## Abstract

We consider how the ability to control quantum effects might give rise to entirely new technologies, present an overview of potential applications and consider some of the key challenges facing quantum control. A general overview of the main techniques that have been employed successfully so far in controlling various quantum phenomena is given and their applications, advantages and shortcomings are discussed. We conclude with an outlook on the future challenges to be overcome to make quantum technologies a reality.

## 1. Introduction: why we need control

The quest to adapt our physical environment and control the forces of nature has played a central role in the development of humankind. From the invention of Stone Age tools and the discovery of fire to the present day, we have always sought to invent new tools and find ways to exert greater control over physical processes, and extend our ability to control to new domains. In essence, our central desire to control destiny is the motivation for the development of technology, and the ability to control physical processes is the enabling force.

One of the major challenges today is extending control to the quantum domain, a regime where the laws of classical physics are no longer accurate, and new effects, quantum effects, become significant. To understand the behaviour of these systems and make *accurate* predictions about them, we require new laws, the laws of quantum physics. Lest there be any confusion, the laws of quantum physics, as far as we know today, apply to everything in the Universe, including cars, planets and galaxies. However, for the latter types of systems, quantum effects are usually negligible, and the laws of classical physics make predictions about their behaviour that are sufficiently accurate for practical purposes.

When we refer to quantum systems, we usually imply systems whose behaviour is dominated by quantum effects and cannot be accurately described by the laws of classical physics. Nature provides us with a multitude of such systems, from subatomic particles, such as electrons to atoms and molecules; thanks to significant recent advances in nanotechnology, we today have the ability to fabricate artificial structures of nanometre-scale dimensions that exhibit quantum behaviour (Hari 2003; see also figure 1). Other technological and scientific breakthroughs have allowed us to actually create macroscopic quantum states of matter, in what is known as Bose–Einstein condensates (BEC; Griffin *et al*. 1995), showing that size does not matter when it comes to quantum effects. Thus, the laws of quantum physics govern the behaviour of a huge and ever expanding range of systems.

The prospect of being able to control the behaviour of systems as diverse as complex molecules, trapped atoms or ions, photons travelling at the speed of light, or electrons in semiconductors—i.e. to be able to not only watch them ‘dance’ and predict their moves, but to actually ‘choreograph’ their dance—is a fascinating prospect in its own right, but it is also crucial from a practical point of view, as the development of many applications and rather possibly entirely new technologies depends on it.

For instance, our relentless quest to build ever smaller and more powerful gadgets, especially in the area of computing and electronics, has made it necessary to make the basic building blocks of electronic equipment such as transistors, ever smaller in an attempt to cram more of them into a smaller area or volume. We are already at a stage where quantum effects have to be taken into account in various applications and we are rapidly approaching system sizes where quantum effects will become dominant. Unless we learn how to control them, we will soon reach the ultimate limits of computing power and system size. On the other hand, if we succeed in learning how to properly control quantum systems, we may not only be able to continue our quest for smaller and more powerful computers, but we may also be able to exploit quantum effects to build entirely new machines, quantum computers (Nielsen & Chuang 2000), which can take advantage of quantum parallelism to efficiently solve some problems that are very hard to solve with conventional computers.

Quantum information processing offers other exciting possibilities such as completely secure communication based on quantum cryptography systems, a technology already commercially available today. However, the applications of quantum technology do not stop at quantum computing and communication. Quantum clocks (Diddams *et al*. 2004), for instance, promise to improve frequency standards and clock accuracy significantly, which is important for many practical applications including computing, communication and perhaps the next generation of super-accurate global positioning systems. Among other exciting potential applications are quantum detectors designed to exploit the nonlinear dynamic response of different types of molecules to tailored laser pulses to discriminate molecules of different species or detect the presence of certain molecules. Unlike conventional spectroscopic techniques, strong laser pulses can have vastly different effects on molecules with similar spectroscopic signatures, and such ‘nonlinear detectors’ might be able to detect traces of chemical or biological substances whose spectroscopic signature is completely obscured by their surroundings. Finally, lasers can be used as photonic reagents to change the outcome of chemical reactions (Rabitz *et al*. 2000). While photonic reagents are currently too expensive to use in industrial-scale chemical engineering, they may lead to the discovery of new molecules and materials with interesting properties and may even be useful for medical applications.

Thus, quantum technology holds tremendous potential, realizing which will require learning how to effectively control quantum systems, i.e. choreograph their behaviour. In this paper, we will look at how we might achieve this ambitious goal, considering the obstacles encountered, the major success stories of quantum control so far and future challenges to be met.

## 2. Obstacles to controlling quantum dynamics

Before we consider how to control quantum systems, we must understand the fundamental concepts of quantum physics, how quantum theory differs from classical physics and what implications this has for control.

To model a dynamical system, i.e. a system whose state changes over time, we need a way to describe the state of the system and a dynamical law that tells us how the state evolves. If we wish to control the system, the dynamical law that governs its evolution has to depend on external factors, such as control fields applied, or more generally the state of a controller. In classical control engineering, it is generally assumed that there is a controller consisting of sensors that we can use to observe the system and gain information about its state, and actuators that can effect changes to the state of the system. In the ideal case, the system would interact only with the controller. Unfortunately, in the real world, most systems are not completely isolated from their environment besides the controller and the latter can have unwanted effects on the former, which we cannot control.

There are two ways in which we can operate our controller. In open-loop mode, the sensors and actuators operate independently, i.e. without communication between them. In closed-loop mode, information acquired by the sensors about the system is regularly (usually continuously) fed back to the actuators and thus can be used to decide what the next action by the actuators should be to achieve a desired outcome of, e.g. steering the system to a certain state or effect a particular evolution. The benefits of feedback are clear; controlling the system without feedback is a bit like driving blind. If we know our initial and target location and the state of the road in-between, and can predict the behaviour of all road users (the entire system) at all times, it is possible to plot a route from our initial position A to the target location B. If we are lucky and our knowledge was sufficiently accurate and nothing unexpected happened, we may reach our destination but it would be rather difficult at best. However, what if watching the road actually altered its state? This is hard to imagine from a classical point of view. Certainly, the act of simply observing the road and vehicles around us will not change their state.

Unfortunately, quantum theory tells us that this is exactly what happens in the quantum case. By simply looking at the system (or any part of it), we disturb it. This effect, known as measurement back-action, has very serious implications for feedback control of quantum systems. The sensors become co-actuators, effectively changing the state of the system, and what is more, the observations by the sensors continuously change the state of the system in a way that usually cannot be reversed by the actuators. Hence, designing and implementing closed-loop control with real-time feedback is a challenging task for quantum systems. Moreover, there are many practical difficulties to implementing feedback in real time for quantum systems, such as limits on how fast we can perform measurements, process data and feed information back to the actuators. If the evolution of the system is considerably faster than the speed at which we can acquire data and process information, feedback may not be very useful, rather like the delayed response of a sleepy driver on a high-speed motorway, who tries to brake after colliding with the vehicle in front, except that in the quantum case there will be a ‘penalty’ for watching the road even if there is no collision. Thus, we will often have to ‘make do’ without real-time feedback.

However, this is still not all. Even if we content ourselves with leaving the sensors switched off and try to steer the system from its initial state to the destination—relying only on our knowledge of our initial and target states, and the roads—or the dynamical law that governs the behaviour of our system, there are still many problems with quantum control. Another peculiar feature of quantum systems is coherence, loosely explained, the ability of quantum states to interfere, rather like classical waves do. Maintaining coherence is fundamental to exploiting quantum effects, since loss of coherence means that the system will behave entirely classically. Thus, we require ‘coherent actuators’. However, even if the actuators interact with the system coherently, random interactions with the environment will still conspire against us and try to destroy the coherence of our system. Therefore, trying to control quantum dynamics is beginning to look a lot like trying to drive blindfolded in difficult terrain and really bad weather. Is there any hope of success? Fortunately, despite these problems, there are a number of success stories.

## 3. Nuclear spin ensembles and geometric control

Perhaps one of the earliest examples of quantum choreography is the control of ensembles of nuclear spins in nuclear magnetic resonance (NMR) spectroscopy (Levitt 2001). The former originated from the desire to understand the structure of complex molecules such as proteins. The nuclei in molecules have a property called spin. Some nuclei have no spin, while others can have two or more different spin states. These different states have (slightly) different energies and by applying electromagnetic fields of the correct frequency, usually in the radio-frequency range, we can induce transitions between these spin states. Conversely, if a nucleus undergoes a transition from a higher to lower energy (spin-) state, it will emit a photon of a certain characteristic wavelength. NMR spectroscopy takes advantage of this to get information about, for example, the structure of organic compounds, where the carbon and hydrogen atoms in a molecule are located, etc. Although the problem was initially not stated in the language of control theory, control has played a very important role in NMR spectroscopy, since precise spectroscopy requires sophisticated tools for coherent control of nuclear spin ensembles. Thus, the desire to understand the structure of complex molecules has led to the development of control techniques that have since been applied in entirely different contexts such as quantum information processing (Vandersypen & Chuang 2005).

NMR systems have several advantages compared with many other quantum systems. First, nuclear spin states generally couple only weakly to the environment and hence their coherence lifetimes are usually long compared with the control time-scales. Therefore, it is often a good approximation to treat a nuclear spin ensemble together with the control and the measurement apparatus (such as a radio-frequency pulse generator and magnetization detectors) as a closed control system. Moreover, since collective measurements on a large ensemble of spins will usually have only minimal effects on a particular spin, and we are usually dealing with large ensembles of identical molecules rather than single spins, there is at least a theoretical possibility for closed-loop control with minimal back-action. While this possibility perhaps should be explored further, most of the control schemes that have been successfully demonstrated in NMR to date are based on rather simple geometric ideas and involve no real-time feedback.

In the simplest case of a spin−1/2 particle such as a nucleus with two spin states say spin-up |↑〉 and spin-down |↓〉, the prototype for a quantum bit or qubit, we can visualize the evolution of the system on what is called the Bloch sphere, the sphere of radius 1 in ordinary Euclidean space . Each point inside or on the Bloch sphere, can be represented by a vector in , the Bloch vector, and corresponds to a particular quantum state. The effect of any unitary evolution of the system can be thought of simply as a rotation of the Bloch vector in . Hence, the free evolution of the system (i.e. what it does when left to its own devices) corresponds to a rotation about a particular axis, usually the *z*-axis. Applying a (constant) control field changes the Hamiltonian that governs the evolution of the system, which manifests itself as a change of the rotation axis, as shown in figure 2. For particularly simple fields, such as a sinusoidally varying field of the form *f*(*t*)=*A* cos (*ω*_{f}*t*+*ϕ*) with constant amplitude *A*, certain simplifying assumptions (such as the rotating wave approximation) allow us to relate the declination angle *θ* of the new rotation axis (with respect to the *z*-axis) to the detuning *Δω*=*ω*_{f}−*ω* of the applied field. If the former is resonant with the transition frequency *ω*, *Δω*=0, then the rotation axis lies in the equatorial plane and the horizontal angle *ϕ* is determined by the initial phase of the pulse, e.g. *ϕ*=0 corresponds to a rotation about the *x*-axis, *ϕ*=−*π*/2 to a rotation about *y*-axis. Thus, we can change the rotation axis by varying the phase *ϕ* and frequency *ω*_{f} of the field. The rotation frequency *Ω* is determined by the amplitude *A* of the field and the strength of the coupling *d*, *Ω*=*A*×*d*. Thus, to perform a rotation by an angle *α*, we in principle simply have to choose the pulse amplitude *A* and pulse length *T* such that *ΩT*=*α*.

Geometric control is based on the fact that we can generate some elementary rotations by applying relatively simple fields and that arbitrary rotations can be implemented simply by combining elementary rotations about a set of fixed axes. For instance, given two orthogonal rotation axes in , we can implement arbitrary rotations by performing at most three rotations (in an alternating sequence) about the two given axes. This idea can be generalized to higher-dimensional systems (Schirmer *et al*. 2002) and systems of interacting qubits (D'Alessandro 2001; Ramakrishna *et al*. 2002). By applying precisely timed sequences of pulses with properly chosen frequencies, phases and amplitudes, we can in principle choreograph the behaviour of complicated systems consisting of many spins simultaneously. Indeed, there have been a number of impressive experimental demonstrations of quantum choreography using geometric control pulse sequences for nuclear spin ensembles, including the demonstration of quantum gates and quantum algorithms, process tomography, and even advanced operations, such as quantum error correction and decoherence-free subspace encodings (Price *et al*. 2000).

Despite its success in NMR, geometric control has its shortcomings. The basic idea that we can *selectively* operate on individual spins or transitions by choosing the frequency of the control pulse to match that of a target spin or transition becomes problematic when there are too many transitions with insufficiently distinct frequencies so that any given control pulse is likely to excite many transitions simultaneously. Although such unwanted excitations can in principle be reduced by choosing weaker pulses, or compensated for by composite pulse sequences, these techniques are not always practical, in particular when fast control is essential, e.g. to reduce the detrimental effects of relaxation and decoherence induced by uncontrollable interactions of the system with the environment.

## 4. Control of atomic states via adiabatic passage

Geometric control can be applied to various atomic or molecular systems. For instance, consider the problem of population transfer from state |1〉 to |3〉 via an intermediate state |2〉 in a three-level atom, assuming that direct transitions between the initial and the target states are prohibited, as shown in figure 3 (left). A possible solution is to apply two control pulses, the first resonantly exciting the 1→2 transition and the second, the 2→3 transition, sequentially, as shown in figure 3 (right). Geometric control tells us in this case that the condition for perfect population transfer (assuming the validity of various approximations) is that both control pulses should have effective pulse area *π* and that the initial pulse phases in this particular case do not matter. In theory, this is a good solution, but in practice the upper level |2〉 normally has a finite lifetime *T*_{1}, i.e. there is a 50% chance that it will randomly decay to one of the ground states by emitting a photon in time *T*_{1}. Populating the excited state |2〉 thus introduces relaxation and decoherence.

Therefore, it would be better if we could transfer the population from state |1〉 to state |3〉 without populating the upper level. Since direct transitions between the initial and the target states are not allowed, this appears impossible at first, but there is a solution. One can show that we *can* get from state |1〉 to |3〉 without populating |2〉, simply by applying both control pulses in what is often termed a ‘counter-intuitive’ sequence, i.e. such that the pulse driving the 2→3 transition precedes the one driving the 1→2 transition, but both pulses overlap. If both pulses are sufficiently strong and timed correctly, the system will undergo adiabatic passage from state |1〉 directly to state |3〉 via a two-photon process, as illustrated in figure 4. Since the upper level |2〉 is never populated, the population transfer by adiabatic passage is robust against decay, which is often the main source of decoherence, especially in atomic physics.

Another advantage of the adiabatic approach is robustness with regard to pulse variations. The probability of successful state transfer is insensitive to variations in the area of the control pulses, unlike pulse schemes based on geometric control, which are necessarily sensitive to pulse area variations as the former determine the rotation angles in geometric control schemes. This feature is very useful when the control pulses are optical fields derived from laser sources, which are prone to random fluctuations, since the former are not as easily tractable as the systematic pulse errors common for radio-frequency pulses in NMR, which can be dealt with using composite pulse sequences. This robustness against decay and insensitivity to pulse area fluctuations has made adiabatic passage techniques the method of choice for control problems involving state transfer in atomic and molecular physics (Vitanov *et al*. 2001), and has prompted recent efforts to generalize adiabatic passage techniques to tackle more complex control problems such as realizing non-stationary quantum states (coherent superpositions of the system's eigenstates) and quantum processes (Vedral 2002)

## 5. Optimal control and molecular physics

Geometric and topological control techniques have proved useful for a number of applications from NMR spectroscopy to control of atomic and molecular states to the implementation of quantum processes for reasonably simple and sufficiently well-isolated quantum systems such as ensembles of a few spins weakly coupled to the environment. However, when it comes to applying these techniques to complex systems with many possible transitions and closely spaced transition frequencies, there are problems, such as limits to frequency-selective excitation of individual transitions and decoherence.

To make this point explicit, consider an ensemble of five quantum dots, perhaps similar to the system shown in figure 1*b*. Owing to variations in size and shape, each dot will have slightly different energy levels. Although this is a gross oversimplification, let us assume for simplicity that we can model each quantum dot as a two-level system with a characteristic transition frequency *ω*_{k}, where *k* ranges from 1 to 5. If we can individually address each dot by focusing a laser of the right frequency onto it (without exciting any of the adjacent dots), we can easily implement arbitrary quantum operations using the geometric control schemes discussed earlier. For example, to achieve complete population inversion in any of the dots, assuming it is initially in its ground state, we simply apply a resonant *π* pulse. The situation is more complicated when the dots are too small or too close together so that it is impossible for us to shine a laser onto a single dot. However, if the transition frequencies of the dots are sufficiently distinct, we can selectively excite individual dots simply by choosing the frequency. For example, to achieve simultaneous population inversion in quantum dots 1 and 3, we could simply apply a sequence two Gaussian pulses with pulse area *π* and frequencies *ω*_{1} and *ω*_{3}, respectively, as shown in figure 5*a*. In theory, this should transfer dots 1 and 3 from their ground state (south pole of Bloch sphere) to the exited state (north pole of Bloch sphere), while having no effect on any of the other dots, provided they have different frequencies, as shown in figure 6*a*.

However, if the quantum dots have similar frequencies and we apply short control pulses to beat decoherence, this is not what happens. For instance, for a system of five model dots, similar to the system considered in Toda *et al*. (2004), with energy gaps *hω*_{k} equal to 1.32, 1.35, 1.375, 1.38 and 1.397 eV, respectively, and two Gaussian control pulses approximately 2 ps each (or 500 time units in units of *ℏ*/1 eV≈4.14 fs) long, as shown in figure 5*a*, the actual effect of the pulses on the quantum dots, shown in figure 6*b* is rather different. The pulses excite not only the target dots 1 and 3, but there is also significant excitation of the other dots, and this for a model that was highly simplified! For real physical quantum dots, the situation would be even worse because there would be many more transitions. Therefore, what can we do instead? One solution to this problem, which is common in molecular physics and chemistry applications, is using optimally shaped pulses as shown in figure 5*b* instead of simple Gaussian wave packets. The shaped pulse shown was designed using model-based optimal control (Schirmer *et al*. 2000) to maximize the simultaneous excitation of dots 1 and 3 while minimizing the excitation of all other dots, at the final time (*t*_{F}=1000 AU≈4.14 ps). Figure 6*c* shows that the optimally shaped pulse indeed achieves this aim almost perfectly (for the model system it was designed for, of course).

The previous example illustrates the potential advantages of coherent control using optimally shaped pulses in particular for ultrafast control. But how can we derive optimally shaped pulses and can we possibly implement pulses as complicated as the example shown in figure 5*b*? We will answer the second question first. There are two main approaches to implement shaped pulses: time-domain and frequency-domain pulse shaping. The former essentially involves modulating the amplitude of the field directly as a function of time. This may be useful for applications involving radio-frequency fields, where amplitude modulation is relatively easy. However, a key application for optimally shaped pulses is ultrafast control using coherent optical fields derived from lasers. In this regime, time-domain pulse shaping is unrealistic, but we can implement even complicated pulses very easily using frequency-domain pulse shaping with a set-up as shown in figure 7. All we require is a laser that produces short, intense pulses. Using a diffraction grating, we can decompose this pulse into its frequency components, which can then be modulated individually using acousto-optic modulators, phase shifters or liquid crystal displays to attenuate the intensity of individual spectral components and introduce phase shifts. The spectrally modulated pulse is then transformed back to the time-domain by another diffraction grating, yielding the desired complex pulse.

As far as the derivation of optimal pulses is concerned, there are two basic approaches: control design using model-based feedback (Schirmer *et al*. 2000; Khaneja *et al*. 2001, 2005; Ferrante *et al*. 2002; Ohtsuki *et al*. 2004) and direct laboratory optimization (Turinici *et al*. 2004). The former generally involves formulation of the control problem in the language of optimal control theory, and the solution of the resulting optimal control problem using various mathematical and computational techniques depending on the nature of the problem, the objective and the constraints involved. This was the approach used to obtain the optimally shaped control pulse for the quantum dot ensemble shown in figure 5*b*. Having computed the optimal pulse in the time domain, we can Fourier transform it and use the magnitudes and phases of the Fourier components (figure 8) to determine the phase shifts and attenuations to program the pulse-shaping equipment. A problem with this approach is that for complicated systems such as complex molecules subject to intense fields, or artificial structures such as quantum dots subject to fabrication tolerances, we often lack sufficiently accurate models to describe the dynamics.

Direct laboratory optimization techniques circumvent the problem of inadequate or non-existing models by using information from a large number of experiments, rather than model-based design, to find control pulses that maximize a target functional or fitness function, which quantifies the fitness-for-purpose of a control pulse, i.e. its success in achieving a stated aim, possibly satisfying certain constraints. There are a number of techniques for identifying suitable control fields using information from experiments only (Turinici *et al*. 2004). A particularly popular approach is the genetic or evolutionary search technique (Judson & Rabitz 1992), which has proved very successful in finding shaped pulses with high fitness in many laboratory settings and control experiments ranging from molecular control problems (Pearson *et al*. 2001) and applications in femtosecond chemistry (Brixner & Gerber 2003) to control of energy flow in simple biological systems (Herek 2002), to mention only a few.

Although learning control techniques rely on feedback from experiments—either repeated on the same system or on many identical copies of the system—they must be strictly distinguished from proper closed-loop control design, in which the information obtained by the sensors is sent back, immediately or with a small delay, to the actuators and used to continuously adjust the control field applied to the *same* system (Wiseman 1994). A great advantage of learning control is that measurement back-action and the resulting complex dynamics can be avoided, since the experiments are performed on identical *copies* of the system or the system is re-initialized after each measurement. It should be noted that this type of ‘feedback control’ is limited to problems that are amenable to open-loop control. For some problems such as system stabilization (Wang & Wiseman 2001), closed-loop control with real-time feedback is essential.

## 6. Conclusions and outlook

Learning how to control quantum phenomena is essential for the development of novel quantum-enhanced technologies with many exciting potential applications. The task appears almost hopeless at first owing to a multitude of problems from decoherence, to sensor back-action and the difficulty of implementing real-time feedback. Nonetheless, we have seen that there are many success stories and there is reason for optimism. Many quantum control problems—from choreographing the behaviour of nuclear spins ensembles to controlling electronic, vibrational and rotational degrees of freedom of atoms, molecules and artificial structures such as quantum dots—can in principle be solved even without real-time feedback using simple open-loop control strategies.

A key prerequisite for most of these open-loop strategies is the availability of accurate models. This presents a practical challenge, as for many systems—from molecules to artificial structures such as quantum dots—sufficiently accurate models have yet to be devised especially for ultrafast control with strong, optimally tailored fields, for instance. However, the need for a model can be circumvented by direct laboratory optimization. A potential limiting factor of this approach is that it generally requires a large number of experiments to succeed, making it best suited for situations where experiments can be performed on large ensembles of identical copies of a target system at high repetition rates and little expense.

An alternative is to use system identification to derive models for quantum systems of interest. Sometimes, such models can be derived directly from the basic laws of physics, but often such models lack sufficient accuracy to use as a basis for open-loop control design, in particular for engineered systems such as quantum dots, which are prone to variations that are at best difficult to predict theoretically. These problems can be addressed using established techniques, such as various forms of spectroscopy (Hollas 2004) and tomography (Kosut *et al*. 2004), although these are not always sufficient, and novel characterization techniques (Schirmer *et al*. 2004) may be necessary to construct accurate models. A particular challenge in this regard is the development of efficient, adaptive techniques to derive and tune control system models based on feedback from experiments.

Another major challenge is to extend the basic control techniques described for more complex realistic systems such as solid-state nano-devices or optically dense media, which are not governed by relatively simple Hamiltonians with linear control dependence etc., and may be subject to non-trivial and generally uncontrollable interactions with the environment for which simple decoherence models are no longer adequate. There are many open problems in this regard ranging from the controllability of open systems to better algorithms for optimal control field design that incorporate both model-based and experimental feedback where available, and fully account for practical limitations, model uncertainty, etc. (Brown & Rabitz 2002). Finally, modelling of quantum measurement processes, in particular for weak indirect measurements (such as electronic readout using single-electron transistors or similar) will be crucial for the design of quantum sensors and effective closed-loop controllers.

## Acknowledgments

Sincere thanks are due to my colleagues at the Centre for Quantum Computing, Cambridge, David Williams and co-workers from Hitachi Cambridge Labs, Herschel Rabitz, Princeton, Tim Havel, MIT and Lloyd Hollenberg, Simon Devitt and co-workers from U. Melbourne for helpful discussions, and CMI and EPSRC for crucial financial support.

## Footnotes

One contribution of 23 to a Triennial Issue ‘Mathematics and physics’.

- © 2006 The Royal Society