Quantum Computing

The following sections are included:

• About the Authors
• Contents
• List of Figures
• List of Tables

The implication is that reality can be computed. The startling premise of quantum computing is that the bizarre quantum mechanical realm that was thought incomprehensible can be computed. Quantum mechanics might remain incomprehensible, but at least it can be computed. The bigger notion is that the link between quantum computing and quantum physics suggests the possibility of delivering on the promise of understanding the true nature of physical reality…

This work aims to establish a deeper connection between physics and information technology. Smart network theory is proposed as a physical theory of network technologies, particularly to encompass a potential expansion into the quantum computing domain. The objective is to elaborate a physical basis for technology theories that is easily deployable in the design, operation, and catalytic emergence of next-generation smart network systems. The general notion of smart network theory as a physical basis for smart network technologies is developed into a smart network field theory (SNFT) and a smart network quantum field theory (SNQFT) relevant to the two scale domains. The intuition is that the way to orchestrate many-particle systems from a characterization, control, criticality, and novelty emergence standpoint is through an SNFT and an SNQFT. Such theories should be able to make relevant predictions about smart network systems.

Quantum computing is a research frontier in physical science with a focus on developing information processing at the quantum scale. Quantum computing involves the use of algorithms to exploit the special properties of quantum mechanical objects (such as superposition, entanglement, and interference) to perform computation. In physics, quantum mechanics is the body of laws that describe the behavior and interaction of electrons, photons, and other subatomic particles that make up the universe. Quantum computing engages the rules of quantum mechanics to solve problems using quantum information. Quantum information is information concerning the state of a quantum system which can be manipulated using quantum information algorithms and other processing techniques. Although quantum computing is farther along than may be widely known, it is an early-stage technology fraught with uncertainty. The overall aim in the longer term is to construct universal fault-tolerant quantum computers.

The special properties of quantum objects (atoms, ions, and photons) are superposition, interference, and entanglement. Superposition refers to particles existing across all possible states simultaneously. Interference is the situation where intervention from noise in the environment damages the quantum object, and also the possibility that the wave functions of particles can either reinforce or diminish each other. Entanglement means that groups of particles are connected and can interact in ways such that the quantum state of each particle cannot be described independently of the state of the others even when the particles are separated by a large distance. One of the most important implications of entanglement is that qubits can be error-corrected, which will likely be necessary for the advent of universal quantum computing. An application of quantum computing that is already available is certifiably random bits, a proven source of randomness, which is used in secure cryptography.

Many developments are underway in blockchains. An overall theme is that a new level of complexity is being abstracted into digital economic networks to accommodate financial transactions beyond payments. The migration to blockchain-based systems appears as a replay of the entirety of world economic history in the space of a few years. Both public and private blockchains are instantiating PrivacyTech and ProofTech through zero-knowledge proof technology, verifiable computing, and user-centric digital identity solutions. There are efficiency, privacy, and security upgrades planned for the large public blockchains (Bitcoin and Ethereum). Layer 2 expansions continue, involving sidechains, payment channels, and off-chain processing to expand the real-time ease of using blockchains. There are new innovations such as stablecoins, smart routing, and next-generation consensus algorithms.

The future of global network communications could include a quantum internet with various features such as quantum key distribution (QKD), secure end-to-end communication, quantum memories, quantum repeaters, and quantum-based applications such as quantum blockchains. Reactions toward the potential quantum information era are first, one of preparing to be quantum-resistant and quantum-secure, and second, being quantum-compatible and quantum-embracing by taking advantage of the new functionality offered by quantum systems. There are implementation suggestions for quantum blockchains, both the overall protocols and individual aspects such as cryptography upgrades. The nearest-term potential application of quantum computing appears to be QKD as an antidote to RSA cryptographic standards possibly being compromised by quantum computers. Specific ways in which blockchains can become more quantum-secure are considered.

One of the biggest transformations underway in cryptography and blockchains is zero-knowledge proofs. The core idea is being able to prove validity without disclosing underlying information. Zero-knowledge proofs are a crucial enabling technology in the overall trend to make the internet more private and secure (zero-knowledge proofs are PrivacyTech and ProofTech). A zero-knowledge proof is a cryptographic method that separates data verification from the data itself. One party (the prover) can prove to another party (the verifier) the possession or existence of some information without revealing the information. Zero-knowledge proofs might be used in an extensive range of applications such as credit card authorization, account confirmation, and identity validation. Current state-of-the-art zero-knowledge proof systems include SNARKs, bulletproofs, STARKs, and Zether.

Zero-knowledge proofs are just the basic concept in proof technology. Many other sophisticated formulations of proofs and related quantum-secure technology are proposed. Earlier forms of zero-knowledge proof technology are not quantum-resistant. Proofs based on generic hash functions as opposed to classical PKI cryptography are thought to be post-quantum secure because the cryptography can be upgraded, whereas classical PKI cryptography is known not to be quantum-secure. STARKs are a new form of zero-knowledge proofs for blockchains with a sophisticated architecture based on error correction codes, random queries, and inconsistency checks. The two main approaches to post-quantum cryptography are lattice-based cryptography and hash function-based cryptography. Lattice-based cryptography is currently the main method being promulgated in next-generation US NIST algorithm development. Keeping public digital infrastructure safe is a key concern. Post-quantum cryptography suggests various approaches to prepare cryptographic infrastructure for a potential future of quantum computing.

Machine learning refers to mechanistic systems that “learn” by modeling high-level abstractions in data and cycling through trial-and-error guesses with feedback to establish an optimally weighted system that can make accurate predictions about new data. The main forms of deep learning networks are convolutional neural nets (CNNs) for image recognition and recurrent neural nets (RNNs) for text and speech recognition. Learning may be supervised (executed on pre-classified datasets) or unsupervised (finding patterns in unlabeled data). Prominent advances include adversarial nets, dark knowledge, and manifold learning. The technical aspects of machine learning include formulating the problem as a logistic regression, running it on a modular processing network, and iterating to find an optimal solution. Although machine learning excels at certain tasks, namely image and text recognition, its expandability to more complicated data analysis problems has been questioned.

One of the first uses of quantum computing is machine learning. Both techniques are aimed at the same kinds of problems in statistical data analysis and optimization. In this sense, quantum computing may be an improved method over machine learning. Machine learning applications such as optimization and simulation are being instantiated in quantum computing formats. Standardized tools are emerging such as the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA). Quantum algorithms are being specified with standard gate logic, including the Hadamard gate (which acts on 1 qubit to put it in a superposition state), the CNOT gate (which acts on 2 qubits to flip one), and the Toffoli gate (which acts on three or more qubits to implement the Boolean operators). Other advances in geometric deep learning and information geometry facilitate a wider range of 3D computational operations in problem-solving.

Smart network field theory (SNFT) is derived from statistical physics (statistical neural field theory and spin-glass models) and information theory (the anti-de Sitter space/conformal field theory, AdS/CFT, correspondence). In this chapter, statistical neural field theory and spin-glass models as two model field theories are elaborated in a step-by-step method with regard to the various physics concepts and methods they employ. The theories are exemplars of the technophysics principle of formulating systems in a way that is analytically solvable. For both field theories, first an initial approximation of the system is made (with a mean field theory and a random energy model). Then, a more detailed analytical formulation is posited that can be solved exactly (statistical neural field theory and p-spherical spin-glass theory). The approximation reflects the system at stable equilibrium, and the solvable model, the system at criticality. Energy minimization and statistical distributions are key organizing principles in both models. The field theories are selected for their comprehensive technophysics focus in complex systems such as the brain and superconducting materials, and because these physical models continue to inspire advances in machine learning. The model theories might be used as a template for formulating field theories in any domain, whether physical or algorithmic.

Smart network field theory (SNFT) is defined as a conceptual and formal tool for the characterization, monitoring, and control of smart networks. Smart networks are complex systems with thousands, millions, or billion many-particle elements, and therefore, a well-grounded analytic system-level theory, with robust foundations in physical science, is needed. The SNFT is derived from statistical physics (statistical neural field theory and spin glass models) and information theory (the anti-de Sitter space/conformal field theory, AdS/CFT, correspondence). The practical use of the SNFT is for fleet-many item orchestration and system criticality detection in smart network systems. In this chapter, SNFT is specified in terms of system structure, operation, and criticality, and potential applications are considered in blockchain and deep learning smart network systems.

The anti-de Sitter space/conformal field theory (AdS/CFT) correspondence is the theory of a proposed duality between a bulk region with (d+1) dimensions and a boundary region with (d) dimensions. The AdS/CFT correspondence (also called gauge/gravity duality) suggests that in any physical system, there is a correspondence between a volume of space and its boundary region such that the interior bulk can be described by a boundary theory in one fewer dimensions. The AdS/CFT correspondence is a formalization of the holographic principle which denotes the possibility of a 3D volume being reconstructed on a 2D surface. There is an information-theoretic interpretation of the AdS/CFT correspondence as a quantum error-correction code, which is formally solved with tensor network models. Many other fields use the AdS/CFT correspondence model to study various phenomena in superconducting materials, condensed matter plasmas, and network theory.

This chapter continues the research discussion that interprets the anti-de Sitter/conformal field theory (AdS/CFT) correspondence as an information-theoretic domain that (among other operations) can be error-corrected with quantum error-correction codes. Harlow & Hayden (2013) identify the AdS/CFT correspondence as an information theory problem, which implies computational complexity as an analysis tool, and the quantum information properties of entanglement and error correction. Almheiri et al. (2015) propose the interpretation of the AdS/CFT correspondence as a quantum error-correcting code. This model is realized by Pastawski et al. (2015), by enumerating a specific quantum error-correcting code. Such holographic codes may be used in a variety of applications, including in the development of technophysics-based quantum error-correcting codes.

The smart network theories are derived from statistical physics (statistical neural field theory and spin glass models) and information theory (the the anti-de Sitter/conformal field theory, AdS/CFT, correspondence). Whereas the smart network field theory (SNFT) is aimed at the next phases of the classical development of smart network systems, the smart network quantum field theory (SNQFT) is intended to facilitate the potential expansion into the quantum domain, in the implementation of quantum smart networks. This chapter develops the SNQFT from the AdS/CFT correspondence, and suggests the possibility of the correspondence supplanting probability as a central organizing principle for certain ways of understanding physical reality. The SNQFT is developed and motivated towards a variety of potential applications. It is suggested that everyday macroscale reality is the boundary to the quantum mechanical bulk. An emphasis is placed on models that integrate quantum mechanical field views and geometric views, including for the exploitation of quantum mechanical systems in macroscale reality. Finally, the work considers risks and limitations and concludes. The content in this chapter is more speculative and conjectural than that in previous chapters.

The following sections are included:

• Glossary
• Index