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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].

Control Theory
1. Classical Control
2. Quantum Control
(i)
Noise Control
(ii)
Measurement Theory and Control
(iii)
Mathematical Methods of Quantum Control
(iv)
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:





1.
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?
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 non-perturbing 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.

Noise Control
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)
(ii)
(iii)
Noise control
Selective addressing noise control
Measurement noise control
Environmental noise control
(i) Selective addressing noise control
Selective addressing of qubits is a theoretical idealization; in practice there are often
cross-talk 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
(i)
(ii)
(i)
Matrix mechanics
Quantum feedback
Matrix mechanics
A quantum computer processes information by mapping input bits to output bits via a
unitary transformation. However, the read-out 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 (non-unitary 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
(i)
(ii)
(iii)
(iv)
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 wave-packets created by adiabatic turn-on and sudden
switch-off ; 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 non-zero 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
time-dependent 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
(v)
(vi)
(vii)
(i)
Permutation Gates
Graph States
Dynamical Systems and Differential Equations
Permutation Gates
The permutation gates combined with the one-bit 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 2-qubit 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 least-action 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.
(ii)
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 multiply-coupled 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.
(iii)
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 least-time. 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.

Discussion
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 back-action 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 path-integral 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 co-ordinates 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 error-correcting 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. Mirror-inversion 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 non-locality 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 non-local 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 non-Boolean causal logic;
the problem arises when Boolean logic is used to analyse a non-Boolean 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, well-founded 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 lag-time or delayed reaction?
What is the time-optimal 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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