Quanta

On the Relation Between Quantum Computational Speedup and Retrocausality

Giuseppe Castagnoli

Published Jan 10, 2016• 8 citations

Abstract

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.

DOI: 10.12743/quanta.v5i1.38

Related papers

An exact relation between number of black box computations required to solve an oracle problem quantumly and quantum retrocausality

Giuseppe Castagnoli

We investigate the reason for the quantum speedup -- quantum algorithms requiring fewer computation steps than their classical counterparts. We extend their representation to the process of setting the problem. The initial measurement selects a setting at random, Bob (the problem setter) unitarily changes it into the desired one. This representation is to Bob and any external observer, it cannot be to Alice (the problem solver). It would tell her the function computed by the black box, which to her should be hidden inside it. We resort to relational quantum mechanics. To Alice, the projection of the quantum state due to the initial measurement is retarded at the end of her problem solving action. To her, 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. We show that we can ascribe to the final measurement the selection of any part -- say the R-th part -- of the random outcome of the initial measurement. This projects the input state to Alice on a state of lower entropy where she knows a corresponding part of the problem setting. 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 an oracle problem and a value of R, this retrocausality model provides the number of black box computations required to solve it. Conversely, given a known quantum algorithm, it yields the value of R that explains its speed up. R = 1/2 always yields the number of black box computations required by an existing quantum algorithm and the order of magnitude of the number required by optimal one.

arXiv:1508.00454Aug 3, 20150 citations

An exact relation between number of oracle queries required to solve an oracle problem quantally and quantum retrocausality

Giuseppe Castagnoli

Aug 3, 20150 citations

A retrocausal model of the quantum computational speedup

Giuseppe Castagnoli

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.

arXiv:1603.02246Mar 7, 20161 citation

Mechanism Of The Quantum Speed-up

Giuseppe Castagnoli, Daniel P. Sheehan

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.

arXiv:1101.4359AIP conference proceedingsJan 1, 20113 citations

Explaining the mechanism of the quantum speed-up in time-symmetric quantum mechanics

Giuseppe Castagnoli

Bob chooses a function from a set of functions and gives Alice the black box that computes it. Alice is to find a characteristic of the function through function evaluations. In the quantum case, the number of function evaluations can be smaller than the minimum classically possible. The fundamental reason for this violation of a classical limit is not known. We trace it back to a disambiguation of the principle that measuring an observable determines one of its eigenvalues. Representing Bob's choice of the label of the function as the unitary transformation of a random quantum measurement outcome shows that: (i) finding the characteristic of the function on the part of Alice is a by-product of reconstructing Bob's choice and (ii) because of the quantum correlation between choice and reconstruction, one cannot tell whether Bob's choice is determined by the action of Bob (initial measurement and successive unitary transformation) or that of Alice (further unitary transformation and final measurement). Postulating that the determination shares evenly between the two actions, in a uniform superposition of all the possible ways of sharing, implies that quantum algorithms are superpositions of histories in each of which Alice knows in advance one of the possible halves of Bob's choice. Performing, in each history, only the function evaluations required to classically reconstruct Bob's choice given the advanced knowledge of half of it yields the quantum speed-up. In all the cases examined, this goes along with interleaving function evaluations with non-computational unitary transformations that each time maximize the amount of information about Bob's choice acquired by Alice with function evaluation.

arXiv:1107.0934Jul 5, 20114 citations

Completing the physical representation of quantum algorithms explains their computational speedup

Giuseppe Castagnoli

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.

May 7, 20170 citations