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arXiv:1005.1493 · Physical Review A

Quantum correlation between the selection of the problem and that of the solution sheds light on the mechanism of the quantum speed-up

Giuseppe Castagnoli

Published Nov 29, 2010• 12 citations
arXivHTMLPDF

Abstract

In classical problem solving, there is, of course, correlation between the selection of the problem on the part of Bob (the problem setter) and that of the solution on the part of Alice (the problem solver). In quantum problem solving, this correlation becomes quantum. This means that Alice contributes to selecting $50%$ of the information that specifies the problem. As the solution is a function of the problem, this gives to Alice advanced knowledge of $50%$ of the information that specifies the solution. Both the quadratic and exponential speed-ups are explained by the fact that quantum algorithms start from this advanced knowledge.

DOI
10.1103/physreva.82.052334

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In classical problem solving, there is, of course, correlation between the selection of the problem on the part of Bob (the problem setter) and that of the solution on the part of Alice (the problem solver). In quantum problem solving, this correlation becomes quantum. This means that Alice contributes to selecting 5050%50 of the information that specifies the problem. As the solution is a function of the problem, this gives to Alice advanced knowledge of 5050%50 of the information that specifies the solution. Both the quadratic and exponential speed-ups are explained by the fact that quantum algorithms start from this advanced knowledge.
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.
We investigate the reason for the quantum speedup (quantum algorithms require fewer computation steps than their classical counterparts). We extend the representation of the quantum algorithm to the process of setting the problem, namely choosing the function computed by the black box. The initial measurement selects a setting at random, Bob (the problem setter) unitarily changes it into the desired one. With reference to the observer dependent quantum states of relational quantum mechanics, this representation is with respect to Bob and any external observer, it cannot be with respect to Alice (the problem solver). It would tell her the function computed by the black box, which to her should be hidden. To Alice, the projection of the quantum state due to the initial measurement is retarded at the end of her problem solving action, so that the algorithm input state remains one of complete ignorance of the setting. By black box computations, she unitarily sends it into the output state that, for each possible setting, encodes the corresponding solution, acquired by the final measurement. Mathematically, we can ascribe to the final measurement the selection of any fraction R of the random outcome of the initial measurement. This projects the input state to Alice on one of lower entropy where she knows the corresponding fraction of the problem setting. Given the appropriate value of R, the quantum algorithm is a sum over classical histories in each of which Alice, knowing in advance one of the R-th parts of the setting, performs the black box computations still required to identify the solution. Given a quantum algorithm, this retrocausality model provides the value of R that explains its speed up; in the major quantum algorithms, R is 1/2 or slightly above it. Conversely, given the problem, R=1/2 always yields the order of magnitude of the number of black box computations required to solve it in an optimal quantum way. Quanta 2016; 5: 34–52.
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.
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.
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.
We analyse the quantum speed-up: the fact that some quantum algorithms are faster than their classical counterparts. The classical formulation of problem solving must be modified. In general a problem statement defines the solution only implicitly, giving a property that a solution must possess, and does not represent a process that constructs the solution explicitly. A solution algorithm must then specify explicitly a finite step-by-step procedure that constructs the solution through a dynamical development. In the quantum speed-up, however, an implicit definition directly determines the solution through an extra-dynamical irreversible non-algorithmic quantum measurement. While in an algorithmic solution, the initial conditions determine the solution in a one-way manner, in a solution by measurement, the initial conditions and a final condition influence the solution jointly. The quantum computation is as implicit as the problem statement. Implicit determination with joint influence is richer than explicit one-way dynamical determination, as it yields the speed-up. Quantum problem solving with speed-up is essentially extra-dynamical and irreversible. Other forms of implicit quantum problem solving may exploit extra-dynamical projections for speed-up.