68 research outputs found
An Inherently Quantum Computation Paradigm: NP-complete=P Under the Hypothetical Notion of Continuous Uncomplete von Neumann Measurement
The topical quantum computation paradigm is a transposition of the Turing
machine into the quantum framework. Implementations based on this paradigm have
limitations as to the number of: qubits, computation steps, efficient quantum
algorithms (found so far). A new exclusively quantum paradigm (with no
classical counterpart) is propounded, based on the speculative notion of
continuous uncomplete von Neumann measurement. Under such a notion, NP-complete
is equal to P. This can provide a mathematical framework for the search of
implementable paradigms, possibly exploiting particle statistics.Comment: 1 figure. From the Quantum Computation and Communication Pathfinder
Meeting in Helsinki (September 26-28, 1998) - extended versio
Completing the physical representation of quantum algorithms provides a quantitative explanation of their computational speedup
The usual representation of quantum algorithms, limited to the process of
solving the problem, is physically incomplete. We complete it in three steps:
(i) extending the representation to the process of setting the problem, (ii)
relativizing the extended representation to the problem solver to whom the
problem setting must be concealed, and (iii) symmetrizing the relativized
representation for time reversal to represent the reversibility of the
underlying physical process. The third steps projects the input state of the
relativized representation, where the problem solver is completely ignorant of
the setting and thus the solution of the problem, on one where she knows half
solution (half of the information specifying it when the solution is an
unstructured bit string). Completing the physical representation shows that the
number of computation steps (oracle queries) required to solve any oracle
problem in an optimal quantum way should be that of a classical algorithm
endowed with the advanced knowledge of half solution. This fits the major
quantum algorithms known today and would solve the quantum query complexity
problem.Comment: Explicitly addressed the controversial character of the work in an
extended discussion,24 page
Discussing the explanation of the quantum speed up
In former work, we showed that a quantum algorithm is the sum over the
histories of a classical algorithm that knows in advance 50% of the information
about the solution of the problem - each history is a possible way of getting
the advanced information and a possible result of computing the missing
information. We gave a theoretical justification of this 50% advanced
information rule and checked that it holds for a large variety of quantum
algorithms. Now we discuss the theoretical justification in further detail and
counter a possible objection. We show that the rule is the generalization of a
simple, well known, explanation of quantum nonlocality - where logical
correlation between measurement outcomes is physically backed by a
causal/deterministic/local process with causality allowed to go backward in
time with backdated state vector reduction. The possible objection is that
quantum algorithms often produce the solution of the problem in an apparently
deterministic way (when their unitary part produces an eigenstate of the
observable to be measured and measurement produces the corresponding eigenvalue
- the solution - with probability 1), while the present explanation of the
speed up relies on the nondeterministic character of quantum measurement. We
show that this objection would mistake the nondeterministic production of a
definite outcome for a deterministic production.Comment: 6 pages,changed content: explanations of quantum speed up and
nonlocality related with one anothe
Non-Mechanism in Quantum Oracle Computing
A typical oracle problem is finding which software program is installed on a
computer, by running the computer and testing its input-output behaviour. The
program is randomly chosen from a set of programs known to the problem solver.
As well known, some oracle problems are solved more efficiently by using
quantum algorithms; this naturally implies changing the computer to quantum,
while the choice of the software program remains sharp. In order to highlight
the non-mechanistic origin of this higher efficiency, also the uncertainty
about which program is installed must be represented in a quantum way.Comment: 9 text pages, 3 figures in one additional ps file, manuscript of the
presentation to be held at the SILFS Workshop on Logic and Quantum
Computation, Cesena, Italy, February 16, 199
Origin of the quantum speed-up
Bob chooses a function from a set of functions and gives Alice the black box
that computes it. Alice is to find a characteristic of the function through
function evaluations. In the quantum case, the number of function evaluations
can be smaller than the minimum classically possible. The fundamental reason
for this violation of a classical limit is not known. We trace it back to a
disambiguation of the principle that measuring an observable determines one of
its eigenvalues. Representing Bob's choice of the label of the function as the
unitary transformation of a random quantum measurement outcome shows that: (i)
finding the characteristic of the function on the part of Alice is a by-product
of reconstructing Bob's choice and (ii) because of the quantum correlation
between choice and reconstruction, one cannot tell whether Bob's choice is
determined by the action of Bob (initial measurement and successive unitary
transformation) or that of Alice (further unitary transformation and final
measurement). Postulating that the determination shares evenly between the two
actions, in a uniform superposition of all the possible ways of sharing,
implies that quantum algorithms are superpositions of histories in each of
which Alice knows in advance one of the possible halves of Bob's choice.
Performing, in each history, only the function evaluations required to
classically reconstruct Bob's choice given the advanced knowledge of half of it
yields the quantum speed-up. In all the cases examined, this goes along with
interleaving function evaluations with non-computational unitary
transformations that each time maximize the amount of information about Bob's
choice acquired by Alice with function evaluation.Comment: 21 pages, 1 figure. Corrected a misleading typing error at point III)
of Section 3: "Do the same with B" replaced by "Do the same with V"; minor
distributed text improvements. arXiv admin note: text overlap with
arXiv:1101.435
An exact relation between number of black box computations required to solve an oracle problem quantumly and quantum retrocausality
We investigate the reason for the quantum speedup -- quantum algorithms
requiring fewer computation steps than their classical counterparts. We extend
their representation to the process of setting the problem. The initial
measurement selects a setting at random, Bob (the problem setter) unitarily
changes it into the desired one. This representation is to Bob and any external
observer, it cannot be to Alice (the problem solver). It would tell her the
function computed by the black box, which to her should be hidden inside it. We
resort to relational quantum mechanics. To Alice, the projection of the quantum
state due to the initial measurement is retarded at the end of her problem
solving action. To her, the algorithm input state remains one of complete
ignorance of the setting. By black box computations, she unitarily sends it
into the output state that, for each possible setting, encodes the
corresponding solution, acquired by the final measurement. We show that we can
ascribe to the final measurement the selection of any part -- say the R-th part
-- of the random outcome of the initial measurement. This projects the input
state to Alice on a state of lower entropy where she knows a corresponding part
of the problem setting. The quantum algorithm is a sum over classical histories
in each of which Alice, knowing in advance one of the R-th parts of the
setting, performs the black box computations still required to identify the
solution. Given an oracle problem and a value of R, this retrocausality model
provides the number of black box computations required to solve it. Conversely,
given a known quantum algorithm, it yields the value of R that explains its
speed up. R = 1/2 always yields the number of black box computations required
by an existing quantum algorithm and the order of magnitude of the number
required by optimal one.Comment: Added a short history of the birth of quantum computation, a
positioning of the work, and various clarifications, 29 pages. arXiv admin
note: text overlap with arXiv:1308.507
Parallel Quantum Computation, the Library of Babel and Quantum Measurement as the Efficient Librarian
The complementary roles played by parallel quantum computation and quantum
measurement in originating the quantum speed-up are illustrated through an
analogy with a famous metaphor by J.L. Borges.Comment: 2 pages, RevTex, no figures. An excerpt from a longer (referenced)
work, with a justification of the quantum speed-u
Quantum problem solving as simultaneous computation
I provide an alternative way of seeing quantum computation. First, I describe
an idealized classical problem solving machine that, thanks to a many body
interaction, reversibly and nondeterministically produces the solution of the
problem under the simultaneous influence of all the problem constraints. This
requires a perfectly accurate, rigid, and reversible relation between the
coordinates of the machine parts - the machine can be considered the many body
generalization of another perfect machine, the bounching ball model of
reversible computation. The mathematical description of the machine, as it is,
is applicable to quantum problem solving, an extension of the quantum
algorithms that comprises the physical representation of the problem-solution
interdependence. The perfect relation between the coordinates of the machine
parts is transferred to the populations of the reduced density operators of the
parts of the computer register. The solution of the problem is reversibly and
nondeterministically produced under the simultaneous influence of the state
before measurement and the quantum principle. At the light of the present
notion of simultaneous computation, the quantum speed up turns out to be
"precognition" of the solution, namely the reduction of the initial ignorance
of the solution due to backdating, to before running the algorithm, a
time-symmetric part of the state vector reduction on the solution; as such, it
is bounded by state vector reduction through an entropic inequality. PACS
numbers: 03.67.Lx, 01.55.+b, 01.70.+wComment: 12 pages, part of a work to be published on IJT
An explanation of the quantum speed up
In former work, we showed that a quantum algorithm requires the number of
operations (oracle's queries) of a classical algorithm that knows in advance
50% of the information that specifies the solution of the problem. We gave a
preliminary theoretical justification of this "50% rule" and checked that the
rule holds for a variety of quantum algorithms. Now, we make explicit the
information about the solution available to the algorithm throughout the
computation. The final projection on the solution becomes acquisition of the
knowledge of the solution on the part of the algorithm. Backdating to before
running the algorithm a time-symmetric part of this projection, feeds back to
the input of the computation 50% of the information acquired by reading the
solution.Comment: 15 pages, resubmitted to Phys Rev A, the 50% rule is now related to
backdating to before running the algorithm a time-symmetric part of the final
projection on the solution
A Quantum Logic Gate Representation of Quantum Measurement: Reversing and Unifying the Two Steps of von Neumann's Model
In former work, quantum computation has been shown to be a problem solving
process essentially affected by both the reversible dynamics leading to the
state before measurement, and the logical-mathematical constraints introduced
by quantum measurement (in particular, the constraint that there is only one
measurement outcome). This dual influence, originated by independent initial
and final conditions, justifies the quantum computation speed-up and is not
representable inside dynamics, namely as a one-way propagation. In this work,
we reformulate von Neumann's model of quantum measurement at the light of above
findings. We embed it in a broader representation based on the quantum logic
gate formalism and capable of describing the interplay between dynamical and
non-dynamical constraints. The two steps of the original model, namely (1)
dynamically reaching a complete entanglement between pointer and quantum object
and (2) enforcing the one-outcome-constraint, are unified and reversed. By
representing step (2) right from the start, the same dynamics of step (1)
yields a probability distribution of mutually exclusive measurement outcomes.
This appears to be a more accurate and complete representation of quantum
measurement. PACS: 03.67.-a, 03.67.Lx, 03.65.BzComment: 17 pages, RevTex, 1 PostScript file with figure, submitted to Int. J.
Theor. Phy
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