arXiv subjects

Giuseppe Castagnoli

Publications and source records attributed to Giuseppe Castagnoli.

At least 19 recordsLinked to original sources

arXiv2026

Quantum mechanics provides the physical basis of teleological evolutions

We show that the quantum computational speedup of quantum algorithms is due to their teleological character, their being evolutions toward a goal (the solution of the problem) with an attractor in the very goal they will produce in the future (the solution of the problem again). We also show that, under the quantum cosmological assumption and for the Fine-tuned Universe version of the Anthropic Principle, the physical basis of the teleological character of quantum algorithms applies as well to the evolutions of the living for which the teleological notion was originally conceived.

physics.gen-ph↗

arXiv2025

Quantum computational speedup and retrocausality

Involving only the measurements of commuting observables - the problem-setting and the corresponding solution - quantum algorithms should be subject to classical logic. This would allow flanking their customary quantum description with a classical logic description, with surprising consequences. In the classical logic description of the quantum algorithm, very simply, it is as if the problem-solver knew in advance, before beginning her problem-solving action, one of the possible halves of the information that specifies the solution of the problem she will produce and measure in the future and could use this knowledge to produce the solution with fewer computation steps. This is a causal loop whose retrocausal character turns out to be implicit in the very notion of quantum state superposition, both an essential ingredient of the quantum computational speedup and one of the pillars of quantum mechanics. Indeed, the key point of the work is that the classical logic description of a quantum state superposition must resort to a logical form of retrocausality that in turn must be physically implicit in the superposition itself. The existence of retrocausality in ordinary quantum physics implies e different way of viewing physical reality. It explains in a unified way all quantum speedups and quantum nonlocality. It highlights the teleological character of quantum algorithms, that is, their being evolutions toward a goal (the solution of the problem) with an attractor in the solution they will produce in the future (the solution again). Under the quantum cosmological assumption, it provides a plausible physical basis to the teleological character of natural evolutions.

quant-ph↗

arXiv2025

The physical basis of teleological evolutions

We show that the sheer existence of a quantum computational speedup logically implies the mutually exclusive or of well-defined causal loops. In each of them, it is as if the problem-solver knew in advance one of the possible halves of the information about the solution she will produce and measure in the future and could use this knowledge to produce the solution with fewer computation steps. Involving only the measurements of commuting observables, quantum algorithms are submitted to the same (classical) logic and should therefore host the quantum superposition of the causal loops in question. However, their ordinary quantum description does not and, being causal in character, cannot describe causal loops. It must therefore be incomplete and is in fact completed by time-symmetrizing it. This leaves the unitary part of the ordinary quantum description of the quantum algorithm mathematically unaltered but changes the behavior of causality along it. The single causal process of the ordinary quantum superposition is replaced by the quantum superposition of the causal loops in question. In their completed quantum description, quantum algorithms respond to the pre-scientific notion of teleological evolution, that is, an evolution toward a goal (the solution) with an attractor in the goal it will produce in the future (the solution again). Once applied to the evolutions of the living, this notion was dismissed with the advent of modern science for the alleged absence of a physical basis. We show that, under a quantum cosmological interpretation of the Anthropic Principle, the same quantum superposition of causal loops underlying the teleological character of quantum algorithms becomes the missing physical basis of the teleological character of natural evolutions.

physics.gen-ph↗

arXiv2023

The quantum mechanical notion of unobservable causal loop and the anthropic principle

It can be argued that the ordinary description of the reversible quantum process between two one-to-one correlated measurement outcomes is incomplete because, by not specifying the direction of causality, it allows causal structures that violate the time symmetry that is required of a reversible process. This also means that it can be completed simply by time-symmetrizing it, namely by requiring that the initial and final measurements evenly contribute to the selection of their correlated pair of outcomes. This leaves the description unaltered but shows that it is the quantum superposition of unobservable time-symmetrized instances whose causal structure is completely defined. Each instance consists of a causal loop: the final measurement that changes backwards in time the input state of the unitary transformation that leads to the state immediately before it. In former works, we have shown that such loops exactly explain the quantum computational speedup and quantum nonlocality. In this work we show that they lead to a completion of the anthropic principle that allows a universe evolution with quantum speedup.

quant-ph↗

arXiv2021

Unobservable causal loops as a way to explain both the quantum computational speedup and quantum nonlocality

We consider the reversible processes between two one-to-one correlated measurement outcomes which underly both problem-solving and quantum nonlocality. In the former case the two outcomes are the setting and the solution of the problem, in the latter those of measuring a pair of maximally entangled observables whose subsystems are space separate. We argue that the quantum description of these processes mathematically describes the correlation but leaves the causal structure that physically ensures it free, also of violating the time-symmetry required of the description of a reversible process. It would therefore be incomplete and could be completed by time-symmetrizing it. This is done by assuming that the two measurements evenly contribute to selecting the pair of correlated measurement outcomes. Time-symmetrization leaves the ordinary quantum description unaltered but shows that it is the quantum superposition of unobservable time-symmetrization instances whose causal structure is completely defined. Each instance is a causal loop: causation goes from the initial to the final measurement outcome and then back from the final to the initial outcome. In the speedup, all is as if the problem solver knew in advance half of the information about the solution she will produce in the future and could use this knowledge to produce the solution with fewer computation steps. In nonlocality, the measurement on either subsystem retrocausally and locally changes the state of both subsystems when the two were not yet spatially separate. This locally causes the correlation between the two future measurement outcomes.

quant-ph↗

arXiv2019

A relational time-symmetric framework for analyzing the quantum computational speedup

The usual representation of quantum algorithms is limited to the process of solving the problem. We extend it to the process of setting the problem. Bob, the problem setter, selects a problem-setting by the initial measurement. Alice, the problem solver, unitarily computes the corresponding solution and reads it by the final measurement. This simple extension creates a new perspective from which to see the quantum algorithm. First, it highlights the relevance of time-symmetric quantum mechanics to quantum computation: the problem-setting and problem solution, in their quantum version, constitute pre- and post-selection, hence the process as a whole is bound to be affected by both boundary conditions. Second, it forces us to enter into relational quantum mechanics. There must be a representation of the quantum algorithm with respect to Bob, and another one with respect to Alice, from whom the outcome of the initial measurement, specifying the setting and thus the solution of the problem, must be concealed. Time-symmetrizing the quantum algorithm to take into account both boundary conditions leaves the representation to Bob unaltered. It shows that the representation to Alice is a sum over histories in each of which she remains shielded from the information coming to her from the initial measurement, not from that coming to her backwards in time from the final measurement. In retrospect, all is as if she knew in advance, before performing her problem-solving action, half of the information that specifies the solution of the problem she will read in the future and could use this information to reach the solution with fewer computation steps (oracle queries). This elucidates the quantum computational speedup in all the quantum algorithms examined.

quant-ph↗

arXiv2018

Completing the physical representation of quantum algorithms provides a quantitative explanation of their computational speedup

The usual representation of quantum algorithms, limited to the process of solving the problem, is physically incomplete. We complete it in three steps: (i) extending the representation to the process of setting the problem, (ii) relativizing the extended representation to the problem solver to whom the problem setting must be concealed, and (iii) symmetrizing the relativized representation for time reversal to represent the reversibility of the underlying physical process. The third steps projects the input state of the relativized representation, where the problem solver is completely ignorant of the setting and thus the solution of the problem, on one where she knows half solution (half of the information specifying it when the solution is an unstructured bit string). Completing the physical representation shows that the number of computation steps (oracle queries) required to solve any oracle problem in an optimal quantum way should be that of a classical algorithm endowed with the advanced knowledge of half solution. This fits the major quantum algorithms known today and would solve the quantum query complexity problem.

quant-ph↗

arXiv2017

Back to the seminal Deutsch algorithm

A bare description of the seminal quantum algorithm devised by Deutsch could mean more than an introduction to quantum computing. It could contribute to opening the field to interdisciplinary research.

quant-ph↗

arXiv2017

Completing the physical representation of quantum algorithms provides a retrocausal explanation of their speedup

In previous works, we showed that an optimal quantum algorithm can always be seen as a sum over classical histories in each of which the problem solver knows in advance one of the possible halves of the solution she will read in the future and performs the computation steps (oracle queries) still needed to reach it. Given an oracle problem, this retrocausal explanation of the speedup yields the order of magnitude of the number of oracle queries needed to solve it in an optimal quantum way. Presently, we provide a fundamental justification for the explanation in question and show that it comes out by just completing the physical representation of quantum algorithms. Since the use of retrocausality in quantum mechanics is controversial, showing that it answers the well accepted requirement of the completeness of the physical description should be an important pass.

quant-ph↗

arXiv2015

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

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.

quant-ph↗

arXiv2015

Highlighting the mechanism of the quantum speedup by time-symmetric and relational quantum mechanics

Bob hides a ball in one of four drawers. Alice is to locate it. Classically she has to open up to three drawers, quantally just one. The fundamental reason for this quantum speedup is not known. The usual representation of the quantum algorithm is limited to the process of solving the problem. We extend it to the process of setting the problem. The number of the drawer with the ball becomes a unitary transformation of the random outcome of the preparation measurement. This extended, time-symmetric, representation brings in relational quantum mechanics. It is with respect to Bob and any external observer and cannot be with respect to Alice. It would tell her the number of the drawer with the ball before she opens any drawer. To Alice, the projection of the quantum state due to the preparation measurement should be retarded at the end of her search; in the input state of the search, the drawer number is determined to Bob and undetermined to Alice. We show that, mathematically, one can ascribe any part of the selection of the random outcome of the preparation measurement to the final Alice's measurement. Ascribing half of it explains the speedup of the present algorithm. This projects the input state to Alice on a state of lower entropy where she knows half of the number of the drawer with the ball in advance. The quantum algorithm turns out to be a sum over histories in each of which Alice knows in advance that the ball is in a pair of drawers and locates it by opening one of the two. In the sample of quantum algorithms examined, the part of the random outcome of the initial measurement selected by the final measurement is one half or slightly above it. Conversely, given an oracle problem, the assumption it is one half always corresponds to an existing quantum algorithm and gives the order of magnitude of the number of oracle queries required by the optimal one.

quant-ph↗

arXiv2012

Origin of the quantum speed-up

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.

quant-ph↗

arXiv2011

Mechanism of the quantum speed-up

We explain the mechanism of the quantum speed-up - quantum algorithms requiring fewer computation steps than their classical equivalent - for 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. Solving this problem explains the speed-up. 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.

quant-ph↗

arXiv2010

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

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.

quant-ph↗

arXiv2010

An explanation of the quantum speed up

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.

quant-ph↗

arXiv2009

Quantum computation and the physical computation level of biological information processing

On the basis of introspective analysis, we establish a crucial requirement for the physical computation basis of consciousness: it should allow processing a significant amount of information together at the same time. Classical computation does not satisfy the requirement. At the fundamental physical level, it is a network of two body interactions, each the input-output transformation of a universal Boolean gate. Thus, it cannot process together at the same time more than the three bit input of this gate - many such gates in parallel do not count since the information is not processed together. Quantum computation satisfies the requirement. At the light of our recent explanation of the speed up, quantum measurement of the solution of the problem is analogous to a many body interaction between the parts of a perfect classical machine, whose mechanical constraints represent the problem to be solved. The many body interaction satisfies all the constraints together at the same time, producing the solution in one shot. This shades light on the physical computation level of the theories that place consciousness in quantum measurement and explains how informations coming from disparate sensorial channels come together in the unity of subjective experience. The fact that the fundamental mechanism of consciousness is the same of the quantum speed up, gives quantum consciousness a potentially enormous evolutionary advantage.

quant-ph↗

arXiv2009

Discussing the explanation of the quantum speed up

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.

quant-ph↗

arXiv2009

Quantum algorithms know in advance 50% of the solution they will find in the future

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.

quant-ph↗