Feynman's algorithm is an algorithm that is used to simulate the operations of a quantum computer on a classical computer. It is based on the Path integral formulation of quantum mechanics, which was formulated by Richard Feynman.
Overview.
An formula_1 qubit quantum computer takes in a quantum circuit formula_2 that contains formula_3 gates and an input state formula_4. It then outputs a string of bits formula_5 with probability formula_6.
In Schrödinger's algorithm, formula_7 is calculated straightforwardly via matrix multiplication. That is, formula_8. The quantum state of the system can be tracked throughout its evolution.
In Feynman's path algorithm, formula_7 is calculated by summing up the contributions of formula_10 histories. That is, formula_11. 
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 formula_12 time and formula_13 space while Feynman's takes formula_14 time and formula_15 space.
Example.
Consider the problem of creating a Bell state. What is the probability that the resulting measurement will be formula_16?
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 formula_17. In that case, formula_18 using Schrödinger's algorithm. So probability resulting measurement will be formula_16 is formula_20.
Using Feynman's algorithm, the Bell state circuit contains formula_21 histories: formula_22 . So formula_23 = |formula_24 + formula_25 + formula_26 + formula_27.
References.
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