Quantum Computation of f(x) = x² with Sub J Array

Core Concept Overview

In quantum computing, the function f(x) = x² is implemented using quantum arithmetic circuits. The "sub j" array represents a quantum register programmed in base 2 (binary) that serves as the output storage for the computation result.

Quantum Implementation Structure

The quantum computer handles this computation through several structured phases. First, the input value x is encoded into a quantum register using binary representation. For an n-bit input, this requires n qubits. The output register "sub j" is allocated with approximately 2n qubits to accommodate the squared result without overflow.

The computation phase uses quantum gates to perform multiplication. Hadamard gates create superpositions for parallel computation, CNOT gates handle linear operations, and Toffoli gates manage the multiplication logic through quantum versions of AND operations. The specific gate sequence implements a quantum multiplication algorithm that computes x² through controlled operations.

Role of the Sub J Array

The sub j array serves as the dedicated output register for storing x² in binary format. After quantum computation, measuring this register collapses the quantum state to classical binary values. The array can be designed as either a single critical qubit indicating specific properties of the result or a multi-qubit register holding the complete squared value.

Quantum Circuit Example

# Simplified quantum circuit for f(x) = x² # Input: 2-qubit register |x⟩ = |x₁x₀⟩ # Output: 4-qubit sub j array |j₃j₂j₁j₀⟩ Initialize |x⟩ to desired input state Initialize |j⟩ to |0000⟩ # Quantum multiplication gates Apply controlled operations based on x-bits: - If x₀=1: Add x to output (shifted appropriately) - If x₁=1: Add 2x to output (shifted appropriately) # Result: |j⟩ contains x² in binary Measure sub j array

Practical Implementation Code

from qiskit import QuantumCircuit, QuantumRegister import numpy as np # Define quantum registers input_reg = QuantumRegister(2, name='x') output_reg = QuantumRegister(4, name='j') # sub j array # Create quantum circuit qc = QuantumCircuit(input_reg, output_reg) # Initialize input (example: x = 2, binary '10') qc.x(input_reg[0]) # Set first bit to |1⟩ # Apply quantum multiplication gates # ... (quantum arithmetic circuit implementation) # Measure sub j array qc.measure_all() print(qc.draw())

Hardware vs. Code Considerations

While the quantum algorithm for computing f(x) = x² is defined independently of hardware, practical implementation faces hardware-specific constraints. Different quantum technologies (superconducting, trapped ions, photonic) exhibit varying qubit counts, connectivity patterns, and error rates that directly effect computation reliability.

The fundamental quantum advantage emerges from superposition and entanglement. When the input register is prepared in superposition, the quantum computer can compute x² for multiple values simultaneously, providing exponential parallelism compared to classical sequential computation.

Note: The sub j array represents the classical binary output after quantum measurement. During computation, it exists in superposition, but measurement collapses it to a definite binary value according to quantum probabilities.