be seen, the effectiveness of the optimization algorithm for different g(t) representations is almost the same. In our simulations we use the BFGS optimization method, and this means that we do not need to calculate the fidelity f for each a on a specially chosen area of the possible values of a with a fixed step. Nevertheless, it is instructive to look at the "landscape" of the fidelity f as a function of two free parameters and make sure that the gradient based BFGS optimization algorithm works correctly. Moreover, such kind of data may be of independent interest. In figure 9, we show the landscapes corresponding to different parametrizations of g(t) and the same physical setting (ring-shaped array, N = 6, T = 0.6, J = 1, B = 2). White lines in figures 9(a),(b) correspond to the results of the BFGS optimization.
We can also consider g(t) as a sequence of the K squared pulses whose amplitudes form the set of free parameters a = {a 1 , . . . , a K }, and the time length of each pulse is ∆t = T /K: where θ(t) is a Heaviside step function. In figures 10(a) -10(c), we show the optimal shape of g(t) for K = 2, and in figures 10(d)-10(f) K = 9 is used. Our setting is the same as in figure 4. By comparing the first row in figure 10 with the shapes in figure 4 (dashed lines), we see the correspondence between the shapes of smooth and pulse controls. Moreover, the optimization time for a pulsed shape with K ≈ 10 is much faster than the parametrized optimization with smooth shape of g(t) even for K = 2. This is because for optimization of a smooth g(t), the algorithm divides the [0, T ] interval into more than 100 pieces to accurately calculate the propagator (2) while for the pulse control, the evolution interval is divided into the given number of K pieces.

Stitching of the quantum system
Now we consider the opposite process: a stitching of the quantum system. In this setting, we have a disentangled initial state of the A+B system which is the ground state of the Hamiltonian H 0 . Then, we switch on the interaction V between A and B systems, and the desirable final state of the joint A+B system is the ground state of the Hamiltonian H 0 + V . We use the fidelity f ′ to describe the effectiveness of such a process (because both states of the A and B parts are important at the start and end of the stitching). Thus we have to satisfy more strict conditions in order to achieve the high fidelity if the stitching is involved: the energy gap between the two lowest states of the Hamiltonian H = H 0 + g(t)V cannot be zero for any g(t) ∈ (0, 1). This requirement is not applicable to the chain cutting, where only the final state of the A system is essential.
Here we consider parametrization of the control function for the stitching process g(t) expressed in the same way as for the cutting (6) just by changing t → T − t: Note that this polynomial parametrization of g(t) does not imply that the control shape for stitching can be reconstructed from the control shape for the corresponding cutting since the optimal sets of parameters are essentially different for these processes. As an example we show in figure 11 the results for the optimization of the stitching process for a ring-shaped array.
Here we make a general comment related to the cutting and stitching processes. The ground states of the initial and final Hamiltonians (with g = 0 and g = 1) can be degenerate for the cutting or stitching process. In such a case we have the ground state subspace, and we have to choose the initial ground state |ψ 0 in such a way that ψ 0 |ψ ′ 0 → 1 for |ψ ′ 0 being a non-degenerate ground state of the Hamiltonian H = H 0 + (g(0) + δg)V , where g(0) = 1 and δg → −0 (g(0) = 0 and δg → +0) for the cutting (stitching) process.

Conclusion
We have shown the feasibility of the unitary non-adiabatic cutting and stitching of complex quantum systems with a demanded high-fidelity output. The key feature of our proposal is the possibility of using only local control instead of the global one. The optimal shape of the control function can be found by the gradient numerical optimization. We show that even a simple two-parametric control can be efficient for cutting and stitching the complex quantum system with a high fidelity. We do not use any approximations and therefore our proposal is useful only for relatively small quantum systems which can be numerically simulated. The time of the process can be decreased significantly in order to achieve the demanded fidelity in comparison with the non-optimized adiabatic control. We show that our proposal is robust against small variations of parameters and noise in the control as well. It was shown that different parametrizations of the control function can be used. Our results can stimulate further investigations in the adiabaticity shortcut problems. From the experimental point of view, they can be applicable to the electrical control of spin states in an ensemble of quantum dots, where the modulation of spin-spin interaction can be done by gating the interdot tunneling on the time scale as short as 10 −2 nanoseconds.