Quantum Computing. Melanie Swan

problems such as the Fermi theory of beta decay. In general relativity, effective field theories have been used to simplify gravitational wave problems, and theorize that general relativity itself may be the low-energy effective field theory of a full theory of quantum gravity (in which the expansion scale is the Planck mass). Particularly relevant for quantum computing is the practical application of effective field theories in the domains of superconductivity and condensed matter physics.

2.6.4 Complex multi-level systems

      A key requirement for a SNFT is that it can be used to manage across diverse scale levels within a complex system. Such a field theory should be able to “identify macroscopic smoothness from microscopic noise” as prescribed by complexity theory (Mitchell, 2009). Various methods, including statistical physics, may be used for linking multiple dimensions within complex systems to obtain signal from noise.

      Some aspects of a system are easier to measure on different scales. For example, computing the energy spectrum of the Hamiltonian at different levels of quantum mechanical systems can be challenging. Such calculations may be straightforward at higher levels of the system abstraction, but more difficult when incorporating the energetic fields in which the particles actually behave. At this scale, it is essentially impossible to compute because there is so much data about particle movement. One strategy is to reinterpret particles as states of a quantized field (Jaffe & Witten, 2001). A field theory helps to reinstantiate or roll the system up to a higher level of abstraction at which such calculations can be made. The method is finding or defining an effective field theory at a scale that renders the system analytically solvable.

      For example, the elliptical orbits of the planets are more easily calculated with Newtonian gravity than with general relativity. This simplification can be all that is necessary for certain applications. The benefit of a field theory is that is provides the ability to focus on a particular scale of a system, emphasizing one aspect while limiting others (Georgi, 1993). The objective is to find the simplest framework that captures the essential physics of the target area. For example, when there is interest in lighter particles (such as bottom quarks), heavier particles (e.g. z-bosons and w-bosons) can be eliminated from the model.

      In complex multi-level systems, identifying a macroscopic term corresponding to microscopic behavior is a key challenge. The analogs to the temperature and pressure terms arising from a room of septillions of moving particles in a model system are not always clear. Hence, an effective field theory is a formal process that can be used to identify a system’s “temperature” term and other system-level metrics. Effective field theories are similar to the renormalization concept (in the sense of mathematically scaling to a different level of the system to focus on a parameter of interest in a simplified manner that can be calculated).

2.7Five Steps to Defining an Effective Field Theory

      Effective field theories are important because there is interesting physics at all scales. Being able to portably travel up and down scale dimensions can make it easier to analyze certain aspects of systems. The idea is to use effective field theories as a tool for isolating parameters of interest within a system and engaging the system at that level. Effective field theories may work best when there is a large separation between the length scale of interest and the length scale of the underlying dynamics.

      A distillation of the steps involved in deriving an effective field theory is outlined in Table 2.5. The aim of an effective field theory is to specify the simplest framework that captures the essential physics of interest. The zeroth step is to confirm that there are no already existing fundamental theories to describe the phenomenon and that an effective field theory is useful. Theories with related aspects could be identified as inspiration.

Table 2.5. Steps in articulating an effective field theory.

Step	Description
1.Define the system	Characterize the overall scope, shape, and levels of the system, including the relevant scales, lengths, and energies.
2.Identify system elements	Identify the constituent elements of the system, the kinds of interactions between them.
3.Isolate variables of interest	Articulate the aspects of interest that the field theory should study.
4.Reduce complexity by eliminating unnecessary system substructure	Identity the degrees of freedom (the aspects of the system that matter for the problem of study), and irrelevant substructure that can be ignored, and note symmetries, anomalies, or other known complexity attributes.
5.Identify quantitative metrics	Articulate the mathematics to measure the system, averaging the underlying behavior to derive a simplified model with a global term such as a Hamiltonian or Lagrangian.

      Source: Adapted from Manohar (2017).

      The first of the five steps is to define the system by characterizing the overall scope and shape of the system to be studied, including the relevant scale levels in terms of lengths or energies that comprise the system. The second step is to identify the system elements, the particles or other elements that constitute the system, and the kinds of interactions between them. The third step is to isolate the particular variables of interest that the theory aims to study. The fourth step is to reduce complexity by eliminating the unnecessary system substructure which can be ignored for studying the variables of interest within the system. More detailed aspects of the subsystem of interest are identified such as the degrees of freedom (system parameters) and any complexity properties such as symmetry and anomalies that may influence the theory application. The fifth step is identifying the relevant quantitative metrics for measuring the system. The available quantities in the system are identified, and averaged over to generate a metric as a system composite measure such as a Hamiltonian or Lagrangian.

      An effective field theory example in a biological neural network is that the system-wide quantity of interest might be the spiking activation (the threshold at which neurons fire), and other data would be superfluous. Another example is a minimal effective field theory that only specifies the fields, the interactions, and the power counting of the system (the dimensions of power counting across scales).

      Beyond the basic steps, effective field theories might include more complicated aspects. There could be additional quantities to measure such as available potential energy, propagation, and the range of system states. Also relevant is identifying the dynamics of the system, the dimensions into which the system is expanding. There could be various degrees of freedom. The term degrees of freedom generally connotes system parameters. More specifically, degrees of freedom is a statistical term that means each of a number of independently variable factors that can affect the range of states in which a system may exist, and the directions in which independent motion can occur. Degrees of freedom can be conceived simply as system parameters, and with greater sophistication as a statistical measure of the number of states and ways in which a dynamic system can move, whether this “motion” is considered in physical space or in an abstract space of configurations.

      Overall, the key steps in specifying an effective field theory consist of defining (1) the system, (2) the system elements and interactions, (3) the variables of interest, (4) the irrelevant structure that can be ignored, and (5) the quantitative metrics that can be averaged over the system to produce a temperature-type term. Applying the effective field theory development technique to the smart network context, the idea is to consider the different levels and dimensions of the system, and identify the elements, interactions, and relevant quantities to calculate in order to obtain the system behavior. This is developed in more detail in Chapters 11 and 12.

References

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      Alemi, A.A. & Fischer, I. (2018). TherML: Thermodynamics of Machine Learning. ICML 2018. Theoretical Foundations and Applications of Deep Generative Models Workshop.

      Bitnodes. https://bitnodes.earn.com/. Accessed June 30, 2019.