{"title":"Conservation Laws in Quantum Database Search","authors":"Li-Yi Hsu, Ching-Hsu Chen","doi":"10.1007/s10701-024-00806-1","DOIUrl":null,"url":null,"abstract":"<div><p>Recently, the correspondence between the air track scenario and quantum database search algorithm was revealed. The conservation laws of linear momentum and nonlinear kinetic energy in the former case, which involve sequential elastic collisions, have their analogs in the latter case. Obviously, probability normalization combined with the Born rule serves as an analog for kinetic energy conservation. Here we explore the linear conservation laws in a generic quantum database search. Regarding the non-uniform distribution of the marked state, the uneven state is initially prepared. In this way, the Grover diffusion operator results in a linear but nonphysical conservation law. On the other hand, in the CTC-assisted database search with the vast reduction of query complexity, the nonlinear instead of linear conservation laws are found. Finally, we conjecture that there are no conservation laws in the generalized Grover’s algorithm including the imaginary number <i>i</i>.</p></div>","PeriodicalId":569,"journal":{"name":"Foundations of Physics","volume":"54 5","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Foundations of Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s10701-024-00806-1","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Recently, the correspondence between the air track scenario and quantum database search algorithm was revealed. The conservation laws of linear momentum and nonlinear kinetic energy in the former case, which involve sequential elastic collisions, have their analogs in the latter case. Obviously, probability normalization combined with the Born rule serves as an analog for kinetic energy conservation. Here we explore the linear conservation laws in a generic quantum database search. Regarding the non-uniform distribution of the marked state, the uneven state is initially prepared. In this way, the Grover diffusion operator results in a linear but nonphysical conservation law. On the other hand, in the CTC-assisted database search with the vast reduction of query complexity, the nonlinear instead of linear conservation laws are found. Finally, we conjecture that there are no conservation laws in the generalized Grover’s algorithm including the imaginary number i.
最近，人们发现了空轨情景与量子数据库搜索算法之间的对应关系。前者涉及连续弹性碰撞的线性动量守恒定律和非线性动能守恒定律在后者中也有类似之处。显然，概率归一化与博恩规则相结合可作为动能守恒的类似物。在这里，我们探讨了通用量子数据库搜索中的线性守恒定律。关于标记态的非均匀分布，不均匀态最初是准备好的。这样，格罗弗扩散算子就会产生一个线性但非物理的守恒定律。另一方面，在 CTC 辅助数据库搜索中，查询复杂度大大降低，找到的是非线性守恒定律而不是线性守恒定律。最后，我们推测广义格罗弗算法中不存在包含虚数 i 的守恒定律。
期刊介绍：
The conceptual foundations of physics have been under constant revision from the outset, and remain so today. Discussion of foundational issues has always been a major source of progress in science, on a par with empirical knowledge and mathematics. Examples include the debates on the nature of space and time involving Newton and later Einstein; on the nature of heat and of energy; on irreversibility and probability due to Boltzmann; on the nature of matter and observation measurement during the early days of quantum theory; on the meaning of renormalisation, and many others.
Today, insightful reflection on the conceptual structure utilised in our efforts to understand the physical world is of particular value, given the serious unsolved problems that are likely to demand, once again, modifications of the grammar of our scientific description of the physical world. The quantum properties of gravity, the nature of measurement in quantum mechanics, the primary source of irreversibility, the role of information in physics – all these are examples of questions about which science is still confused and whose solution may well demand more than skilled mathematics and new experiments.
Foundations of Physics is a privileged forum for discussing such foundational issues, open to physicists, cosmologists, philosophers and mathematicians. It is devoted to the conceptual bases of the fundamental theories of physics and cosmology, to their logical, methodological, and philosophical premises.
The journal welcomes papers on issues such as the foundations of special and general relativity, quantum theory, classical and quantum field theory, quantum gravity, unified theories, thermodynamics, statistical mechanics, cosmology, and similar.
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