Basis for a neuronal version of Grover's quantum algorithm

Abstract

Grover's quantum (search) algorithm exploits principles of quantum information theory and computation to surpass the strong Church-Turing limit governing classical computers. The algorithm initializes a search field into superposed N (eigen)states to later execute nonclassical "subroutines" involving unitary phase shifts of measured states and to produce root-rate or quadratic gain in the algorithmic time (O(N (1/2))) needed to find some "target" solution m. Akin to this fast technological search algorithm, single eukaryotic cells, such as differentiated neurons, perform natural quadratic speed-up in the search for appropriate store-operated Ca(2+) response regulation of, among other processes, protein and lipid biosynthesis, cell energetics, stress responses, cell fate and death, synaptic plasticity, and immunoprotection. Such speed-up in cellular decision making results from spatiotemporal dynamics of networked intracellular Ca(2+)-induced Ca(2+) release and the search (or signaling) velocity of Ca(2+) wave propagation. As chemical processes, such as the duration of Ca(2+) mobilization, become rate-limiting over interstore distances, Ca(2+) waves quadratically decrease interstore-travel time from slow saltatory to fast continuous gradients proportional to the square-root of the classical Ca(2+) diffusion coefficient, D (1/2), matching the computing efficiency of Grover's quantum algorithm. In this Hypothesis and Theory article, I elaborate on these traits using a fire-diffuse-fire model of store-operated cytosolic Ca(2+) signaling valid for glutamatergic neurons. Salient model features corresponding to Grover's quantum algorithm are parameterized to meet requirements for the Oracle Hadamard transform and Grover's iteration. A neuronal version of Grover's quantum algorithm figures to benefit signal coincidence detection and integration, bidirectional synaptic plasticity, and other vital cell functions by rapidly selecting, ordering, and/or counting optional response regulation choices.

Keywords: biotechnology; calcium-induced calcium reactions (CICRs); cellular decision making; classical and quantum computation; inositol 1,4,5-trisphosphate receptors (IP3Rs); intracellular calcium; neuronal plasticity; quantum molecular networks and memory.

Figure 1

Calcium-induced calcium reactions (CICRs) emulate Grover's quantum algorithm in neuronal information processing. Left panel portrays major characteristic substrate (e.g., receptors, organelles, etc.) involved in Ca²⁺-mediated response regulation of arbitrary glutamatergic neurons, including, but not limited to, substrate critical for synaptic plasticity, cellular energetics, immunoprotection, homeostasis, gene expression, biosynthesis, molecular trafficking, cytoskeletal organization, and cell fate. Similar mechanisms affect both pre- and post-synaptic neurons, but, for descriptive purposes, post-synaptic cell activity is emphasized. Ca²⁺ entry into the post-synaptic neuron through voltage-gated receptor (VGC), ligand-gated receptor (LGC), and transient potential receptor (TRP) channels and stimulated inositol 1,4,5-trisphosphate (IP₃) production by activated G-protein coupled receptors (GCR) help initiate cytosolic CICRs from integral IP₃ receptors (IP₃R) located along the endoplasmic reticulum (ER) membrane. CICRs may cause traveling waves of varying velocities and patterns which emulate search routines capable of eliciting/suppressing appropriate response regulation from different cellular compartments. Lower right panel illustrates CICR saltatory and continuous waves. Saltatory Ca²⁺ waves and the information they carry conduct at velocities (V) proportional to the classical Ca²⁺ diffusion coefficient (D). Whereas, faster continuous Ca²⁺ waves and the information they transmit move at velocities proportional to the square-root of the classical Ca²⁺ diffusion coefficient. Coefficient D of continuous waves for either intercluster or intracluster diffusion is assumed to be up to orders of magnitude greater than that for saltatory waves. The quadratic disparity in the velocities of saltatory and continuous waves corresponds to the root-rate increase of information processing by Grover's quantum algorithm over classical algorithms. Upper right panel shows schematic of Grover's quantum algorithm. The algorithm takes as input n qubits, upon which it performs Hadamard transformations (H^⊗n) and Grover's operation (GO) to find a target m of M solutions stored in database N. Regardless of whether one or more consultations of the Oracle are needed, Grover's quantum algorithm finds the target solution within O = N^1/2 algorithmic steps or operations O. Additional abbreviations: arachidonic acid (AA), Ca²⁺ binding molecule (CBM), Ca²⁺ uniporter (Uni), diacylgycerol (DG), Golgi apparatus (Golgi), L-glutamate (L-Glu), nucleus (Nucl), mitochondria (Mito), nitric oxide (NO), nitric oxide synthase (NOS), phospholipase A₂ (PLA₂), phospholipase C (PLC), plasma-membrane Ca²⁺ ATPase (PMCA), ryanodine receptor (RyR), sarcoplasmic-endoplasmic-reticulum Ca²⁺ ATPase (SERCA), Na⁺/Ca²⁺ exchanger (Exch), synaptic vesicle (SV).

Figure 2

Model of conformation, ion permeability, and corresponding Grover's quantum-algorithm function of an inositol 1,4,5-trisphosphate receptor channel (IP₃R). Each cross-section contains two of four complete IP₃R subunits. When only cytosolic IP₃ (blue sphere) binds, the receptor lumen stays closed and inactive. Cobinding of cytosolic IP₃ and Ca²⁺ (red sphere) to separate high-affinity sites proximal to the IP₃-binding domain dissociates suppressor (Suppr), calmodulin (CaM), and gatekeeper (Keeper) regions, repositioning the transmembrane gate (Gate) and activating Ca²⁺ conductance. In absence of IP₃ binding, low-affinity binding of cytosolic Ca²⁺ to one of two calmodulin heads occludes the ion channel as calmodulin crosslinks with suppressor and gatekeeper regions of adjacent receptor subunit. No ligand binding is accompanied by a small leaking Ca²⁺ conductance. Free cytosolic proteins, nucleotides, and other substances can facilitate or impair IP₃R gating by interacting with the IP₃ binding-core, suppressor, and gate-keeper regions. Free endoplasmic-reticulum proteins and Ca²⁺ may also further modulate pore activity (not shown) via selectivity filters (small blue cylinders) located near pore helices (small rose cylinders). In the superposition state |ψ〉, the IP₃R samples all possible index values Γ_max marking D_max, returning an output x denoting a successful or unsuccessful search for shortest times or fastest rates m needed for store-released concentrations of free cytosolic Ca²⁺ to continuously diffuse and autocatalytically activate a nearest neighbor receptor. This superposition state may be regarded indefinitely stable in saturating IP₃ concentrations. A phase shift by O_IP3R reversibly inactivates the receptor channel with high free cytosolic Ca²⁺ concentrations. Another subsequent phase shift reactivates the channel, confirming solution m has been found. See Equation 10 and relevant text for additional details. IP₃R conformation representations adapted from Clark and Eisenstein (2013) with permission.