#  The physical basis of teleological evolutions

G. Castagnoli
$Formerly:ElsagBaileyICTDivisionandQuantumLaboratory$

###### Abstract

Optimal quantum algorithms have been shown to be quantum superpositions of
causal loops in each of which, logically, 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 used this knowledge to produce
the solution with a correspondingly reduced number of computation steps. This
also answers the pre-scientific notion of teleological evolution, that is, an
evolution toward a goal with an attractor in the goal it will produce in the
future. This notion, formerly applied to the evolutions of the living, was
dismissed with the advent of modern science for the alleged absence of a
physical basis. In this work, we show that the quantum causal loops at the
physical basis of the teleological character of quantum algorithms could also
be at the physical basis of that of the evolutions of life. A quantum
cosmological interpretation of the Anthropic Principle yields a formal
identity between a random search for life by a quantum evolution of the
universe under the fundamental physical laws and constants and a random search
for the solution by quantum search algorithms. According to it, the evolution
of the universe toward life should be endowed with quantum speedup. We show
that the evolution of a living system, in principle obtainable from that of
the universe by tracing out all the rest, even if decohered, should have the
same quantum speedup of its image in the universe’s evolution. It would
therefore be teleological in character.

##  1 Extended summary

This work is concerned with quantum causal loops, of a special time-symmetric
kind that should be immune from the paradoxes that plague the usual kind. They
would be commonplace in non-relativistic quantum mechanics, indeed always
present in quantum correlations. Of course, this is an unconventional view. It
clashes against current opinion and practice. Generally speaking, there is a
deeply rooted prevention toward the notion of rerocausality and the ordinary
quantum description of quantum algorithm, where causality always goes in the
same time direction, cannot describe causal loops. The “extended summary”
format is to explain in sufficient detail a new way of viewing the quantum
description of quantum processes.

Time-symmetric quantum causal loops provide a unified quantitative explanation
of the quantum computational speedups, a local explanation of quantum non-
locality, and the physical basis to the notion of teleological evolution, a
well known pre-scientific notion of natural philosophy that was dismissed with
the advent of modern science for the alleged absence of a physical basis.

To start with, let us show that the causal loops we are dealing with must be
present in quantum algorithms, where they explain their quantum computational
speedups. As this work aims to be interdisciplinary, let us explain the latter
expression. It denotes the fact that quantum algorithms solve their problems
with fewer logical operations (computation steps) than their classical
counterparts, sometimes demonstrably fewer than what is possible in the
classical case.

Let us provide an example of quantum computational speedup right away.

Consider the following problem. Bob, the problem-setter, hides a ball in a
chest of four drawers; the number of the drawer in which to hide the ball is
selected at random among the four possible drawer numbers. Alice, the problem-
solver, is to locate the ball by opening drawers. In the classical case, she
has to open from one to three drawers – if the ball is only in the third
drawer opened, she has solved the problem; if not even in that, it must be in
the only drawer not yet opened and she has solved the problem of locating the
ball as well. In the quantum case, with Grover quantum search algorithm
$\left[1\right]$, she always locates the ball by opening just one drawer, in a
quantum superposition of the four possible drawer numbers. Always solving the
problem by opening just one drawer would obviously be impossible in the
classical case.

In view of what will follow, let us observe that Grover quantum search
algorithm is optimal, in Long’s version $\left[2\right]$ always provides the
solution with certainty, and delivers a quadratic quantum speedup. This means
that it always solves the problem of locating the ball hidden in $N$ drawers
by opening $\operatorname*{O}\left(\sqrt{N}\right)$ drawers against the
$\operatorname*{O}\left(N\right)$ of the best classical algorithm. It is also
a most general quantum problem-solving mechanism, as it can solve with a
quadratic quantum speedup any NP problem.

We go to the heart of the matter and provide a surprisingly simple
demonstration that quantum speed of (the otimal) Grover algorithm logically
implies the mutually exclusive $or$ of well-defined causal loops. We refer
again to the four-drawer case (the extension to any number of drawers is
straightforward). The logic is as follows.

a) The fact that Alice always solves the problem of locating the ball hidden
in one of four drawers by opening just one drawer logically implies that it is
as if she always knew in advance, before beginning her problem-solving action,
that the ball is in a mutually exclusive way in one of the three pairs of
drawers formed by pairing the drawer with the ball with another drawer.
Indeed, in any of these three cases, she opens either drawer; if the ball is
in it, she has solved the problem; if not, it must be in the other drawer and
she has solved the problem of locating the ball as well.

b) The question is then: from where does Alice always get this advanced
knowledge about the solution? Since, to her, the only possible source of it is
the solution she will produce and measure in the future, the only possible
answer is that it comes to her from her future measurement of the solution.
She then avails herself of this knowledge to produce the solution by opening
just one drawer.

c) Summarizing, the logic is as if, in a mutually exclusive way, the problem-
solver knew in advance one of the possible halves of the information that
specifies the solution of the problem she will produce and measure in the
future and used this knowledge to produce the solution with fewer logical
operations (drawer openings in the present case). This is obviously a causal
loop. There must be (in a mutually exclusive way) one causal loop for each of
the possible ways of taking half of the information that specifies the
solution of the problem.

More in general, there are $N$ drawers and, in each causal loop, Alice knows
in advance one of the possible halves of the information about the number of
the drawer with the ball. This means that she know that the ball in one of the
possible $\sqrt{N}$ tuples each of $\sqrt{N}$ drawers, she then solves the
problem by opening $\operatorname*{O}\left(\sqrt{N}\right)$drawers.

Let us shortly discuss the above argument. There is a contrast between its
simplicity and its consequences, which, as we will soon see, are relevant.
Moreover, so simple, it has never been noticed in the nearly four decades
since the discovery of the quantum speedup222 by Deutsch $\left[3\right]$, in
1985, after Feynman’s $\left[4\right]$ observation that the classical
simulation of a quantum process can require an exponentially higher amount of
resources than those employed in the quantum process itself. . All this might
make it dubious; after all, it is in words and words can be deceiving. So, let
us defend it. First, we said we should refer to the logic underlying the
narrative, which is clear. If you doubted it, try to specify where and why
there is the error. To us, the present argument is inescapable.

If the argument has never been noticed before, this might be ascribed to the
already mentioned prevention toward the notion of retocausality333 There are
exceptions of course, for example the retrocausal approach to the quantum
speedup $\left[5,9\right]$, of which the present work is the further
development, and $\left[10\right]$, on the possibility that a future choice
affects a past measurement outcome. . After all, physical processes have been
considered to be causal since Galileo and Newton, whereas other kinds of
processes were ascribed to non-scientific creeds. That with the advent of
modern science that was a classical science. We hope we will provide
sufficient evidence about the fact that, after the advent of quantum
mechanics, in particular the discovery of the quantum computational speedup,
should be hight time to change mind.

We have seen that the simple existence of a (optimal) quantum speedup
logically involves the mutually exclusive $or$ of well defined causal loops.
These causal loops, for the time being, are in an abstract world.

The question now is where, physically, those causal loops are. Of course, they
should be implicit in the quantum algorithms. However, since the latter only
involves the measurements of commuting observables, it is submitted to the
same classical logic. This means that it should host, as they are, the causal
loops in question. More precisely, since the logical mutual exclusivity of
possibilities in the quantum world becomes their quantum superposition, it
must host a quantum superposition of these causal loops.

This naturally clashes with the fact that the ordinary quantum description of
quantum algorithms is a causal description that cannot describe any causal
loop. Coherently with our line of thinking, we must conclude that, in the case
of quantum algorithms, the quantum description is deficient. We will see that
this deficiency is due to the fact that it is incomplete. In fact, it vanishes
when the ordinary quantum description is is completed as follows:

i) A first incompleteness is trivial. The usual quantum description of quantum
algorithms is limited to the process of solving the problem where the quantum
speedup resides. It consists of a unitary evolution that computes the solution
of the problem, naturally without changing the problem-setting, and a final
measurement that reads the solution. The initial measurement is missing. Of
course, the canonical description of a quantum process must consist of initial
measurement, unitary evolution, and final measurement. The remedy is also
trivial. We should extend the usual quantum description of the process of
solving the problem to the upstream process of setting the problem. An initial
measurement selects the problem-setting (the number of the drawer with the
ball in Grover algorithm) out of a quantum superposition (or, indifferently,
mixture) of all the possible problem-settings. This is followed by the unitary
evolution that computes the corresponding solution (without changing the
problem-setting) and the final measurement of the latter.

Let us call this “less incomplete” quantum description the ordinary quantum
description of the quantum algorithm. It is ordinary since it is still the
customary causal description that cannot describe causal loops. In this
description, the initial measurement determines both the problem-setting and
the corresponding solution (both the number of the drawer with the ball in
Grover algorithm), the final measurement just reads the solution without
determining anything. Causality goes from the initial to the final measurement
outcome. For, simplicity, here and in the following, we assume that the
solution is an invertible function of the problem-setting444 If the solution
is not an invertible function of the problem-setting, in order to have one-to-
one correlation between the two measurement outcomes it is enough that the
problem-solver, at the end of her problem-solving action, simultaneously
measures the solution and the problem-setting. . In any case, we are dealing
with a unitary process between two one-to-one correlated measurement outcomes,
the basic form of quantum correlation. We will see that this is a key feature
of quantum algorithms, of course not explicit in their usual representation.

ii) The fact that the ordinary, causal, quantum description in question cannot
describe causal loops is the incompleteness of a fundamental kind. We will see
that the only possible remedy is to time-symmetrize the quantum description of
the reversible process between the initial and final one-to-one correlated
measurement outcomes; note that this process is reversible despite comprising
the final measurement given that the latter does not change the quantum state.
We should require that the selection of the information that specifies the
sorted out pair of correlated measurement outcomes among all the possible
pairs evenly shares between the initial and final measurements, of course in a
quantum superposition of all the possible ways. Then, in each time-
symmetrization instance, the initial measurement selects one of the possible
halves of that information. The initial measurement outcome unitarily
propagates forward in time until becoming the state immediately before the
final measurement. The final measurement selects the remaining half
information. Time-symmetrically, the final measurement outcome propagates
backward in time, by the inverse of the time-forward unitary transformation,
until replacing the previous outcome of the initial measurement. The latter
propagation, which inherits both selections, is an instance of the time-
symmetrized description of the quantum process.

One can see that, mathematically, i.e. disregarding the time-direction of
causality, it is the unitary process between the two correlated measurement
outcomes back again. In fact, its final measurement outcome, which inherits
both selections, is the same of the ordinary quantum description of the
quantum algorithm. It is naturally preceded by the outcome of the initial
measurement and the unitary transformation between the two measurement
outcomes of the ordinary quantum description of the quantum algorithm.

Taking instead the time-direction of causality into account, the above forward
and backward in time propagations, together, constitute a causal loop that
changes its own past, in fact the outcome of its initial measurement.
Eventually, we should take a quantum superposition of all the time-
symmetrization instances. This, of course, mathematically, still yields the
unitary process between the two correlated measurement outcomes but also shows
that the latter is a quantum superposition of causal loops. Summing up, the
time-symmetrization in question leaves the mathematical description of the
unitary evolution between the two one-to-one correlated measurement outcomes
unaltered but changes the structure of causality along it.

iii) The causal loops in question are not yet those whose mutually exclusive
$or$ is implied by the sheer fact that the optimal Grover algorithm delivers a
quantum speedup. This is because they come from time-symmetrizing the ordinary
quantum description of the quantum algorithm, which is with respect to the
customary external observer. In the formalizations of the next section, we
will see that this quantum description would tell the problem-solver the
solution of the problem (e. g. the number of the drawer with the ball)
immediately after the initial measurement, therefore before she begins her
problem solving action. The causal loops whose mutually exclusive $or$ is
logically implied by the sheer existence of the quantum speedup, in which the
problem-solver knows in advance one of the possible halves of the information
about the solution, are instead with respect to the problem-solver. Therefore,
we should relativize with respect to the problem-solver the causal loops
obtained so far. We should just postpone after her problem-solving action the
projection of the quantum state associated with the initial measurement. This
yields exactly the causal loops whose mutually exclusive $or$ is logically
implied by the sheer existence of the quantum speedup.

Most importantly, the fact that this time symmetrization is imposed by logic
and quantum mechanics, as long as the latter are consistent, should make the
resulting causal loops free of the paradoxes that plague the usual kind. To
distinguish them from the usual kind, we call them time-symmetric. The reason
for the name is that they are generated by a time-symmetrization procedure.

These time-symmetric causal loops complete the ordinary quantum description of
the (optimal) quantum algorithm with respect to the problem-solver, providing
a unified quantitative description of the quantum computational speedups. We
have already seen that, in each of them, logically, it is as if the problem-
solver knew in advance one of the possible halves of the information that
specifies the solution she will produce and measure in the future and made use
of this knowledge to produce the solution with correspondingly fewer
computation steps. The number of computation steps needed in the optimal
quantum case is that needed in the optimal logical case benefiting of the
advanced knowledge in question. Note that there is nothing like that in the
mainstream – one-way causality – theory of quantum computation. Here, given a
(oracle) problem, one does not even know whether it can be solved with quantum
speedup. If one finds a quantum algorithm that solves it with quantum speedup,
he or she does not know whether it is optimal in character.

As we will see in Section 2.5, this time-symmetrization is mandatory also in
the case that the reversible process in question is not a quantum algorithm.
This because of the intrinsic symmetry of the processes between two one-to-one
correlated measurement outcomes. Therefore, these processes turn out to always
be quantum superpositions of time-symmetric causal loops.

This allows to export the causal loops we are dealing with outside quantum
computation, in particular to quantum nonlocality. Here, there is one-to-one
correlation between the outcomes of measuring in the same basis two maximally
entangled observables in a state of spatial separation of the respective
subsystems. The unitary process between the two correlated measurement
outcomes turns out to be a quantum superposition of causal loops. The latter
locally go from one to the other measurement outcome via the time the two
subsystems were not yet spatially separate, when, always locally, change the
state of both observables in a way that assures the appropriate correlation
between the future measurement outcomes $\left[9\right]$. Therefore, they play
the role of the hidden variables envisaged by Einstein and the others in their
famous EPR paper $\left[11\right]$, that of locally explaining quantum
nonlocality. Obviously, this also means that the incompleteness we are
presently ascribing to the quantum description of quantum algorithms is the
same that Einstein and the others famously ascribed to the quantum description
of quantum nonlocality.

By the way, as is well known, the EPR paper – the first to put the finger on
the puzzling character of quantum entanglement (the spooky action at a
distance) – inspired discoveries of great theoretical and practical interest,
like Bell’s inequalities $\left[12\right]$ and Ekert’s entanglement based
quantum cryptography $\left[13\right]$. However, the fact that the paper also
asserted the incompleteness of the quantum description apparently was removed
by the scientific community. As is well known, in the Einstein-Bohr dispute
originated by that paper, Bohr, who argued for the completeness of the quantum
description, was decreed the winner. Evidently, that was once and for all. Of
course, according to the present work, the winner should have been Einstein
instead.

This concludes the first part of the present work, a review and further
development of the retrocausal approach to the quantum computational speedup
$\left[5-9\right]$.

The second part is about the special notion of quantum evolution implicit in
the causal loops description of quantum algorithms. We have seen that in it,
logically, it is as if the problem-solver knew in advance, before beginning
her problem solving action, one of the possible halves of the information
about the solution she will produce and measure in the future and used this
knowledge to produce the solution with correspondingly fewer computation steps
(obviously a causal loop). In other words, the process of producing the
solution of the problem has an attractor in the very solution it will produce
in the future. This is the definition of teleological evolution, in fact, that
of an evolution toward a goal with an attractor in the goal it will produce in
the future. It is a pre-scientific notion of natural philosophy often applied
to the evolution of the living before the scientific revolution. It was
dismissed with the advent of the latter for the alleged absence of a physical
basis. That was of course under the notion of causal (one-way causality)
evolution dominant in science also today and that the present work is
challenging.

It is then natural to ask ourselves whether the causal loops we are dealing
with, besides being the physical basis of the teleological character of
quantum algorithms, are the missing physical basis of the teleological
character of life evolutions.

Obviously, this would require that evolutions that can be the size of the
Earth and last billions of years host quantum features, in particular a
quantum speedup and the consequent teleological character. In recent years,
evidence has emerged that quantum mechanics should play a role in the
fundamental processes of life, but how it can be so is still considered a
mystery. The extremal difficulty of the present case suggests a possible
solution. Instead of trying to scale up an autonomous quantum evolution from
the microscopic world typical of quantum processes, we should scale down a
quantum behavior from the most macroscopic and enduring process there is,
namely the evolution of the universe.

To this end, we should adopt quantum cosmology, namely the assumption that the
evolution of the universe, a perfectly isolated system since by definition
there is nothing outside it, is a unitary quantum evolution
$\left[14,15\right]$. This allows us, in principle, to scale down from the
quantum description of the universe to the description of any other system: it
suffices to trace out of the quantum description of the universe all the rest.

At this point, we should call into play the Anthropic Principle
$\left[16-20\right]$. The latter states that the evolution of the universe
under the fundamental physical laws and constants builds up a correlation
between the value (for values) of the fundamental physical constants since the
beginning of time and the existence, or non-existence, of sentient life today.
Its significance lies in the fact that the least change of those values would
bring to a different universe most probably unable to host life.

Indeed, a quantum cosmological interpretation of the Anthropic Principle
yields a formal identity between:

(i) the building up, by the unitary quantum evolution of the universe under
the fundamental physical laws and constants, of the correlation between the
value of the fundamental physical constants since the beginning of time and
the existence or non-existence of sentient life today and

(ii) the building up, by the unitary part of quantum search algorithms, of the
correlation between the value of the problem-setting since their beginning and
the value of the solution at their end. Note that this occurs with quantum
speedup even in the case that the search for the solution is random
$\left[21\right]$.

According to this formal identity, the evolution of the universe toward life
should be endowed with quantum speedup and would thus be teleological in
character even if the search for sentient life by the universe’s evolution
under the fundamental physical laws and constants is random. By the way, that
there has been a search for sentient life by the evolution of the universe we
are in is obvious. Note that also the inanimate part of this evolution can be
considered evolution toward life. In fact, it just produces the building
blocks of the living, i.e. atoms and molecules, and the environment for their
aggregation, i.e. planets with liquid water.

In Section 3.2, we will provide a toy model of the universe’s evolution toward
life mapped on Grover quantum search algorithm.

We can go now to scaling down from the evolution of the universe toward life
to the evolution of the living on Earth. Since the liquid water stage of the
latter, this evolution should have been almost completely independent from
that of the rest of the universe. If the independence were complete, it could
be factorized out of the universe’s evolution: it would thus be quantum,
endowed with quantum speedup, and therefore teleological in character. Anyhow,
even in the case of almost complete independence, it could be obtained in
principle from the evolution of the universe toward life by tracing out all
the rest. Even if decohered, it should at least inherit the quantum speedup of
its image in the evolution of the universe toward life. In fact, evolution
times must be identical, and the physical resources employed by the evolution
of the living on Earth obviously could not exceed those employed by its image
in the evolution of the universe that also includes them. Also the evolution
of life on Earth would therefore be teleological in character. Naturally, also
the evolutions of species, or groups of interacting species, would share this
character.

Let us show that a decohered evolution of the living endowed with quantum
speedup is not necessarily a contradiction. This is because the evolutions of
the living must be described in a probabilistic way. We can make reference to
the stochastic description of a Darwinian evolution, as an alternation of
random transitions in the species DNA pool each followed by natural selection
of the changes more suited to the environment.

In a classical stochastic process, the transition probabilities can depend on
the past of the process, of course not on the future of it. Time-symmetrizing
a process of this kind yields that the transition probabilities time-
symmetrically depend on the past and the future of the process. This would
make the transitions toward those changes that will survive in the future of
the process because more suited to the environment more probable with respect
to the same transitions in the classical stochastic process. It would give the
process a quantum speedup and therefore a teleological character. Moreover,
since this character would be hidden in transition probabilities, the process
would be difficult to distinguish from a classical stochastic process.

In the next Section 2, we provide the formalized quantum causal loops
description of quantum search algorithms, always focusing on the four drawer
instance of Grover algorithm.

In Section 3, we show that the time-symmetric causal loops that constitute the
physical basis of the teleological character of quantum algorithms could also
be the physical basis of the teleological character of natural evolutions.

Section 4 provides the prospects for further research.

Section 5 gives the conclusions.

##  2  The time-symmetric causal loops

We have seen that the sheer existence of an optimal quantum speedup logically
implies the mutually exclusive $or$ of well defined causal loops whose quantum
superposition must be hosted in the quantum algorithm that delivers it. In
fact, involving only the measurements of commuting observables, the latter is
submitted to the same (classical) logic. This immediately faces the difficulty
that the customary quantum description of quantum algorithms, being causal,
cannot describe any causal loop. As we shall see, it will after completing it
through the following steps.

(1) Extend the usual quantum description, limited to the process of solving
the problem, to the upstream process of setting the problem. The resulting
description is that between two one-to-one correlated measurement outcomes,
namely the basic description of quantum correlation. We call it the ordinary
quantum description of the quantum algorithm, ordinary because it is still an
ordinary causal (one-way causality) description.

(2) The ordinary quantum description of the quantum algorithm is with respect
to the customary external observer. It would tell the problem-solver, before
she begins her problem-solving action, the solution of the problem. To conceal
her from the solution of the problem the former quantum description must be
relativized to her.

(3) Complete the ordinary quantum description by time-symmetrizing it. This
produces the causal loops we are looking for but with respect to the external
observer.

(4) Relativize the causal loops in question with respect to the problem-
solver.

###  2.1  Extending the usual quantum description of the quantum algorithm to
the process of setting the problem

The usual quantum description of quantum algorithms is limited to the process
of solving the problem, where the quantum speedup resides, and lacks the
initial measurement. Of course, it is trivially incomplete: the canonical
quantum description of a quantum process must consist of initial measurement,
unitary evolution, and final measurement. We should extend the quantum
description to the upstream process of setting the problem. An initial
measurement of the problem-setting, performed in a quantum superposition (or,
indifferently, mixture) of all the possible problem-settings, selects one of
them at random. This is followed by the unitary transformation that computes
the corresponding solution and the final measurement of the latter.

Let us introduce the notation. We make reference to the simplest four drawer
instance of Grover algorithm:

We number the four draws in binary notation: $00,01,10,11$.

We will need two quantum registers of two quantum bits555 The notion of
quantum bit and that computation is possible in the quantum world was
introduced by Finkelstein in 1969 $\left[22\right]$.  each:

A register $B$ under the control of the problem-setter Bob contains the
problem-setting (the number of the drawer in which the ball is hidden). The
content of register $B$ is the problem-setting observable $\hat{B}$ of
eigenstates and corresponding eigenvalues:

$|00\rangle_{B}$ and $00$; $|01\rangle_{B}$ and $01$; $|10\rangle_{B}$ and
$10$; $|11\rangle_{B}$ and $11$.

A register $A$ under the control of the problem-solver Alice contains the
number of the drawer opened by her. The content of register $A$ is the
observable $\hat{A}$ of eigenstates and corresponding eigenvalues:

$|00\rangle_{A}$ and $00$; $|01\rangle_{A}$ and $01$; $|10\rangle_{A}$ and
$10$; $|11\rangle_{A}$ and $11$;

we call it the search for the solution observable. Measuring $\hat{A}$ at the
end of the quantum algorithm yields the number of the drawer with the ball
selected by the initial measurement, namely the solution of the problem. Note
that $\hat{B}$ and $\hat{A}$ commute.

The quantum description of the extended quantum algorithm, namely its ordinary
quantum description is:

$\begin{array}[]{ccc}\begin{array}[]{c}\;\text{time }t_{1}\text{, meas.
of}\;\hat{B}\end{array}&t_{1}\rightarrow t_{2}&\text{time }t_{2}\text{, meas.
of}\;\hat{A}\\\ &&\\\
\begin{array}[]{c}\left(|00\rangle_{B}+|01\rangle_{B}+|10\rangle_{B}+|11\rangle_{B}\right)\\\
\left(|00\rangle_{A}+|01\rangle_{A}+|10\rangle_{A}+|11\rangle_{A}\right)\end{array}&&\\\
\Downarrow&&\\\
|01\rangle_{B}\left(|00\rangle_{A}+|01\rangle_{A}+|10\rangle_{A}+|11\rangle_{A}\right)&\Rightarrow\hat{\mathbf{U}}_{1,2}\Rightarrow&|01\rangle_{B}|01\rangle_{A}\end{array}$
(Table I)

Here and in the following we disregard normalization.

We should start from the quantum state in the upper-left corner of Table I
diagram and then follow the vertical and horizontal arrows. Initially, at time
$t_{1}$, both registers, each by itself, are in a quantum superposition of all
their basis vectors. By measuring $\hat{B}$ (the content of register $B$) in
this initial state, Bob selects the number of the drawer with the ball at
random, say it comes out $01$ – see the state of register $B$ under the
vertical arrow666 By the way, Bob would be free to unitarily change at will
the sorted out number of the drawer with the ball. Nothing else would change.
.

It is now Alice’s turn. The unitary part of her problem-solving action,
$\hat{\mathbf{U}}_{1,2}$, never changes the problem-setting selected by the
initial measurement. We represent it by the generic unitary transformation
that sends that initial into that final measurement outcome. We do not need to
know it, it is sufficient to know that there can always be a unitary
transformation between the measurement outcomes in question. At the end of
Alice’s problem-solving action, at time $t_{2}$, register $A$ contains the
solution of the problem, namely the number of the drawer with the ball –
bottom-right corner of the diagram.

Eventually, by measuring $\hat{A}$ (the content of register $A$), which is
already in one of its eigenstates, Alice acquires the solution with
probability $1$ – the quantum state in the bottom-right corner of the diagram
naturally remains unaltered throughout this measurement. Correspondingly,
there is a unitary transformation between the initial and final measurement
outcomes; the process between them is reversible despite comprising the final
measurement.

Note that leaving $\hat{\mathbf{U}}_{1,2}$ unspecified makes of Table I
diagram the description of the generic mechanism that one-to-one correlates
two initially uncorrelated observables without changing the observable
selected by the initial measurement (its extension to any number of drawers is
immediate).

###  2.2  Relativizing the ordinary quantum description of the quantum
algorithm with respect to the problem-solver

The ordinary quantum description of the quantum algorithm of Table I is the
description of what goes on with respect to the customary external observer,
namely an observer external to the quantum process. It is not a description
good for the problem-solver (Alice). In fact, it would say that, immediately
after the initial measurement of $\hat{B}$, Alice knows the number of the
drawer with the ball selected by Bob, which is also the solution of the
problem – see the state of register $B$ in the bottom-left corner of Table I
diagram. Of course, in the description of the quantum algorithm with respect
to Alice, the number of the drawer with the ball selected by Bob must be
concealed.

Since Alice’s problem-solving action never changes the number of the drawer
with the ball selected by Bob and the observables $\hat{B}$ and $\hat{A}$
commute, describing this concealment is simple. It suffices to postpone after
the end of Alice’s problem-solving action the projection of the quantum state
associated with the initial measurement of $\hat{B}$ (or, indifferently, to
postpone the very measurement of $\hat{B}$). The result is the following
description of the quantum algorithm with respect to Alice:

$\begin{array}[]{ccc}\begin{array}[]{c}\text{time}\text{ }t_{1}\text{, meas.
of}\;\hat{B}\end{array}&t_{1}\rightarrow t_{2}&\text{\text{time
}\text{}}t_{2}\text{, meas. of}\;\hat{A}\\\ &&\\\
\begin{array}[]{c}\left(|00\rangle_{B}+|01\rangle_{B}+|10\rangle_{B}+|11\rangle_{B}\right)\\\
\left(|00\rangle_{A}+|01\rangle_{A}+|10\rangle_{A}+|11\rangle_{A}\right)\end{array}&\Rightarrow\hat{\mathbf{U}}_{1,2}\Rightarrow&\begin{array}[]{c}|00\rangle_{B}|00\rangle_{A}+|01\rangle_{B}|01\rangle_{A}+\\\
|10\rangle_{B}|10\rangle_{A}+|11\rangle_{B}|11\rangle_{A}\end{array}\\\
&&\Downarrow\\\ &&|01\rangle_{B}|01\rangle_{A}\end{array}$ (Table II)

To Alice, the initial measurement of$\hat{B}$ leaves the initial state of
register $B$ unaltered. At the beginning of her problem-solving action she is
completely ignorant of the number of the drawer with the ball selected by Bob
– see the state of register $B$ on the left of the horizontal arrows.

The unitary transformation $\hat{\mathbf{U}}_{1,2}$ is then performed for a
quantum superposition of the four possible numbers of the drawer with the
ball. The state at the end of it – on the right of the horizontal arrows – is
a quantum superposition of tensor products, each a possible number of the
drawer with the ball in register $B$ tensor product the corresponding solution
(that same number) in register $A$. Since $\hat{B}$ and $\hat{A}$ commute, the
final measurement of $\hat{A}$ must project this superposition on the tensor
product of the number of the drawer with the ball initially selected by Bob
and the corresponding solution – bottom-right corner of Table II diagram.

Note that this relativization necessarily relies on relational quantum
mechanics $\left[23,24\right]$, where the quantum state is relative to the
observer. In fact, we are left with two distinct quantum descriptions of the
same quantum algorithm, the ordinary one with respect to the external observer
(Table I) and another one with respect to the problem-solver (Table II). In
the following, we will continue to call the former quantum description the
ordinary quantum description of the quantum algorithm and always specify that
the latter quantum description is the one with respect to the problem-solver.

###  2.3  Time-symmetrizing the ordinary quantum description of the quantum
algorithm

As we have already seen, in order to obtain the quantum superposition of the
causal loops whose mutually exclusive $or$ is implied by the sheer existence
of an optimal quantum speedup, we should first time-symmetrize the ordinary
quantum description of the quantum algorithm that delivers the quantum
speedup, then relativize the causal loops obtained with respect to the
problem-solver.

Then, first, we should require that the selection of the information that
specifies the sorted out pair of correlated measurement outcomes among all the
possible pairs time-symmetrically shares between the initial and final
measurements. The half information selected by the initial measurement should
propagate forward in time, by $\hat{\mathbf{U}}_{1,2}$ until becoming the
state immediately before the final measurement. Time-symmetrically, the
complementary selection performed by the final measurement should propagate
backward in time, by $\hat{\mathbf{U}}_{1,2}^{{\dagger}}$, until replacing the
previous outcome of the initial measurement.

We consider the time-symmetrization instance where the initial measurement of
$\hat{B}$ reduces to the measurement of $\hat{B}_{l}$, the left bit of the
two-bit number contained in register $B$; say that it randomly selects left
bit $0$. The final measurement of $\hat{A}$ should correspondingly reduce to
that of $\hat{A_{r}}$ the right bit of the two-bit number contained in
register $A$; say that it randomly selects right bit $1$. The randomly
selected number of the drawer with the ball is thus $01$.

In this particular time-symmetrization instance, the ordinary quantum
description of the quantum algorithm with respect to the external observer of
Table I becomes:

$\begin{array}[]{ccc}\begin{array}[]{c}\;\text{time }t_{1}\text{,
}\text{}\text{meas.
of}\;\hat{B}_{l}\end{array}&t_{1\rightleftarrows}t_{2}&\text{time
}\text{}t_{2}\text{, meas. of}\;\hat{A_{r}}\\\ &&\\\
\begin{array}[]{c}\left(|00\rangle_{B}+|01\rangle_{B}+|10\rangle_{B}+|11\rangle_{B}\right)\\\
\left(|00\rangle_{A}+|01\rangle_{A}+|10\rangle_{A}+|11\rangle_{A}\right)\end{array}&&\\\
\Downarrow&&\\\
\begin{array}[]{c}\left(|00\rangle_{B}+|01\rangle_{B}\right)\\\
\left(|00\rangle_{A}+|01\rangle_{A}+|10\rangle_{A}+|11\rangle_{A}\right)\end{array}&\Rightarrow\hat{\mathbf{U}}_{1,2}\Rightarrow&|00\rangle_{B}|00\rangle_{A}+|01\rangle_{B}|01\rangle_{A}\\\
&&\Downarrow\\\
|01\rangle_{B}\left(|00\rangle_{A}+|01\rangle_{A}+|10\rangle_{A}+|11\rangle_{A}\right)&\Leftarrow\hat{\mathbf{U}}_{1,2}^{{\dagger}}\Leftarrow&|01\rangle_{B}|01\rangle_{A}\end{array}$
(Table III)

The initial measurement of $\hat{B}_{l}$, selecting the $0$ of $01$, projects
the initial quantum superposition of all the basis vectors of register $B$ on
the superposition of those beginning with $0$ – vertical arrow on the left of
the diagram. Under $\hat{\mathbf{U}}_{1,2}$, the latter superposition causally
propagates into the superposition of the two tensor products on the right of
the right looking horizontal arrows. Then the final measurement of
$\hat{A_{r}}$, selecting the $1$ of $01$, projects the latter superposition on
the term ending with $1$ under the right vertical arrow. Note that this
outcome of the final measurement, which inherits both selections, coincides
with that of the ordinary quantum description – see the bottom-right corner of
Table I diagram. Time-symmetrically, this final measurement outcome causally
propagates backward in time by $\hat{\mathbf{U}}_{1,2}^{{\dagger}}$ until
becoming the definitive outcome of the initial measurement. This backward in
time propagation (which inherits both selections) is an instance of the time-
symmetrized description of the quantum algorithm – bottom line of the diagram.
Note that, since the symbol
“$\Leftarrow\hat{\mathbf{U}}_{1,2}^{{\dagger}}\Leftarrow$” is mathematically
equivalent to “$\Rightarrow\hat{\mathbf{U}}_{1,2}\Rightarrow$”, this instance,
up to the time-direction of causality, is the unitary process between the two
one-to-one correlated measurement outcomes back again.

Also note that causality performs a causal loop that changes its own past; the
outcome of its initial measurement of $\hat{B}_{l}$ is changed into the
outcome of the initial measurement of $\hat{B}$ of the ordinary quantum
description; compare the state of register $B$ under the left vertical arrow
with the state in the bottom-left corner of the diagram.

We should eventually take the quantum superposition of all the time-
symmetrization instances. We can see that, no matter how we evenly share the
selection between initial and final measurements, the final measurement
outcome, which inherits both selections, is always that of the final
measurement of $\hat{B}$ of the ordinary quantum description of the quantum
algorithm. Correspondingly, its subsequent backward in time propagation by
$\hat{\mathbf{U}}_{1,2}^{{\dagger}}$ (bottom line of the diagram) is always
the ordinary quantum description of the unitary process between the two
correlated measurement outcomes, namely the bottom line of Table I diagram.

Summing up, time-symmetrization leaves the unitary process between the two
measurement outcomes unaltered but shows that it is the common (backward in
time) part of a quantum superposition of causal loops that describe the
behavior of causality along it in all the possible time-symmetric ways.

###  2.4  Relativizing the causal loops with respect to the problem-solver

The causal loops generated by time-symmetrizing the ordinary quantum
description of the quantum algorithm, which is relative to the external
observer, do not yet map on those logically implied by the sheer existence of
the quantum speedup. This is because the latter causal loops, concerning the
problem-solver advanced knowledge of one of the halves of the information
about the solution, must be relative to the problem-solver.

Therefore, we should relativize with respect to the problem-solver the causal
loop of Table III by postponing after her problem-solving action the
projection of the quantum state due to the initial measurement of
$\hat{B}_{l}$ (whose outcome should be concealed from Alice). This yields the
causal loop in the following table:

$\begin{array}[]{ccc}\begin{array}[]{c}\;\text{time }t_{1}\text{, meas.
of}\;\hat{B}_{l}\end{array}&t_{1\rightleftarrows}t_{2}&\text{time
}t_{2}\text{, meas. of}\;\hat{A_{r}}\\\ &&\\\
\begin{array}[]{c}\left(|00\rangle_{B}+|01\rangle_{B}+|10\rangle_{B}+|11\rangle_{B}\right)\\\
\left(|00\rangle_{A}+|01\rangle_{A}+|10\rangle_{A}+|11\rangle_{A}\right)\end{array}&\Rightarrow\hat{\mathbf{U}}_{1,2}\Rightarrow&\begin{array}[]{c}|00\rangle_{B}|00\rangle_{A}+|01\rangle_{B}|01\rangle_{A}+\\\
|10\rangle_{B}|10\rangle_{A}+|11\rangle_{B}|11\rangle_{A}\end{array}\\\
&&\Downarrow\\\
\begin{array}[]{c}\left(|01\rangle_{B}+|11\rangle_{B}\right)\\\
\left(|00\rangle_{A}+|01\rangle_{A}+|10\rangle_{A}+|11\rangle_{A}\right)\end{array}&\Leftarrow\hat{\mathbf{U}}_{1,2}^{{\dagger}}\Leftarrow&|01\rangle_{B}|01\rangle_{A}+|11\rangle_{B}|11\rangle_{A}\end{array}$
(Table IV)

The projection of the initial state of register $B$ due to the measurement of
$\hat{B}_{l}$ – top-left corner of the diagram – is postponed after Alice’s
problem-solving action, namely outside the present table which is limited to
that action. With this postponement, the initial state of register $B$ goes
unaltered through the initial measurement of $\hat{B}_{l}$ becoming, by
$\hat{\mathbf{U}}_{1,2}$, the quantum superposition of tensor products above
the vertical arrow. Alice’s final measurement of $\hat{A_{r}}$, which in
present assumptions selects the value $1$ of the right bit of the number
contained in register $A$, projects the state above the vertical arrow on the
state below it. Then (in causal order), this final measurement outcome
propagates backward in time by $\hat{\mathbf{U}}_{1,2}^{{\dagger}}$ until
becoming the definitive outcome of the initial measurement. This backward in
time propagation is an instance of the time-symmetrized description of the
quantum algorithm with respect to the problem-solver.

Note that we would have obtained the identical diagram by time-symmetrizing
the description of the quantum algorithm relativized to the problem-solver of
Table II.

Also note that, for the mathematical equivalence between
$\Leftarrow\hat{\mathbf{U}}_{1,2}^{{\dagger}}\Leftarrow$ and
$\Rightarrow\hat{\mathbf{U}}_{1,2}\Rightarrow$, the bottom line of the diagram
of Table IV can also be read in the usual left to right way:

$\begin{array}[]{ccc}\text{time }t_{1}&t_{1}\rightarrow t_{2}&\text{time
}t_{2}\\\ &&\\\ \begin{array}[]{c}(|01\rangle_{B}+|11\rangle_{B})\\\
\left(|00\rangle_{A}+|01\rangle_{A}+|10\rangle_{A}+|11\rangle_{A}\right)\end{array}&\Rightarrow\hat{\mathbf{U}}_{1,2}\Rightarrow&|01\rangle_{B}|01\rangle_{A}+|11\rangle_{B}|11\rangle_{A}\end{array}$
(Table V)

We can see that the diagram of Table IV describes one of the causal loops
logically implied by the sheer existence of the quantum speedup: the one where
Alice knows in advance that the ball is in drawer either $01$ or $11$. This
causal loop changes backward in time Alice’s initial complete ignorance of the
solution into knowledge of half of the information about it: compare the
states of register $B$ in the top-left and bottom-left corners of table IV
diagram.

Summing up, the state of register $B$ in the bottom-left corner of the diagram
of Table IV, or in the left part of the diagram of Table V, tells us that the
problem-solver Alice, before beginning her problem-solving action, knows that
the ball is in drawer either $01$ or $11$. In other words, the computational
complexity of the problem to be solved by her reduces. The problem of locating
the ball in one of $4$ drawers – as for the state of register $B$ in the
upper-left corner of the diagram of Table IV – becomes that of locating it in
one of $2=\sqrt{4}$ drawers – as for the state of register $B$ in the bottom-
left corner of Table IV diagram (or the left of Table V diagram).

In equivalent terms, the dimension of the Hilbert space in which the search
for the solution takes place reduces from the $4$ dimensions of the ordinary
quantum description of the quantum algorithm to $2=\sqrt{4}$ dimensions. Of
course, solving a less complex problem, or searching for something in a
smaller space, can be quicker; in the present case it takes opening only one
drawer.

In conclusion, the TS causal loops complete the ordinary quantum description
of the quantum algorithm telling that the computational complexity of the
problem to be solved reduces because of the advanced knowledge, on the part of
the problem-solver, of one of the possible halves of the information about the
solution she will produce and measure in the future; this of course in a
quantum superposition of all the possible ways of taking the half.

Correspondingly, the number of computation steps (oracle queries) needed to
solve the (oracle) problem in the optimal quantum way is that needed to solve
the reduced complexity problem in the optimal logical way.

###  2.5  Discussion

The time-symmetrization of the ordinary quantum description of the quantum
algorithm is an unconventional operation with destabilizing consequences. Of
course, it must be justified.

We have seen that the causal loops whose mutually exclusive $or$ is logically
implied by the sheer existence of the quantum speedup must be hosted in the
quantum algorithm that delivers it since the latter is submitted to the same
(classical) logic. The justification for time-symmetrizing the quantum
algorithm is that this is the only way of generating the causal loops that
must be hosted in it.

This is easily seen. To have a significant causal loop, it is necessary that
both the initial and final measurements select a proper part of the
information that specifies the sorted out pair of measurement outcomes. If
either measurement selected no information – did not change the quantum state
– we would remain with an insignificant causal loop, just the same unitary
evolution between the two correlated measurement outcomes read first forward
and then backward in time; certainly not one of the causal loops we are
looking for. Once ascertained that both measurements should select a proper
part of the information, for reasons of time-symmetry they should select the
same amount. This exactly defines the time-symmetrization in question.

Interestingly, this time-symmetrization has other justifications independent
of the fact that the unitary evolution between the two correlated measurement
outcomes is a quantum algorithm. One is as follows.

Since the value of $\hat{B}$ selected by the initial measurement remains
unaltered along the unitary process in question, we are free to postpone the
subject measurement at will, nothing else should change. Making it
simultaneous with the final measurement ($\hat{B}$ and $\hat{A}$ commute)
makes the above time-symmetrization an atemporal symmetrization necessary to
satisfy the perfect symmetry between the two simultaneous measurements. For
nothing to change with the postponement, when there is none there must be the
time-symmetry we are dealing with. Note that, otherwise, a lot would change:
the causal loops would be there when the two measurements are simultaneous and
would disappear when they are not.

Still another justification is that the time-symmetrization in question is
also the only way of avoiding a preferred time-direction of causality along
the time-symmetric unitary evolution between two one-to-one correlated
measurement outcomes.

Either justification is important since allows to export the causal loops to
any unitary process between two one-to-one correlated measurement outcomes, in
particular, to quantum nonlocality (see the following subsection).

On another issue, we should note that the ordinary quantum description is
incomplete only from the physical standpoint; that is, it does not completely
describe the behavior of causality within the quantum algorithm. It is instead
compete from the mathematical standpoint. Although changing from one quantum
algorithm to the other, the unitary transformation between the initial and
final measurement outcomes, mathematically, is a complete description of the
quantum computation of the solution, in particular, of course, it yields the
number of computation steps taken by the quantum algorithm.

Let us then ask ourselves what are the advantages of this physical completion
of the ordinary quantum description of the unitary processes between two one-
to-one correlated measurement outcomes.

A first advantage is the unification of the mathematical explanations of the
quantum speedup (all different from each other). Given a (oracle) problem –
only the problem – the causal loops in question tell us the number of
computation steps (oracle queries) needed to solve it in an optimal quantum
way. Summarizing, this is expressed by the following (unifying) advanced
knowledge rule:

Given any (oracle) problem, the number of computation steps (oracle queries)
needed to solve it in an optimal quantum way is that needed to solve it in an
optimal logical way given the advanced knowledge of one of the possible halves
of the information that specifies the solution among all the possible
solutions.

By the way, in $\left[8\right]$, we have checked that the above rule is
satisfied by all the major quantum algorithms.

Instead, without the present physical completion of the quantum description,
thus in the mainstream approach to quantum computation, given a (oracle)
problem, one does not even know whether it can be solved with a quantum
speedup. Once, and if, found a quantum algorithm that solves it with quantum
speedup, one does not know whether it solves it in an optimal quantum way.

The question then becomes why the same advanced knowledge rule could not be
obtained in a purely mathematical way without resorting to the physical notion
of causality. The answer is that it should, but finding how could be
difficult. For example, the demonstration of the optimality of Grover
algorithm alone relies on very subtle mathematics (see what follows).

If it is legitimate to compare the little with the big, an analogy with the
Copernican revolution might help clarify the kind of advantage we should be
dealing with. The geocentric and heliocentric views of planetary system are
kinematically equivalent, but the latter is much simpler and paved the way to
Newton’s discovery of celestial bodies’ dynamics. Although not impossible, it
would of course be more difficult and unnatural to associate a dynamics to a
geocentric kinematics. Mutatis mutandis, the TS causal loops we are dealing
with might provide this kind of simplification.

It should be noted that it is a simplification already envisaged by Grover. In
$\left[25\right]$, he writes:

“It has been proved that the quantum search algorithm cannot be improved at
all, that is, any quantum mechanical algorithm will need at least
$\operatorname*{O}\sqrt{N}$ steps to perform an exhaustive search of $N$
items. Why is it not possible to search in fewer than
$\operatorname*{O}\sqrt{N}$ steps? The arguments used to prove this are very
subtle and mathematical. What is missing is a simple and convincing two-line
argument that shows why one would expect this to be the case.”

Our “two-line” argument would be:

The reason any optimal quantum search algorithm will need at least
$\operatorname*{O}\sqrt{N}$ steps is that, in it, it is as if the problem-
solver knew in advance half of the information about the solution she will
produce and measure in the future and could use this knowledge to produce the
solution with a correspondingly reduced number of computation steps.

It is understood that the reduced number in question is that given by
(classical) logic in the case that the problem-solver knows in advance half of
the information about the solution. In the optimal logical case, it is of
course $\operatorname*{O}\sqrt{N}$ steps.

###  2.6  Other applications of the time-symmetric causal loops to the quantum
domain

The time-symmetric (TS) causal loops we are dealing with are not limited to
quantum computation. They are much more pervasive in quantum mechanics since
they are present in most ubiquitous quantum correlation.

In this broader context, of course, we are not the first to argue that the
ordinary quantum description is incomplete. As is well known, Einstein,
Podolsky, and Rosen argued for its incompleteness in their EPR paper
$\left[11\right]$. Here the incompleteness was ascribed to the fact that the
ordinary quantum description does not describe quantum nonlocality locally –
i.e. physically. We have already seen in the Introduction that the time-
symmetric causal loops play the role of the hidden variables envisaged by
Einstein and the others to explain quantum nonlocality locally. They locally
go back and forth in time from one measurement outcome to the other via the
time the two subsystems were not yet space separate when, always locally,
change the state of both observables in a way that ensures the appropriate
correlation between the two future measurement outcomes, of course regardless
of the time separation between the two measurements and the spatial separation
between the two subsystems.

By the way, the present causal loop explanation of quantum nonlocality has
another kind of precedent. In a 1953 publication $\left[26\right]$, Costa de
Beauregard assumed that the entire outcome of the first performed measurement
(of one of the two maximally entangled observables) locally goes backward in
time, by the inverse of the time-forward unitary transformation that preceded
the measurement, until the time the two subsystems were not yet space-separate
when, always locally, it would change the state of both observables. The
changed state, going then (in causal order) forward in time again, would
ensure the appropriate correlation between the outcomes of the two future
measurements.

There are the following differences with respect to the present work. In the
first place, in our work the time-symmetric causal loops are imposed by the
symmetry of the process they describe. In the second, under the assumption of
the consistency of quantum mechanics that imposes them, they should be exempt
from the paradoxes that plague the ordinary notion of causal loop. Sending
backward in time an entire measurement outcome is instead arbitrary, violates
the symmetry of the process, and can produce paradoxical consequences. By the
way, Costa de Beauregard’s explanation of quantum nonlocality, an insight in
the right direction according to the present work, despite the author’s
notoriety never took off in mainstream science. Since 70 years have passed
since then, this fact should testify to the persistence of the negative
attitude of mainstream physics toward causal loops.

##  3  Applying the notion of the time-symmetric causal loops to the
teleological view of natural evolutions

As we have seen, according to the causal loop description of quantum
algorithms, the quantum search for the solution of the problem has an
attractor in the solution it will produce in the future. In the case of
Grover’s quantum search algorithm, of the quantum search algorithms in
general, this holds even if the quantum search is random $\left[21\right]$.

Put in this form, the present causal loop description of quantum algorithms
has a conceptual precedent. It is the concept of teleological evolution, an
evolution toward a goal that has an attractor in the goal it will produce in
the future. It is a new notion from the scientific standpoint, not from that
of natural philosophy. It was first applied to the formation of the functional
structure of the living by Plato and Aristotle – teeth were created for the
purpose of chewing $\left[27\right]$ – and then, by various schools of
thought, to natural evolutions. This, until the advent of modern science, when
the teleological notion was dismissed for the alleged absence of a physical
basis. That was, of course, under the notion of causal evolution that is
dominant in science also today and that the present work is challenging.

The fact that quantum algorithms are “scientific” examples of teleological
evolutions naturally raises the question of whether the causal loops that
constitute their physical basis could also be the missing physical basis of
teleological natural evolutions.

Let us mention other motivations for addressing this question.

One is that the teleological view has been banished from the natural sciences
despite the fact that it responds much better to empirical evidence; of course
this should be why it was conceived in the first place. This tension is
captured by the following sentence by the evolutionary biologist Haldane
$\left[28\right]$:

Teleology is like a mistress to a biologist: he cannot live without her, but
he is unwilling to be seen with her in public.

A further motivation is having to address the physical foundations common to
quantum mechanics and the natural sciences. There may be cross-fertilizations
– there will be.

Of course, the above raises a seemingly insurmountable problem. How could an
evolution of the living that can be the size of the Earth and have duration of
billions of years host the needed quantum features?

###  3.1  Looking for quantum features in the evolutions of the living

Looking for quantum features in the evolutions of the living should be a
timely problem. In recent years, evidence has emerged that quantum mechanics
should play a role in the fundamental processes of life, but how it can be so
is still considered a mystery. The problem, naturally, is how to have a
quantum behavior in the mesoscopic, sometimes macroscopic, and always long-
lasting regime of life processes. As far as we know, so far people have faced
the hard problem of trying to scale up the quantum character typical of the
microscopic world to find an autonomous quantum quantum behavior in the life
processes. This strategy would obviously be inapplicable to life processes of
the size of the Earth and of durations of billions of years. However, it is
precisely the enormity of the difference that suggests the possible way out:
abandon the attempt to find an autonomous quantum behavior in the life process
by scaling up from the microscopic world to look at the possibility of finding
a non-autonomous quantum behavior by scaling down from the most macroscopic
and long lasting process there is, namely the evolution of the universe.

If the evolution of the universe as a whole were quantum, in principle the
evolution of everything would be derivable from it by tracing out all the
rest. Furthermore, if the quantum evolution in question were endowed with
quantum speedup and therefore teleological in character, there could be the
possibility that the evolutions of life, as the traces of the universe’s
evolution over all the rest, inherit both.

Let us show that all the above is physically plausible.

Since we are talking of the teleological character of an evolution, we should
first identify the goal this evolution is heading to. Let us say that it is
a-posteriori empirically evident that the evolution of the universe we are in
is toward life. In fact, think of the formation of: subatomic particles out of
pure energy in the Big Bang, lighter and heavier atoms in first and second-
generation stars, molecules, planets with liquid water, prebiotic molecules,
the first self-replicating molecular assembly, and, eventually, sentient life.
As also the inanimate part of the universe’s evolution is directed toward the
production of life – it produces the bricks of the living beings (atoms and
molecules) and the environment for their aggregation (planets with liquid
water) – it should be justified to say that the evolution of life is evolution
of the universe toward life. Naturally, we should comprise the case that the
quantum search for life by the universe’s evolution under the fundamental
physical laws and constants is random.

Then, we should justify the assumption that the evolution of the universe
toward life is quantum. Of course, for this, we should adopt quantum cosmology
$\left[14,15\right]$, namely the assumption that the unitary dynamics of
quantum mechanics (the Schrödinger equation or its analogue in relativistic
quantum field theory) applies everywhere and at all times in the universe.

Under this assumption, the search for life on the part of the universe’s
evolution should be a unitary quantum evolution. Of course, this brings the
search for life by the universe’s evolution and the quantum search algorithms
closer together: now they are both quantum.

A further step calls into play the Anthropic Principle $\left[16-20\right]$.
This principle asserts that there is correlation between the “value” (for set
of values) of the fundamental physical constants since the beginning of time
and the existence – or non-existence – of sentient life today. Its
significance lies in the fact that the slightest variation in these values
would result in a significantly different universe most likely incapable of
hosting life.

Of course, the correlation in question was not there at the beginning of time:
no matter the value of the fundamental physical constants, there could have
been no life in the Big Bang. It must have been built up by the (unitary)
universe’s evolution toward life.

Combining together quantum cosmology and the Anthropic Principle produces a
formal identity between: (i) the building up, by the unitary evolution of the
universe toward life, of the correlation between the value of the physical
constants since the beginning of time and the existence (or non-existence) of
sentient life today and (ii) the building up, by the unitary part of the
quantum search algorithms, of the correlation between the value of the
problem-setting since its beginning and the value of the solution at its end.
In the next subsection, we will provide a formalized toy model of the search
for life by the universe’s evolution. It will just be a straightforward
interpretation of Grover quantum search algorithm.

In conclusion, the evolution of the universe toward life would be quantum,
could occur with quantum speedup and would in the case that its search for
life under the fundamental physical laws and constants is random. It would
thus be teleological in character.

By the way, here and in the following we leave in the background the case that
the value of the fundamental physical constants is unsuitable to the
development of life.

Let us go now to the evolution of the living on Earth. Since the Earth’s
liquid water stage, this evolution should have been almost completely
independent from that of the rest of the universe. If it were completely
independent, it would be factorizable out of the universe’s quantum evolution,
thus quantum and teleological in character itself. In any case, also in the
case of almost complete independence, it can be obtained in principle from the
evolution of the universe by tracing out all the rest. Of course, although
decohered, the evolution obtained should fully inherit the quantum speedup and
thus the teleological character of its (almost completely independent from the
rest) image in the evolution of the universe toward life. In fact, evolution
times must be the same and the amount of resources employed by the evolution
of life on Earth, obviously, cannot exceed those employed by the corresponding
part of the universe’s evolution that includes them. This leaves us with a
decohered evolution endowed with quantum speedup, naturally a questionable
result. However, the fact that the evolutions of life are necessarily modeled
as stochastic processes can offer the way out.

For example, consider a Darwinian species’ evolution, an alternation of random
mutations in the species DNA pool each followed by natural selection of the
mutations more suited to the environment. Of course, in the case of a
classical stochastic process, transition probabilities can depend only on the
past of the process. Now, we could accommodate a quantum speedup in this
classical picture by simply requiring that transition probabilities time-
symmetrically depend on the past and the future of the process. The
transitions toward those mutations that will survive in the future because
more suited to the environment would become more probable with respect to the
case that transition probabilities depend only on the past of the process. In
other words, the mutations surviving in the future would become attractors for
the evolution, hence the quantum speedup and consequent teleological character
of the latter. Note that, since this mechanism would be hidden in transition
probabilities, the corresponding stochastic evolution would be difficult to
distinguish from a classical stochastic evolution.

Another problem is how to model a decohered evolution endowed with quantum
speedup,

The canonical way – obviously holding only in principle – would be to obtain
this evolution from that of the universe by tracing out all the rest. Of
course, this could be done only at the level of toy models.

It would be interesting to develop a toy model of the situation we are dealing
with. One should identify a teleological, purely quantum, evolution of a
system where the achievement of the goal is mostly localized in a subsystem
thereof. Then, one should formally check that the evolution of the subsystem,
although decohered, inherits the quantum speedup and thus the teleological
character of the evolution of the system. Note that the interaction of the
subsystem with the rest of the system here would do the opposite than
decoherence. It would be what makes the evolution of the subsystem toward the
goal to behave in a quantum way as a part of the purely quantum evolution of
the system toward the same goal.

Another possibility is just to ignore the fact that the evolution is decohered
and model it in a quantum way, as if it were completely independent of the
rest of the universe and factorizable out of the universe’s evolution. There
are encouraging precedents to this. For example, it has been shown that
modeling in a quantum way the interplay between species of predators competing
for the same preys allows to explain the empirically ascertained species’
asymptotic behavior in a way otherwise difficult to explain $\left[29\right]$.

As we have just seen, a further possibility is to model the decohered
evolution as a stochastic process with transition probabilities time-
symmetrically depending on both the past and the future of the process.

###  3.2  Teleological evolution of the universe toward life – a toy model

Let us avail ourselves of the formalizations of the previous section to
provide a toy model of the teleological evolution of the universe toward life.
It will just be a trivial interpretation of Grover algorithm.

We should imagine a value of the fundamental physical constants observable
$\hat{B}$ and a search for the value of sentient life observable $\hat{A}$.
Say that the eigenvalue $01$ of $\hat{B}$ is the value of the fundamental
physical constants compatible with the development of sentient life and the
eigenvalues $00,10,11$ the values of those incompatible. We assume that,
correspondingly, the eigenvalue $01$ of the measurement of $\hat{A}$ at the
present time would mean existence of sentient life and the eigenvalues
$00,10,11$ absence of life.

Initially, $\hat{B}$ should be in a quantum superposition (or, indifferently,
mixture) of all its eigenstates (i.e of all the possible universes each with
its fundamental physical constants). We can assume that also $\hat{A}$, the
search for the value of sentient life observable, is initially in a quantum
superposition of all its eigenstates. The universe’s unitary evolution under
the fundamental physical laws and constants would have finally, at the present
time, built up a quantum superposition of tensor products, each an eigenstate
of $\hat{B}$ tensor product the corresponding eigenstate of $\hat{A}$. In our
universe, at the present time one with observers, the measurement of $\hat{A}$
at the present time would select the eigenvalue $01$ of it, namely “existence
of sentient life” and the corresponding eigenvalue $01$ of the value of the
fundamental physical constants observable $\hat{B}$. Since the universe’s
evolution does not change the value of the fundamental physical constants, it
is as if the eigenvalue of $\hat{B}$ was selected by a measurement of
$\hat{B}$ at the beginning of time, when there could be no observers.

Note that this is exactly the quantum search algorithm that we have described
in the previous section, the version with respect to the problem-solver of
Table II. As shown by its time-symmetrization (Table IV), it can undergo with
a quantum speedup, and does even if the quantum search for the value of
sentient life is random. It is thus teleological in character.

The above is of course reminiscent of Wheeler’s Participatory Anthropic
Principle $\left[30,31\right]$, in that both assume a mutual causation between
the observer today and the fundamental physical laws and constants since the
beginning of time mediated by a unitary evolution of the universe in between.

However, we would note two significant differences. Wheeler’s hypothesis is
based on the notion of delayed choice experiment, then on the backward in time
propagation of an entire measurement outcome. This is the form of
retrocausality that can lead to the well-known paradoxes. Our hypothesis is
based on the time-symmetric causal loops that should be immune to the
paradoxes in question.

Moreover, in our case, the mutual causality between observer today and
fundamental physical laws and constants since the beginning of time is
formally related to the quantum speedup and thus the teleological character of
the evolution of the universe toward sentient life. The sentient life stage of
this evolution would then be an attractor for the evolution. Of course, this
is not formalized in Wheeler’s model that was formulated before the discovery
of the quantum computational speedup.

Let us eventually mention what could be an emblematic example of teleological
evolution. It is known that, for many millions of years, the ancestors of the
flying birds were feathered dinosaurs who did not fly. Very light, strong but
flexible, cave, and air-proof, feathers are of course ideal for flight. So,
why anticipating something designed for flight millions of years in advance?

The usual explanation is of course a causal one. At first, feathers appeared
at random and survived as a good thermal insulator, later they proved useful
for speeding escape from predators, perhaps for gliding from trees to the
ground, and only eventually they turned out to be what was needed to conquer
the air.

According to the present teleological (mutual causality) view of the
evolutions of the living, even during the flightless period of the feathered
dinosaurs, those random DNA mutations that would have led in the future to the
conquest of the air – obviously an extremely successful outcome from the point
of view of survival – would have more likely occurred.

In the following subsections, we will discuss a variety of impacts that the
present teleological view of natural evolutions could have on: the natural
sciences, the way of modeling natural evolutions, and eventually our own
worldview.

###  3.3  Impact on the natural sciences

The position of the natural sciences toward teleology can be considered in
some measure conflicting. On the one hand, the teleological view would seem to
allow a deeper understanding of natural evolutions and the formation of the
functional structure of the living. On the other, as this vision is considered
without physical basis and therefore misleading, there is outright hostility
to it, even to the use of an otherwise convenient teleological language.

It would seem that, today, no evolutionary biologist supports the teleological
view, some explicitly assert that it is unworkable because it implies
retrocausality.

No biologist, of course, does not mean no scientist. A paradigmatic example is
the American philosopher Nagel $\left[32\right]$, who argues that the
principles that explain the emergence of life should be teleological, rather
than materialistic or mechanistic in character.

We would put together Nagel’s view in support of the teleological character of
the evolutions of the living, Haldane’s assertion that a biologist cannot live
without teleology, and the constant temptation to adopt a teleological view
and language in describing the evolutions and the functional structure of the
living, to affirm that the notion of teleological natural evolution is more
“scientific” than the mainstream one of causal natural evolution. This in the
sense that it better answers empirical evidence – of course the first canon of
science. The only reason biologists would be against teleology in spite of
empirical evidence would then come from another science. We mean the notion –
current also in today’s physics – that a teleological evolution is physically
impossible.

Of course, the sheer notion that quantum computation is teleological in
character may already put a strain on that position of today’s biological
science: teleological evolutions are physically possible. This strain should
naturally increase with the notion that, under quantum cosmology and the
Anthropic Principle, the evolution of the universe toward sentient life should
be teleological in character. It should further increase with the consequent
notion that also the evolution of life on Earth, obtainable in principle from
that of the universe by tracing out all the rest, even if decohered, should
inherit the teleological character of the universe’s evolution.

Summing up, given the above, the impact of the present teleological view of
natural evolutions on today’s evolutionary biology would seem to be almost a
head on collision.

Almost because, on the one side, in recent years a group of evolutionary
biologists $\left[33\right]$ expressed the need to rethink Darwin’s theory of
evolution. As already said, that was after the discovery that some species’
evolution steps occurred far too quickly to be solely ascribed to random
changes in the species’ DNA pool. Of course, the present teleological view of
evolutions, with the associated quantum speedups, would offer a way out.

On the other side, there are scientists who have began to apply essentially
quantum mechanical features like quantum entanglement and nonlocality to the
macroscopic game of species, showing that this can explain some empirically
ascertained features of it otherwise difficult to explain $\left[29\right]$.

###  3.4  Impact on philosophy

Let us call the present teleological view of evolutions and the causal view of
today’s physics and the natural sciences respectively the mutual causality
view777 Mutual causality between the initial and final measurement outcomes is
what generates the quantum superposition of the time-symmetric causal loops.
and the causal view.

The causal view and the mutual causality view seem to have subtended precisely
two opposing views throughout the entire history of philosophy. The causal
view would lie under the materialist, mechanistic, and reductionist views and
the mutual causality view under the idealist, teleological and monist views.

Consider the opposition between materialism and idealism. The former, about
the observer being causally created by physical reality, is a causal view; the
latter, about the mutual creation between observer and physical reality, like
in the Barrow and Tipler’s $\left[14\text{}\right]$ anthropic cosmological
principle, is a mutual causality view.

We find a similar situation in the opposition between the mechanistic and
teleological views of evolutions, which obviously respond to the causal and
mutual causality views of evolutions, respectively.

Similarly, the causal view is compatible with reductionism and the mutual
causality view with monism.

Eventually, it would be no accident that the idealist, teleological, and
monist views fell out of favor with the advent of modern science. It would
always be for the idea that they lack a physical basis under the causal view
of physical evolutions dominant also in contemporary science.

There are other theories in the history of philosophy reminiscent of the
mutual causality between sentient life today and the physical laws with their
physical constants since the beginning of time:

(i) For Plato, “Ideas [of our mind] are objective perfections that exist in
themselves and for themselves. At the same time, they are the causes of
natural phenomena and constitute their unity” $\left[34\right]$. In modern
language, the cause of natural phenomena that constitute their unity are the
fundamental physical laws. The identity between the fundamental physical laws
since the beginning of time and the ideas of our mind today is reminiscent of
the identity (and mutual causality) between the number of the drawer with the
ball selected by Bob’s measurement at the beginning of the quantum algorithm
and the number of the drawer with the ball selected by Alice’s measurement at
the end of it. This should also go along with Fritjov Capra’s
$\left[35\right]$ notion that there is identity between the fundamental states
of consciousness described by eastern theosophies, and the fundamental laws of
quantum physics.

(ii) We do not need to be anthropocentric. This mutual causality would also be
between physical laws and spatial intelligence, often superior to our in non-
human animals. The assumption that spatial intelligence is achieved with
quantum speedup excludes that it is only the result of “ordinary” natural
evolution (obeying the principle of causality). Indeed a quantum speedup in
the development of spatial intelligence would imply the mutual causality in
question.

(iii) The present notion of teleological evolution, namely of an evolution
toward a goal with an attractor in the goal it will reach in the future, could
also be seen in terms of a “systematically lucky” evolution, lucky from the
standpoint of the objective of reaching its goal, indeed because attracted by
it. Interestingly, also the notion of a systematically lucky evolution seems
to have a precedent in pre-scientific thinking. It could be the Stoic notion
of $\pi\rho o\nu o\iota\alpha$, foreknowledge, sometimes also translated into
providence and, in the present context, possibly translatable into advanced
knowledge. Foreknowledge, or advanced knowledge, would have pushed the
evolution toward its goal in a luckier and thus faster way than what is
possible in the case of a causal evolution, exactly as the advanced knowledge
about the solution does in quantum algorithms. This $\pi\rho o\nu
o\iota\alpha$ view would also fit well with the notion of teleological
evolution understood as a stochastic processes with transition probabilities
time-symmetrically affected by the past and future of the process.

(iv) It is natural to wander whether the present time-symmetric causal loops
view of the universe’s evolution toward sentient life, where the sentient life
stage is an attractor of the evolution of the universe toward it, could be
related to Nietzsche’s intuition of an eternal return.

(v) Michelangelo’s “Creation of Adam” on the ceiling of the Sistine Chapel
could be the manifesto of the present theory. There is a mirror symmetry
between creator and created $\left[36\right]$. Could this be interpreted as
the time-symmetry – and the related mutual causation – between the fundamental
physical laws with their physical constants since the beginning of time and
the existence of sentient life today? Furthermore, the fact that the figures
and shapes portrayed behind the creator figure would seem to be an
anatomically accurate portrait of the human brain $\left[37\right]$, could it
allude to the teleological character of the evolution of our intelligent
species: the human brain being an attractor of the search for it? We would not
feel like dismissing these analogies as sheer coincidences. As a matter of
fact, in idealistic philosophy – in Shelling – aesthetic insight would
reproduce the fundamental mechanisms of the production of life.

##  4  Research prospects

The notion that the time-symmetric causal loops are commonplace in non-
relativistic quantum mechanics is somewhat revolutionary; we have seen that it
leads to a different way of seeing both quantum physics and the natural
sciences. Future work on the subject could therefore have a broad spectrum.

For what concerns quantum physics:

(i) Try to derive analytically the advanced knowledge rule synthetically
(axiomatically) derived in the present work. Given a (oracle) problem, this
rule allows to compute the number of computation steps (oracle queries) needed
to solve it in an optimal quantum way. The present synthetic derivation could
constitute the thread to derive it in an analytical (i.e. purely mathematical)
way.

(ii) Look for a synthesis between the present notion of the time-symmetric
causal loops and time-symmetric quantum mechanics $\left[10,38-41\right]$ to
which it is obviously inspired.

(iii) Look for formal models that capture the mechanism for which the
evolution of life on Earth, although decohered, inherits the quantum speedup
and consequently the teleological character of the evolution of the universe
toward life.

(iv) Look for the possible impact of the teleological view on quantum
cosmology. According to present assumptions, also the inanimate part of it,
anyway an evolution toward life, should be teleological in character. Mutual
causality between what was the universe at the beginning of time and what is
today could have an impact on how we see its evolution.

(v) The Anthropic Principle is limited to stating the correlation between “the
existence or non-existence of sentient life today” and “the value of the
fundamental physical constants since the beginning of time”. One could look
also for a correlation between the former and to the fundamental physical laws
since the beginning of time. Of course, also these would have an attractor in
the goal of creating life.

About the natural sciences, looking for:

(I) A reconsideration of the ban on applying the teleological view and
language to natural evolutions. Restoring the dismissed notion of teleological
evolution would obviously be a paradigm shift for today’s natural sciences.

(II) Toy models of teleological Darwinian processes, i.e. teleological
construction of forms that answer some needs. These evolutions could be
modeled either as pure quantum processes endowed with quantum speedup or as
decohered stochastic processes with transition probabilities time-
symmetrically affected by the past and the future of the process.

(III) The possible role of the teleological character of the evolution in the
formation of a self-replicating system. There seem to be an enormous jump of
complexity in the transition from life-precursor molecules to a self-
replicating system. The idea that this transition has an attractor in the full
fledged self-replicating system can therefore be attractive.

(IV) More in general, an application of the notion of the time-symmetric
causal loops to self-organization theory.

Finally, opposing schools of thought throughout the history of philosophy
could be reviewed in the light of the opposition between the causal and mutual
causality views, as exemplified in the previous subsection.

##  5 Conclusions

As we have navigated uncharted waters, it may be useful to recap the route
followed.

The notion of causal loop is controversial and mostly opposed. We have seen
that a special – time-symmetric – type of quantum causal loop, which should be
immune to the paradoxes that plague the usual type, is instead always present
in quantum correlation, more specifically the unitary evolutions between two
one-to-one correlated measurement outcomes. Are of this kind quantum
algorithms, where the two outcomes are those of measuring the setting and
respectively the solution of the problem. Here the causal loops in question
provide a unified quantitative explanation of their quantum speedups. Let us
recall the related argument.

From a purely logical standpoint, the optimal quantum speedup of Grover
algorithm implies the mutually exclusive $or$ of well-defined causal loops, in
each of which the logic 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 used this knowledge to produce the solution with
correspondingly fewer computation steps.

Since quantum algorithms only involve the measurements of commuting
observables (the setting and the corresponding solution of the problem) they
are submitted to the same (classical) logic and therefore must host the
quantum superposition of the causal loops in question. Logical mutual
exclusivity of possibilities, in the quantum case, naturally becomes their
quantum superposition.

This is at clash with the fact that the ordinary quantum description of
quantum algorithms is a causal description that obviously cannot describe
causal loops. We have seen that this is so because this quantum description is
incomplete. In fact, completed by time-symmetrizing it, it turns out to be the
quantum superposition of the causal loops we are dealing with.

What should be time-symmetrized is the quantum description of the quantum
algorithm with respect to the problem-solver. We should require that the
selection of the information that specifies the sorted out pair “problem-
setting and corresponding solution” evenly shares between the initial and
final measurements in a quantum superposition of all the possible ways. In
each time-symmetrization instance, the outcome of the initial measurement
unitarily propagates forward in time until becoming the state immediately
before the final measurement; time-symmetrically, the outcome of the final
measurement propagates backward in time by the inverse of the time-forward
unitary transformation until replacing the previous outcome of the initial
measurement. This is a causal loop that changes its own past. We call it time-
symmetric because generated by a time-symmetrization.

These time-symmetric causal loops complete the quantum description of quantum
algorithms explaining their quantum speedups in a unified quantitative way.
Given a (oracle) problem, the number of computation steps (oracle queries)
needed to solve it in the optimal quantum way is that needed to solve it in
the optimal logical way benefiting of the advanced knowledge of any one of the
possible halves of information about the solution.

The time-symmetrization in question is the only possible way of generating the
causal loops whose mutually exclusive $or$ is logically implied by the sheer
existence of the quantum speedup and that the quantum algorithm must host.
Therefore, as long as logic and quantum mechanics are consistent, these causal
loops should be exempt from the paradoxes that plague the ordinary notion of
causal loop.

As we have seen, also when the process between the two correlated measurement
outcomes is not a quantum algorithm, it must be a quantum superposition of
time-symmetric causal loops. In fact, the time-symmetrization we are dealing
with is also imposed, in a redundant way, by the symmetry of the processes in
question. Quantum nolocality, when the two maximally entangled observables are
measured in the same basis (in a state of spatial separation of the respective
subsystems) is also a process of this kind. Here the causal loops, locally
going from one to the other measurement outcome via the time the two
subsystems were not yet spatially separate, play the role of the hidden
variables envisaged by Einstein and the others to locally explain quantum
nonlocality.

The role the time-symmetric causal loops play in quantum algorithms makes of
the latter scientific examples of teleological evolutions, namely evolutions
toward a goal (the solution) with an attractor in the goal they will produce
in the future (the solution again). As well known, the teleological one is a
pre-scientific notion of natural philosophy that, before the advent of modern
science, was often applied to the evolutions of the living. With that advent,
it was dismissed for the alleged absence of a physical basis. That was of
course under the notion of causal evolution that has been dominant in science
since Galileo and Newton and that the present work is challenging.

At this point it is natural to ask ourselves whether the time-symmetric causal
loops that are at the basis of the teleological character of quantum
algorithms could be the missing physical basis of the teleological character
of natural evolutions.

Of course, here we are dealing with evolutions that can be the size of the
Earth and have durations of billions of years. Scaling up to these levels the
quantum character of the microscopic world, as usually done in the search for
possible quantum characters in the mesoscopic processes of the living, would
obviously be hopeless. The strategy we have applied is kind of opposite, it is
scaling down an assumed quantum character of the most macroscopic and of
longest duration evolution that exists, that of the universe.

To this end, we must assume, with quantum cosmology, that the evolution of the
universe, a perfectly isolated system since by definition there is nothing
outside it, is quantum. There is the following chain of consequences:

(i) A quantum interpretation of the Anthropic Principle. This principle
asserts that the evolution of the universe under the fundamental physical laws
and constants builds up a correlation between the value of the physical
constants since the beginning of time and the existence – or non-existence –
of sentient life today. The assumption that the evolution of the universe is
quantum makes this building up of correlation a quantum building up.

(ii) This creates a formal identity with the quantum search algorithms that
build up a quantum correlation between the value of the problem-setting since
their beginning and the value of the solution at their end. As a consequence,
in the universe we are in, the evolution of the universe toward life occurs
with quantum speedup even if its search for life under the fundamental
physical laws and constants is random. It is thus teleological in character.

(iii) The evolution of life on Earth, in principle obtainable from that of the
universe by tracing out all the rest, should inherit the quantum speedup and,
consequently, the teleological character of the latter evolution. In fact, its
evolution times should be the same of its image in the universe’s evolution
and the resources employed cannot exceed those employed by an image that
comprises them. The evolution of life on Earth is therefore teleological; of
course, the same must consequently apply to the evolutions of species and so
on.

In conclusion, the notion of the time-symmetric causal loops turns out to be
the common physical basis to such diverse physical phenomena as the quantum
computational speedup, quantum nonlocality, and – under quantum cosmology –
the teleological character of natural evolutions. Of course, it is a notion
that goes against common beliefs both in quantum physics and the natural
sciences. If it constituted a paradigm shift, it would be that for the all of
science.

##  Acknowledgments

The author thanks Eliahu Cohen, Artur Ekert, Avshalom Elitzur, and David
Finkelstein for helpful discussions along the many years of the present
retrocausal approach to the quantum speedup, and Daniel Sheehan for organizing
the San Diego AAAS Pacific Division conferences on retrocausality, a far-
looking forum for the discussion of frontier, also unorthodox, physics. Thanks
are due to Mario Rasetti for organizing, with the sponsoring of the author’s
company Elsag Bailey, the first series of international workshops on quantum
communication and computation, the Elsag Bailey-ISI Turin Villa Gualino
workshops of the years 1991-96.

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