A universal quantum gate is introduced for tensors of vector spaces. By using integer powers of s... more A universal quantum gate is introduced for tensors of vector spaces. By using integer powers of such a gate and by using classical reversible gates one can approximate any element of the unitary group to any accuracy needed. The proof uses a version of Kronecker's theory and the structure of the Bloch sphere for tensors.
In this chapter we give a map of classical game theory (see also Appendix C), the main ideas and ... more In this chapter we give a map of classical game theory (see also Appendix C), the main ideas and motivations. The text aims the readers who are familiar of basic notions in game theory. For an elaborate discussion of the topic we refer the reader to Neuman et al. [281], Osborne [212], Fudenberg [106], Heifetz [139].
A vectorial distance measure for trees is presented. Given two trees, we align the trees from the... more A vectorial distance measure for trees is presented. Given two trees, we align the trees from their centers outwards, starting from the root-branches, to make the next level as similar as possible. The algorithm is recursive; condition on the alignment of the root-branches we align the sub-branches, thereafter each alignment is conditioned on the previous one. We define a minimal alignment under a lexicographic order which follows the intuition that the differences between the two trees closer to their cores dominate their differences at a higher level. Given such a minimal alignment, the difference in the number of branches calculated at any level defines the entry of the distance vector at that level. We compare our algorithm to other well-known tree distance measures in the task of clustering sets of phylogenetic trees. We use the TreeSimGM simulator for generating stochastic phylogenetic trees. The vectorial tree distance can successfully separate symmetric from asymmetric trees, and hierarchical from non-hierarchical trees.
A vectorial distance measure for trees is presented. Given two trees, we define a Tree-Alignment ... more A vectorial distance measure for trees is presented. Given two trees, we define a Tree-Alignment (T-Alignment). We T-align the trees from their centers outwards, starting from the root-branches, to make the next level as similar as possible. The algorithm is recursive; condition on the T-alignment of the root-branches we T-align the sub-branches, thereafter each T-alignment is conditioned on the previous one. We define a minimal T-alignment under a lexicographic order which follows the intuition that the differences between the two trees constitutes a vector. Given such a minimal T-alignment, the difference in the number of branches calculated at any level defines the entry of the distance vector at that level. We compare our algorithm to other well-known tree distance measures in the task of clustering sets of phylogenetic trees. We use the TreeSimGM simulator for generating stochastic phylogenetic trees. The vectorial tree distance (VTD) can successfully separate symmetric from as...
Abstract. We briefly review various computational methods for the solution of optimization proble... more Abstract. We briefly review various computational methods for the solution of optimization problems. First, several classical methods such as Metropolis algorithm and simulated annealing are discussed. We continue with a description of quantum methods, namely adiabatic quantum computation and quantum annealing. Next, the new D-Wave computer and the recent progress in the field claimed by the D-Wave group are discussed. We present a set of criteria which could help in testing the quantum features of these computers. We conclude with a list of considerations with regard to future research. 1
In this chapter we give very brief introduction to some of the interesting very new developments ... more In this chapter we give very brief introduction to some of the interesting very new developments in this field. In particular, we have discussed some of the extensions of Kolkata Paise Restaurant problem for dynamic settings or development of efficient strategies employing reinforced learning or applications of such strategies to other social problems like modelling income distributions in societies etc.
In this paper we introduce an easy to compute upper bound on the Tsallis entropy of a density mat... more In this paper we introduce an easy to compute upper bound on the Tsallis entropy of a density matrix describing a system coupled to a noise source. This suggests that the Tsallis entropy is most natural in the context of quantum information theory. Similarly we redefine entropy exchange and use the same bound.
We discuss in this chapter the basics of adiabatic computation, as well as some physical implemen... more We discuss in this chapter the basics of adiabatic computation, as well as some physical implementations. After a short introduction of the quantum circuit model, we describe quantum adiabatic computation, quantum annealing, and the strong relations between the three. We conclude with a brief presentation of the D-Wave computer and some future challenges.
Prediction in natural environments is a challenging task, and there is a lack of clarity around h... more Prediction in natural environments is a challenging task, and there is a lack of clarity around how a myopic organism can make short-term predictions given limited data availability and cognitive resources. In this context, we may ask what kind of resources are available to the organism to help it address the challenge of short-term prediction within its own cognitive limits. We point to one potentially important resource: ordinal patterns, which are extensively used in physics but not in the study of cognitive processes. We explain the potential importance of ordinal patterns for short-term prediction, and how natural constraints imposed through (i) ordinal pattern types, (ii) their transition probabilities and (iii) their irreversibility signature may support short-term prediction. Having tested these ideas on a massive dataset of Bitcoin prices representing a highly fluctuating environment, we provide preliminary empirical support showing how organisms characterized by bounded ra...
International Journal of Quantum Information, 2014
On May 2011, D-Wave Systems Inc. announced "D-Wave One", as "the world's first... more On May 2011, D-Wave Systems Inc. announced "D-Wave One", as "the world's first commercially available quantum computer". No wonder this adiabatic quantum computer based on 128-qubit chip-set provoked an immediate controversy. Over the last 40 years, quantum computation has been a very promising yet challenging research area, facing major difficulties producing a large scale quantum computer. Today, after Google has purchased "D-Wave Two" containing 512 qubits, criticism has only increased. In this work, we examine the theory underlying the D-Wave, seeking to shed some light on this intriguing quantum computer. Starting from classical algorithms such as Metropolis algorithm, genetic algorithm (GA), hill climbing and simulated annealing, we continue to adiabatic computation and quantum annealing towards better understanding of the D-Wave mechanism. Finally, we outline some applications within the fields of information and image processing. In addition...
Quantum Studies: Mathematics and Foundations, 2015
We investigate the power of weak measurements in the framework of quantum state discrimination. F... more We investigate the power of weak measurements in the framework of quantum state discrimination. First, we define and analyze the notion of weak consecutive measurements. Our main result is a convergence theorem whereby we demonstrate when and how a set of consecutive weak measurements converges to a strong measurement. Second, we show that for a small set of consecutive weak measurements, long before their convergence, one can separate close states without causing their collapse. We thus demonstrate a tradeoff between the success probability and the bias of the original vector towards collapse. Next, we use post-selection within the two-state vector formalism and present the non-linear expansion of the expectation value of the measurement device's pointer to distinguish between two predetermined close vectors.
Consider a function where its entries are distributed among many parties. Suppose each party is a... more Consider a function where its entries are distributed among many parties. Suppose each party is allowed to transmit only a limited amount of information to a net. One can use a classical protocol to guess the value of the global function. Is there a quantum protocol improving the results of all classical protocols? In [3] Brukner et al. showed the deep connection between such problems and the theory of Bell's inequalities. Here we generalize the theory to trits. There, the best classical protocol fails whereas the quantum protocol yields the correct answer.
Why the genetic code has a fixed length? Protein information is transferred by coding each amino ... more Why the genetic code has a fixed length? Protein information is transferred by coding each amino acid using codons whose length equals 3 for all amino acids. Hence the most probable and the least probable amino acid get a codeword with an equal length. Moreover, the distributions of amino acids found in nature are not uniform and therefore the efficiency of such codes is sub-optimal. The origins of these apparently non-efficient codes are yet unclear. In this paper we propose an a priori argument for the energy efficiency of such codes resulting from their reversibility, in contrast to their time inefficiency. Such codes are reversible in the sense that a primitive processor, reading three letters in each step, can always reverse its operation, undoing its process. We examine the codes for the distributions of amino acids that exist in nature and show that they could not be both time efficient and reversible. We investigate a family of Zipf-type distributions and present their efficient (non-fixed length) prefix code, their graphs, and the condition for their reversibility. We prove that for a large family of such distributions, if the code is time efficient, it could not be reversible. In other words, if pre-biotic processes demand reversibility, the protein code could not be time efficient. The benefits of reversibility are clear: reversible processes are adiabatic, namely, they dissipate a very small amount of energy. Such processes must be done slowly enough; therefore time efficiency is non-important. It is reasonable to assume that early biochemical complexes were more prone towards energy efficiency, where forward and backward processes were almost symmetrical.
Weak measurement devices resemble band pass filters: they strengthen average values in the state ... more Weak measurement devices resemble band pass filters: they strengthen average values in the state space or equivalently filter out some 'frequencies' from the conjugate Fourier transformed vector space. We thereby adjust a principle of classical communication theory for the use in quantum computation. We discuss some of the computational benefits and limitations of such an approach, including complexity analysis, some simple examples and a realistic not-so-weak approach.
In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can... more In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can find a single marked object in a database of size N by using only O(√ N) queries of the oracle that identifies the object. His result was generalized to the case of finding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either M or K, our task is to determine which is the case. We show how to solve this problem with a high probability of success using iterations of Grover's basic step only, and no other algorithm. Let m be the required number of iterations; we prove that under certain restrictions on the sizes of M and K the estimation m ≤ 2 √ N √ K− √ M obtains. This bound reproduces previous results based on more elaorate algorithms, and is known to be optimal up to a constant factor. Our method involves simultaneous Diophantine approximations, so that Grover's algorithm is conceptualized as an orbit of an ergodic automorphism of the torus. We comment on situations where the algorithm may be slow, and note the similarity between these cases and the problem of small divisors in classical mechanics.
Consider a function where its entries are distributed among many parties. Suppose each party is a... more Consider a function where its entries are distributed among many parties. Suppose each party is allowed to transmit only a limited amount of information to a net. One can use a classical protocol to guess the value of the global function. Is there a quantum protocol improving the results of all classical protocols? In [3] Brukner et al. showed the deep connection between such problems and the theory of Bell’s inequalities. Here we generalize the theory to trits. There, the best classical protocol fails whereas the quantum protocol yields the correct answer.
We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kull... more We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kullback-Leibler divergence. Suppose Alice is sending classical information to Bob using a quantum channel, while Bob is performing some projective measurement. We bound the classical mutual information in terms of the Hilbert-Schmidt norm by its quantum Hilbert-Schmidt counterpart. This constitutes a Holevo-type upper bound on the classical information transmission rate via a quantum channel. The resulting inequality is rather natural and intuitive relating classical and quantum expressions using the same measure.
Motivated by successful classical models for noise reduction, we suggest a quantum technique for ... more Motivated by successful classical models for noise reduction, we suggest a quantum technique for filtering noise out of quantum states. The purpose of this paper is twofold: presenting a simple construction of quantum cross-correlations between two wave-functions, and presenting a scheme for a quantum noise filtering. We follow a well-known scheme in classical communication theory that attenuates random noise, and show that one can build a quantum analog by using non-trace-preserving operators. By this we introduce a classically motivated signal processing scheme to quantum information theory, which can help reducing quantum noise, and particularly, phase flip noise.
A universal quantum gate is introduced for tensors of vector spaces. By using integer powers of s... more A universal quantum gate is introduced for tensors of vector spaces. By using integer powers of such a gate and by using classical reversible gates one can approximate any element of the unitary group to any accuracy needed. The proof uses a version of Kronecker's theory and the structure of the Bloch sphere for tensors.
In this chapter we give a map of classical game theory (see also Appendix C), the main ideas and ... more In this chapter we give a map of classical game theory (see also Appendix C), the main ideas and motivations. The text aims the readers who are familiar of basic notions in game theory. For an elaborate discussion of the topic we refer the reader to Neuman et al. [281], Osborne [212], Fudenberg [106], Heifetz [139].
A vectorial distance measure for trees is presented. Given two trees, we align the trees from the... more A vectorial distance measure for trees is presented. Given two trees, we align the trees from their centers outwards, starting from the root-branches, to make the next level as similar as possible. The algorithm is recursive; condition on the alignment of the root-branches we align the sub-branches, thereafter each alignment is conditioned on the previous one. We define a minimal alignment under a lexicographic order which follows the intuition that the differences between the two trees closer to their cores dominate their differences at a higher level. Given such a minimal alignment, the difference in the number of branches calculated at any level defines the entry of the distance vector at that level. We compare our algorithm to other well-known tree distance measures in the task of clustering sets of phylogenetic trees. We use the TreeSimGM simulator for generating stochastic phylogenetic trees. The vectorial tree distance can successfully separate symmetric from asymmetric trees, and hierarchical from non-hierarchical trees.
A vectorial distance measure for trees is presented. Given two trees, we define a Tree-Alignment ... more A vectorial distance measure for trees is presented. Given two trees, we define a Tree-Alignment (T-Alignment). We T-align the trees from their centers outwards, starting from the root-branches, to make the next level as similar as possible. The algorithm is recursive; condition on the T-alignment of the root-branches we T-align the sub-branches, thereafter each T-alignment is conditioned on the previous one. We define a minimal T-alignment under a lexicographic order which follows the intuition that the differences between the two trees constitutes a vector. Given such a minimal T-alignment, the difference in the number of branches calculated at any level defines the entry of the distance vector at that level. We compare our algorithm to other well-known tree distance measures in the task of clustering sets of phylogenetic trees. We use the TreeSimGM simulator for generating stochastic phylogenetic trees. The vectorial tree distance (VTD) can successfully separate symmetric from as...
Abstract. We briefly review various computational methods for the solution of optimization proble... more Abstract. We briefly review various computational methods for the solution of optimization problems. First, several classical methods such as Metropolis algorithm and simulated annealing are discussed. We continue with a description of quantum methods, namely adiabatic quantum computation and quantum annealing. Next, the new D-Wave computer and the recent progress in the field claimed by the D-Wave group are discussed. We present a set of criteria which could help in testing the quantum features of these computers. We conclude with a list of considerations with regard to future research. 1
In this chapter we give very brief introduction to some of the interesting very new developments ... more In this chapter we give very brief introduction to some of the interesting very new developments in this field. In particular, we have discussed some of the extensions of Kolkata Paise Restaurant problem for dynamic settings or development of efficient strategies employing reinforced learning or applications of such strategies to other social problems like modelling income distributions in societies etc.
In this paper we introduce an easy to compute upper bound on the Tsallis entropy of a density mat... more In this paper we introduce an easy to compute upper bound on the Tsallis entropy of a density matrix describing a system coupled to a noise source. This suggests that the Tsallis entropy is most natural in the context of quantum information theory. Similarly we redefine entropy exchange and use the same bound.
We discuss in this chapter the basics of adiabatic computation, as well as some physical implemen... more We discuss in this chapter the basics of adiabatic computation, as well as some physical implementations. After a short introduction of the quantum circuit model, we describe quantum adiabatic computation, quantum annealing, and the strong relations between the three. We conclude with a brief presentation of the D-Wave computer and some future challenges.
Prediction in natural environments is a challenging task, and there is a lack of clarity around h... more Prediction in natural environments is a challenging task, and there is a lack of clarity around how a myopic organism can make short-term predictions given limited data availability and cognitive resources. In this context, we may ask what kind of resources are available to the organism to help it address the challenge of short-term prediction within its own cognitive limits. We point to one potentially important resource: ordinal patterns, which are extensively used in physics but not in the study of cognitive processes. We explain the potential importance of ordinal patterns for short-term prediction, and how natural constraints imposed through (i) ordinal pattern types, (ii) their transition probabilities and (iii) their irreversibility signature may support short-term prediction. Having tested these ideas on a massive dataset of Bitcoin prices representing a highly fluctuating environment, we provide preliminary empirical support showing how organisms characterized by bounded ra...
International Journal of Quantum Information, 2014
On May 2011, D-Wave Systems Inc. announced "D-Wave One", as "the world's first... more On May 2011, D-Wave Systems Inc. announced "D-Wave One", as "the world's first commercially available quantum computer". No wonder this adiabatic quantum computer based on 128-qubit chip-set provoked an immediate controversy. Over the last 40 years, quantum computation has been a very promising yet challenging research area, facing major difficulties producing a large scale quantum computer. Today, after Google has purchased "D-Wave Two" containing 512 qubits, criticism has only increased. In this work, we examine the theory underlying the D-Wave, seeking to shed some light on this intriguing quantum computer. Starting from classical algorithms such as Metropolis algorithm, genetic algorithm (GA), hill climbing and simulated annealing, we continue to adiabatic computation and quantum annealing towards better understanding of the D-Wave mechanism. Finally, we outline some applications within the fields of information and image processing. In addition...
Quantum Studies: Mathematics and Foundations, 2015
We investigate the power of weak measurements in the framework of quantum state discrimination. F... more We investigate the power of weak measurements in the framework of quantum state discrimination. First, we define and analyze the notion of weak consecutive measurements. Our main result is a convergence theorem whereby we demonstrate when and how a set of consecutive weak measurements converges to a strong measurement. Second, we show that for a small set of consecutive weak measurements, long before their convergence, one can separate close states without causing their collapse. We thus demonstrate a tradeoff between the success probability and the bias of the original vector towards collapse. Next, we use post-selection within the two-state vector formalism and present the non-linear expansion of the expectation value of the measurement device's pointer to distinguish between two predetermined close vectors.
Consider a function where its entries are distributed among many parties. Suppose each party is a... more Consider a function where its entries are distributed among many parties. Suppose each party is allowed to transmit only a limited amount of information to a net. One can use a classical protocol to guess the value of the global function. Is there a quantum protocol improving the results of all classical protocols? In [3] Brukner et al. showed the deep connection between such problems and the theory of Bell's inequalities. Here we generalize the theory to trits. There, the best classical protocol fails whereas the quantum protocol yields the correct answer.
Why the genetic code has a fixed length? Protein information is transferred by coding each amino ... more Why the genetic code has a fixed length? Protein information is transferred by coding each amino acid using codons whose length equals 3 for all amino acids. Hence the most probable and the least probable amino acid get a codeword with an equal length. Moreover, the distributions of amino acids found in nature are not uniform and therefore the efficiency of such codes is sub-optimal. The origins of these apparently non-efficient codes are yet unclear. In this paper we propose an a priori argument for the energy efficiency of such codes resulting from their reversibility, in contrast to their time inefficiency. Such codes are reversible in the sense that a primitive processor, reading three letters in each step, can always reverse its operation, undoing its process. We examine the codes for the distributions of amino acids that exist in nature and show that they could not be both time efficient and reversible. We investigate a family of Zipf-type distributions and present their efficient (non-fixed length) prefix code, their graphs, and the condition for their reversibility. We prove that for a large family of such distributions, if the code is time efficient, it could not be reversible. In other words, if pre-biotic processes demand reversibility, the protein code could not be time efficient. The benefits of reversibility are clear: reversible processes are adiabatic, namely, they dissipate a very small amount of energy. Such processes must be done slowly enough; therefore time efficiency is non-important. It is reasonable to assume that early biochemical complexes were more prone towards energy efficiency, where forward and backward processes were almost symmetrical.
Weak measurement devices resemble band pass filters: they strengthen average values in the state ... more Weak measurement devices resemble band pass filters: they strengthen average values in the state space or equivalently filter out some 'frequencies' from the conjugate Fourier transformed vector space. We thereby adjust a principle of classical communication theory for the use in quantum computation. We discuss some of the computational benefits and limitations of such an approach, including complexity analysis, some simple examples and a realistic not-so-weak approach.
In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can... more In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can find a single marked object in a database of size N by using only O(√ N) queries of the oracle that identifies the object. His result was generalized to the case of finding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either M or K, our task is to determine which is the case. We show how to solve this problem with a high probability of success using iterations of Grover's basic step only, and no other algorithm. Let m be the required number of iterations; we prove that under certain restrictions on the sizes of M and K the estimation m ≤ 2 √ N √ K− √ M obtains. This bound reproduces previous results based on more elaorate algorithms, and is known to be optimal up to a constant factor. Our method involves simultaneous Diophantine approximations, so that Grover's algorithm is conceptualized as an orbit of an ergodic automorphism of the torus. We comment on situations where the algorithm may be slow, and note the similarity between these cases and the problem of small divisors in classical mechanics.
Consider a function where its entries are distributed among many parties. Suppose each party is a... more Consider a function where its entries are distributed among many parties. Suppose each party is allowed to transmit only a limited amount of information to a net. One can use a classical protocol to guess the value of the global function. Is there a quantum protocol improving the results of all classical protocols? In [3] Brukner et al. showed the deep connection between such problems and the theory of Bell’s inequalities. Here we generalize the theory to trits. There, the best classical protocol fails whereas the quantum protocol yields the correct answer.
We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kull... more We prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kullback-Leibler divergence. Suppose Alice is sending classical information to Bob using a quantum channel, while Bob is performing some projective measurement. We bound the classical mutual information in terms of the Hilbert-Schmidt norm by its quantum Hilbert-Schmidt counterpart. This constitutes a Holevo-type upper bound on the classical information transmission rate via a quantum channel. The resulting inequality is rather natural and intuitive relating classical and quantum expressions using the same measure.
Motivated by successful classical models for noise reduction, we suggest a quantum technique for ... more Motivated by successful classical models for noise reduction, we suggest a quantum technique for filtering noise out of quantum states. The purpose of this paper is twofold: presenting a simple construction of quantum cross-correlations between two wave-functions, and presenting a scheme for a quantum noise filtering. We follow a well-known scheme in classical communication theory that attenuates random noise, and show that one can build a quantum analog by using non-trace-preserving operators. By this we introduce a classically motivated signal processing scheme to quantum information theory, which can help reducing quantum noise, and particularly, phase flip noise.
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Papers by Boaz Tamir