In former work, we showed that a quantum algorithm requires the number of operations (oracle's queries) of a classical algorithm that knows in advance 50% of the information that specifies the solution of the problem. We gave a preliminary theoretical justification of this "50% rule" and checked that the rule holds for a variety of quantum algorithms. Now, we make explicit the information about the solution available to the algorithm throughout the computation. The final projection on the solution becomes acquisition of the knowledge of the solution on the part of the algorithm. Backdating to before running the algorithm a time-symmetric part of this projection, feeds back to the input of the computation 50% of the information acquired by reading the solution.
In former work, we showed that a quantum algorithm requires the number of operations (oracle's queries) of a classical algorithm that knows in advance 50% of the information that specifies the solution of the problem. We gave a preliminary theoretical justification of this "50% rule" and checked that the rule holds for a variety of quantum algorithms. Now, we make explicit the information about the solution available to the algorithm throughout the computation. The final projection on the solution becomes acquisition of the knowledge of the solution on the part of the algorithm. Backdating to before running the algorithm a time-symmetric part of this projection, feeds back to the input of the computation 50% of the information acquired by reading the solution.
In former work, we showed that a quantum algorithm is the sum over the histories of a classical algorithm that knows in advance 50% of the information about the solution of the problem - each history is a possible way of getting the advanced information and a possible result of computing the missing information. We gave a theoretical justification of this 50% advanced information rule and checked that it holds for a large variety of quantum algorithms. Now we discuss the theoretical justification in further detail and counter a possible objection. We show that the rule is the generalization of a simple, well known, explanation of quantum nonlocality - where logical correlation between measurement outcomes is physically backed by a causal/deterministic/local process with causality allowed to go backward in time with backdated state vector reduction. The possible objection is that quantum algorithms often produce the solution of the problem in an apparently deterministic way (when their unitary part produces an eigenstate of the observable to be measured and measurement produces the corresponding eigenvalue - the solution - with probability 1), while the present explanation of the speed up relies on the nondeterministic character of quantum measurement. We show that this objection would mistake the nondeterministic production of a definite outcome for a deterministic production.
Quantum algorithms require less operations than classical algorithms. The exact reason of this has not been pinpointed until now. Our explanation is that quantum algorithms know in advance 50 % of the solution of the problem they will find in the future. In fact they can be represented as the sum of all the possible histories of a respective ”advanced information classical algorithm”. This algorithm, given the advanced information (50 % of the bits encoding the problem solution), performs the operations (oracle’s queries) still required to identify the solution. Each history corresponds to a possible way of getting the advanced information and a possible result of computing the missing information. This explanation of the quantum speed up has an immediate practical consequence: the speed up comes from comparing two classical algorithms, with and without advanced information, with no physics involved. This simplification could open the way to a systematic exploration of the possibilities of speed up. 1
The oracle chooses a function out of a known set of functions and gives to the player a black box that, given an argument, evaluates the function. The player should find out a certain character of the function (e. g. its period) through function evaluation. This is the typical problem addressed by the quantum algorithms. In former theoretical work, we showed that a quantum algorithm requires the number of function evaluations of a classical algorithm that knows in advance 50 % of the information that specifies the solution of the problem. This requires representing physically, besides the solution algorithm, the oracle’s choice. Here we check that this 50 % rule holds for the main quantum algorithms. In the structured problems, a classical algorithm with the advanced information, to identify the missing information should perform one function evaluation. The speed up is exponential since a classical algorithm without advanced information should perform an exponential
We extend the representation of quantum algorithms to the preparation of the problem setting. An initial measurement serves to set the problem, the final one as usual to read the solution. This extension highlights the possibility of ascribing the selection of a variable fraction R of the information that specifies the random outcome of the initial measurement to the final measurement. The input state of the quantum algorithm to the problem solver (Alice), of maximal ignorance of the problem setting that to her should be hidden, would consequently be projected on a state of lower entropy where she knows in advance an R-th part of the setting (i. e. of the information that specifies it). Given the appropriate value of R, the quantum algorithm turns out to be a sum over classical histories in each of which Alice knows in advance one of the possible R-th parts of the problem setting and performs the Q(R) computations (oracle queries) still needed to find the solution. Grover's search problem can be solved with any value of R from 0 to about 1/2, when the solution algorithm becomes the optimal Grover algorithm. In all the optimal quantum algorithms examined, R is close to 1/2 and Q(1/2) yields the order of magnitude of the number of queries. If this held in general, as it seems to be plausible, given any oracle problem we would know the order of magnitude of the number of queries needed for its optimal quantum solution.
We explain the mechanism of the quantum speed-up -quantum algorithms requiring fewer computation steps than their classical equivalentfor a family of algorithms.Bob chooses a function and gives to Alice the black box that computes it.Alice, without knowing Bob's choice, should find a character of the function (e. g. its period) by computing its value for different arguments.There is naturally correlation between Bob's choice and the solution found by Alice.We show that, in quantum algorithms, this correlation becomes quantum.This highlights an overlooked measurement problem: sharing between two measurements the determination of correlated (thus redundant) measurement outcomes.All is like Alice, by reading the solution at the end of the algorithm, contributed to the initial choice of Bob, for half of it in quantum superposition for all the possible ways of taking this half.This contribution, back evolved to before running the algorithm, where Bob's choice is located, becomes Alice knowing in advance half of this choice.The quantum algorithm is the quantum superposition of all the possible ways of taking half of Bob's choice and, given the advanced knowledge of it, classically computing the missing half.This yields a speed-up with respect to the classical case where, initially, Bob's choice is completely unknown to Alice.
The usual representation of quantum algorithms, limited to the problem solving process, is incomplete as it lacks the initial measurement. Completing it involves three steps: (i) representing the problem setting process, (ii) relativizing this representation to Alice, the problem solver, to whom the problem setting must be concealed, (iii) representing the reversibility of the computation process by symmetrizing it for time reversal. The latter step shows that all is as if Alice, before beginning her problem solving action, knew half of the information that specifies the solution of the problem she will read in the future. An optimal quantum algorithm is a sum over classical histories in each of which Alice knows in advance one of the possible halves of the solution and performs the computation steps (oracle queries) necessary to identify the missing half. This explains the quantum computational speedup and allows to predict the speedup offered by the optimal quantum solution of any oracle problem.