An qubit quantum computer takes in a quantum circuit that contains gates and an input state . It then outputs a string of bits with probability .
In Schrödinger's algorithm, is calculated straightforwardly via matrix multiplication. That is, . The quantum state of the system can be tracked throughout its evolution.[2]
In Feynman's path algorithm, is calculated by summing up the contributions of histories. That is, . [3]
Schrödinger's takes less time to run compared to Feynman's while Feynman's takes more time and less space.More precisely, Schrödinger's takes time and space while Feynman's takes time and space.[4]
Example
Consider the problem of creating a Bell state. What is the probability that the resulting measurement will be ?
Since the quantum circuit that generates a Bell state is the H (Hadamard gate) gate followed by the CNOT gate, the unitary for this circuit is . In that case, using Schrödinger's algorithm. So probability resulting measurement will be is .
Using Feynman's algorithm, the Bell state circuit contains histories: . So = | + + + .