Quantum Circuit Simulation (Qiskit/Cirq)

Qiskit, developed by IBM, is a comprehensive quantum computing framework that allows you to simulate circuits and run them on real IBM quantum hardware via the cloud. Cirq, developed by Google, is primarily designed for creating, editing, and simulating quantum circuits, specifically targeting near-term quantum computer architectures. Both use Python and are excellent for learning quantum gates, but Qiskit currently has a larger ecosystem for actual hardware execution.

You can run a simulated version of Shor's algorithm on a local laptop using Qiskit, but you are severely limited by the exponential scaling of classical memory. Simulating just a few dozen qubits requires massive amounts of RAM, making it impossible to factor large numbers on a standard machine. For educational purposes, simulating a 4-qubit version of Shor's algorithm to factor the number 15 is very common and easy to do locally.

In quantum simulators, quantum gates are represented as unitary matrices operating on the state vector of the qubits. When you apply a gate to multiple qubits, the simulator calculates the tensor product of those matrices to determine the overall transformation. This linear algebra approach is why simulating complex, error-corrected quantum circuits on classical computers requires so much computational power.

Yes, Cirq includes modules and integrations that allow researchers to simulate and study quantum error-corrected circuits. You can build logical qubits using surface code frameworks and inject faults to test the system's resilience. This is a highly advanced topic necessary for understanding how future fault-tolerant quantum computers will operate in reality.