Quantum Hall states are a form of topological order that falls outside Landau symmetry breaking. In particular, the “fractional quantum Hall effect” is of interest to fault-tolerant quantum computing, and is obtained by applying a strong magnetic field perpendicular to a two-dimensional electron system at low temperature. In fractional quantum Hall states, electrons create quasiparticles (collective states) that have a fraction of the charge of a single electron and obey anyonic statistics [40].

The Hall effect is the production of a voltage difference (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 (per Edwin Hall 1879). The idea is that a current of electrons in a thin conducting strip (two-dimensional plane) is subject to a constant magnetic field in the normal direction, while the Lorentz force perpendicular to the current causes a buildup of charge on the edge of the strip that induces a voltage across the width of the strip. The quantum Hall effect is the quantum version of the Hall effect, observed in two-dimensional electron systems at low temperature as magnetic fields are applied and the Hall conductance takes on quantized values.

The “fractional quantum Hall effect” indicates 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 [5,6]. (The 1998 Nobel prize was awarded for the discovery of a new form of quantum fluid with fractionally charged excitations.) The integer quantum Hall effect indicates quantized plateaus at integer values of charge. The result is quantized tiers (“Hall plateaus”) that persist when electron density is varied; there is a finite density of states that are localized (pinned, as in the Anderson localization), which is useful in computational devices as electrons can be pinned (localized). A further advance is quantum spin Hall states, instantiating the quantum Hall effect based on the flow of spin currents (as opposed to charge currents), for potential application to next-generation quantum computing methods. A matrix mechanics formalism (allowing the ability to diagonalize multiple matrices to aid in solving many-body problems) has been extended to quantum Hall states, to characterize entanglement and emergent structure [41].

The main way that long-range entanglement is engaged in quantum matter systems is with quantum spin liquids. Quantum spin liquids are quantum matter phases in which the elementary degrees of freedom are magnetic spins (which can be instantiated as qubits in a quantum computational system) [42]. In general, magnetic systems are ordered in one of three ways: with all spins pointing in the same direction (ferromagnet, as a refrigerator magnet), disordered with neighboring spins (on different sublattices) pointing in opposite directions (antiferromagnet), or a frustrated order that is a combination of both (spin liquid or spin glass). Quantum spin liquids are the quantum version of a spin liquid, a “liquid” of disordered spins, that, is a phase of matter formed by quantum spins interacting in magnetic materials. Specifically, a quantum spin liquid is a magnetic system that does not settle into a large-scale ordered configuration, even at zero temperature, and resides in a nontrivial quasi-disordered ground state, which can be manipulated. Quantum spin liquids are typically characterized by topological order, long-range entanglement, and fractionalized (anyon) excitations.

In the usual situation of regular magnets at low temperature, electrons stabilize and form large-scale patterns (such as domains, stripes, or checkerboards) with magnetic properties. However, in a quantum spin liquid, the electrons do not stabilize when cooled and preserve their disorder much in the way liquid water exists in a disordered state. The electrons are constantly changing and fluctuating (like a liquid) in a highly entangled quantum state. Hence, the quantum spin liquid is called a liquid because, like a liquid, the fluctuating elements (electrons) do not settle into in a regular lattice as in a solid. Quantum spin liquids are attractive in quantum computing for the possibility of creating topological qubits made with quantum spin liquid matter phases (by instantiating the quantum spin liquids in a geometrical array).

Several different physical models have a disordered ground state that can be described as a quantum spin liquid. The real-life mineral, Herbertsmithite (named after mineralogist Herbert Smith), was discovered in Chile in 1972, and subsequently in many other locations (Iran, Chile, Arizona, and Greece). Herbertsmithite is a mineral with quantum spin liquid magnetic properties (neither ferromagnet nor antiferromagnet). The magnetic particles of the material have constantly fluctuating, scattered orientations in a kagome (triangle–hexagon) lattice. The mineral is comprised of Zinc, Copper, Oxygen, Hydrogen, and Carbon. In the laboratory setting, a specific kind of proposed quantum spin liquid formulation, the Kitaev honeycomb, was measured experimentally in 2015 with the excitation of a spin liquid on a honeycomb lattice with neutrons in a graphene-like material (ruthenium) (Oak Ridge National Laboratory [43]). Measurements confirmed the expected properties of the quantum spin liquid, namely, strong spin orbit coupling and low-temperature magnetic order.

In 2021, two projects created quantum spin liquids from scratch, specifically demonstrating the long-range entanglement property, by using a coupled superconducting circuit and an optical atom array [44]. The former team (from Google Quantum AI) used a 32-qubit quantum processor to study the ground state and excitations of the toric code [45]. The latter team (from the Lukin laboratory at Harvard) detected signatures of a toric code-type quantum spin liquid in a two-dimensional array of Rydberg atoms held in optical tweezers (lasers) [46]. The central achievement for both projects was engineering the topological order known as the toric code, an archetypical two-dimensional lattice model that exhibits the exotic properties of topologically ordered states and is proposed for quantum error correction [2].

In more detail, the first team realized topologically ordered states using a 32-qubit superconducting quantum processor (Sycamore). The ground state of the toric code Hamiltonian was prepared using an efficient quantum circuit on the Sycamore processor. The topological nature of the state was experimentally established by measuring the topological entanglement entropy (topology-based measure of quantum entanglement entropy) and by simulating anyon interferometry to extract the braiding statistics of the emergent excitations. The second team used a 219-atom programmable quantum simulator to probe quantum spin liquid states. Arrays of atoms were placed on the links of a kagome lattice (lattice comprised of equilateral triangles and hexagons). The onset of the toric-type quantum spin liquid phase was detected using topological string operators (which indicate the signatures of topological order and quantum correlations). A class of dimer models (molecules with identical molecules linked together) was implemented as a promising candidate to host quantum spin liquid states.

One result of the quantum spin liquid demonstrations is new understandings of the bulk–boundary relationship in condensed matter physics. The bulk–boundary relationship usually entails some range of unrestricted boundary behavior, within the context of boundary symmetries that are linked to and protected by bulk invariants. However, having an experimental platform for the more rigorous creation and manipulation of quantum spin liquids is revealing new things such as that, strikingly, under certain open boundary conditions, the boundary itself undergoes a second-order quantum phase transition, independent of the bulk [47]. Future work could tackle creating even more precise atomic quantum spin liquids, assembled from scratch by building lattices of magnetic atoms from the bottom up, literally atom-by-atom, with the probe tip of a scanning tunneling microscope positioning the atoms on the surface [48].

The reason quantum matter phases with long-range entanglement are attractive as potential topological qubits in quantum computing is due to the error-correction possibility afforded by non-locality. Working with quantum spin liquid phases entails accessing non-local observables, through, for example, topological string operators. The non-local nature of quantum spin liquid states makes them attractive platforms for fault-tolerant quantum computation, as quantum information encoded in locally indistinguishable ground states is robust to local perturbation. The principle underlying topological quantum error-correcting codes is that, in the quantum spin liquid model, the logical codespace corresponds to the degenerate ground state subspace of a lattice model. The key benefit of long-range entanglement is being able to perform quantum error correction (through non-local measurements).

Entanglement is an important aspect of being able to manage system criticality and phase transition. A phase transition between different quantum phases can be triggered by a change in physical parameters such as magnetic field or pressure [49]. Whereas classical phase transition is triggered by varying a macroscopic physical parameter such as temperature or density, quantum phase transition is caused by changing a quantum physical parameter at zero temperature, tuning a non-temperature variable such as magnetic field, pressure, or chemical composition (via a Hamiltonian term). Although a classical phase transition is often temperature-based (often called thermal phase transition), a quantum phase transition is not carried out by varying the temperature, because the system remains at zero temperature throughout; the quantum system is at zero temperature before and after the phase transition.

Topological entanglement entropy is a topology-based measure of entanglement entropy specific to quantum matter and quantum phase transition. The phases on either side of a quantum critical point may be characterized by different kinds of topological order. The quasiparticle excitations or quantum correlations among the microscopic degrees of freedom might have qualitatively different properties in the two phases. Since it may not be possible to distinguish the phases based on a local order parameter, a non-local parameter is needed such as a topology-based measure of the long-range system entanglement.

Topological entanglement entropy incorporates aspects that the usual quantum entropy measures (e.g., von Neumann entropy and Rényi entropy) do not, that are specific to measuring entanglement entropy in quantum many-body states with topological order [50]. The topological entanglement entropy is calculated either from the quasiparticle excitations of the many-body state or in a comparison between the system and the von Neumann entropy. (Specifically, topological entanglement entropy is computed as the logarithm of the total quantum dimension of the quasiparticle excitations of the state, or by comparing the von Neumann entropy between a spatial block and the rest of the system).

In fact, since it treats topological invariance, topological entanglement entropy constitutes a topological quantum field theory, in three dimensions (two space and one time), in the general formulation. A topological quantum field theory is a quantum field theory that emphasizes topological invariants and in which the correlation functions do not depend on the metric of spacetime. This means that the theory is not sensitive to changes in the shape of spacetime; if spacetime warps or contracts, the correlation functions do not change, and they are topologically invariant. Just as any topological object which can bend and be deformed but not cut and the invariant properties persist, so too in a topological quantum field theory, spacetime can warp or contract but the correlation functions do not change and remain topologically invariant. This topological entanglement entropy formulation is operationalized as tripartite information (equations involving two space and one time dimension) [51].