The concrete realization of topological quantum computing using low-dimensional quasiparticles, known as anyons, remains one of the important challenges of quantum computing. A topological quantum computing platform promises to deliver more robust qubits with additional hardware-level protection against errors that could lead to the desired large-scale quantum computation. We propose quasicrystal materials as such a natural platform and show that they exhibit anyonic behavior that can be used for topological quantum computing. Different from anyons, quasicrystals are already implemented in laboratories. In particular, we study the correspondence between the fusion Hilbert spaces of the simplest non-abelian anyon, the Fibonacci anyons, and the tiling spaces of the one-dimensional Fibonacci chain and the two-dimensional Penrose tiling quasicrystals. A concrete encoding on these tiling spaces of topological quantum information processing is also presented by making use of inflation and deflation of such tiling spaces. While we outline the theoretical basis for such a platform, details on the physical implementation remain open.The concrete realization of topological quantum computing using low-dimensional quasiparticles, known as anyons, remains one of the important challenges of quantum computing. A topological quantum computing platform promises to deliver more robust qubits with additional hardware-level protection against errors that could lead to the desired large-scale quantum computation. We propose quasicrystal materials as such a natural platform and show that they exhibit anyonic behavior that can be used for topological quantum computing. Different from anyons, quasicrystals are already implemented in laboratories. In particular, we study the correspondence between the fusion Hilbert spaces of the simplest non-abelian anyon, the Fibonacci anyons, and the tiling spaces of the one-dimensional Fibonacci chain and the two-dimensional Penrose tiling quasicrystals. A concrete encoding on these tiling spaces of topological quantum information processing is also presented by making use of inflation and deflation of such tiling spaces. While we outline the theoretical basis for such a platform, details on the physical implementation remain open.

arXiv:2207.08928SymmetryAug 26, 20227 citations

Periodic functions and quantum computing

Karl E. Friedl

Jan 1, 20030 citations

Quantum Computation

Xiaoyu Li, Junjie Wu, Jialin Zhang

Communications in computer and information scienceJan 1, 20261 citation

Spacetime Quasicrystals and Computational Complexity Theory in the Road Towards Quantum Gravity

Sotirios Mygdalas

Part I: Semiclassical Gravity Efficiently Solves $\mathsf{NP}$ -Complete Problems Assuming the gravitational field is classical and that it couples to quantum fields via the semiclassical Einstein field equations (EFEs), we show that the weak-field dynamics of a massive and non-relativistic qubit can in principle be used to solve an $\mathsf{NP}$ -complete problem in polynomial time. We attribute this vast computational power to the non-linear qubit dynamics afforded by the semiclassical EFEs. Consequently, the above two assumptions entail a violation of the Physical Extended Church--Turing Thesis, which we regard as evidence for the quantization of gravity. Part II: Spacetime Quasicrystals We generalize self-similar quasicrystals, such as the Penrose tilings, from Euclidean space to Minkowski spacetime. We construct the first examples of Lorentzian quasicrystals and identify their key novel features. We then explore possible connections to fundamental physics. First, we propose a speculative scenario in which our $(3+1)$ D universe is embedded in a particularly symmetric $(9+1)$ D torus $\mathbb{T}^{9,1}$ , previously identified as the most symmetric toroidal compactification of the superstring. We suggest that this construction may shed light on the seesaw relation $M_{\rm Pl}M_{\rm vac} \approx M_{\rm EW}^{2}$ between the Planck, vacuum-energy, and electroweak scales. Second, we replace the random causal sets of causal set theory with ordered, highly symmetric quasi-crystalline causal sets. We find suggestive evidence that our quasicrystals outperform Poisson random sets in the number-volume correspondence in both $(1+1)$ D and $(3+1)$ D, where such behaviour has been conjectured to be impossible. Finally, extending hyperuniformity from space to spacetime, we show that our $(1+1)$ D quasicrystal is the first demonstrated example of a spacetime-hyperuniform structure.Part I: Semiclassical Gravity Efficiently Solves $\mathsf{NP}$-Complete Problems Assuming the gravitational field is classical and that it couples to quantum fields via the semiclassical Einstein field equations (EFEs), we show that the weak-field dynamics of a massive and non-relativistic qubit can in principle be used to solve an $\mathsf{NP}$-complete problem in polynomial time. We attribute this vast computational power to the non-linear qubit dynamics afforded by the semiclassical EFEs. Consequently, the above two assumptions entail a violation of the Physical Extended Church--Turing Thesis, which we regard as evidence for the quantization of gravity. Part II: Spacetime Quasicrystals We generalize self-similar quasicrystals, such as the Penrose tilings, from Euclidean space to Minkowski spacetime. We construct the first examples of Lorentzian quasicrystals and identify their key novel features. We then explore possible connections to fundamental physics. First, we propose a speculative scenario in which our $(3+1)$D universe is embedded in a particularly symmetric $(9+1)$D torus $\mathbb{T}^{9,1}$, previously identified as the most symmetric toroidal compactification of the superstring. We suggest that this construction may shed light on the seesaw relation $M_{\rm Pl}M_{\rm vac} \approx M_{\rm EW}^{2}$ between the Planck, vacuum-energy, and electroweak scales. Second, we replace the random causal sets of causal set theory with ordered, highly symmetric quasi-crystalline causal sets. We find suggestive evidence that our quasicrystals outperform Poisson random sets in the number-volume correspondence in both $(1+1)$D and $(3+1)$D, where such behaviour has been conjectured to be impossible. Finally, extending hyperuniformity from space to spacetime, we show that our $(1+1)$D quasicrystal is the first demonstrated example of a spacetime-hyperuniform structure.

hep-thCross-listedarXiv:2610.09855Oct 8, 2026

Thermodynamics and Melting of a Quantum Quasicrystal

Anonymous, Thomas G. Kiely, Leon Balents +1 more

A quantum quasicrystal was proposed to exist by mean-field, variational arguments in two-dimensional Rashba spin-orbit coupled BECs with dipolar interactions. Despite this remarkable prediction, there is little known about the superfluid character or stability of this quasicrystalline state against thermal and quantum fluctuations, whose importance is implied by the massive single-particle degeneracy of the Rashba Hamiltonian. Here, we apply field-theoretic numerical simulations based on a boson coherent state path integral representation to simulate the finite-temperature behavior of a periodic approximant of an octagonal quasicrystal. We find that the quasicrystal state maintains a 50% superfluid fraction down to low temperature, suggesting supersolid character. We compute a phase diagram that depicts a region of thermodynamic stability for the octagonal quasicrystal phase at low temperature. At intermediate temperature, the quasicrystal undergoes a first-order transition into a crystalline BEC with hexagonal symmetry. Our findings support a possible experimental realization of the octagonal quasicrystal in ^{164}Dy quantum gases below a temperature of 75 nK.A quantum quasicrystal was proposed to exist by mean-field, variational arguments in two-dimensional Rashba spin-orbit coupled BECs with dipolar interactions. Despite this remarkable prediction, there is little known about the superfluid character or stability of this quasicrystalline state against thermal and quantum fluctuations, whose importance is implied by the massive single-particle degeneracy of the Rashba Hamiltonian. Here, we apply field-theoretic numerical simulations based on a boson coherent state path integral representation to simulate the finite-temperature behavior of a periodic approximant of an octagonal quasicrystal. We find that the quasicrystal state maintains a 50% superfluid fraction down to low temperature, suggesting supersolid character. We compute a phase diagram that depicts a region of thermodynamic stability for the octagonal quasicrystal phase at low temperature. At intermediate temperature, the quasicrystal undergoes a first-order transition into a crystalline BEC with hexagonal symmetry. Our findings support a possible experimental realization of the octagonal quasicrystal in ^{164}Dy quantum gases below a temperature of 75 nK.

Physical Review LettersJul 1, 20260 citations

Quantum computation

Dawei Ding

XRDS Crossroads The ACM Magazine for StudentsSep 20, 20160 citations

Part I: Semiclassical Gravity Efficiently Solves

\mathsf{NP}

-Complete Problems Assuming the gravitational field is classical and that it couples to quantum fields via the semiclassical Einstein field equations (EFEs), we show that the weak-field dynamics of a massive and non-relativistic qubit can in principle be used to solve an

\mathsf{NP}

-complete problem in polynomial time. We attribute this vast computational power to the non-linear qubit dynamics afforded by the semiclassical EFEs. Consequently, the above two assumptions entail a violation of the Physical Extended Church--Turing Thesis, which we regard as evidence for the quantization of gravity. Part II: Spacetime Quasicrystals We generalize self-similar quasicrystals, such as the Penrose tilings, from Euclidean space to Minkowski spacetime. We construct the first examples of Lorentzian quasicrystals and identify their key novel features. We then explore possible connections to fundamental physics. First, we propose a speculative scenario in which our

(3+1)

D universe is embedded in a particularly symmetric

(9+1)

D torus

\mathbb{T}^{9,1}

, previously identified as the most symmetric toroidal compactification of the superstring. We suggest that this construction may shed light on the seesaw relation

M_{\rm Pl}M_{\rm vac} \approx M_{\rm EW}^{2}

between the Planck, vacuum-energy, and electroweak scales. Second, we replace the random causal sets of causal set theory with ordered, highly symmetric quasi-crystalline causal sets. We find suggestive evidence that our quasicrystals outperform Poisson random sets in the number-volume correspondence in both

(1+1)

D and

(3+1)

D, where such behaviour has been conjectured to be impossible. Finally, extending hyperuniformity from space to spacetime, we show that our

(1+1)

D quasicrystal is the first demonstrated example of a spacetime-hyperuniform structure.