QUBIT TYPE

The type models quantum bits as spectra in stable homotopy theory, parameterized by a coefficient type (e.g., complex numbers) and a space (e.g., a Hilbert space or smooth set). Qubits formalize quantum systems in the cohesive type system, supporting gates and entanglement.

In Urs, is a type in , with operations like state formation , gate application , entanglement , superposition, and tensor products, potentially over smooth sets .

Formation

(-Formation). The type is a spectrum in , formed for each , encoding a quantum system.

def Qubit (C H : U_0) : Sp
(* Qubit as a spectrum *)

Introduction

(-Introduction). Terms of type are quantum states, introduced via constructors for state formation, gate application, and entanglement.

def qubit (C H : U_0) (s : C * H) : Qubit C H
def app_qubit (C H : U_0) (A : H -> H) (Q : Qubit C H) (x : H) : Qubit C H
def fuse_qubit (C H : U_0) (q_1 q_2 : Qubit C H) (chan : H ⊗ H) : Qubit C (H ⊗ H)
(* Constructors for qubits *)

Elimination

(-Elimination). The eliminator for maps quantum states to properties in , analyzing their stable homotopy structure.

def qubit_ind (C H : U_0) (beta : Qubit C H -> Sp) (h : Π (Q : Qubit C H), beta Q) : Π (Q : Qubit C H), beta Q
(* Eliminator for qubits *)

Theorems

(Superposition). For , states in support linear combinations, forming superpositions.

def superpose (C H : U_0) (Q_1 Q_2 : Qubit C H) (a b : C) : Qubit C H
(* Linear combination of states *)

(Tensor Product). Qubits and form composite systems via tensor products.

def tensor (C H_1 H_2 : U_0) : Qubit C H_1 -> Qubit C H_2 -> Qubit C (H_1 ⊗ H_2)
(* Tensor product of qubits *)

(Smooth Qubits). If , then inherits a smooth structure, enabling quantum fields.

def smooth_qubit (C : U_0) (H : SmthSet) : SheafStr (Qubit C H) ℝ^m
(* Smooth structure for qubits *)