Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2012.13250

The duality-character Solution-Information-Carrying (SIC) unitary propagators

Xijia Miao

Published Dec 24, 2020• 0 citations
arXivHTMLPDF

Abstract

The HSSS quantum search process owns the dual character that it obeys both the unitary quantum dynamics and the mathematical-logical principle of the unstructured search problem. It is essentially different from a conventional quantum search algorithm. It is constructed with the duality-character oracle operations of unstructured search problem. It consists of the two consecutive steps: (1) the search-space dynamical reduction and (2) the dynamical quantum-state-difference amplification (QUANSDAM). The QUANSDAM process is directly constructed with the SIC unitary propagators, while the latter each are prepared with the basic SIC unitary operators. Here the preparation for the SIC unitary propagators of a single-atom system is concretely carried out by starting from the basic SIC unitary operators. The SIC unitary propagator of a quantum system may reflect the quantum symmetry of the quantum system, while the basic SIC unitary operators may not. The quantum symmetry is considered as the fundamental quantum-computing-speedup resource in the quantum-computing speedup theory. The purpose for the preparation is ultimately to employ the quantum symmetry to speed up the QUANSDAM process. The preparation is a solution-information transfer process. It is unitary and deterministic. It obeys the information conservation law. In methodology it is based on the energy eigenfunction expansion and the multiple-quantum operator algebra space. Furthermore, a general theory mainly based on the Feynman path integration technique and also the energy eigenfunction expansion method is established to treat theoretically and calculate a SIC unitary propagator of any quantum system in the coordinate representation, which may be further used to construct theoretically an exponential QUANSDAM process in future.

DOI
10.48550/arxiv.2012.13250

Related papers

The unitary dynamical state-locking process, the HSSS quantum search process, and the quantum-computing speedup theory

Xijia Miao

A unitary dynamical state-locking (UNIDYSLOCK) process is a unitary process that can transform two orthogonal states to two non-orthogonal states. Its inverse process, i.e., the QUANSDAM process could realize an exponential QC speedup in solving an unstructured search problem in the quantum-computing (QC) speedup theory (X. Miao, arXiv:1105.3573 [quant-ph] (2011)). In this paper the principle of how a UNIDYSLOCK process works is described on the basis of the QC speedup theory. A UNIDYSLOCK process does not exist at all in pure quantum mechanics. That it can change the quantum-state difference is original from the fundamental interaction between the quantum physical laws (i.e., the unitary quantum dynamics and the Hilbert-space symmetric structure) and the mathematical-logical principle of the search problem. It is characteristic for the QC speedup theory. Even if a usual quantum computational (Qc) process that is reversible or unitary contained a UNIDYSLOCK (or QUANSDAM) process, the contribution of the UNIDYSLOCK (or QUANSDAM) process to the QC speedup of the usual Qc process would be secondary or negligible. Whether or not a UNIDYSLOCK (or QUANSDAM) process can make an essential contribution to a QC speedup can distinguish the QC speedup theory from any usual Qc theory. In the paper the QC speedup mechanism of the HSSS quantum search process is also studied. The search process is essentially different from a usual quantum search algorithm. It works in both the usual Hilbert space and the math Hilbert space. Due to the math Hilbert space a mathematical-parallel operation is characteristic for the QC speedup theory, while a quantum-parallel operation is characteristic for the usual Qc theory. The fundamental interaction mentioned above and the fundamental quantum-computing resource play the main role in speeding up the search process.

arXiv:1612.05969•Dec 18, 2016•0 citations

Universal construction for the unsorted quantum search algorithms

Xijia Miao

The multiple-quantum operator algebra formalism has been exploited to construct generally an unsorted quantum search algorithm. The exponential propagator and its corresponding effective Hamiltonian are constructed explicitly that describe in quantum mechanics the time evolution of a multi- particle two-state quantum system from the initial state to the output of the unsorted quantum search problem. The exponential propagator usually may not be compatible with the mathematical structure and principle of the search problem and hence is not a real quantum search network, but it can be further decomposed into a product of a series of the oracle unitary operations such as the selective phase-shift operations and the nonselective unitary operations which can be expressed further as a sequence of elementary building blocks such as one-qubit quantum gates and the two-qubit diagonal phase gates, resulting in that the decomposed propagator is compatible with the mathematical structure and principle of the search problem and thus, becomes a real quantum search network. The decomposition for the propagator can be achieved with the help of the operator algebra structure and symmetry of the effective Hamiltonian, and the properties of the multiple-quantum operator algebra spaces, especially the characteristic transformation behavior of the multiple-quantum operators under the z-axis rotations. It has been shown that the computational complexity of the search algorithm is dependent upon that of the numerical multidimensional integration and hence it is believed that the search algorithm could solve efficiently the unsorted search problem. An NMR device is also proposed to solve efficiently the unsorted search problem in polynomial time.

arXiv:quant-ph/0101126•Jan 26, 2001•4 citations

The universal quantum driving force to speed up a quantum computation -- The unitary quantum dynamics

Xijia Miao

It is shown in the paper that the unitary quantum dynamics in quantum mechanics is the universal quantum driving force to speed up a quantum computation. This assertion supports strongly in theory that the unitary quantum dynamics is the fundamental and universal principle in nature. On the other hand, the symmetric structure of Hilbert space of a composite quantum system is the quantum-computing resource that is not owned by classical computation. A new quantum-computing speedup theory is set up on the basis of the unitary quantum dynamics. Both the unitary quantum dynamics and the symmetric structure and property of the Hilbert space of the quantum system are mainly responsible for an exponential quantum-computing speedup for a general efficient quantum algorithm. The inherent importance for the unitary quantum dynamics to speed up a quantum computation lies in the unique ability of the unitary quantum dynamics to build the effective interaction between the symmetric structure of the Hilbert space of the quantum system and the mathematical symmetric structure of a problem to be solved on the quantum system. This unique ability could result in an essential difference of computational power between quantum and classical computations by combining the symmetric structure and property of the Hilbert space. The new quantum-computing speedup theory also provides reasonable mechanisms for exponential quantum-computing speedup for the existing efficient quantum algorithms based on the quantum parallel principle. These existing quantum algorithms including the hidden-subgroup-problem quantum algorithms and conventional quantum search algorithms have the common character that the symmetric structure of the Hilbert space does not have any effective effect on these quantum algorithms. This could be the main reason why these quantum algorithms are quite special and considered to be semiclassical.

arXiv:1105.3573•May 18, 2011•2 citations

Quantum Computing

Dong Ruan, Gui‐Lu Long, Shi-Jie WEI +1 more

Quantum computing exploits quantum mechanical properties to perform computations. It enables quantum parallelism and provides much more powerful data processing capabilities than classical computers. With a quantum computer, one can exponentially accelerate quantum system simulation and accelerate some important classical algorithms. Traditional quantum algorithms use unitary evolution to process information. A quantum computing process is the product of a series of unitary operators. In 1994, Shor invented a quantum prime factorization algorithm that exponentially accelerates the factorization of an integer. In 1996, Grover proposed a quantum search algorithm that accelerates the search of an unsorted database using square-root steps. Afterwards, the development of quantum algorithms slowed down, leading to a subsequent query by Shor in 2003 regarding why no more significant quantum algorithms have been found. Since 2009, many important new quantum algorithms have been proposed, such as a quantum algorithm for solving linear equations with the capability of exponential acceleration, a quantum algorithm for sparse Hamiltonian simulation using linear combinations of unitary operators, and a novel algorithm for Hamiltonian simulation, which provides exponential improvements in precision. In this paper, we first describe the basic principles of quantum computation. We then describe the Shor algorithm and Grover/Long search algorithm. These quantum algorithms are the general quantum algorithms that use unitary operations in their computational processes. Next, we introduce the basic principles of duality quantum computing, which was proposed in 2002. In contrast to traditional quantum computing, duality quantum computing allows the use of linear combinations of unitary operators in the computation process, meaning the multiplication, division, addition, and subtraction of unitary operators are all possible in duality quantum computing. Thus, duality quantum computing provides more flexibility for constructing quantum algorithms and the techniques used in classical algorithm design can be directly used to construct quantum algorithms. We then review recent work regarding newer quantum algorithms that enable the linear combination of unitary operators, which are actually duality quantum algorithms. Additionally, a duality quantum simulation algorithm for open quantum systems is introduced. This algorithm not only reduces computational complexity, but also improves accuracy exponentially. Finally, a summary and the future prospects of quantum algorithms are provided.

Scientia Sinica Informationis•Sep 25, 2017•8 citations

Quantum search processes in the cyclic group state spaces

Xijia Miao

The hardness to solve an unstructured quantum search problem by a standard quantum search algorithm mainly originates from the low efficiency to amplify the amplitude of the marked state by the oracle unitary operation associated with other known quantum operations. In order to bypass the square speedup limitation of a standard quantum search algorithm it is necessary to develop other type of quantum search algorithms. It is described in detail in the paper for a quantum dynamical method to solve the quantum search problems in the cyclic group state space. The binary dynamical representation for a quantum state in the Hilbert space of the n-qubit quantum system is generalized to the multi-base dynamical representation for a quantum state in the cyclic group state space. Thus, any quantum state in the cyclic group state space may be described completely in terms of a set of dynamical parameters that are closely related to the symmetric property and structure of the cyclic group. The quantum search problem therefore could be solved by determining the set of dynamical parameters that describe completely the unknown marked state of the search problem instead by directly measuring the marked state which is a necessary step in the standard quantum search algorithm. An unstructured quantum search problem in the Hilbert space is inevitably affected greatly by the symmetric property and structure of a group. The main attempt of the paper is to make use of the symmetric properties and structures of groups to help solving the quantum search problems in the group state spaces. It is shown how the quantum search process could be reduced from the cyclic group state space to these cyclic group state subspaces with the help of the symmetric property and structure of the cyclic group on a universal quantum computer.

arXiv:quant-ph/0507236•Jul 25, 2005•3 citations

Solving the quantum search problem in polynomial time on an NMR quantum computer

Xijia Miao

The quantum search problem is an important problem due to the fact that a general NP problem can be solved efficiently by an unsorted quantum search algorithm. Here it has been shown that the quantum search problem could be solved in polynomial time on an NMR quantum computer. The NMR ensemble quantum computation is based on the quantum mechanical unitary dynamics that both a closed quantum system and its ensemble obey the same quantum mechanical unitary dynamics instead of on the pseudopure state or the effective pure state of the classical NMR quantum computation. Based on the new principle the conventional NMR multiple-quantum spectroscopy has been developed to solve experimentally the search problem. The solution information of the search problem is first loaded on the unitary evolution propagator which is constructed with the oracle unitary operation and oracle-independent unitary operations and is used to excite the multiple-quantum coherence in a spin ensemble. Then the multiple- quantum spectroscopy is used to extract experimentally the solution information. It has been discussed how to enhance the output NMR signal of the quantum search NMR multiple-quantum pulse sequence and some approaches to enhancing the NMR signal are also proposed. The present work could be helpful for conventional high field NMR machines to solve efficiently the quantum search problem.

arXiv:quant-ph/0206102•Jun 16, 2002•4 citations
The HSSS quantum search process owns the dual character that it obeys both the unitary quantum dynamics and the mathematical-logical principle of the unstructured search problem. It is essentially different from a conventional quantum search algorithm. It is constructed with the duality-character oracle operations of unstructured search problem. It consists of the two consecutive steps: (1) the search-space dynamical reduction and (2) the dynamical quantum-state-difference amplification (QUANSDAM). The QUANSDAM process is directly constructed with the SIC unitary propagators, while the latter each are prepared with the basic SIC unitary operators. Here the preparation for the SIC unitary propagators of a single-atom system is concretely carried out by starting from the basic SIC unitary operators. The SIC unitary propagator of a quantum system may reflect the quantum symmetry of the quantum system, while the basic SIC unitary operators may not. The quantum symmetry is considered as the fundamental quantum-computing-speedup resource in the quantum-computing speedup theory. The purpose for the preparation is ultimately to employ the quantum symmetry to speed up the QUANSDAM process. The preparation is a solution-information transfer process. It is unitary and deterministic. It obeys the information conservation law. In methodology it is based on the energy eigenfunction expansion and the multiple-quantum operator algebra space. Furthermore, a general theory mainly based on the Feynman path integration technique and also the energy eigenfunction expansion method is established to treat theoretically and calculate a SIC unitary propagator of any quantum system in the coordinate representation, which may be further used to construct theoretically an exponential QUANSDAM process in future.
A unitary dynamical state-locking (UNIDYSLOCK) process is a unitary process that can transform two orthogonal states to two non-orthogonal states. Its inverse process, i.e., the QUANSDAM process could realize an exponential QC speedup in solving an unstructured search problem in the quantum-computing (QC) speedup theory (X. Miao, arXiv:1105.3573 [quant-ph] (2011)). In this paper the principle of how a UNIDYSLOCK process works is described on the basis of the QC speedup theory. A UNIDYSLOCK process does not exist at all in pure quantum mechanics. That it can change the quantum-state difference is original from the fundamental interaction between the quantum physical laws (i.e., the unitary quantum dynamics and the Hilbert-space symmetric structure) and the mathematical-logical principle of the search problem. It is characteristic for the QC speedup theory. Even if a usual quantum computational (Qc) process that is reversible or unitary contained a UNIDYSLOCK (or QUANSDAM) process, the contribution of the UNIDYSLOCK (or QUANSDAM) process to the QC speedup of the usual Qc process would be secondary or negligible. Whether or not a UNIDYSLOCK (or QUANSDAM) process can make an essential contribution to a QC speedup can distinguish the QC speedup theory from any usual Qc theory. In the paper the QC speedup mechanism of the HSSS quantum search process is also studied. The search process is essentially different from a usual quantum search algorithm. It works in both the usual Hilbert space and the math Hilbert space. Due to the math Hilbert space a mathematical-parallel operation is characteristic for the QC speedup theory, while a quantum-parallel operation is characteristic for the usual Qc theory. The fundamental interaction mentioned above and the fundamental quantum-computing resource play the main role in speeding up the search process.
The multiple-quantum operator algebra formalism has been exploited to construct generally an unsorted quantum search algorithm. The exponential propagator and its corresponding effective Hamiltonian are constructed explicitly that describe in quantum mechanics the time evolution of a multi- particle two-state quantum system from the initial state to the output of the unsorted quantum search problem. The exponential propagator usually may not be compatible with the mathematical structure and principle of the search problem and hence is not a real quantum search network, but it can be further decomposed into a product of a series of the oracle unitary operations such as the selective phase-shift operations and the nonselective unitary operations which can be expressed further as a sequence of elementary building blocks such as one-qubit quantum gates and the two-qubit diagonal phase gates, resulting in that the decomposed propagator is compatible with the mathematical structure and principle of the search problem and thus, becomes a real quantum search network. The decomposition for the propagator can be achieved with the help of the operator algebra structure and symmetry of the effective Hamiltonian, and the properties of the multiple-quantum operator algebra spaces, especially the characteristic transformation behavior of the multiple-quantum operators under the z-axis rotations. It has been shown that the computational complexity of the search algorithm is dependent upon that of the numerical multidimensional integration and hence it is believed that the search algorithm could solve efficiently the unsorted search problem. An NMR device is also proposed to solve efficiently the unsorted search problem in polynomial time.
It is shown in the paper that the unitary quantum dynamics in quantum mechanics is the universal quantum driving force to speed up a quantum computation. This assertion supports strongly in theory that the unitary quantum dynamics is the fundamental and universal principle in nature. On the other hand, the symmetric structure of Hilbert space of a composite quantum system is the quantum-computing resource that is not owned by classical computation. A new quantum-computing speedup theory is set up on the basis of the unitary quantum dynamics. Both the unitary quantum dynamics and the symmetric structure and property of the Hilbert space of the quantum system are mainly responsible for an exponential quantum-computing speedup for a general efficient quantum algorithm. The inherent importance for the unitary quantum dynamics to speed up a quantum computation lies in the unique ability of the unitary quantum dynamics to build the effective interaction between the symmetric structure of the Hilbert space of the quantum system and the mathematical symmetric structure of a problem to be solved on the quantum system. This unique ability could result in an essential difference of computational power between quantum and classical computations by combining the symmetric structure and property of the Hilbert space. The new quantum-computing speedup theory also provides reasonable mechanisms for exponential quantum-computing speedup for the existing efficient quantum algorithms based on the quantum parallel principle. These existing quantum algorithms including the hidden-subgroup-problem quantum algorithms and conventional quantum search algorithms have the common character that the symmetric structure of the Hilbert space does not have any effective effect on these quantum algorithms. This could be the main reason why these quantum algorithms are quite special and considered to be semiclassical.
Quantum computing exploits quantum mechanical properties to perform computations. It enables quantum parallelism and provides much more powerful data processing capabilities than classical computers. With a quantum computer, one can exponentially accelerate quantum system simulation and accelerate some important classical algorithms. Traditional quantum algorithms use unitary evolution to process information. A quantum computing process is the product of a series of unitary operators. In 1994, Shor invented a quantum prime factorization algorithm that exponentially accelerates the factorization of an integer. In 1996, Grover proposed a quantum search algorithm that accelerates the search of an unsorted database using square-root steps. Afterwards, the development of quantum algorithms slowed down, leading to a subsequent query by Shor in 2003 regarding why no more significant quantum algorithms have been found. Since 2009, many important new quantum algorithms have been proposed, such as a quantum algorithm for solving linear equations with the capability of exponential acceleration, a quantum algorithm for sparse Hamiltonian simulation using linear combinations of unitary operators, and a novel algorithm for Hamiltonian simulation, which provides exponential improvements in precision. In this paper, we first describe the basic principles of quantum computation. We then describe the Shor algorithm and Grover/Long search algorithm. These quantum algorithms are the general quantum algorithms that use unitary operations in their computational processes. Next, we introduce the basic principles of duality quantum computing, which was proposed in 2002. In contrast to traditional quantum computing, duality quantum computing allows the use of linear combinations of unitary operators in the computation process, meaning the multiplication, division, addition, and subtraction of unitary operators are all possible in duality quantum computing. Thus, duality quantum computing provides more flexibility for constructing quantum algorithms and the techniques used in classical algorithm design can be directly used to construct quantum algorithms. We then review recent work regarding newer quantum algorithms that enable the linear combination of unitary operators, which are actually duality quantum algorithms. Additionally, a duality quantum simulation algorithm for open quantum systems is introduced. This algorithm not only reduces computational complexity, but also improves accuracy exponentially. Finally, a summary and the future prospects of quantum algorithms are provided.
The hardness to solve an unstructured quantum search problem by a standard quantum search algorithm mainly originates from the low efficiency to amplify the amplitude of the marked state by the oracle unitary operation associated with other known quantum operations. In order to bypass the square speedup limitation of a standard quantum search algorithm it is necessary to develop other type of quantum search algorithms. It is described in detail in the paper for a quantum dynamical method to solve the quantum search problems in the cyclic group state space. The binary dynamical representation for a quantum state in the Hilbert space of the n-qubit quantum system is generalized to the multi-base dynamical representation for a quantum state in the cyclic group state space. Thus, any quantum state in the cyclic group state space may be described completely in terms of a set of dynamical parameters that are closely related to the symmetric property and structure of the cyclic group. The quantum search problem therefore could be solved by determining the set of dynamical parameters that describe completely the unknown marked state of the search problem instead by directly measuring the marked state which is a necessary step in the standard quantum search algorithm. An unstructured quantum search problem in the Hilbert space is inevitably affected greatly by the symmetric property and structure of a group. The main attempt of the paper is to make use of the symmetric properties and structures of groups to help solving the quantum search problems in the group state spaces. It is shown how the quantum search process could be reduced from the cyclic group state space to these cyclic group state subspaces with the help of the symmetric property and structure of the cyclic group on a universal quantum computer.
The quantum search problem is an important problem due to the fact that a general NP problem can be solved efficiently by an unsorted quantum search algorithm. Here it has been shown that the quantum search problem could be solved in polynomial time on an NMR quantum computer. The NMR ensemble quantum computation is based on the quantum mechanical unitary dynamics that both a closed quantum system and its ensemble obey the same quantum mechanical unitary dynamics instead of on the pseudopure state or the effective pure state of the classical NMR quantum computation. Based on the new principle the conventional NMR multiple-quantum spectroscopy has been developed to solve experimentally the search problem. The solution information of the search problem is first loaded on the unitary evolution propagator which is constructed with the oracle unitary operation and oracle-independent unitary operations and is used to excite the multiple-quantum coherence in a spin ensemble. Then the multiple- quantum spectroscopy is used to extract experimentally the solution information. It has been discussed how to enhance the output NMR signal of the quantum search NMR multiple-quantum pulse sequence and some approaches to enhancing the NMR signal are also proposed. The present work could be helpful for conventional high field NMR machines to solve efficiently the quantum search problem.