Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
quantum mechanical behavior of small objects in devising computers, instead of
neutralizing it.
(ii) As another motivation, one can invoke highly speculative, but intriguing,
conjectures that our brain is in fact a quantum computer. For example, the recent
progress in writing efficient chess playing software (Deep Blue) shows that to sim-
ulate the world championship level using only classical algorithms, one has to be
able to analyze about 106 positions/sec and use about 1010 memory bytes. Since
the characteristic time of neuronal processing is about 10−3 sec, it is very difficult
1Talk at the Bourbaki Seminar, June 1999.
1
2
to explain how the classical brain could possibly do the job and play chess as suc-
cessfully as Kasparov does. A less spectacular, but not less resource consuming
task, is speech generation and perception, which is routinely done by billions of hu-
man brains, but still presents a formidable challenge for modern computers using
classical algorithms.
Computational complexity of cognitive tasks has several sources: basic variables
can be fields; a restricted amount of small blocks can combine into exponentially
growing trees of alternatives; databases of incompressible information have to be
stored and searched.
Two paradigms have been developed to cope with these difficulties: logic–like
languages and combinatorial algorithms, and statistical matching of observed data
to an unobserved model (see D. Mumford’s paper [Mu] for a lucid discussion of the
second paradigm.)
In many cases, the second strategy efficiently supports an acceptable perfor-
mance, butusuallycannotachieveexcellencyoftheDeepBluelevel. Bothparadigms
requirehugecomputationalresources, anditisnotclear, howtheycanbeorganized,
unless hardware allows massive parallel computing.
The idea of “quantum parallelism” (see sec. 2 below) is an appealing theoretical
alternative. However, it is not at all clear that it can be made compatible with
the available experimental evidence, which depicts the central nervous system as a
distinctly classical device.
The following way out might be worth exploring. The implementation of effi-
cient quantum algorithms which have been studied so far can be provided by one,
or several, quantum chips (registers) controlled by a classical computer. A very
considerable part of the overall computing job, besides controlling quantum chips,
is also assigned to the classical computer. Analyzing a physical device of such
architecture, we would have direct access to its classical component (electrical or
neuronal network), whereas locating its quantum components might constitute a
considerable challenge. For example, quantum chips in the brain might be rep-
resented by macromolecules of the type that were considered in some theoretical
models for high temperature superconductivity.
The difficulties are seemingly increased by the fact that quantum measurements
produce non–deterministic outcomes. Actually, one could try to use this to one’s
advantage, because there exist situations where we can distinguish the quantum
randomness from the classical one by analyzing the probability distributions and
using the Bell–type inequalities. With hindsight, one recognizes in Bell’s setup
the first example of the game–like situation where quantum players can behave
demonstrably more efficiently that the classical ones (cf. the description of this
setup in [Ts], pp. 52–54).
It would be extremely interesting to devise an experimental setting purporting
to show that some fragments of the central nervous system relevant for information
3
processing can in fact be in a quantum superposition of classical states.
(iii) Finally, we turn to mathematics. One can argue that nowadays one does
not even need additional motivation, given the predominant mood prescribing the
quantization of “everything that moves”. Quantum groups, quantum cohomology,
quantum invariants of knots etc come to mind. This actually seemed to be the
primary motivation before 1994, when P. Shor ([Sh]) devised the first quantum
algorithm showing that prime factorization can be done on quantum computers in
polynomialtime, that is, considerably faster than by any known classical algorithm.
(P. Shor’s work was inspired by the earlier work [Si] of D. Simon). Shor’s paper
gave a new boost to the subject. Another beautiful result due to L. Grover ([Gro])
is that a quantum search among N objects can be done in c√N steps. A. Kitaev
[Ki1] devised new quantum algorithms for computing stabilizers of abelian group
actions; his work was preceded by that of D. Boneh and R. Lipton [BoL], who
treated the more general problem by a modification of Shor’s method (cf. also
[Gri]). At least as important as the results themselves, are the tools invented by
Shor, Grover, and Kitaev.
Shor’s work is the central subject of this lecture. It is explained in sec. 4. This
explanation follows the discussion of the general principles of quantum computing
and massive quantum parallelism in sec. 2, and of four quantum subroutines,
including Grover’s searching algorithm, in sec. 3. The second of these subroutines
involving quantum computations of classical computable functions shows how to
cope with the basic issue of quantum reversibility vs classical irreversibility. For
more on this, see [Ben1] and [Ben2]. The opening sec. 1 contains a brief report on
the classical theory of computability. I made some effort to express certain notions
of computer science, including P/NP, in the language of mainstream mathematics.
The last section 5 discusses Kolmogorov complexity in the context of classical and
quantum computations.
Last, but not least, the hardware for quantum computing does not exist as yet:
see 3.3 below for a brief discussion of the first attempts to engineer it. The quantum
algorithmsinvented and studied up to now will stimulatethe search of technological
implementationwhich–ifsuccessful –willcertainlycorrectourpresent understand-
ing of quantum computing and quantum complexity.
Acknowledgements. IamgratefultoAleshaKitaev, DavidMumford, andDimitri
Manin for their interest and remarks on the earlier version of this report. Many of
their suggestions are incorporated in the text.
1. Classical theory of computation
1.1. Constructive universe. In this section I deal only with deterministic
computations, which can be modelled by classical discrete time dynamical systems
and subsequently quantized.
4
Alan Turing undertook the microscopic analysis of the intuitive idea of algorith-
mic computation. In a sense, he found its genetic code. The atom of information
is one bit, the atomary operators can be chosen to act upon one/two bits and to
produce the outputs of the same small size. Finally, the sequence of operations is
strictly determined by the local environment of bounded size, again several bits.
For a change, I proceed in the reverse direction, and start this section with a
presentation of the macrocosm of the classical theory of computation. Categorical