2026-09-19T06:28:51Zhttps://dspace.mit.edu/server/oai/request

oai:dspace.mit.edu:1721.1/772572022-01-13T07:54:41Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022

Edward H. Farhi. Yoshida, Beni Massachusetts Institute of Technology. Dept. of Physics. Massachusetts Institute of Technology. Department of Physics 2013-03-01T14:34:02Z 2013-03-01T14:34:02Z 2012 2012 http://hdl.handle.net/1721.1/77257 827730325 Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2012. This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections. Cataloged from student-submitted PDF version of thesis. Includes bibliographical references (p. 133-140). The emerging closeness between correlated spin systems and error-correcting codes enables us to use coding theoretical techniques to study physical properties of many-body spin systems. This thesis illustrates the use of classical and quantum coding theory in classifying quantum phases arising in many-body spin systems via a systematic study of stabilizer Hamiltonians with translation symmetries. In the first part, we ask what kinds of quantum phases may arise in gapped spin systems on a D-dimensional lattice. We address this condensed matter theoretical question by giving a complete classification of quantum phases arising in stabilizer Hamiltonians at fixed points of RG transformations for D = 1; 2; 3. We found a certain dimensional duality on geometric shapes of logical operators where m-dimensional and (D m)-dimensional logical operators always form anti-commuting pairs (m is an integer). We demonstrate that quantum phases are completely classified by topological characterizations of logical operators where topological quantum phase transitions are driven by non-analytical changes of geometric shapes of logical operators. As a consequence, we argue that topological order is unstable at any nonzero temperature and self-correcting quantum memory in a strict sense may not exist where the memory time is upper bounded by some constant at a fixed temperature, regardless of the system size. Our result also implies that topological field theory is the universal theory for stabilizer Hamiltonians with continuous scale symmetries. In the second part, we ask the fundamental limit on information storage capacity of discrete spin systems. There is a well-known theoretical limit on the amount of information that can be reliably stored in a given volume of discrete spin systems. Yet, previously known systems were far below this theoretical limit. We propose a construction of classical stabilizer Hamiltonians which asymptotically saturate this limit. Our model borrows an idea from fractal geometries arising in the Sierpinski triangle, and is a rare manifestation of limit cycle behaviors with discrete scale symmetries in real-space RG transformations, which may be beyond descriptions of topological field theory. by Beni Yoshida. Ph.D. 140 p. eng Massachusetts Institute of Technology M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 Physics. Studying many-body physics through quantum coding theory Thesis application/pdf false Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="1db3368b-f1bf-429d-830c-6978ea16e01a"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Language>eng</Language> <Title>Studying many-body physics through quantum coding theory</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2012</PublicationDate> <Authors> <Author> <DisplayName>Yoshida, Beni</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <DisplayName>Massachusetts Institute of Technology</DisplayName> <OrgUnit /> </Publisher> </Publishers> <License>http://dspace.mit.edu/handle/1721.1/7582</License> <Keyword>Physics.</Keyword> <Abstract>The emerging closeness between correlated spin systems and error-correcting codes enables us to use coding theoretical techniques to study physical properties of many-body spin systems. This thesis illustrates the use of classical and quantum coding theory in classifying quantum phases arising in many-body spin systems via a systematic study of stabilizer Hamiltonians with translation symmetries. In the first part, we ask what kinds of quantum phases may arise in gapped spin systems on a D-dimensional lattice. We address this condensed matter theoretical question by giving a complete classification of quantum phases arising in stabilizer Hamiltonians at fixed points of RG transformations for D = 1; 2; 3. We found a certain dimensional duality on geometric shapes of logical operators where m-dimensional and (D m)-dimensional logical operators always form anti-commuting pairs (m is an integer). We demonstrate that quantum phases are completely classified by topological characterizations of logical operators where topological quantum phase transitions are driven by non-analytical changes of geometric shapes of logical operators. As a consequence, we argue that topological order is unstable at any nonzero temperature and self-correcting quantum memory in a strict sense may not exist where the memory time is upper bounded by some constant at a fixed temperature, regardless of the system size. Our result also implies that topological field theory is the universal theory for stabilizer Hamiltonians with continuous scale symmetries. In the second part, we ask the fundamental limit on information storage capacity of discrete spin systems. There is a well-known theoretical limit on the amount of information that can be reliably stored in a given volume of discrete spin systems. Yet, previously known systems were far below this theoretical limit. We propose a construction of classical stabilizer Hamiltonians which asymptotically saturate this limit. Our model borrows an idea from fractal geometries arising in the Sierpinski triangle, and is a rare manifestation of limit cycle behaviors with discrete scale symmetries in real-space RG transformations, which may be beyond descriptions of topological field theory.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>