QC derives its theoretical foundations from quantum mechanics, which is based on fundamental properties of atomic and sub-atomic particles. While classical computers represent information using binary bits that can assume values of either 0 or 1, QC represents information using qubits that can assume an infinite number of values resulting from combinations of 0 and 1.
The immense power of QC emerges from three fundamental properties of qubits, viz. superposition, entanglement, and interference.
Superposition, Entanglement, and Interference
Superposition: While in principle, qubits can assume an infinite number of values between 0 and 1, when measured, they either give a value of 0 or 1. This phenomenon is represented using the “basis state” of a qubit, which can either be |0〉 and |1〉. Superposition is the phenomenon of qubits existing in a linear combination of the states |0〉 and |1〉, unlike classical bits which can only be either 0 or 1 at a time. More specifically, the state of a qubit can be written as a|0〉+b|1〉, where a and b are complex numbers. Upon measurement, the qubit "collapses" to the state |0〉 with a probability of |a|^2, or to the state |1〉 with a probability of |b|^2. The process of collapse to |0〉 or |1〉 is truly stochastic—a property which finds use in the generation of truly random number through QC.
Entanglement: Without getting into the how and why of entanglement, for the sake of our understanding, when two qubits are entangled, if one obtains a state of one of the qubits, the state of the other qubit can be predicted with certainty, without having to measure it at all, regardless of the distance between them. Similarly, any change made to the state of one of the entangled qubits has a ripple effect on the state of the other. This strong non-classical correlation extends to multiple qubits. Consequently entanglement, has huge implications in quantum technologies, including computing, communication, and sensing. In QC, every quantum algorithm which shows "exponential speed up" compared to classical algorithms, must exploit entanglement. A follow-up point of it is that the phenomenon of entanglement cannot be simulated classically at a large scale.
Interference: This refers to the wave functions either reinforcing or diminishing each other. Leveraging the capability of interference, the probability of obtaining certain states is amplified at the cost of others, such that the amplified states occur with higher probabilities and correspond to the sought-after solutions to the problem being solved.
Due to physical limitations on the number of transistors that can be packed on a microchip, there are limits to scaling the performance of chips that power classical computers. QC promises a way to break through these limitations and provide the power needed to solve complex problems that current supercomputers struggle to handle. These problems, generally referred to as intractable or NP complete (nondeterministic polynomial), occur in a wide a variety of use cases such as vehicle routing, portfolio optimization, and molecular simulation