# Quantum Matter Overview

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## Abstract

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## 1. Introduction

#### 1.1. Quantum Matter

#### The Relation of Quantum Matter and Quantum Information Science

#### 1.2. Symmetry and Topology

#### 1.2.1. Quasiparticles (Fermions) and Collective Excitations (Bosons)

#### 1.2.2. Anyons: Third Kind of Particle

#### 1.3. Hyperbolic Space (Negative Curvature)

#### 1.3.1. Hyperbolic Bloch Theorem and Hyperbolic Band Theory

#### 1.3.2. Magnetics

#### 1.3.3. Condensates

## 2. Short-Range Protected Topological Materials

#### 2.1. Introduction to Short-Range Protected Topological Order

#### 2.2. Topological Semimetals

#### 2.2.1. Weyl and Dirac Topological Semimetals

#### 2.2.2. Nodal-Line Topological Semimetals

_{3}P

_{2}) to elaborate on how the bulk–boundary relationship gives rise to the topological protection of line nodes [30]. Various forms of symmetry are implicated including reflection, time-reversal symmetry, SU(2) spin-rotation (special unitary group), and inversion symmetry. A related projects extends the Dirac semimetal formulation in Dirac nodal-line semimetals to investigate one-dimensional Dirac nodal rings that are protected by a combination of inversion and time-reversal symmetry [32]. A small break in the inversion symmetry allows a Hall-like current to be created in which carriers at opposite sides of the Dirac nodal ring flow to opposite surfaces when an electric field is applied. The result is the formulation of a Berry-phase (geometric phase) supported topological transport theory in inversion and time-reversal invariant nodal-line semimetals. In addition to transport properties, the team also focuses on topological band degeneracies in nodal-line semimetals and provides a classification map based on this [33]. A complete taxonomy of nonsymmorphic (non-comparable by symmetry) band degeneracies in hexagonal materials with strong spin-orbit coupling is elaborated upon. (Nonsymmorphic symmetries are symmetries that not comparable by the usual symmetry properties because symmetry operations do not have a common point on the lattice.)

#### 2.2.3. Magnetic Topological Semimetals

#### 2.2.4. Chiral Topological Semimetals

## 3. Long-Range Entangled Topological Materials

#### 3.1. Quantum Hall States

#### 3.2. Quantum Spin Liquids

#### 3.2.1. Initial Discovery of Quantum Spin Liquids

#### 3.2.2. Creating Quantum Spin Liquids from Scratch

#### 3.2.3. Topological Qubits, Non-Locality, and Quantum Error Correction

#### 3.3. Entanglement Entropy and Quantum Phase Transition

#### 3.3.1. Topological Entanglement Entropy

#### 3.3.2. Topological Quantum Field Theory

## 4. Codes (Toric, etc.)

#### 4.1. Stabilizer Codes

#### 4.1.1. The Toric Code

#### 4.1.2. Quantum Error Correction

#### 4.1.3. Stabilizer Codes in Quantum Error Correction

#### 4.2. Bosonic Codes

## 5. Application Example: Quantum Neuroscience

#### 5.1. Quantum Neuroscience

#### 5.1.1. Neural Signaling Phase Transitions

#### 5.1.2. AdS/Brain Theory (Anti-de Sitter Space)

#### 5.2. Quantum Biology of Deformable Soft Solids

## 6. Discussion

#### 6.1. Quantum Matter and Quantum Information Science Integration

#### 6.1.1. Universal Quantum Computing

#### 6.1.2. Quantum Machine Learning

#### 6.1.3. Quantum Simulation Platforms

#### 6.1.4. Quantum Nanoscience

#### 6.2. Future Outlook for Quantum Matter Science

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Abelian | Commuting (order of terms does not matter) |

AdS/Brain | Multi-tier holographic theory of neural signaling based on the AdS/CFT correspondence |

AdS/CFT correspondence | (Anti-de Sitter space/Conformal Field Theory) Theory positing that a physical system with a bulk volume can be described by a boundary theory in one fewer dimensions |

Angular momentum | Measure of a body’s tendency to roll, spin, or orbit; measured by amount (magnitude), direction (projection), and intrinsic angular momentum (spin) |

Anyon | Third type of particle between a fermion and a boson, not fundamental but emerges in many-body systems |

Anyonic exchange statistics | Computable measure of particles changing places, exchanging wavefunctions by ‘braiding’ |

Artificial lattice | Atomic-scale structure designed to confine electrons or spins on a chosen lattice (made with scanning tunneling microscopy or electron beam) |

Band gap | Prohibited range of energy in a system |

Berry curvature | Gauge-invariant geometrical property of a band; invariant under changes in the phase of the wavefunction |

Bloch theorem | Solution to the Schrödinger equation for periodic systems |

Boson | Force particle (photon, gluon, graviton) with integer spin |

Bosonic code | Photon-based system in which physical and protective logical qubits can be encoded in a self-contained continuous value system in a single bosonic mode (state) |

Bosonic mode | Photon state controllable with standard Gaussian operations such as squeezing, displacement, phase rotation, and beam splitting |

Brillouin zone | Geometric zone inside a reciprocal lattice or crystal |

Cat code | (Schrödinger’s cat) error correction based on superpositioned coherent states |

Charge | Electrical valence of a particle (particle properties: spin, charge, angular momentum) |

Chern number | Topological invariant of the Berry curvature flux over a closed momentum surface (global system property) |

Chern–Simons theory | Mathematical model of topological invariance |

Chiral topological semimetals | Topological semimetals in crystals with a chiral structure (handedness due to lack of mirror and inversion symmetries) |

cQED Circuit quantum electrodynamics | Experimental platform for modeling and synthesizing quantum matter phases in hyperbolic space |

Code (error correction) | Allowed values (or structure of values) for data in a system, may include parameters re: how many ancilla (extra) bits protect one logical bit over what distance |

Codespace | Error-correction domain; possibly denoted by lattice grid states or graph states |

Coherent state | Oscillatory quantum state (the quantum state of the harmonic oscillator) |

Correlation function | The average (expectation value) of field operators at different positions; the amplitude for propagation of a particle or excitation between two points |

Crystal | Atoms organized in a regular array (lattice); has discrete translation symmetry |

Crystal lattice | Symmetrical three-dimensional arrangement of atoms inside a crystal |

Dimer | Molecule of two identical molecules linked together |

Discrete time crystal | Non-equilibrium state of matter that breaks time translation symmetry (repeating time structure) |

Eigenvalues | Values at allowable scale tiers in a system, levels; characteristic system values |

Electron holes | Positively charged quasiparticles denoting the lack of an electron in a state in the valence band of a semiconductor |

Energy band theory | Allowed/prohibited energy bands/band gaps in systems |

Entanglement | Quantum property of correlated physical attributes among particles (position, momentum, spin, polarization) |

Fermion | Matter particle (electron, quark) with half-odd integer (1/2, 3/2, etc.) spin |

Fractional quantum Hall effect | Quantized plateaus at fractional values of charge, giving rise to quasiparticles (collective states) in which electrons bind magnetic flux lines to make new quasiparticles that have a fractional charge and obey anyonic statistics |

Floquet engineering | Control of periodically driven time cycles in quantum matter systems |

Floquet theorem | Periodic system (Bloch) transform to solvable linear differential equations |

Frontier orbitals | Highest occupied/lowest unoccupied orbitals of a molecule |

Fuchsian model | Hyperbolic Riemannian surface model |

Gapless | No band gap in spacing between energy levels in a system |

Gapped | Gaps in energy bands in a system |

GKP codes (Gottesman, Kitaev, Preskill) | Error-corrected qubit encoding in an oscillator with superpositions of squeezed states, protected against shifts in position and amplitude damping |

Hall effect (Hall conductance) | Production of a voltage difference (the Hall voltage) across an electrical conductor that is transverse (perpendicular) to an electric current in the conductor and to an applied magnetic field perpendicular to the current (Hall 1879) |

Hamiltonian | Operator (function) used to calculate the energy levels of a quantum system |

Herbertsmithite | Mineral with quantum spin liquid magnetic properties (neither ferromagnet nor antiferromagnet); magnetic particles with constantly fluctuating, scattered orientations on a kagome lattice; (Zinc, Copper, Oxygen, Hydrogen, Chlorine); (Iran, Chile, Arizona, Greece) |

Hilbert space | Infinite-dimensional space of quantum mechanics (vs 3D Euclidean space) |

Honeycomb lattice | Standard hexagonal/triangular lattice (e.g., graphene) (generally looks the same from any direction) |

Hyperbolic band theory | Energy band theory in hyperbolic space |

Hyperbolic lattice | Synthetic quantum matter in which particles hop on a discrete tessellation of two-dimensional hyperbolic space |

Hyperbolic space | Geometric space with negative curvature (vs Euclidean space (zero curvature) and elliptic space (positive curvature) |

Josephson junction | Quantum tunneling superconducting device used in quantum computing |

Josephson junction-based superconducting circuits | Superconducting qubits controlled with microwave photons (quantized electromagnetic fields stored in the superconducting circuits) |

K-space | Wave vector space of possible values of momentum for a particle (also the spatial frequency domain of a Fourier transform or a compactly generated topological space) |

Kagome lattice | Uniform tiling of equilateral triangles and hexagons |

Kitaev honeycomb lattice | Exactly solvable spin model in two dimensions; spins are on the vertices of a honeycomb lattice with nearest-neighbor interactions |

Lattice | Regular array geometric arrangement of matter in a space (e.g., crystal); scaffolding |

Lattice surgery | Switching between error-correcting codes on the fly |

Magnon | Collective excitation of electron spins in a crystal lattice |

Many-body problem | Physical systems with many interacting particles (three/more) |

Majorana fermions | Exotic fermions that are their own antiparticles |

Many-body localization (MBL) | Many-body interactions causing quantum particles to be localized and maintained in an out-of-equilibrium state |

Multi-scale entanglement renormalization ansatz (MERA) | Tensor network representation for ground states of critical quantum spin chains (with a network that extends in an additional dimension corresponding to scale) |

Mode | Characteristic state, normal frequency, allowable value |

Molecular code | GKP codes for asymmetric bodies (molecules) in free space |

Nodal-line semimetals | Topological semimetals with energy band-touching manifolds in the shape of closed loops |

Non-abelian | Non-commuting (order of terms matters) (v. abelian) |

Nonsymmorphic | Not comparable on a symmetry basis (symmetry operations do not have a common point on the lattice) |

Paramagnet | Material weakly attracted by an external magnetic field |

Particle properties | Spin, charge, angular momentum, polarization |

Phonon | Collective excitation of atoms in a rigid crystal structure |

Plasmon | Collective excitation of electrons simultaneously oscillating with respect to ions |

Quantum droplet | Quantum matter phase defined by properties emerging from the interactions of bosonic or fermionic constituents |

Quantum error-correcting code | Logical codespace corresponding to the physical subspace of a lattice |

Quantum Hall effect | Quantum version of the Hall effect; obtained by applying a strong magnetic field perpendicular to a two-dimensional electron system |

Quantum matter | Novel phases of matter at zero temperature with emergent order and exotic properties, possibly including the emergence of quasiparticles (collective excitations) with anyonic exchange statistics, gauge theory, quantum phase transitions, and low-energy effective theories of topologically ordered states |

Quantum nanoscience | Nanostructure fabrication that exploits quantum effects |

Quantum phase transition | Phase transition between different quantum phases via parameter change such as magnetic field or pressure |

Quantum spin Hall states | Quantum Hall effect based on the flow of spin currents (as opposed to charge currents) |

Quantum spin liquid | Quantum matter phase with magnetic spins (qubits) degrees of freedom; a magnetic system that does not settle into a long-range ordered configuration, even at zero temperature, residing in a nontrivial quasi-disordered ground state |

Quantum topology | Novel properties of topological shapes in quantum systems |

Quasiparticle | Long-lived, low-energy excitation of a many-body state in fermions (collective excitations in bosons); examples: electron holes, phonons, plasmons, magnons |

Rényi entropy | Composite of Shannon, Hartley, collision, and minimum entropy |

“Schrödinger cat” states | Superpositioned quantum states |

Soliton | Stable solitary wavepacket in nonlinear systems |

Spin | Intrinsic form of angular momentum carried by elementary particles (depicted as an axis of rotation, but actual particles do not rotate); particles with spin may possess a magnetic dipole moment (exploited in electronic devices) |

Spin chain | Linear collection of magnetic moments with spin–spin coupling interactions |

Spin engineering | Control of spin systems in devices and materials, including the precise arrangement of magnetic atoms as a probe |

Spinors | More complicated version of vectors and tensors needed to describe the rotations of particle spin |

Spintronics (spin electronics) | Using electron spins as the internal degree of freedom to store 0 s/1 s for information processing |

Squeezed (coherent) state | Pinched oscillatory quantum state (to reduce the quantum noise (environmental interference)) |

Stabilizer code | Quantum error-correction code (quantum version of linear codes), based on X-, Y-, Z-axis Pauli operators to measure entangled states and correct (bit flip, spin flip) a corrupt quantum state to its original state; commuting operators |

Subsystem code | Quantum error code with non-commuting operators |

Surface code | Stabilizer code, topology-based, defined on a two-dimensional spin lattice, taking various shapes |

Symmetry | Features of particles and spacetime which are unchanged under some transformation; property of looking the same from different points of view (face, cube, laws of physics) |

Symmetry breaking | Phase transition, rupturing a system’s symmetry (e.g., time-reversal, particle-hole, chiral) |

Symmetry-protected topological (SPT) order | Quantum matter phases with trivial topological order (short-range entanglement), symmetry, and a finite energy gap (e.g., topological insulator) |

Tessellation | Tiling of a plane using geometric shapes (tiles) |

Time-reversal symmetry breaking | System property: the dynamics of a process remain well-defined when the sequence of time-states is reversed |

Topological entanglement entropy | Topology-based measure of entanglement entropy specific to quantum matter and quantum phase transition calculated from the quasiparticle excitations of the many-body state or in a comparison between the system and the von Neumann entropy (tripartite information; two time, one space dimension) |

Topological insulator | Material with a conducting surface and an insulating interior; surface states are symmetry-protected (e.g., time-reversal, particle-hole, chiral symmetry) |

Topological quantum field theory | Quantum field theory emphasizing topological invariants and impervious to spacetime contraction; explains quantum matter phases |

Topological qubits | Computational qubits made with quantum matter phases (e.g., by putting quantum spin liquids into a geometrical array) |

Topological semimetals | Material with energy band-touching manifolds (at zero-dimensional points or one-dimensional nodal lines/rings) |

Topological strings | Strings linking atoms entangled in a quantum spin liquid |

Topology | The property of geometric form being preserved under deformation (such as bending, stretching, twisting, and crumpling, but not cutting or gluing) |

Toric code | Stabilizer code defined on a two-dimensional lattice with periodic boundary conditions (torus-shaped); stabilizer operators on the spins around vertex and plaquette (face) |

Trotterization | Operation to simulate the evolution of a Hamiltonian |

Valleytronics (valley electronics) | Using valleys in the electronic band structure of the first Brillouin zone of a crystal as the internal degree of freedom to store 0 s/1 s for information processing |

von Neumann entropy | (Quantum mechanical entropy) minimum over all measurement bases of Shannon entropy |

Wavefunction | Quantum system state description (positions or speeds (momenta) of entire system configurations); generally intractable Schrödinger equation applied (complex-valued probability amplitudes with real and imaginary wave-shaped components) |

Weyl and Dirac topological semimetals | Topological semimetals created by two energy bands crossing at a single node (Weyl node), with two-fold (Weyl) and four-fold (Dirac) degenerate Fermi points |

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Building Blocks | Short-Range Protection | Long-Range Entanglement |
---|---|---|

Symmetry and topology | Topological insulators | Quantum spin liquids |

Anyons/quasiparticles | Topological superconductors | Quantum Hall states |

Hyperbolic space | Topological semimetals | Entanglement entropy |

Code | Description | |
---|---|---|

1 | Code (general) | Allowed values (or value structure) for data or other parameters |

2 | QEC code | Logical codespace corresponding to a physical lattice model space |

3 | Stabilizer code | Topology-based Pauli operators (X, Y, Z) to restore bit/spin flip |

4 | Toric code | Stabilizer codes defined on a 2D torus-shaped spin lattice |

5 | Surface code | Stabilizer codes defined on a 2D spin lattice in any shape |

6 | Bosonic code | Self-contained photon-based oscillator system with bosonic modes |

7 | GKP code | Bosonic code: squeezed states protect position-amplitude shifts |

8 | Molecular code | GKP codes extended to asymmetric bodies (molecules) in free space |

9 | Cat code | Superpositioned states (Schrödinger) used as error-correction codes |

Model Parameter | Quantum Information | Quantum Matter | |
---|---|---|---|

1 | Multiscalar model | AdS/CFT correspondence | Condensed matter bulk–boundary |

2 | Phase transition | Neural signal integration | Topological entanglement entropy |

3 | Symmetry | Symmetry rebalancing | Short-range: topological semimetal |

4 | Entanglement | Non-local measurement | Long-range: quantum spin liquid |

5 | Renormalization | bMERA tensor network | MERA (wavefunction) TPUs |

6 | Floquet dynamics | Neural signal periodicity | Floquet circuits with NISQ QC |

7 | Matrix mechanics | Diagonalize matrices | Quantum Hall matrix mechanics |

8 | Quantum walks | Faster lattice search | Molecular code rotation |

9 | Optimization | Chern–Simons curvature | High Chern numbers (invariance) |

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**MDPI and ACS Style**

Swan, M.; Dos Santos, R.P.; Witte, F.
Quantum Matter Overview. *J* **2022**, *5*, 232-254.
https://doi.org/10.3390/j5020017

**AMA Style**

Swan M, Dos Santos RP, Witte F.
Quantum Matter Overview. *J*. 2022; 5(2):232-254.
https://doi.org/10.3390/j5020017

**Chicago/Turabian Style**

Swan, Melanie, Renato P. Dos Santos, and Frank Witte.
2022. "Quantum Matter Overview" *J* 5, no. 2: 232-254.
https://doi.org/10.3390/j5020017