# Quantum control robust with respect to coupling with an external environment

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## Abstract

We study coherent quantum control strategy that is robust with respect to coupling with an external environment. We model this interaction by appending an additional subsystem to the initial system and we choose the strength of the coupling to be proportional to the magnitude of the control pulses. Therefore, to minimize the interaction, we impose \(L_1\) norm restrictions on the control pulses. In order to efficiently solve this optimization problem, we employ the BFGS algorithm. We use three different functions as the derivative of the \(L1\) norm of control pulses: the signum function, a fractional derivative \(\frac{\mathrm {d}^\alpha |x|}{\mathrm {d}x^\alpha }\), where \(0<\alpha <1\), and the Fermi–Dirac distribution. We show that our method allows to efficiently obtain the control pulses which neglect the coupling with an external environment.

## Keywords

Quantum information Quantum computation Control in mathematical physics## 1 Introduction

The ability to manipulate the dynamics of a given complex quantum system is one of the fundamental issues of the quantum information science. It has been an implicit goal in many fields of science such as quantum physics, chemistry, or implementations of quantum information processing [1, 2, 3]. The usage of experimentally controllable quantum systems to perform computational task is a very promising perspective. Such usage is possible only if a system is controllable. Thus, the controllability of a given quantum system is an important issue of the quantum information science, since it concerns whether it is possible to drive a quantum system into a previously fixed state.

When manipulating quantum systems, a coherent control strategy is a widely used method. In this case, the application of semiclassical potentials, in a fashion that preserves quantum coherence, is used to manipulate quantum states. If a given system is controllable, it is interesting to obtain control sequence that drives a system to a desired state and simultaneously minimize the value of the disturbance caused by imperfections of practical implementation. In the realistic implementations of quantum control systems, there can be various factors which disturb the evolution [4]. One of the main issues in this context is *decoherence*—the fact that the systems are very sensitive to the presence of the environment, which often destroys the main feature of the quantum dynamics. Other disturbance can be a result of the restriction on the frequency spectrum of acceptable control parameters [5]. In the case of such systems, it is not accurate to apply piecewise-constant controls. In an experimental setup that utilizes an external magnetic field [6, 7], such restrictions come into play and cannot be neglected.

In many situations, the interaction with the control fields causes an undesirable coupling with the environment, which can lead to a destruction of the interesting features of the system [8]. In such situations, it is reasonable to seek a control field with minimal total influence on a system. Depending on a type of interaction with an environment, the influence differs. In this article, we consider an interaction that is proportional to the magnitude of a control field. To minimize the influence of an environment in such case, when the control field performs the desired evolution, the \(L_1\) norm should be minimized.

A different dynamical method for beating decoherence in open quantum systems is dynamical decoupling [9, 10, 11, 12]. In this case, additional perturbation on a system is added, which protects the evolution against the effects of the environment influence at the same time driving the system to the desired state. In our case, the interaction with the environment is in strict relation to the control strategy, since it emerges only if the control pulses are applied. On the other hand, in a typical dynamical decoupling scheme, the coupling to the environment is constant, given by some Hamiltonian \(H_{SE}\) acting on the system and environment. Another approach to robust quantum control is quantum sliding mode control [13]. This model combines unitary control and periodic projective measurements. First, the initial state is driven into a *sliding mode* and then a periodic projective measurement is performed. Finally, there is risk-sensitive quantum control [14, 15] that is a robust control method with a feedback loop.

The paper is organized as follows. In Sect. 2, we introduce the model used for simulations. Section 3 describes the simulation setup. In Sect. 4, we show results of numerical simulations, and in Sect. 5, we draw the final conclusions.

## 2 Our model

The system described above is operator controllable, as it was shown in [18] and follows from a controllability condition using a graph infection property introduced in the same article. The controllability of the described system can be also deduced from a more general condition utilizing the notion of hypergraphs [19].

In the case of disturbed system, we will measure the quality of the control by a trace distance between Choi–Jamiołkowski states, which gives an estimation of a diamond norm.

- The
*signum*function:$$\begin{aligned} \frac{\mathrm {d}|x|}{\mathrm {d}x} = \mathrm {sgn}(x). \end{aligned}$$(9) - A fractional derivative:where \(\Gamma (x)=(x-1)!\) and we set \(\alpha =0.99\).$$\begin{aligned} \frac{\mathrm {d}^\alpha |x|}{\mathrm {d}x^\alpha } = \pm \frac{\Gamma (2)}{\Gamma (2-\alpha )}x^{1-\alpha }, \end{aligned}$$(10)
- Rescaled Fermi–Dirac distributionwhere we set \(kT = 0.01\).$$\begin{aligned} \frac{\mathrm {d}|x|}{\mathrm {d}x} \approx 2 \left( \frac{-1}{\exp \left( \frac{x}{kT}\right) + 1} + 0.5\right) \!, \end{aligned}$$(11)

## 3 Simulation setup

We provide an explicit example in which we set the duration of the control pulse to \(\Delta t = 0.2\) and the total number of pulses in each direction to \(n=64\) for the three-qubit chain and \(n=256\) in the four-qubit case, although the presented method may be applied for arbitrary values of \(\Delta t\) and \(n\). The weight of fidelity in Eq. (8) is set to \(\mu =0.2\) in the three-qubit scenario and to \(\mu =0.4\) in the four-qubit scenario.

## 4 Results

The fidelity obtained in both cases is \(F > 0.99\), and the value of \(P\) has the order of \(10^{-2}\).

Summary of the value of Eq. (20) for the studied cases

Without additional qubit | With additional qubit | |||
---|---|---|---|---|

\(\mu =1\) | \(\mu < 1\) | \(\mu =1\) | \(\mu < 1\) | |

\(\hbox {NOT}_3\) | 0.0000 | 0.0000 | 0.0975 | 0.0086 |

\(\hbox {NOT}_4\) | 0.0000 | 0.004 | 0.9788 | 0.0142 |

\(\hbox {SWAP}_3\) | 0.0000 | 0.0001 | 0.0135 | 0.0133 |

\(\hbox {SWAP}_4\) | 0.0000 | 0.0020 | 0.0843 | 0.0064 |

## 5 Conclusions

In this work, we introduced a method of obtaining a piecewise-constant control field for a quantum system with an additional constrain of minimizing the \(L_1\) norm. To demonstrate the beneficialness of our approach, we have shown results obtained for a spin chain, on which we implemented two quantum operations: negation of the last qubit of the chain and swapping the states of the two last qubits of the chain. Our results show that it is possible to obtain control fields which have minimal energy and still give a high fidelity of the quantum operation. Our method may be used in situations where the interaction with the control field causes additional coupling to the environment. As our method allows one to minimize the number of control pulses, it also minimizes the amount of coupling to the environment. It is important to note that our model differs from known in the literature dynamical decoupling, in which additional perturbation on a system is added, which protects the evolution against the effects of the environment influence. In our case the interaction with the environment is related to the control strategy, and it emerges only if the control is applied. Our model allows to optimize high fidelity control pulses for the cases with and without external environment, as shown in Table 1 as long as the coupling is induced by the control pulses themselves. Other possible usage of our method includes systems, in which it is possible to use rare, but high value of control pulses, for example, superconducting magnets with high impulse current.

## Notes

### Acknowledgments

Ł. Pawela was supported by the Polish National Science Centre under the Grant Number N N514 513340. Z. Puchała was supported by the Polish Ministry of Science and Higher Education under the Project Number IP2011 044271.

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