Introduction
Complex systems have many behaviors that operate independently across scales. One of the universal phenomena of complex systems are the so-called Chimera States, the metastable states that exist in between the states of chaos and order, in systems of classical and quantum oscillators.
Chimera states emerge in many different domains of complex systems science, from biological synchronization, the physics of quantum phase transitions, the factors that influence social dynamics and the patterns that exist in so-called emergent phenomena in many different domains of reality.
Taking its name from the mythological beast from Greek Mythology, the Chimera is a fusion of beings that manifests from chaos, is difficult to track and disappears when being examined or pursued by a hunter. This legend has an ironic resonance when we use this creature as an analogy of the fusion of order and disorder in systems which can be treated as systems of oscillators.
The main system of oscillators we are concerned with are the so-called pure quantum systems that we have discussed in previous videos, which obey precise and ordered quantum transition rules but nevertheless have chaotic behavior.
Due to the universality of the chimera states across many different fields of study, from quantum to classical systems, the pure quantum systems being used in networks for computation and optimization problems offer a way to simulate and solve many equivalent behaviors in the real world. Its an astonishing but true fact that many of the behaviors of chimera states are scale independent and operate in complex classical, quantum and even relativistic systems.
Not taking this amazement for granted we may have to think about many of the seemingly separate fields of science as being under the shadow of the underlying quantum chimera. This may be the key to solving many of the recalcitrant problems in the world today and allow us to reach new levels of understanding and emulation with computer technology built around the operation and use of chimera states.
The Quantum Chimera:
The observable universe hosts a vast array of complex systems across many different fields of interest.
These can be modelled using theories that create a layer of abstraction to separate the complexity from the
fundamental nature of the individual interactions.
Describing complex systems as "systems of oscillators" may sound obscure, but in a strange convenience it describes, in a very general way,
all sorts of physical systems not just in the context of standard classical and quantum harmonics.
As examples, there are lots of biological systems can be reduced to populations of harmonic and anharmonic oscillators.
The heartbeat is just oscillating heart cells that a wave propagates on.
And synchronizing neurons in the brain are oscillators as well, and have been treated with these methods to give rich understanding
of the kinds of patterns we see experimentally.
Rather than just simply observing and recording the patterns from such reductionist systems, abstract generative models must be constructed if we are to enrich our
understanding beyond simply measuring outputs from abstract models.
In higher level modelling we can say that Ensembles of globally coupled systems of oscillators synchronization can manifest
themselves into the appearance of a macroscopic mean field. In quantum field theory therefore an ensemble of particles, highly coupled through entanglement say
, can give rise to a scalar field or a scalar field can itself can mediate a strong coupling between ensembles of particles. Both Views happen to be equivalent
or have what we call a duality of description.
Natural systems of synchronized particles can form macroscopic mean fields, such as atoms in Bose-Einstein Condensates or electrons in Superconductors.
In a condensate we see spontaneous breaking of symmetry of the particle where it does not matter if one adds a particle or subtracts it from the condensate,
a mechanism which may appear odd in terms of charge transfer but in terms of representing this in terms of momentum it appears obvious as in the newtons cradle
model.
The core reason why we have the separation of the different nodes in an ensemble of particles is due to the uncertainty principle
and in condensates of bosons this is the cause of the separation that we see.
Quite often clusters of synchronized elements are observed in between regions of unsynchronized elements.
In a type-II superconductor we see an intermediate phase of ordinary conductivity by unsynchronized electrons
mixed with superconductivity mediated by the synchronization of electrons in cooper pairs.
this effectively means we have a mixture of a synchronous condensate with asynchronous individual charge carriers
at intermediate temperature and fields above the superconducting phases.
Synchronous and asynchronous might seem dichotomous conditions of a functioning system, yet both states can, in fact,
exist simultaneously and durably within a system of oscillators, in what's called a chimera state.
Chimera states are patterns where synchronous and asynchronous domains coexist, taking its name from a composite creature in Greek mythology.
According to the myth, the main power of the chimera is that the closer someone gets to it, as they are pursuing it, The further away it is from you in actuality.
This has an ironic resonance with the chimeras we are talking about in the context of systems of quantum oscillators and chaos theory.
This exotic state still holds a lot of mystery, but its fundamental nature offers potential in understanding governing dynamics across many scientific fields.
Of particular interest in this work is the effect symmetries of a complex system can have on the emergence of chimera states.
for example, the effect of having the same versus different coupling strengths of the
outer regions of the field to the center regions. this causes breaking in the symmetry of synchronization and therefore drives the
system to adapt and perform a kind of emergent error correction or annealing. Such annealing can occur in systems of harmonic oscillators,
classical and quantum. Chimera states are therefore an integral part of any emergent complex adaptive system. Such systems can
be used to simulate other complex systems with equivalent behaviour and can be used to solve optimization problems.
Several optimization problems can be represented as paths or logical decision trees, which themselves
can be reduced to the Boolean satisfiability problem, or SAT problem. Basically, it's an algebraic or Boolean logic expression
(that looks like ^ means AND, v means OR,means NOT)
Think for example of trying to find the shortest path on a map from one city to another.
If you wanted to do a brute-force search, you'd be trying every single possibility that existed.
Another example is the knapsack problem, where you have a bunch of items but you can only carry a certain amount.
Suppose you're working for NASA and you're building a rocket, and you're trying to figure out how much fuel to put on the rocket.
But remember fuel adds weight, so what are the optimal fuel to payload ratios?
What's the value of each item versus its weight where there are n objects with weights w_i and values v_i?
Or Perhaps you are studying protein folding, and you need to fold in the minimum energy conformation
so that you can understand how diseases work. Now in these scenarios you might also realize nature doesn't always trend towards the global minimum energy conformation,
and asynchronous affects have to be considered existing with synchronous effect so we are already entering the territory so to speak of the chimera.
Because this is an optimization problem, the notion of solution is not entirely adequate, backtracking is rather designed for decision problems,
in which one should answer questions of the type "is there a feasible solution?" or "is there a feasible solution achieving a value of at least V?".
In this case the extension of a solution might again be a solution.
These are all massive problems with tons of simulated parameters but the common The goal is to maximize the value of the objects selected,
respecting a limit W on the sum of the weights of the selected objects.
Using the Adiabatic Theorem in the design of a Quantum Network can be primarily used to solve the Boolean satisfiability problem,
like the clique network problem, a clique being a complete subgraph of a total graph. The size of a clique is the number of vertices it contains.
The clique problem is the optimization problem of finding a clique of maximum size in a graph.
Using the adiabatic Hamiltonian, where the solutions are the maxima and minima in
terms of a graph where there is a collection of nodes in a grid. This graph is analogous to a "program" in an AQC,
where the initial state of qubits are connected in a certain way.
As a side note it so happens that even though adiabatic annealers and gate-based quantum computers are vastly different paradigms,
the Wigner Jordan transformation allows one to map fermionic problems onto an Ising spin model.
This spin model can of course be implemented on an annealer, or through the use of a Variational Quantum Algorithm, be implemented on a gate based computer.
The "annealing" part of the Variational Quantum Algorithm comes from using classical machine learning to find the parameters of the gates that minimize
the ground state of the Ising Hamiltonian.
One thing that is easy for the Adiabatic Quantum Computation to do is quantum error correction.
The adiabatic theorem tells you that the longer you wait for your system to reach its final state,
the more likely you are to have stayed in the ground state (or, said another way, the less likely you are to have excitations,
which basically translate to errors). Therefore, in order to reduce error, all you need to do is run the algorithm in
the adiabatic quantum computer for a longer time. If you ran it for T = infinity, you'd be 100% accurate.
Mapping a problem to a basic Boolean satisfiability problem can difficult or can create added work in the translation
(imagine a problem that requires the number of qubits to scale with the number of real variables by a polynomial of high degree, or worse, exponentially).
You might also find that an approximate solver is not good when you have hard constraints that must not be violated
(these machines operate at a finite time scale, therefore there is always some noise in the adiabatic quantum computer). Therefore it is key to understand
how such systems work with a mixture of synchronized and unsynchronized states, to understand our chimera state for a given system in effect.
As discussed, chimera states can form naturally in systems that can be treated as systems of oscillators, like the quantum systems we have discussed.
This is really the main reason why quantum systems are being considered applicable for solving these kinds of Boolean SAT problems in the first place.
Although, as we have seen in a previous video, the quantum systems can be very sensitive to initial conditions and in effect create chaos
even in controlled (and isolated) quantum systems, we see from looking at the mixture of asynchronous and synchronous elements in chimera states in such systems as
in fact beneficial to driving an adiabatic quantum network to achieve quantum error correction but at the same time allowing for it
to behave as a meta-heuristic state so that we never have the system becoming trapped in solutions, corresponding to local minima, which are not the globally
optimized solution to the problem.
In effect our quantum chimeras really offer the best signature for the quantum system we are interested in using
for solving the kinds of problems we may want a quantum network to solve.
The system’s response to the presence of a chimera state is a sharp transition at a critical value of a variable p,
above which percolation occurs but below which it doesn’t occur.
Near this critical value, the system is very sensitive to minor perturbations,
and a number of intriguing phenomena (such as the formation of self-similar fractal patterns as seen in our discussion of chaotic systems)
are found to take place at or near this transition point, which are called critical behaviours or percolation thresholds.
Many complex systems, including biological and physical networks, are considered to be utilizing such critical behaviours
for their self-organizing and information processing purposes. For example, there is a conjecture that animal nervous systems
tend to dynamically maintain critical states in their neural dynamics in order to maximize their sensitivity responses and information processing
capabilities. Such self-organized criticality in natural systems has been a fundamental research topic
in complex systems science and relates intimately to the study of percolation thresholds with these percolation thresholds corresponding to chimera states.
Moreover, the link between the percolation dynamics of complex networks has not been lost on those that
employ statistics in the analysis of such networks, as renormalization group theory, a technique straight from
the quantum field theory toolkit, is sometimes used to quantify the percolation thresholds of complex systems.
The real amazing thing is that so many of the problems that are solvable by such quantum networks,
in the classical world, should be solvable in the first place.
Not taking this amazement for granted, we may have to think about the emergent non-linear behaviour and dynamics of the real world as merely an echo of the balance
of synchronization and chaos that exist in the quantum mechanical foundations of the whole universe,
an echo of the quantum chimera that happens to shows up into our reality.
