Literature Review: Quantum Information Science
This review seeks to outline the significant research discussions and topics that are currently being conducted in the field of quantum information science, especially the field of quantum control. The document focuses on possible areas of shortfalls in theoretical modelling, in particular the analysis of quantum noise control, time optimisation and decoherence.
Quantum Information Science is the study and design of quantum computational systems. These computers are different to those currently in operation in many ways, and potentially could unravel many of the difficulties that face modern science today. Quantum computers have many interesting attributes that relate to both of the welldeveloped fields of classical control and error correction, and also pose many new questions about the physical implementation and behaviour of such devices.
The goal of quantum control can be stated concisely as:
The goal is to find the shortest path between the identity transformation and the point in the Hilbert space corresponding with the desired quantum gate [4].

Classical Control

Quantum Control

Noise Control

Measurement Theory and Control

Mathematical Methods of Quantum Control

Physical Implementations of Quantum Control
Control Theory seeks to develop models of generalised systems on the basis of linear systems analysis. Using control theory it is possible to answer such questions as:

Will the system reach the desired state given enough time?

Is it possible to control the system with a given interaction?

Is a particular system controllable?

Does a given control achieve a target goal?

Is a particular method of control “optimal” in some sense?
1. Classical Control
Classical control theory is the application of linear systems theory to operations management. The science of classical control has been incredibly successful, with its major achievements including the design of control systems for a group of diverse areas including factories and production lines, electronics, electricity generation and servo mechanisms. Although classical control is primarily based on feedback principles the methods of control theory will have to be modified and adapted in order to be useful for quantum engineering design as nonperturbing measurements are not possible in the atomic region.
2. Quantum Control
Quantum control is an emerging scientific field in which methods and techniques from engineering, mathematics, physics and computer science are used in order to analyse the behaviour of quantum systems with reference to the measurement instrument. This involves specific analysis of the system, experimenter and their interaction.
In order for controllability to be achieved it will be necessary to design ways in which a desired unitary transformation can be implemented in the presence of noise and imperfections while using minimal resources [4]. It has been shown that the solution of minimal time operation can benefit from the use of the optimal unitary operator and the maximal transfer efficiency functionals [36]. Using Optimal Control Theory (OCT) it is possible to calculate a field to effect a desired unitary transformation, with possible restrictions on resources [38].

Measurement Theory and Control
The degree of control it is possible to exert over a quantum system is a question that has generated considerable interest [1]. Also, given that there exist certain parameters or boundary conditions on our degree of control, the time optimality of operation is extremely important. Indeed, steering the system from the initial state to the target in minimal time appears to be a key problem [1]. This is because it is particularly desirable that the pulse sequences be as short as possible to minimise decoherence effects [2].

Mathematical Methods of Quantum Control
It is possible to break up each control problem into a sequence of excitation and relaxation steps [37]. By this means it may be possible that some of the methods of physics can be used to analyse the quantum mechanics of the computation, as the reduction of the computation to such processes can allow a variety of techniques of Quantum Field Theory (QFT) to be adapted for this purpose.
The application of control theory to quantum systems is relatively new [31]. It is therefore important to consider the possible ways in which classical control theory may have to be modified in order to fulfil certain conditions of quantum mechanics. It will be interesting to consider the behaviour of quantum systems operating under incompletely specified goal functions. These generalised questions are tractable, while different in nature to deterministic problems [32].

Physical Implementations of Quantum Control
It is also important to note that there exist various different methods of quantum computation which can all be analysed under the umbrella of quantum control theory. These include chemical reaction control, NMR quantum control, electromagnetic quantum control and adiabatic quantum control.
(i) Selective addressing noise control
(ii) Measurement noise control

Environmental noise control
(i) Selective addressing noise control
Selective addressing of qubits is a theoretical idealization; in practice there are often crosstalk effects which cause correlations between qubits to deviate from the expected values [4]. The major difficulty is to address and control the qubits sufficiently well to remove undesired couplings [21]. Conducting quantum computations requires precise operational techniques and low noise conditions. If these conditions are not met, the quantum gates that are produced will differ from the intended transformations [6]. Analysing the departure of the achieved unitary from the expected value as a function of the noise will be an interesting and technically useful problem. As sensitivity functions can be used to assess performance and give measures of decoherence [18] it will be particularly relevant to consider these functionals and examine which methods can be adapted and extended in this area.
(ii) Measurement noise control
Coherent quantum control is used by controlling the interference between different paths that lead to the same final state. As it is impossible to separate between degenerate paths on a scale less than that of the accuracy of the measurement device, this can lead to difficulties in the outcomes of quantum computations [34]. Instrumental imperfections and other experimental artefacts are related to Hamiltonian terms which are neglected in the ideal control formalism [4]. The analysis of such factors, perhaps from a perturbation theory viewpoint, could be quite productive.
(iii) Environmental noise control
DiVincenzo’s five criteria for quantum computers provide a useful outline to experimental limitations to the realisation of such a device [22]. However, with the exception of the ion trap and NMR systems, many potential candidates do not appear to qualify as useful quantum computation systems by these criteria. Experimentally, it is very difficult to isolate the quantum computer from the environment [6].
The maximum coherent operation time is limited by how well the device can be shielded from unwanted interactions with the environment [33]. Faulty controls and decoherence errors are the primary obstacles to any realisation of quantum information processing [19].

Measurement theory and control

Matrix mechanics

Quantum feedback
(i) Matrix mechanics
A quantum computer processes information by mapping input bits to output bits via a unitary transformation. However, the readout process is an irreversible process that collapses the system into an eigenstate [8]. As this cannot be represented by a unitary operator, it may be useful to consider whether these collapsing measurements can be modelled as the results of idempotent (nonunitary but Hermitian) operators. Methods for transferring the information about unitary operators into physical states also need to be examined in greater depth, as it is possible to store the information relating to a particular unitary operator in a quantum state [20].
(ii) Quantum feedback
Feedback cannot be directly implemented for quantum control due to the measurement postulate [18]. This directly means that much of classical control theory, which relies on feedback principles will have to be modified in order to be useful in this area. The division of the classical world from the quantum system is the most difficult problem in modern physics [29]. Whether this restriction can be relaxed or the classical fields can be reworked into quantised impulses remains an open problem. As classical information about the results of measurements may be modelled as additional randomness in the quantum state [29] perhaps the classical fields could be analysed in a similar fashion.

Specific Physical Implementations of Quantum Control

Chemical Reaction Control

NMR Quantum Control

Electromagnetic Quantum Control

Adiabatic Quantum Control
(i) Chemical Reaction Control
It has been demonstrated that the distribution of the products of a chemical reaction involving sodium iodide can be controlled; this is achieved by varying the phase between two competing reaction paths. By using specially designed electromagnetic pulses this procedure can be implemented in practice [28]. This is an area in which it is quite straightforward to check the predictions of quantum control theory in an experimental situation.
(ii) NMR Quantum Control
The control of RF phases, amplitudes and frequencies lies at the heart of control of NMR systems [4]. NMR systems provide a useful testing ground for quantum computation, as the analysis of experimental factors can help with the theoretical modelling of such systems. There are many boundaries and parameters that are important to the operation of such a physical device, and at present the description of relaxation using two parameters neglects many factors, especially for those systems where coupled relaxation mechanisms appear [4].
It is both possible to phase lock a field to the system so as to eliminate unwanted excitations [38] and to design NMR pulse sequences such that any erroneous terms accumulated are cancelled out before the conclusion of the calculation [22]. These innovative techniques allow some of the potential problems with implementing a quantum computer to be partially solved.
(iii) Electromagnetic Quantum Control
The dynamic Stark effect provides a second way to demonstrate quantum control. This method uses switched wavepackets created by adiabatic turnon and sudden switchoff ; these are different in nature to those created via excitation or ionisation [30]. The use of different pulses of electromagnetic radiation in quantum control can result in quite a wide variety of experimental outcomes. As these pulses always have a finite nonzero bandwidth it is important to investigate quantum computation with limited control over individual qubits.
(iv) Adiabatic Quantum Control
The adiabatic quantum computation model is described by the time evolution of a timedependent Hamiltonian which satisfies several physical conditions [35]. Unitary quantum gates and Boolean functions can be implemented in this system using simple adiabatic Hamiltonians [35].

Mathematical Methods of Quantum Control

Permutation Gates
The permutation gates combined with the onebit gates form a complete set of logic gates [3]. They are particularly straightforward to apply to quantum logic, as some of the permutation matrices are directly related to classical (reversible) logic operations.
It has been shown that using the permutation gates keeps the overall operating time short when compared with the decoherence time of the system, as well as avoiding introducing and tracking phase rotations [3]. This general method for the construction of permutation pulse sequences can be extended from the 2qubit case to more complex spin systems [3].
Also, as it is possible to represent quantum gate arrays as sums over classical gate arrays in a similar fashion to the path integral method of Feynman [5], it would be interesting to see whether it is possible to formulate a leastaction principle over the logic gates and Hamiltonian in order to solve the time optimal problem in QCT. It is likely that the true power of quantum computation may arise from harnessing massively parallel computational resources [5]. As it is possible to study certain symmetries of quantum graph states in terms of connectivity matrices [9] and a sequence of mirror inversions can generate any permutation of a quantum state [16], it would be interesting to consider further the vital role that these matrices play in the field of Quantum Information Science.

Graph States
Since many models used for quantum computation require simplifications, such as only nearest neighbour interactions are allowed, or only single coupling is permissible, the problem of controllability of multiplycoupled quantum systems is a signature topic for quantum information science [4]. Coupled systems with nearestneighbour interactions may be represented by graph states, where vertices represent physical systems and edges represent interactions between them [26]. However for general graphs of mixed valence, it is not obvious how to define the discrete time unitary operator without knowledge of qlobal properties of the graph [9]. It also may be possible to extend the graph states to weighted graphs, where a number associated with each vertice characterises the strength of the interaction [26]. As it is possible to map each and every graph state onto a spin system in a crystal, perhaps it may be useful to examine whether any general principles of spin dynamics can be deduced using graph states, or vice versa.

Dynamical Systems and Differential Equations
Many of the results of classical control theory hinge on the use of linearized systems of differential equations. These methods use properties of functions around critical points in order to analyse the behaviour of the system as a function of the measurement apparatus and appropriate feedback conditions.
It is possible to use methods of optimal control theory to find the minimum time cost of a quantum computing scheme. However, the challenge seems to be to extract a principle for a scalable control scheme [17]. Optimization problems can often be stated in terms of a least action principle, in this case leasttime. It will be interesting to see how the quantum action principle shall apply to quantum computers, especially with respect to the path integral representation of quantum mechanics.
At the most basic level of quantum mechanics one must respect the uncertainty principle, and hence the classical control theory cannot be naively extended without modification [27]. Given appropriate experimental testing grounds, it should become feasible to develop a technically useful quantum control theory on an iterative basis. It is well known (and practiced) by many physicists that theories that directly describe the effects of active observation and interaction are more powerful that theories that neglect such factors [12]. For this reason it is important that the experimental and theoretical research groups work particularly closely in order to find the important factors in the theory of quantum control.
Quantum computers do not necessarily follow a linear computational path in order to achieve a solution [21]. Perhaps a more physical picture is to consider a computational trajectory that spreads out along many paths, each contributing an amplitude to the total computation. This would be consistent with the Feynman path integral representation in the computational basis. Developing such a model would be useful in the future design of quantum computer programs.
Implementing quantum fourier transforms and exponentiated unitary operations is difficult as they require production of static entangled states and dynamic quantum control [21]. It may be possible that the relation between past and future is so interconnected that the present becomes difficult to define in a deterministic fashion [14]. Both of these problems are intimately related, for what is the present without a time to define it? Static states do not change in time, and exponentiated unitaries require precise times of evolution. The introduction of a “quantum clock” and its possible backaction would be an interesting physical example to consider with respect to quantum control theory.
The physical representation of Huygen’s Principle in terms of the wave function was not solved until the pathintegral method was developed by Feynman [13]. Perhaps we will be able to develop a method for showing how classical binary logic is a limiting case of quantum logic, in terms of a path integral over all possible computations. This would be a very useful theory.
Feynman stated that “It is suggestive that perhaps coordinates and the spacetime they represent may in some future theory be replaced completely by an analysis of ordered quantities in some hypercomplex algebra” [15]. An interesting property of the permutation gates is that they can be derived from the interaction between an atom and an entangled photon pair. Whether it is possible to map a linear combination of Dirac matrices to a combination of permutation gates via a linear transformation remains to be solved. If it is possible to do so, this could perhaps aid in the analysis of Feynman’s problem.
It is also important to consider alternatives to the current popular methods in use. Dynamical decoupling provides an economical alternative to quantum errorcorrecting codes [19]. As such, their use in combination with errorcorrecting codes could perhaps be examined to see whether there are any situations to which the use of one, the other or both of these methods simultaneously could be used to gain a particular advantage. Other alternatives to current methods of quantum computing include mirror inversion. Mirrorinversion quantum computing data transfer is attractive in that it requires no control over individual qubits [16].
Using multiple energy levels of ions stored in an ion trap could be useful for quantum computation, in that it would reduce the total number of ions needed to be contained within the trap [8]. However, the physics is likely to be different, and it is important to calculate the properties of such a system before experiments are conducted.
As nonlocality appears to be experimentally correct [7] it is important to examine whether the restriction of causality that exists within the standard relativistic quantum theory could be relaxed to take into account the presence of nonlocal entangled photon pairs. The paradoxes that appear so abundant in quantum mechanics may in fact exist because quantum science itself may be based on a nonBoolean causal logic; the problem arises when Boolean logic is used to analyse a nonBoolean system. [12]

Conclusion and possible research questions
It has been demonstrated that there exist many fruitful areas of research within the field of quantum control. The development of a concise, wellfounded science of quantum control is perhaps one of the more important milestones that will be reached on the way to construction of a workable quantum mechanical computing device.
The science of quantum control is perfectly placed to answer a wide variety of questions that are of relevance to the physical sciences and engineering. These questions include:

How is it possible to represent a perturbing measurement in quantum control?

Does it matter if the measurement device has a lagtime or delayed reaction?

What is the timeoptimal strategy given a fixed set of Hamiltonians? Does it always exist?

Is it possible to represent the evolution of the combined system/apparatus in Lagrangian form instead of Hamiltonian form?

What is the matrix for transferring between a basis over the Dirac matrices to a basis over the permutation gates?

What is the optimal strategy to follow to reach a given goal in Hilbert space?
The principal areas of investigation that have been considered in this document are the fields of noise control, measurement, mathematics of quantum control and the physical implementation of quantum control strategies. These areas of research are particularly important in that they are points of convergence of methodologies from physics, mathematics, engineering and computer science.
In considering the field of quantum control as a whole, it is apparent that this is a very new science, with many years of development to come. As such it is crucial that the development of the discipline of quantum control proceed on a firm theoretical and experimental basis.
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