# Quantum control with spectral constraints

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## Abstract

Various constraints concerning control fields can be imposed in the realistic implementations of quantum control systems. One of the most important is the restriction on the frequency spectrum of acceptable control parameters. It is important to consider the limitations of experimental equipment when trying to find appropriate control parameters. Therefore, in this paper, we present a general method of obtaining a piecewise-constant controls, which are robust with respect to spectral constraints. We consider here a Heisenberg spin chain; however, the method can be applied to a system with more general interactions. To model experimental restrictions, we apply an ideal low-pass filter to numerically obtained control pulses. The usage of the proposed method has negligible impact on the control quality as opposed to the standard approach, which does not take into account spectral limitations.

## Keywords

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

One of the fundamental issues of the quantum information science is the ability to manipulate the dynamics of a given complex quantum system. Since the beginning of quantum mechanics, controlling a quantum system has been an implicit goal of quantum physics, chemistry, and implementations of quantum information processing.

If a given quantum system is controllable, i.e., it is possible to drive it into a previously fixed state, it is desirable to develop a control strategy to accomplish the required control task. In the case of finite dimensional quantum systems, the criteria for controllability can be expressed in terms of Lie-algebraic concepts [1, 2, 3]. These concepts provide a mathematical tool, in the case of closed quantum systems, i.e., systems without external influences.

It is an important question whether the system is controllable when the control is performed only on a subsystem. This kind of approach is called a *local-controllability* and can be considered only in the case when the subsystems of a given system interact. As examples may serve coupled spin chains or spin networks [2, 4, 5, 6]. Local-control has a practical importance in proposed quantum computer architectures, as its implementation is simpler and the effect of decoherence is reduced by decreased number of control actuators [7, 8].

A widely used method for manipulating a quantum system is a coherent control strategy, where the manipulation of the quantum states is achieved by applying semi-classical potentials in a fashion that preserves quantum coherence. In the case when a system is controllable, it is a point of interest what actions must be performed to control a system most efficiently, bearing in mind limitations imposed by practical restrictions. Various constraints concerning control fields can be imposed in the realistic implementations of quantum control systems. One of the most important is the restriction on the frequency spectrum of acceptable control parameters. Such restrictions come into play, for example, in an experimental setup that utilizes an external magnetic field [9]. In the case of such systems, due to various limitations, the application of piecewise-constant controls is not accurate. The real realization of controls is somehow smoothed by some filter induced by an experimental limitations. Thus, it is reasonable to seek control parameters in the domain imposed by the experimental restrictions.

In article [10], there has been discussed how the low-pass filtering, i.e., eliminating high-frequency components in a Fourier spectra, on a numerically obtained optimal control pulses affects a quality of performed control. This approach makes a contact with experimental realizations, since it implements the limitations or real quantum control systems.

In this paper, we present a general method of obtaining a piecewise-constant controls, which is robust with respect to low-pass filtering. The above means that elimination of high frequencies in a Fourier spectra reduces the fidelity only by a small amount. We utilize this approach to obtain numerically control pulses on a Heisenberg spin chain [10, 11, 12]; however, it can be applied to a quantum system with more general interactions.

This paper is organized as follows. In Sect. 2, we provide a general description of a quantum mechanical control system. In Sect. 3, we provide the description of the simulation setup used to test our model. Section 4 contains results obtained from numerical simulations and their discussion. In Sect. 5, we provide a summary of the presented work and give some concluding remarks.

## 2 Our model

The system described above is operator controllable, as it was shown in [5] 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 [6].

## 3 Simulation setup

For each of these cases, we find two sets of control parameters. One with frequency constraints and one without. Next, we calculate an appropriate filter and using these filtered values of control parameters, we calculate the fidelity of the quantum operation. In each case, 120 independent sets of control parameters were found.

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 \(n=128\) for the three-qubit chain and \(n=512\) 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. (5) is set to \(\mu =1\) in the unconstrained case and to \(\mu =0.05\) in the constrained case. Although the weight of the fidelity is small, the optimization still yields high fidelity values while maintaining low contribution of high frequencies in the power spectrum. We set the cutoff frequency in Eq. (7) to \(\Delta =\frac{n}{4}\).

We conduct our simulations using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method [16]. This method is commonly used in quantum control theory for optimization of control pulses [10, 11, 17]. We first choose an initial guess for the control field vectors. The algorithm then generates iteratively new control field vectors such that at each iteration point, the fidelity is increased. The algorithm terminates after a desired accuracy is reached. This procedure ensures convergence to a local maximum, but does not guarantee the convergence to a globally optimal sequence. Hence, we perform a number of simulations, using different initial conditions.

## 4 Results

Next, in Fig. 1b, we show the plot of the control parameters for the target gate \(U_T=\hbox {NOT}_3\), before and after applying the frequency filter. Only this time, the controls were found using the value of the weight \(\mu = 0.05\) resulting in a penalty for high-frequency terms. A short glance reveals that the filtered parameters are almost the same as the original ones. This is reflected by the fidelity of the operation. Before filtering, the fidelity is \(F > 1-10^{-9}\), and after filtering, it drops only to \(F>1-10^{-2}\), which is still a satisfactory value. Hence, these sets of control parameters are well-suited for use in computations.

Results obtained for the four-qubit chain (not shown here) qualitatively resemble the results for the three-qubit chain. In this case, a penalty for high-frequency terms of the control parameters also leads to higher fidelities after filtering.

## 5 Conclusions

We investigated the impact of spectral constraints imposed on the control parameters of a quantum operation on the fidelity of the quantum operation which they implement. In order to compare our approach with the unconstrained case, we apply an ideal low-pass filter to the control parameters.

We have shown that imposing spectral constraints on the control parameters leads to higher average fidelity of the quantum operation after appropriate filtering, than in the unconstrained case. These results are independent of the type of quantum operation and the number of qubits in the system under consideration.

Furthermore, the requirement for smooth control parameters does not result in the increase in time necessary to conduct a quantum operation. Comparing with other research in the field [11], our times are on the same order.

Further work on this subject might take into account more subtle parameters of the experimental setup than the frequency cutoff of signal sources. For instance, one could wish to find control parameters that are far from transient characteristics of the experimental setup. This may lead to an enhancement of fidelities of operations achieved experimentally by eliminating unwanted signal roughness.

## Notes

### Acknowledgments

Work by Ł. Pawela was partially supported by the Polish National Science Centre under the Grant Number N N514 513340. Z. Puchała was partially supported by the Polish Ministry of Science and Higher Education under the project number IP2011 044271. The development of numerical methods was supported by the Grant N N514 513340. Analytical studies were supported by the project IP2011 044271. Numerical simulations presented in this work were performed on the “Leming” and “Świstak” computing systems of The Institute of Theoretical and Applied Informatics, Polish Academy of Sciences.

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