In this letter we discuss computing in terms of Gaisi Takeuti's quantum set theory using an explicit example of a rigid body of cuboid form. The universe V(L) of Takeuti will be determined. A set of real numbers in this Universe are also explicitly described including a set of binary numbers. Thus we arrive at the foundations of von Neumann' theory of computing in terms of ordinary binary numbers. Then we see that this extension of computing to the Universe V(L) provides a sound, mathematically well defined theory of quantum computing.In this letter we discuss computing in terms of Gaisi Takeuti's quantum set theory using an explicit example of a rigid body of cuboid form. The universe V(L) of Takeuti will be determined. A set of real numbers in this Universe are also explicitly described including a set of binary numbers. Thus we arrive at the foundations of von Neumann' theory of computing in terms of ordinary binary numbers. Then we see that this extension of computing to the Universe V(L) provides a sound, mathematically well defined theory of quantum computing.

Journal of Artificial Intelligence & Cloud ComputingSep 24, 20240 citations

Gate Based Implementation of the Laplacian with BRGC Code for Universal Quantum Computers

Ermal Rrapaj, Kenneth McElvain, Chia Cheng Chang +2 more

We study the gate-based implementation of the binary reflected Gray code (BRGC) and binary code of the unitary time evolution operator due to the Laplacian discretized on a lattice with periodic boundary conditions. We find that the resulting Trotter error is independent of system size for a fixed lattice spacing through the Baker-Campbell-Hausdorff formula. We then present our algorithm for building the BRGC quantum circuit. For an adiabatic evolution time $t$ with this circuit, and spectral norm error $ε$ , we find the circuit cost (number of gates) and depth required are $\mc{O}(t^2 n A D /ε)$ with $n-3$ auxiliary qubits for a system with $2^n$ lattice points per dimension $D$ and particle number $A$ ; an improvement over binary position encoding which requires an exponential number of $n$ -local operators. Further, under the reasonable assumption that $[T,V]$ bounds $Δt$ , with $T$ the kinetic energy and $V$ a non-trivial potential, the cost of QFT (Quantum Fourier Transform ) implementation of the Laplacian scales as $\mc{O}\left(n^2\right)$ with depth $\mc{O}\left(n\right)$ while BRGC scales as $\mc{O}\left(n\right)$, giving an advantage to the BRGC implementation.We study the gate-based implementation of the binary reflected Gray code (BRGC) and binary code of the unitary time evolution operator due to the Laplacian discretized on a lattice with periodic boundary conditions. We find that the resulting Trotter error is independent of system size for a fixed lattice spacing through the Baker-Campbell-Hausdorff formula. We then present our algorithm for building the BRGC quantum circuit. For an adiabatic evolution time $t$ with this circuit, and spectral norm error $ε$, we find the circuit cost (number of gates) and depth required are $\mc{O}(t^2 n A D /ε)$ with $n-3$ auxiliary qubits for a system with $2^n$ lattice points per dimension $D$ and particle number $A$; an improvement over binary position encoding which requires an exponential number of $n$-local operators. Further, under the reasonable assumption that $[T,V]$ bounds $Δt$, with $T$ the kinetic energy and $V$ a non-trivial potential, the cost of QFT (Quantum Fourier Transform ) implementation of the Laplacian scales as $\mc{O}\left(n^2\right)$ with depth $\mc{O}\left(n\right)$ while BRGC scales as $\mc{O}\left(n\right)$, giving an advantage to the BRGC implementation.

arXiv:2207.11647Jul 24, 20220 citations

The Quantum Numbers

Mathematics across the curriculumSep 1, 20060 citations

THE ENTANGLED UNIVERSE

Thomas L. Wilson, Hans‐Joachim Blome

Jan 1, 20041 citation

The Pure Number Substrate: A Computational Model of Reality Generation

Tipping, Lee Anthony

We present a computational model where reality emerges from the iterative application of a specific pure number $P$ . Building upon relational collapse frameworks (including the Binary Soul theory), we show that the universe can be modeled as a deterministic computation: $F_t = \Phi_P(H_{t-1})$ , where $F_t$ is the current state (the "Now"), $H_{t-1}$ is the complete history, and $\Phi_P$ is a computation determined by $P$ . This model makes three testable predictions: (1) minute computable fluctuations in fundamental constants, (2) laser-mediated coherence enhancement following specific functional forms, and (3) entanglement-enhanced gravitational interactions. We propose experiments to distinguish between candidate pure numbers $P$ , potentially identifying our universe's computational substrateWe present a computational model where reality emerges from the iterative application of a specific pure number $P$. Building upon relational collapse frameworks (including the Binary Soul theory), we show that the universe can be modeled as a deterministic computation: $F_t = \Phi_P(H_{t-1})$, where $F_t$ is the current state (the "Now"), $H_{t-1}$ is the complete history, and $\Phi_P$ is a computation determined by $P$. This model makes three testable predictions: (1) minute computable fluctuations in fundamental constants, (2) laser-mediated coherence enhancement following specific functional forms, and (3) entanglement-enhanced gravitational interactions. We propose experiments to distinguish between candidate pure numbers $P$, potentially identifying our universe's computational substrate

Dec 4, 20250 citations

t

with this circuit, and spectral norm error

ε

, we find the circuit cost (number of gates) and depth required are $\mc{O}(t^2 n A D /ε)$ with

n-3

auxiliary qubits for a system with

2^n

lattice points per dimension

D

and particle number

A

; an improvement over binary position encoding which requires an exponential number of

n

-local operators. Further, under the reasonable assumption that

[T,V]

bounds

Δt

, with

T

the kinetic energy and

V

a non-trivial potential, the cost of QFT (Quantum Fourier Transform ) implementation of the Laplacian scales as $\mc{O}\left(n^2\right)$ with depth $\mc{O}\left(n\right)$ while BRGC scales as $\mc{O}\left(n\right)$, giving an advantage to the BRGC implementation.

We present a computational model where reality emerges from the iterative application of a specific pure number

P

. Building upon relational collapse frameworks (including the Binary Soul theory), we show that the universe can be modeled as a deterministic computation:

F_t = \Phi_P(H_{t-1})

, where

F_t

is the current state (the "Now"),

H_{t-1}

is the complete history, and

\Phi_P

is a computation determined by

P

. This model makes three testable predictions: (1) minute computable fluctuations in fundamental constants, (2) laser-mediated coherence enhancement following specific functional forms, and (3) entanglement-enhanced gravitational interactions. We propose experiments to distinguish between candidate pure numbers

P

, potentially identifying our universe's computational substrate