Summary: More Quantum Algorithms

Here are the key takeaways:

• Quantum algorithms are being developed in many fields, outperforming and working alongside classical computing methods.

• Searching for strings of numbers or text is commonplace. We do it all the time in our personal lives, searching for specific words on web pages or values on spreadsheets. The Grover algorithm implements a search among basis states by manipulating a superposition state so that the desired state coefficient increases while the others decrease. Because the probability of measurement increases with the square of the coefficient, this process quickly increases the probability of finding the desired state. Geometrically, this process reflects a pointer state vector across other vectors in state space. This state space is defined with the desired state as one of the axes, with the other axis being the sum of all the other states’ vectors. First, the uniform superposition vector is reflected across the non-target axis and is then reflected across the initial superposition. This brings the state vector closer to the target state axis, which means that the target state coefficient has increased in the superposition. This process is then repeated until the target state coefficient is close to $1$, implying that the probability of getting the target state upon observation is close to $1$.