The Berry phase is a fundamental quantity in the classification of topological phases of matter. In this paper, we present a new quantum algorithm and several complexity-theoretical results for the Berry phase estimation (BPE) problems. Our new quantum algorithm achieves BPE in a more general setting than previously known quantum algorithms, with a theoretical guarantee. For the complexity-theoretic results, we consider three cases. First, we prove $\mathsf{BQP}$ -completeness when we are given a guiding state that has a large overlap with the ground state. This result establishes an exponential quantum speedup for estimating the Berry phase. Second, we prove $\mathsf{UQMA} \cap \mathsf{co}$ - $\mathsf{UQMA}$ -completeness when we have an $\textit{a priori}$ bound for the ground-state energy. Here, $\mathsf{UQMA}$ is the unique witness version of $\mathsf{QMA}$ , and $\mathsf{UQMA} \cap \mathsf{co}$ - $\mathsf{UQMA}$ precisely captures the complexity of BPE without the known guiding state. Remarkably, this problem is, to our knowledge, the first natural problem complete for $\mathsf{UQMA} \cap \mathsf{co}$ - $\mathsf{UQMA}$ . Third, we show $\mathsf{P}^{\mathsf{UQMA} \cap \mathsf{co}\text{-}\mathsf{UQMA}\mathsf{[log]}}$ -hardness and containment in $\mathsf{P}^{\mathsf{PGQMA[log]}}$ when we have no additional assumption. These results advance the role of quantum computing in the study of topological phases of matter and provide a pathway for clarifying the connection between topological phases of matter and computational complexity.The Berry phase is a fundamental quantity in the classification of topological phases of matter. In this paper, we present a new quantum algorithm and several complexity-theoretical results for the Berry phase estimation (BPE) problems. Our new quantum algorithm achieves BPE in a more general setting than previously known quantum algorithms, with a theoretical guarantee. For the complexity-theoretic results, we consider three cases. First, we prove $\mathsf{BQP}$-completeness when we are given a guiding state that has a large overlap with the ground state. This result establishes an exponential quantum speedup for estimating the Berry phase. Second, we prove $\mathsf{UQMA} \cap \mathsf{co}$-$\mathsf{UQMA}$-completeness when we have an $\textit{a priori}$ bound for the ground-state energy. Here, $\mathsf{UQMA}$ is the unique witness version of $\mathsf{QMA}$, and $\mathsf{UQMA} \cap \mathsf{co}$-$\mathsf{UQMA}$ precisely captures the complexity of BPE without the known guiding state. Remarkably, this problem is, to our knowledge, the first natural problem complete for $\mathsf{UQMA} \cap \mathsf{co}$-$\mathsf{UQMA}$. Third, we show $\mathsf{P}^{\mathsf{UQMA} \cap \mathsf{co}\text{-}\mathsf{UQMA}\mathsf{[log]}}$-hardness and containment in $\mathsf{P}^{\mathsf{PGQMA[log]}}$ when we have no additional assumption. These results advance the role of quantum computing in the study of topological phases of matter and provide a pathway for clarifying the connection between topological phases of matter and computational complexity.

arXiv:2509.13423Sep 16, 20250 citations

Topological Quantum Computation

Jiannis K. Pachos

Lecture notes in computer scienceJan 1, 20137 citations

TOPOLOGICAL PHASES OF MATTER

Mohamad Mousa

Topological phases of matter have been the focus of intense research for the past decades. Their intrinsic topological nature exhibits perturbation-resilient effects in macroscopic materials. These properties have applications that span sensing, electronics, spintronics, lasers, and quantum computing. They have also inspired the development of new mathematical models to study their underlying topological quantum field theory. In this work, we visit three aspects of topological phases of matter. First, we use tensor-network tools, along with the aid of Purdue's high-performance computing clusters, to simulate symmetry-protected topological phases in 1D. We update works on the realization of the Haldane phase in Cu $_2$ O excitons. In addition, we related the group-cohomological classification of these phases to their order parameters. Second, we design 2D topological insulators with higher Chern numbers. This is achieved through distant physical interactions in accordance with algebraic geometry arguments. Third, we study 2D topological quantum error-correcting codes. In particular, we construct the boundaries, domain walls, and $0$ D defects of Abelian twisted quantum doubles leveraging anyon condensation. We discuss these codes' advantages in correcting errors for quantum computing. We also give microscopic and macroscopic methods for computing their ground state degeneracy. These distinct but related studies illustrate the power of using physical arguments and mathematical proofs in the simulation, design, and computing aspects of topological phases.Topological phases of matter have been the focus of intense research for the past decades. Their intrinsic topological nature exhibits perturbation-resilient effects in macroscopic materials. These properties have applications that span sensing, electronics, spintronics, lasers, and quantum computing. They have also inspired the development of new mathematical models to study their underlying topological quantum field theory. In this work, we visit three aspects of topological phases of matter. First, we use tensor-network tools, along with the aid of Purdue's high-performance computing clusters, to simulate symmetry-protected topological phases in 1D. We update works on the realization of the Haldane phase in Cu$_2$O excitons. In addition, we related the group-cohomological classification of these phases to their order parameters. Second, we design 2D topological insulators with higher Chern numbers. This is achieved through distant physical interactions in accordance with algebraic geometry arguments. Third, we study 2D topological quantum error-correcting codes. In particular, we construct the boundaries, domain walls, and $0$D defects of Abelian twisted quantum doubles leveraging anyon condensation. We discuss these codes' advantages in correcting errors for quantum computing. We also give microscopic and macroscopic methods for computing their ground state degeneracy. These distinct but related studies illustrate the power of using physical arguments and mathematical proofs in the simulation, design, and computing aspects of topological phases.

Mar 4, 20260 citations

Introduction to Topological Quantum Matter & Quantum Computation

Tudor D. Stanescu

What is "topological" about topological quantum states? How many types of topological quantum phases are there? What is a zero-energy Majorana mode, how can it be realized in a solid-state system, and how can it be used as a platform for topological quantum computation? What is quantum computation and what makes it different from classical computation? Addressing these and other related questions, Introduction to Topological Quantum Matter & Quantum Computation provides an introduction to and a synthesis of a fascinating and rapidly expanding research field emerging at the crossroads of condensed matter physics, mathematics, and computer science. Providing the big picture and emphasizing two major new paradigms in condensed matter physics – quantum topology and quantum information – this book is ideal for graduate students and researchers entering this field, as it allows for the fruitful transfer of ideas amongst different areas, and includes many specific examples to help the reader understand abstract and sometimes challenging concepts. It explores the topological quantum world beyond the well-known topological insulators and superconductors and unveils the deep connections with quantum computation. It addresses key principles behind the classification of topological quantum phases and relevant mathematical concepts and discusses models of interacting and noninteracting topological systems, such as the toric code and the p-wave superconductor. The book also covers the basic properties of anyons, and aspects concerning the realization of topological states in solid state structures and cold atom systems. Topological quantum computation is also presented using a broad perspective, which includes elements of classical and quantum information theory, basic concepts in the theory of computation, such as computational models and computational complexity, examples of quantum algorithms, and key ideas underlying quantum computation with anyons. This new edition has been updated throughout, with exciting new discussions on crystalline topological phases, including higher-order topological insulators; gapless topological phases, including Weyl semimetals; periodically-driven topological insulators; and a discussion of axion electrodynamics in topological materials. Key Features: · Provides an accessible introduction to this exciting, cross-disciplinary area of research. · Fully updated throughout with new content on the latest result from the field. · Authored by an authority on the subject. Tudor Stanescu is a professor of Condensed Matter Theory at West Virginia University, USA. He received a B.S. in Physics from the University of Bucharest, Romania, in 1994 and a Ph.D. in Theoretical Physics from the University of Illinois at Urbana Champaign in 2002. He was a Postdoctoral Fellow at Rutgers University and at the University of Maryland from 2003 to 2009. He joined the Department of Physics and Astronomy at West Virginia University in Fall 2009. Prof. Stanescu’s research interests encompass a variety of topics in theoretical condensed matter physics including topological insulators and superconductors, topological quantum computation, ultra-cold atom systems in optical lattices, and strongly correlated materials, such as, for example, cuprate high-temperature superconductors. His research uses a combination of analytical and numerical tools and focuses on understanding the emergence of exotic states of matter in solid state and cold atom structures, for example, topological superconducting phases that host Majorana zero modes, and on investigating the possibilities of exploiting these states as physical platforms for quantum computation.What is "topological" about topological quantum states? How many types of topological quantum phases are there? What is a zero-energy Majorana mode, how can it be realized in a solid-state system, and how can it be used as a platform for topological quantum computation? What is quantum computation and what makes it different from classical computation? Addressing these and other related questions, Introduction to Topological Quantum Matter & Quantum Computation provides an introduction to and a synthesis of a fascinating and rapidly expanding research field emerging at the crossroads of condensed matter physics, mathematics, and computer science. Providing the big picture and emphasizing two major new paradigms in condensed matter physics – quantum topology and quantum information – this book is ideal for graduate students and researchers entering this field, as it allows for the fruitful transfer of ideas amongst different areas, and includes many specific examples to help the reader understand abstract and sometimes challenging concepts. It explores the topological quantum world beyond the well-known topological insulators and superconductors and unveils the deep connections with quantum computation. It addresses key principles behind the classification of topological quantum phases and relevant mathematical concepts and discusses models of interacting and noninteracting topological systems, such as the toric code and the p-wave superconductor. The book also covers the basic properties of anyons, and aspects concerning the realization of topological states in solid state structures and cold atom systems. Topological quantum computation is also presented using a broad perspective, which includes elements of classical and quantum information theory, basic concepts in the theory of computation, such as computational models and computational complexity, examples of quantum algorithms, and key ideas underlying quantum computation with anyons. This new edition has been updated throughout, with exciting new discussions on crystalline topological phases, including higher-order topological insulators; gapless topological phases, including Weyl semimetals; periodically-driven topological insulators; and a discussion of axion electrodynamics in topological materials. Key Features: · Provides an accessible introduction to this exciting, cross-disciplinary area of research. · Fully updated throughout with new content on the latest result from the field. · Authored by an authority on the subject. Tudor Stanescu is a professor of Condensed Matter Theory at West Virginia University, USA. He received a B.S. in Physics from the University of Bucharest, Romania, in 1994 and a Ph.D. in Theoretical Physics from the University of Illinois at Urbana Champaign in 2002. He was a Postdoctoral Fellow at Rutgers University and at the University of Maryland from 2003 to 2009. He joined the Department of Physics and Astronomy at West Virginia University in Fall 2009. Prof. Stanescu’s research interests encompass a variety of topics in theoretical condensed matter physics including topological insulators and superconductors, topological quantum computation, ultra-cold atom systems in optical lattices, and strongly correlated materials, such as, for example, cuprate high-temperature superconductors. His research uses a combination of analytical and numerical tools and focuses on understanding the emergence of exotic states of matter in solid state and cold atom structures, for example, topological superconducting phases that host Majorana zero modes, and on investigating the possibilities of exploiting these states as physical platforms for quantum computation.

CRC Press eBooksMay 3, 20246 citations

Topological quantum computation

Tapio Simula

Morgan & Claypool Publishers eBooksJul 1, 201913 citations

\mathsf{BQP}

-completeness when we are given a guiding state that has a large overlap with the ground state. This result establishes an exponential quantum speedup for estimating the Berry phase. Second, we prove

\mathsf{UQMA} \cap \mathsf{co}

\mathsf{UQMA}

-completeness when we have an

\textit{a priori}

bound for the ground-state energy. Here,

\mathsf{UQMA}

is the unique witness version of

\mathsf{QMA}

, and

\mathsf{UQMA} \cap \mathsf{co}

\mathsf{UQMA}

precisely captures the complexity of BPE without the known guiding state. Remarkably, this problem is, to our knowledge, the first natural problem complete for

\mathsf{UQMA} \cap \mathsf{co}

\mathsf{UQMA}

. Third, we show

\mathsf{P}^{\mathsf{UQMA} \cap \mathsf{co}\text{-}\mathsf{UQMA}\mathsf{[log]}}

-hardness and containment in

\mathsf{P}^{\mathsf{PGQMA[log]}}

when we have no additional assumption. These results advance the role of quantum computing in the study of topological phases of matter and provide a pathway for clarifying the connection between topological phases of matter and computational complexity.

_2

O excitons. In addition, we related the group-cohomological classification of these phases to their order parameters. Second, we design 2D topological insulators with higher Chern numbers. This is achieved through distant physical interactions in accordance with algebraic geometry arguments. Third, we study 2D topological quantum error-correcting codes. In particular, we construct the boundaries, domain walls, and

0

D defects of Abelian twisted quantum doubles leveraging anyon condensation. We discuss these codes' advantages in correcting errors for quantum computing. We also give microscopic and macroscopic methods for computing their ground state degeneracy. These distinct but related studies illustrate the power of using physical arguments and mathematical proofs in the simulation, design, and computing aspects of topological phases.