Quantum bio informatics v

Quantum

Bio-Informatics V

Proceedings of Quantum Bio-Informatics 2011


QP–PQ: Quantum Probability and White Noise Analysis*
Managing Editor: W. Freudenberg
Advisory Board Members: L. Accardi, T. Hida, R. Hudson and
K. R. Parthasarathy

QP–PQ: Quantum Probability and White Noise Analysis
Vol. 30:

Quantum Bio-Informatics V
eds. L. Accardi, W. Freudenberg and M. Ohya

Vol. 29:

Quantum Probability and Related Topics
eds. L. Accardi and F. Fagnola

Vol. 28:

Quantum Bio-Informatics IV
From Quantum Information to Bio-Informatics
eds. L. Accardi, W. Freudenberg and M. Ohya

Vol. 27:

Quantum Probability and Related Topics
eds. R. Rebolledo and M. Orszag

Vol. 26:

Quantum Bio-Informatics III
From Quantum Information to Bio-Informatics
eds. L. Accardi, W. Freudenberg and M. Ohya

Vol. 25:

Quantum Probability and Infinite Dimensional Analysis
Proceedings of the 29th Conference
eds. H. Ouerdiane and A. Barhoumi

Vol. 24:

Quantum Bio-Informatics II
From Quantum Information to Bio-informatics
eds. L. Accardi, W. Freudenberg and M. Ohya

Vol. 23:

Quantum Probability and Related Topics
eds. J. C. García, R. Quezada and S. B. Sontz

Vol. 22:

Infinite Dimensional Stochastic Analysis

eds. A. N. Sengupta and P. Sundar

Vol. 21:

Quantum Bio-Informatics
From Quantum Information to Bio-Informatics
eds. L. Accardi, W. Freudenberg and M. Ohya

Vol. 20:

Quantum Probability and Infinite Dimensional Analysis
eds. L. Accardi, W. Freudenberg and M. Schürmann

Vol. 19:

Quantum Information and Computing
eds. L. Accardi, M. Ohya and N. Watanabe

Vol. 18:

Quantum Probability and Infinite-Dimensional Analysis
From Foundations to Applications
eds. M. Schürmann and U. Franz

*For the complete list of the published titles in this series, please visit:
www.worldscientific.com/series/qp-pq


QP–PQ
Quantum Probability and White Noise Analysis

Volume XXX

Quantum

Bio-Informatics V

Proceedings of Quantum Bio-Informatics 2011
Tokyo University of Science, Japan

7 – 12 March 2011

Editors

Luigi Accardi
Università di Roma “Tor Vergata”, Italy

Wolfgang Freudenberg
Brandenburgische Technische Universität Cottbus, Germany

Masanori Ohya
Tokyo University of Science, Japan

World Scientific
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QP–PQ: Quantum Probability and White Noise Analysis — Vol. 30
QUANTUM BIO-INFORMATICS V
Proceedings of the Quantum Bio-Informatics 2011
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Printed in Singapore by Mainland Press Pte Ltd.


v

PREFACE
This volume is based on the fifth international conference of quantum
bio-informatics held at the QBI Center of Tokyo University of Sciences.
The purpose of the conference is towards new stage making interdisciplinary bridges in mathematics, physics, information and life sciences, in
particular, research for new paradigm for information science and life science on the basis of quantum theory.
More than 100 researchers in various fields such as mathematics, physics,
information and biology come from all over the world. The conference was
held for nearly one week, and we had a lot of fruitful discussion. In this fifth

conference, particular attention is come up on quantum entanglement, simulation of bio-systems, brain function, quantum like dynamics and adaptive
systems. Most of speakers gave care to the relation between their own topics
and the mystery of life.
The papers submitted in this volume are all refereed, whose contents
are related to one of the following subjects:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)

Mathematics of Cryptography and its related topics
Quantum algorithm and computation
Quantum entanglement
Quantum entropy and information dynamics
Quantum dynamics and time operator
Stochastic dynamics and white noise analysis
Brain activity
Quantum like models and PD game
Quantum physics and superconductivity
Quantum tomography and sufficiency
Adaptation in Plants
Alignment of sequences

Luigi Accardi
Wolfgang Freudenberg
Masanori Ohya


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vii

Five years of QBIC
Masanori Ohya
Department of Information Sciences,
Tokyo University of Science, Japan

1. Aims of QBIC
The quantum bio-informatics center (QBIC) was founded in 2006 towards
new stage making interdisciplinary bridges in philosophy, mathematics,
physics, information and life sciences. Our research center (QBIC) tries to
nd a new paradigm for information science and life science on the basis of
quantum mathematics. Several researchers more than 100 on mathematics,
physics, information theory and biology who are interested in mathematical
study worked together in QBIC during this ve years from 2006 to 2010.
To solve the mystery of life is one of the most interesting problems in
21 century. After discovery of DNA, people believes that one key to read
the riddle will be hidden in the process how information of life is stored
and its change and transmission are made. Concerning the information
transmission (communication), quantum information opened a new door
and is expected to understand new aspects of existence in-itself.
More concretely, the immensely long DNA, sequence of four bases in

the genome, contains information on life, and decoding or changing this
sequence is involved in the expression and control of life. In quantum
information, meanwhile, we produce various “information” by sequences of
two quantum states, and think of ways of processing, communicating and
controlling them. It is thought that the problems we can process in time
“T” using a conventional computer can be processed in time nearly “log T”
using a quantum computer. However, the transmission and processing of
information in the living body might be much faster than those of quantum
computer and communication.
Seen from this very basic viewpoint, developing the mathematical principles that have been found in quantum information should be useful in
constructing mathematical principles for life sciences, which have not been
established yet. The mechanism of processing information in life is also
expected to be useful for the further growth of quantum information.


viii

Figure 1.

QBIC Research Project

The way of our research is
(1) to return to the starting point of bio-informatics and quantum information, elds that and to solve these fundamental problems, and
(2) to seriously attempt mutual interaction between the two, with a view
to enumerating and solving the many fundamental problems they entail.
In our view, there is no similar research center in the world to return
to the basics of bio-information and quantum information and to focus on
the correlation between the two with a view to new development of each.
Our way with targets and goal are described in the gure below:
We had more than 200 papers published in this ve years. Most of them

have rst published in the ve proceedings of International Conference held
in Tokyo University of Science.


ix

I will here review basic results of some achievements in this ve years
of QBIC.
2. Solving the mystery of life
Solving the mystery of life requires several stages (1) Metaphysical, (2)
Biological & Physical, (3) Mathematical. The works of the stages (1) and
(2) have been done for a long time even in the "new" life science, that is,
many philosophical considerations and various experiments have been done,
and several (tentative) theories have been made. However it is also true we
had not a basic mathematical rule (theory) in the life science so that many
researchers could accept it as quantum mechanics.
In order to make such a theory, we have to try to develop fundamentals
in various elds (mathematics, physics, information theory) with intention
to the goal, i.e., nding the rst principle understanding the life itself.
Biological systems are open systems. Biological systems are multi—
component and context dependent. Biological subsystems in a biological
system are locally interacting each other. Therefore the state of the biological systems depends on its surrounding and the eve of itself. These
observations entail that biological systems are adaptive. We have to nd a
mathematical rule to describe all of those.
However, in order to make our dream realize, we have to develop each
eld such as mathematics, quantum physics, information, structural biology
and bio-informatics so that we can use the fructication to achieve new
paradigm as discussed above.
3. Some works appeared in conference of QBIC
I itemize some works appeared in the conferences of QBIC during this ve

years. It is beyond the introduction and my ability to review all works
appeared in the ve years conferences, so that I only mention some mathematical trials somehow related to life sciences. The fruitful results of
various works can been seen in the series of the ve proceedings of QBIC
conferences.
3.1. Examples of researches in QBIC
3.1.1. Concerning <I> of the Figure above
• White noise, stochastic analysis and some applications to DDS
(Hida, Streit, SiSi, Accardi, Volovich, Smolyanov, Fichtner, In-


x

•

•
•
•

•

oue, Iriyama, Hara, Ohya); Further developments of Hida calculas
and its various applications have been considered, e.g., analysis of
drug delivery systems.
Mathematical physics and noncommutative analysis (Araki, Accardi, Arai, Belavkin, Jamiolkowski, Ojima, Petz, Hiai) have been
developed so that we van apply to life sciences, for examples, quantum tomograhy, micro-macro duality.
Statistics with symmetry (Tomizawa, Miyamoto, Tahata), which
enables to analyze several fuctions of human body.
Supercoductor =, KS model and its development (Kamimura,
Sakata, Ushio), which will be related to a realization of qubit.
Fundamental problems in quantum physics - Bell’s inequality,

adaptive dynamics, quantum like systems, micro-macro duality,
nonequilibrium dynamics (Accardi, Asano, Khrennikov, Volovich,
Ojima, Suzuki, Oryu, Ohya)
Ulitimate secure and fast crypto-algorithm can be found (Acradi,
Regoli, Iriyama, Ohya).

3.2. Concerning <II>
• Protein folding and simulation with brownian molecular dynamics
(Yamato, Ando, Takeda, Im).
• Code structure of genes and works of cis-elements (Miyazaki, Sato,
Khrennikov, Regoli, Wanke).
• Signal network of envitonmental sensing and adaptation in plants
(Kuchitsu).
• Study of biosystems by information measures (R. Belavkin, A. Accardi, Im, Sato, Hara, Ohya), e.g., the most accurate alignment
method is founded in QBIC.
3.3. Concerning <III>
• Quantum teleportation (Fichtner, Freudenberg, Kosakowski,
Asano, Tanaka, Ohya), in which new mathematics and its realization are discussed, where the teleportation process can be linear
and can use all entangled states.
• Quantum entanglement (Belavkin, Jamiolkowski, Accardi, Matsuoka, Kosakowski, Chruscinski, Majewski, Michalski, Hirota,
Ohya), e.g., (1) we can treat even innite systems; (2) we can
treat all correlations including claasical one; (3) .


xi

• Quantum algorithm solving the NPC problem (Volovich, Accardi,
Iriyama, Ohya) and its development; e.g., Our SAT alogithm can
be used to solve factoring problem of Shor.
• Realization of qubit (Takayanagi, Morinaga).

• Quantum entropy and some applications (Accardi, Belavkin, Petz,
Hiai, Araki, Iriyama, Matsuoka, Kosakowski,Watanabe, Suzuki,
Ohya), e.g., Entropy production in linear respose dynamics, entropy production in photosythesis.
3.4. Concerning <IV>
• Generalized Turing machine (Volovich, Iriyama, Ohya) is proposed.
• New description of chaos(Kosakowski, Togawa, Volovich, Inoue,
Ohya) =, Adaptive dynamics=, Application of chaos dynamics
to the classication of HIV-1 and In uenza A viruses (Sato, Tanabe, Hara)
• Non-Kolmogorov probability and its applications; Adaptive dynamics and lifting are applied to nd new probability law (Khrennikov, Accardi, Asano, Basieva, Tanaka, Ohya, Yamato)
• Mathematical model explaining the fuctions of brain are proposed
in Fock space (K.-H. Fichtner, L. Fichtner, Freudenberg, Inoue,
Ohya).
• Quantum tomography and su!ciency (Jamiolkowski, Petz, Matsuoka, Watanabe, Ohya)
• Quantum algorithm solving the protein folding is studied (Goto,
Iriyama, Yamato, Ohya)
• New alignment (MTRAP) of amino acids was proposed in terms of
entanglement (Sato, Hara, Ohya)
• Alignment by means of quantum algorithm was made (Iriyama,
Sato,Ohya)
• Game theory in non-Kolmogorovian probanility theory has been
proposed (Khrennikov, Basieva, Asano, Tanaka, Ohya)
Remark New science must be based on new philosophy. Science (theory) without philosophy is fragile. The 21 century is the era not for new
technology but for new philosophy crossing our existence.
Finally, I like to ask all of you interested in the QBIC conference: In
order to understand the mystery of life and various existence, you dare to
have will and intention to use all materials you obtained in several dierent
elds.


xii

References
1. Accardi, L., Freudenberg, W., Ohya, M., Quantum Bio-Informatics (Quantum
Probability and White Noise Analysis, Vol. 21), World Scientic, 2008
2. Accardi, L., Freudenberg, W., Ohya, M., Quantum Bio-Informatics II (Quantum Probability and White Noise Analysis, Vol. 24), World Scientic, , 2009
3. Accardi, L., Freudenberg, W., Ohya, M., Quantum Bio-Informatics III (Quantum Probability and White Noise Analysis, Vol. 26), World Scientic, 2010
4. Accardi, L., Freudenberg, W., Ohya, M., Quantum Bio-Informatics IV (Quantum Probability and White Noise Analysis, Vol. 28), World Scientic, 2011
5. Accardi, L., Freudenberg, W., Ohya, M., Quantum Bio-Informatics V (Quantum Probability and White Noise Analysis), World Scientic, this volume
6. Ohya, M., Volovich, I., Mathematical Foundations of Quantum Information
and Computation and Its Applications to Nano- and Bio-systems, Springer,
2011


xiii

CONTENTS
Preface
Five Years of QBIC
Complexity Considerations Quantum Computation
Luigi Accardi

v
vii
1

Quantum Markov Chains and Ising Model on Cayley Tree
Luigi Accardi, Farrukh Mukhamedov and Mansoor Saburov

15

Mathematical Aspects of Conserved Quantities in a General
Class of Quantum Systems
Asao Arai

25

Oscillations and Rolling for Duffing’s Equation
Irina Ya. Aref ’eva, Evgeny V. Piskovskiy and Igor V. Volovich

37

General Formalism of Decision Making Based on Theory of Open
Quantum Systems
Masanori Asano, Masanori Ohya, Irina Basieva and
Andrei Khrennikov

49

Quantum-Like Representation of Non-Bayesian Inference
Masanori Asano, Masanori Ohya, Irina Basieva,
Andrei Khrennikov and Yoshiharu Tanaka

57

A Mathematical Treatment of Joint and Conditional Probability
Masanori Asano, Masanori Ohya, Yoshiharu Tanaka,
Ichiro Yamato, Irina Besieva and Andrei Khrennikov

69

Entangled States Preparation in Clusters of Three Resonantly
Interacting Fluorescent Particles
Irina Basieva

85


xiv

Minimum of Information Distance Criterion for Optimal Control of
Mutation Rate in Evolutionary Systems
Roman V. Belavkin

95

On Non-Markovian Quantum Evolution
Dariusz Chru´sci´
nski and Andrzej Kossakowski

117

High Density Limit of the Distribution of the Outcome of
EEG-Measurements
Karl-Heinz Fichtner, Lars Fichtner, Wolfgang Freudenberg
and Masanori Ohya

127

Internal Noise of EEG-Measurements and Certain Boson Systems
Karl-Heinz Fichtner, Lars Fichtner, Kei Inoue and

Masanori Ohya

143

Skew Information and Uncertainty Relation
Shigeru Furuichi and Kenjiro Yanagi

159

Multiple-Photon Absorption Attack on Entanglement-Based
Quantum Key Distribution Protocols
Guillaume Adenier, Noboru Watanabe, Irina Basieva and
Andrei Khrennikov

171

Protein Sequence Alignment Taking the Structure of Peptide Bond
Toshihide Hara, Keiko Sato and Masanori Ohya

181

Space - Time - Noise (Raum - Zeit - Rauschen)
Takeyuki Hida

187

Quantum Algorithm for Protein Folding and Its Computational
Complexity
Satoshi Iriyama, Masanori Ohya and Ichiro Yamato

193

On Effective Procedures in Analyzing of Quantum Operations and
Processes
Andrzej Jamiolkowski

203

On Numerical Ranges of Operators
Jacek Jurkowski

217


xv

Partial Roc Reveals Superiority of Mutual Rank of Pearson’s
Correlation Coefficient as a Coexpression Measure to Elucidate
Functional Association of Genes
Takeshi Obayashi and Kengo Kinoshita

229

QFT and Hadronic World as Dynamical Bases of Natural History
Izumi Ojima

237

Long-Range Property in Time-Dependent Interaction with
Three-Body Structure and New Aspect

Shinsho Oryu

253

Kinetic Isotope Effect on Transport Mediated by
CLC-Type H+ /Cl− Exchangers
Alessandra Picollo, Mattia Malvezzi and Alessio Accardi

271

On Positive Maps; Finite Dimensional Case
Wladyslaw A. Majewski

281

A New Noise Depending on a Space Parameter and Its Application
Si Si and Win Win Htay

291

Schr¨odinger Type Semigroups via Feynman Formulae and All That
Oleg G. Smolyanov

301

Entropy Production and Non-equilibrium Steady States
Masuo Suzuki

315

Test and Measure on Difference of Asymmetry Between Several
Square Tables and Application to Medical Data
Kouji Tahata, Kouji Yamamoto, Nobuko Miyamoto and
Sadao Tomizawa

327

Functional Mechanics and Kinetic Equations
Anton S. Trushechkin

339

Implications of DNA-Nanostructures by Hoogsteen-Dinucleotides on
Transcription Factor Binding
Dierk Wanke, Luise H. Brand, Nina M. Fischer,
Florian Peschke, Joachim Kilian and Kenneth W. Berendzen

351


xvi

On Treatment of Gaussian Communication Process by Quantum
Entropies
Noboru Watanabe

363

Importance of Excluded Volume and Hydrodynamic Interactions on
Macromolecular Diffusion in vivo

Tadashi Ando and Jeffrey Skolnick

375

Self-Repelling Fractional Brownian Motion - A Generalized
Edwards Model for Chain Polymers
Jinky Bornales, Maria Jo˜
ao Oliveira and Ludwig Streit

389

Signaling Networks Involving Reactive Oxygen Species and Ca2+
in Plants
Kazuyuki Kuchitsu

403

A Novel Measure for Finding Disease-Specific Genes from the
Biomedical Literature
Yeondae Kwon, Hideaki Sugawara, Shogo Shimizu and
Satoru Miyazaki

409

Three-Tangle and Three-π for a Class of Tripartite Mixed States
Teng Ma and Shao-Ming Fei

425

Energy Flow and Information Flow in Superconducting Qubit

Measurement Process
Hayato Nakano

435

How Can Steganography be an Interpretation of the Redundancy
in pre–mRNA Ribbon?
Massimo Regoli

447

Counter-factual Phenomenon in Quantum Mechanics
Yutaka Shikano

463

From Structure and Function of Proteins Toward in Silico Biology
Ichiro Yamato

473


Quantum Bio-Informatics V
c 2013 World Scientific Publishing Co. Pte. Ltd.
pp. 1–13

COMPLEXITY CONSIDERATIONS QUANTUM
COMPUTATION

LUIGI ACCARDI

Centro Vito Volterra,
Università degli Studi di Roma “Tor Vergata”, Roma, Italy,
E-mail:
It is usually calimed that quantum computer can outperform classical computer.
Is this statement true? We discuss this issue, not in general, but in the context of
the most famous algorithm of quantum computation: Shor’s algorithm.

1. Introduction
Shor’s algorithm is supposed to achieve integer factorization faster than
classical algorithms. In order to discuss this issue we shortly review Shor’s
algorithm and the strictly related Simon’s period-nding algorithm (see
section (2)). Then we argue that, since quantum computer is an analogical
machine, the complexity estimates on quantum algorithms should involve
the analysis of the concrete implementation of the operations whose use is
required by these algorithms. An outline of this analysis is done in section
(3).
Finally, in order to compare the performance of Shor’s algorithm with some
classical probabilistic factorization algorithms, the latter ones are shortly
reviewed in section (4).
The essence of the factorization problem can be described as follows:
Given a natural integer N = pq, which is the product of two primes p 6= q,
nd p and q. If p and q are large and satisfy additional diophantine conditions, the problem is hard and this di!culty has been exploited by a famous
cryptographic algorithm.
A classical argument of number theory reduces the factorization problem
to the problem of nding the period of the function a 7$ y a (mod N ) (see
section (4)).
At the moment there is no classical algorithm that can nd the period of
the function a 7$ ya (mod N ) (hence the factorization problem) in a num-

1

2

ber of steps of order O(log N )
It was however known a classical probabilistic algorithm that achieves this
goal, not exactly, but with probability of order O(1/ log N ).
D. Simon [Sim94] proposed a quantum algorithm that allows to nd, using
O(log N ) operations and with probability of order O(1/ log N ), the period
of an arbitrary function f : {0, . . . , N  1} $ {0, . . . , N  1} each value of
which can be calculated with an algorithm of complexity of order O(log N )
(with respect to some standard measure of complexity).
P. W. Shor [Sho94a] applied Simon’s period nding algorithm to the function a 7$ y a (mod N ) to construct a quantum factorization algorithm which
needs a number of steps of the same order of magnitude as the classical
probabilistic algorithm and achieves the same result with a probability of
the same order of magnitude.
Contrarily to the classical probabilistic algorithm, Simon’s (hence
Shor’s) algorithm is based on additional physical assumptions the experimental verication of some of which is at the moment not available.
The goal of the present note is to point out some of these assumptions.
Some of the considerations in the present notes are contained in the unpublished lectures of the author at the Volterra-CIRM International School
"Quantum Computer and Quantum Information", Trento, July 25-31, 2001.
2. Simon’s period-nding quantum algorithm
Given N 5 N let f : {0, 1, . . . , N  1} $ {0, 1, . . . , N  1} be a periodic
function with period r, i.e. r is the smallest number in the domain of f
such that
f (x) = f (x +N r)

;

;x 5 {0, 1, . . . , N  1}

(1)

where the symbol +N denotes addition modulo N .
Since addition is taken modulo N , if (1) is satised by r, then it is also
satised by N  r. Thus by denition of period, one must have
r  N  r / r  N/2
Suppose that we know that f is an e!ciently computable function, i.e.
that, for each x, f (x) can be e!ciently computed (i.e. in a number of steps
which is polynomial in the number of digits of x).
If these are the only informations on f , the only way to nd exactly the
period is to carry out an exhaustive search. This requires to calculate f (x)
for a set of x of cardinality N/2. This algorithm is exponential in the


3

number of bits required to specify N , which is of order log N .
In absence of exact results one turns to probabilistic algorithms, either
classical or quantum.
As we have seen the performances of the two are essentially the same.
2.1. Ingredients of Simon’s quantum period nding
algorithm (QPFA)
The state space of this algorithm is
n

n

H2
H2  (C2 )

n
(C2 )
n

(2)

where H := C2 is the so—called q—bit space (the reason why, in (2), one
n
uses two copies of the space H2 is explained in Step (3) of the algorithm
described in section (2.2)).
In the space C2 we x the computational basis,
 ¶
 ¶
0
1
|0i :=
;
|1i :=
1
0
which induces the basis (still called computational) in (C2 )
n
|%1 i
· · ·
|%n i =: |%1 , . . . , %n i ; %j 5 {0, 1}

(3)

Identifying the binary string (%1 , . . . , %n ) to the binary expansion of a natural integer through the formula
x=

N
X
j=1

%j 2j1

; x 5 {0, . . . , N  1 = 2n  1}; %j 5 {0, 1}

(4)

and extending this notation to the corresponding vectors:
|xi = |%1 , . . . , %n i ; x 5 {0, . . . , N  1}; %j 5 {0, 1}

(5)

we will use both the binary and the decimal notation so that the vectors of
the form
|xi
|yi = |%1 , . . . , %n i
|1 , . . . , n i ; x, y 5 {0, . . . , N 1};
dene the computational basis for the state space C2

n

%j , j 5 {0, 1}
(6)
2n
C .

4

2.2. Steps of Simon’s quantum period nding algorithm
(QPFA)
Step (1).
The initial state of the quantum system is,
n

n

|0in
|0in 5 C2
C2  CN
CN

(7)

i.e. all 2n q—bits are in the state |0i.
Step (2).
Apply to the initial state the unitary operator
UH := H
n
1
where H is the discrete Fourier (or Hadamard) transform on C2 dened by
linear extension of the map:
1
1
|0i 7$ s (|0i + |1i) ; |1i 7$ s (|0i  |1i)
2

2
and
H
n := H
H
· · ·
H

n—times

Since
N1
1 X
|xi
H
n |0in = s
N x=0

the action of UH brings the initial state to

N 1
1 X
# o := UH |0in = s
|xi|0in
N x=0

(8)

Step (3a).
Among the unitary extensions of the partial isometry dened by

|xi|0i 7$ |xi|f (x)i

;

x 5 {0, . . . , N  1}

(9)

choose one, denoted Uf , that can be physically realized.
Step (3b).
Realize the physical implementation of Uf .
Step (3c).
Apply to the state (8) the unitary operator Uf . This gives
N1
1 X
Uf #o =: # = s
|xi|f (x)i
N x=0

(10)


5

Step (4a).
Fix arbitrarily u 5 {0, . . . , N  1} and construct the lter dened by the
projection
P := 1n
|uihu|

(11)

Step (4b).
Apply the lter (11) to the quantum state described by the vector (10). This
amounts to lter all the elements of the ensemble (10) for which f (x) = u
and to suppress all the remaining ones.
Theoretical conclusion from Step 4b
By by applying the Luders—Zumino formula of the quantum theory of measurement quantum information theorists conclude that the new quantum
state of the total system is the one associated to the vector:
|!ih!| :=

P |#ih#|P
|P #i hP #|
=
T r(P |#ih#|)
kP #k kP #k

(12)

where # is dened by (10) so that:
N 1
1
1 X
hu|f (x)i|xi|ui = s
P# = s
N x=0
N

X

x5{0,1,...,N1},f (x)=u

|xi|ui

(13)

Notice that
|{x 5 {0, 1, . . . , N  1}, f (x) = u}|
|f 1 (u)|
=
(14)
N
N
We will discuss only the case in which f satises the following additional
conditions:
kP #k2 =

Assumption 2.1. If f is injective on the interval [0, r).
Assumption 2.2. r divides N exactly, i.e.
{0, . . . , N  1}

independently of u 5

|f 1 (u)| =: M = N/r 5 N

(15)

In this case from (13), (14) one deduces that
!=

N 1
M1
1 X hu|f (x)i
1 X
P#
|du + jri|ui
=s
|xi|ui = s
k P# k
N x=0 k P # k
M j=0

(16)

where du + jr, for j = 0, 1, 2 . . . M  1, are all the values of x for which
f (x) = u and du < r.
Step (5a).


6

Construct an apparatus implementing physically the unitary operator
UF T
1n , where UF T is the discrete Fourier transform, given by:
N1
1 X i2kx/N
e
|ki
UF T |xi = s
N k=0

Step (5b).
Apply to ! the unitary operator UF T
1n . This leads to the state
N/r1
X
1
p
UF T |du + jri|ui
N/r j=0

N/r1 N 1
X X
1
1
= s p
ei2kdu /N ei2kjr/N |ki|ui
N
N/r j=0 k=0
3
4
N/r1
N1
X
1 XC 1
p
ei2jrk/N D ei2kdu /N |ki|ui
= s
r
N/r

j=0
k=0

= (UF T
1n )!

(17)

Since, if kr/N is not an integer, then
N/r1

X
j=0

ei2jrk/N =

ei2(N/r)rk/N  1
=0
ei2rk/N  1

the non zero terms in the j—sum are precisely those for which
kr/N 5 N
i.e. those for which k is a multiple of M = N/r. Summing up: at the end
of the 5—th step the state of the quantum system is:
X
1
 := (UF T
1y )! = s
|ki|ui (18)
r

{k5{0,...,N 1}:k is a multiple of M=N/r}
Step (6).
The nal step of the algorithm is usually described in the quantum computer literature as follows (see [St97]):
. . . The nal state of the x register is now measured, and we see that the
value obtained must be a multiple of w/r . . . (In our notations w/r = N/r)
In other words, as a result of a measurement, one obtains an integer k satisfying
k = N/r = M / k/N = /r

(19)


7

for some unknown integer  5 {0, 1, . . . , r  1}. Thus, if we make many of
these measurements, we have a non zero probability to nd a  which is
coprime to r.
Because of (18), in the relation (19), all these multiples will arise with equal
probability (1/r). Therefore one can apply the estimate (29), with  and r
replacing y and N respectively, and deduce that
P ({ 5 {2, . . . , r  1} :  is coprime to r} ) 

1
log r

(20)

If  is coprime to r, we reduce the fraction k/N to an irreducible fraction
and this gives  and r separately.
If we repeat the measurement of the |ki—basis h = O(log r)  O(log N )
times, this will give h possible candidates, r1 , . . . , rh , for the period and the

estimate (20) shows that, with high probability, one of them should be the
desired period.
3. Complexity considerations on Simon’s quantum period
nding algorithm (QPFA)
Step (1).
The initial state of the quantum system must be physically prepared so
that all 2n q—bits are in the state |0i.
An interesting n has an order of a few thousands bits.
Step (2).
The unitary operator UH := H
n
1 must be:
— constructed
— applied to the initial state
Step (3a).
One can appeal to a theorem of K.R. Parthasarathy [KRP01a] to conclude
that, for any given function f , all the unitary extensions of the partial isometry dened by (9) can be physically realized by means of quantum gates,
i.e. unitary operators acting only on a single pair of q—bits.
However the same theorem gives an upper estimate, on the number of gates
to be used, which is exponential in the number of factors. In our case this
number is 2n.
Therefore, in absence of a proof that, among all the unitary extensions of
the partial isometry dened by (9), there exists at least one that can be
physically realized by a number of quantum gates which is polynomial in
2n, it makes no sense to speak of the practical realizability of the algorithm.
Step (3b).


8

Even in presence of such a proof the actual physical implementation of the
unitary operator might be a formidable task, given the fact that the q—bits
involved are of order of thousands.
An alternative way could be the discovery of a physically realizable interaction (Hamiltonian) embedding the given unitary in a continuous time
evolution. But, even supposing that this can be done, the continuous time
evolution will create serious problems due to the extreme non robustness
of the algorithm against small perturbations of the unitary operator Uf .
Step (3c).
Even supposing that the above problems can be solved, the concrete application of the unitary operator Uf to the state (8) is a problem whose
solution requires additional costs in terms of time and of experimental work
to be done.
Step (4a).
The lter dened by the projection (11) must be constructed.
Step (4b).
The above comment, on the cost of the concrete realization of Step (3c),
also holds for the application of the lter (11) to the quantum state described by the vector (10).
Theoretical conclusion from Step 4b
This conclusion heavily depends on the application of the Luders—Zumino
formula of quantum measurement theory. This is quite dierent from the
original von Neumann formula and implies that, after an incomplete measurement on a quantum system in a pure state, the system will still remain
in a pure state.
Although not logically impossible, such a situation is against physical intuition because an incomplete measurement by denition does not produce
maximal information while, in quantum mechanics, a pure state denes a
situation of maximal information.
Only some very strong experimental evidence could prove that this natural
intuition is wrong.
Step (5a).
One must construct an apparatus implementing the discrete Fourier transform on arbitrary quantum states (see above comments to Step (3c)).
Step (5b).
One must apply the above apparatus to the quantum state given by (16)

(see above comments to Step (3c)).
Step (6).
Taken literally, the statement . . . The nal state of the x register is now