Quantum Computing for the Brain

The following sections are included:

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

Quantum Neuroscience has the possibility of bringing the modeling of the human brain’s 86 billion neurons and 242 trillion synapses within reach, even with existing quantum systems (for example, a 53-qubit system has nine quadrillion states (2⁵³)). Available cloud-based quantum computing services could extend their offerings to tools for the study of the brain such as quantum machine learning, high-dimensional photonic entanglement, and spiking neural networks. Contemporary advances in foundational physics and information theory are informing the development of standardized neural circuits and quantum neuroscience applications in wavefunction modeling, quantum biology, and neuroscience physics.

Chemical synapses are the brain’s neural signaling processes between axon, presynaptic terminal, synaptic cleft, postsynaptic density, and den-dritic spiking potentials from dendrite to soma (at the scale tiers of network, neuron, synapse, and molecule). Electrical signals from the action potential are converted to chemical signals in the presynaptic terminal, cross the synaptic cleft as neurotransmitters, are received at dendritic arbors, and are then reconverted to electrical signals as dendritic spikes. Synaptic integration is the process by which the brain integrates thousands of dendritic spikes and other incoming signals.

The AdS/CFT correspondence is a model of physical systems equating a complicated bulk volume to a boundary region with one less dimension. The model surpassed 20,000 journal citations in early 2021 and is applied in various areas of physics, information science, and neuroscience. A novel multi-tier AdS/Brain correspondence is proposed. AdS/DIY mathematics are elaborated with the four equations of the metric, operator, action, and Hamiltonian or Lagrangian.

This chapter describes a photonic particle accelerator-on-a-chip, quantum gravity testing with Rydberg atoms, a black hole on a superconducting chip solution, and quantum simulators of the Sachdev–Ye–Kitaev (SYK) model. The SYK model constitutes a minimal realization of the anti-de Sitter space/conformal field theory (AdS/CFT) correspondence. Due to information-theoretic formulations, foundational physics findings from black holes and the holographic correspondence can be applied in manageable tabletop experiments to study a variety of complex quantum systems including the brain.

This chapter studies symmetry, the property that systems look the same from different points of view (whether a face, a cube, or the laws of nature), and symmetry breaking (phase transition). A neuronal gauge theory interprets the brain as a multiscalar system with a global symmetry (invariant property). In the process of neural signaling, the global symmetry is maintained by the application of gauge fields with energy minimization as the gauge-invariant property of the brain system.

This chapter describes information-theoretic formulations as a control lever for quantum mechanical processes such as quantum computing. Entropy is a key measure of the interconnectedness of subsystems within a system and can be manipulated in the information compression and quantum teleportation of the state information between locations. Quantum walks and out-of-time-order correlation (OTOC) functions manage forward and backward time evolution in quantum systems.

This chapter discusses writing quantum algorithms and quantum circuits to run on quantum hardware platforms. Qubits are encoded into various physical systems such as electron spins, superconducting current, and optical lattices. Input data are modulated (written) onto qubits with basis (one-to-one) encoding or amplitude (many) encoding. Quantum programs are rerun many times to obtain probabilistic confidence of the results. Quantum computation proceeds with unitary transformations (one-off steps) to evolve quantum systems forward in time.

This chapter discusses neural signaling models that progress beyond those that are primarily focused on electrical action potentials, to a broader picture that also includes the action of glial cells, neurotransmitters, and synapse proteins. The connectome and the synaptome could be the “killer applications” of quantum computing for the brain. The connectome is the wiring diagram of all neural connections in the brain, and the synaptome is the similar map for synapses.

This chapter discusses black holes as a model system and informationtheoretic format for foundational physics. The AdS/CFT correspondence was proposed to study black holes and might likewise be applied to the brain. The information-theoretic correspondence formulation uses entropy to study the UV–IR (short-range and long-range) correlations in a system. A geodesic (shortest-length curve) through the bulk provides a measure of boundary region entanglement.

This chapter discusses quantum computing in global networks as opposed to standalone machines. Global networks are built on photonic transfer, an expertise honed over decades. Optical qubits can be encoded and entangled in many ways, leading to the high-dimensional entanglement of qudits (quantum information digits) and the sending of GHZ (Greenberger–Horne–Zeilinger) states (entangled quantum states involving at least three qubits or particle states). Gaussian boson sampling is an optical NISQ device, and can be interpreted with graph theory.

This chapter discusses quantum optical machine learning as the corresponding version of quantum machine learning for optical platforms to test and develop quantum algorithms. Quantum networks as the centerpiece of global quantum computing are presented in the context of a full-stack protocols proposal including heralded (confirmed) entanglement generation, quantum teleportation, and end-to-end qubit delivery. A global quantum clock network of entangled network-wide GHZ (Greenberger–Horne–Zeilinger) states is proposed.

This chapter discusses neuroscience as a “big data” field. Since the fruit fly connectome was only finished in 2018, next-generation methods such as quantum computing are likely necessary for the human connectome and a potential future era of personalized connectomics and disease management. Imaging relies on electron microscopy as the workhorse technique, light sheet microscopy (nonliving samples), and light field microscopy and calcium signaling (living samples), including wholebrain activity recording in behaving organisms. A significant advance is molecular-scale resolution in expansion light field microscopy.

This chapter discusses brain networks as having similar design principles to those seen in human-made electronic networks for computing devices and communications networks. Neural design principles are specified in the areas of wiring and circuit layout, connectivity, energy consumption, signal processing, signal-to-noise ratio, and network reconfiguration (synaptic plasticity). Seeing cortical relations as brain networks has led to the development of network neuroscience and facilitates the AdS/Brain multi-tier model of neural signaling.

This chapter discusses the foundational physics advances in the understanding of dynamics in quantum mechanical systems. Operator size and distribution growth are new levers for measuring the distribution of expectation values of operators acting on observables that produce dynamical behavior. Out-of-time-order correlation (OTOC) functions are applied to evolve thermofield double states (two entangled copies of a quantum state) back and forward in time to understand quantum information scrambling (diffusion). The holographic Sachdev–Ye–Kitaev (SYK) model interprets operator growth as a solvable SYK model in the boundary to calculate aspects of particle momentum in the bulk. Any system with time translation symmetry breaking can be interpreted as a time superfluid, and possibly as a spacetime crystal.

This chapter discusses multiscalar neural dynamics models. Large-scale models of brain behavior, including neural ensemble, neural mass, and neural field theories, populated with empirical data, have a strong need for more capacious computational platforms such as those offered by quantum computing. Probability distributions of large-scale neural signaling activity cannot be calculated with traditional reaction-diffusion methods applied with linear and nonlinear Fokker-Planck equations. In neural state dynamics, seizures indicate a straightforward chaotic dynamics, but resting states suggest a more complex mix of multistability and Hopf bifurcation, and active states in behaving brains are also likely to have complex dynamical behavior.

This chapter discusses quantum machine learning as one of the first mainstream applications of quantum computing. Modeling classical data as quantum wavefunction states allows hidden information in the underlying system to be accessed through wavefunction amplitude. A greater amount of information can be incorporated in the compact and complex format by encoding an exponentially large number of values into a set of qubits. Wavefunction approximation and quantum state simulation (with transformer neural networks) are central applications in using machine learning to study problems in quantum mechanics (ML/Q).

This chapter discusses probabilistic methods for quantum machine learning. One model is the Born machine which employs the Born rule to determine output probabilities (as opposed to the Boltzmann distribution-based energy function traditionally used in machine learning). Born machines are relevant to generative unsupervised (unlabeled data) learning, extending discriminative supervised learning approaches. Discriminative algorithms learn directly from data, but generative algorithms learn the distribution of the data to produce new samples. Reduced density matrices are used to rewrite classical probability distributions in quantum terms. A key insight is that pixel = spin (qubit) for point data, and wavelet = spin (qubit) for sequential data; an image pixel or wavelet equates to a spin value in a quantum system (qubit).

This chapter discusses how quantum methods are used to improve machine learning techniques (Q/ML). The main approaches to machine learning are neural networks, tensor networks, and kernel learning, each of which has a quantum counterpart. Quantum kernel learning methods obeying the RKHS (reproducing kernel Hilbert space) formalism can be used to compute kernel functions at other scale dimensions of the system and have applications in quantum finance. The point is to have an analytical model that grows more slowly than the underlying system. Modeling classical data with wavefunctions finds hidden correlations, namely, how the system is entangled, which can be used as quantum system design principles.

This chapter discusses brain modeling approaches including compartmental neuroscience simulation, classical machine learning, and spiking neural networks in neuromorphic architectures. It is clear that quantum methods and platforms could extend these efforts. Of particular interest is the need to model the effects of nonlinear calcium signaling in synaptic integration which requires the diffusion-reaction mathematics of partial differential equations (PDEs). Spiking neural network methods propose useful alternatives to classical machine learning’s backpropagation as real-life temporal models cannot go backward to iteratively update network weights as permutations of the model cycle.

This chapter argues that quantum neuroscience applications are needed not only given the complexity of the domain but also to obtain a causal understanding of disease. The AdS/Brain theory is introduced as the first multi-tier renormalized interpretation of the AdS/CFT correspondence, for implementation with bMERA (brain) random tensor networks with Floquet periodicity dynamics. Bulk-boundary pairs are outlined in the multi-tier composite neural signaling model that includes the scales of network, neuron, synapse, and molecule. Standard neuroscience quantum circuits are proposed in a list of Millennium Prize-type challenges, and risks and limitations are discussed.

The following sections are included:

• Glossary
• Index