quantum algorithms:
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
What is called the toric code [Kitaev 2003] is a quantum error correcting code whose underlying Hilbert space is that of a collection of qubits (“spins” as in the Ising model) arranged on the vertices of a rectangular 2-dimensional lattice with periodic boundary conditions, hence a lattice on a torus. (If the periodic boundary conditions are disregarded one speaks of a planar code or surface code and here specifically of the Kitaev surface code).
The code subspace of Kitaev’s toric code (the subspace whose elements are regarded as error-free quantum states) is the kernel of an operator that may be thought of as the Hamiltonian of a hypothetical quantum system (a lattice-spin model as often considered in solid state physics akin to the Ising model) featuring next-to-nearest neighbour interactions of the qubits/spins via Pauli operators. Therefore one may think of the physical qbits of the toric code as forming the ground state of this system and of errors to them as the excitations above the ground state.
(The actual physical realization of the toric code Hamiltonian has remained elusive even besides the question of modelling periodic boundary conditions, cf. e.g. Nussinov & Ortiz 2008, VLV21.)
This hypothetical quantum system exhibits a simple form of topological order in that the local errors happen to look like the creation of anyons out of the vacuum, the error correction looks like moving these anyons back intoeach other for them to annihilate, and the failure of error correction through the code corresponds to these anyons undergoing non-trivial braiding before re-annihilation. For this reason, the toric code has attracted much interest also as a theoretical toy model for anyonic topological order in addition to or independently of its use in quantum error correction.
The original article:
Review:
Maria F. Araujo de Resende: A pedagogical overview on 2D and 3D Toric Codes and the origin of their topological orders, Reviews in Mathematical Physics 32 02 (2020) 2030002 [doi:10.1142/S0129055X20300022, arXiv:1712.01258]
Héctor Bombín, around Fig. 9 in: An Introduction to Topological Quantum Codes, in: Quantum Error Correction, Cambridge University Press (2013) [ISBN:9780521897877, arxiv:1311.0277]
Paul Herringer: The Toric Code (2020) [pdf]
See also:
Wikipedia, Toric code
errorcorrectionzoo.org: Toric code
On the problem of its experimental realization:
Zohar Nussinov, Gerardo Ortiz, pp. 5 in: Autocorrelations and Thermal Fragility of Anyonic Loops in Topologically Quantum Ordered Systems, Phys. Rev. B 77 x (2008) Phys. Rev. B 77, 064302 [doi:10.1103/PhysRevB.77.064302, arXiv:0709.2717, arXiv:0709.2717]
“the existence of a gap in this system may not protect a finite expectation value of the Toric code operators […] The physical reason behind this result is the proliferation of topological defects (solitons) at any finite $T$.”
Ruben Verresen, Mikhail D. Lukin, Ashvin Vishwanath: Prediction of Toric Code Topological Order from Rydberg Blockade, Phys. Rev. X 11 (2021) 031005 [doi:10.1103/PhysRevX.11.031005]
“Unfortunately, the experimental realization of such phases […] has been exceedingly difficult.”
The theory of anyonic topologically ordered quantum materials is often discussed assuming periodic boundary conditions, making the space of positions of a given anyon a torus. (While this is a dubious assumption for position-space anyons in actual experiment, the intended ground state-degeracy crucially depends on this assumption.)
In fact, the notion of topological order was introduced already assuming torus-shaped materials:
Xiao-Gang Wen: Vacuum Degeneracy of Chiral Spin State in Compactified Spaces, Phys. Rev. B 40 7387 (1989) [doi:10.1103/PhysRevB.40.7387]
Xiao-Gang Wen: Topological Orders in Rigid States, Int. J. Mod. Phys. B 4 239 (1990) [doi:10.1142/S0217979290000139, pdf]
Xiao-Gang Wen, Qian Niu: Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces, Phys. Rev. B 41 9377 (1990) [doi:10.1103/PhysRevB.41.9377]
E. Keski-Vakkuri, Xiao-Gang Wen: Ground state structure of hierarchical QH states on torus and modular transformation, Int. J. Mod. Phys. B 7 4227 (1993) [doi:10.1142/S0217979293003644, arXiv:hep-th/9303155]
Further discussion along these lines:
Zhu-Xi Luo, Yu-Ting Hu, Yong-Shi Wu: On Quantum Entanglement in Topological Phases on a Torus, Phys. Rev. B 94 075126 (2016) [doi:10.1103/PhysRevB.94.075126, arXiv:1603.01777]
Zhuan Li, Roger S. K. Mong: Detecting topological order from modular transformations of ground states on the torus, Phys. Rev. B 106 (2022) 235115 [doi:10.1103/PhysRevB.106.235115, arXiv:2203.04329]
Explicit discussion of anyons on tori:
Xiao-Gang Wen, E. Dagotto, Eduardo Fradkin: Anyons on a torus, Phys. Rev. B 42 (1990) 6110 [doi:10.1103/PhysRevB.42.6110]
Roberto Iengo, Kurt Lechner: Quantum mechanics of anyons on a torus, Nuclear Physics B 346 2–3 (1990) 551-575 [doi:10.1016/0550-3213(90)90292-L, spire:28316]
Roberto Iengo, Kurt Lechner: Exact results for anyons on a torus, Nuclear Physics B 364 3 (1991) 551-583 [doi:10.1016/0550-3213(91)90277-5]
Yasuhiro Hatsugai, Mahito Kohmoto, Yong-Shi Wu: Anyons on a torus: Braid group, Aharonov-Bohm period, and numerical study, Phys. Rev. B 43 (1991) 10761 [doi:10.1103/PhysRevB.43.10761]
Yutaka Hosotani, Choon-Lin Ho: Anyons on a Torus, AIP Conf. Proc. 272 (1992) 1466–1469 [doi:10.1063/1.43444, arXiv:hep-th/9210112]
Choon-Lin Ho, Yutaka Hosotani: Anyon equation on a torus, International Journal of Modern Physics A 07 23 (1992) 5797-5831 [doi:10.1142/S0217751X92002647]
Ikuo Ichinose, Toshiyuki Ohbayashi: Exactly soluble model of multispecies anyons and the braid group on a torus, Nucl.Phys. B 419 (1994) 529-552 [doi:10.1016/0550-3213(94)90343-3]
Alexei Kitaev: Fault-tolerant quantum computation by anyons, Annals of Physics 303 1 (2003) 2-30 [doi:10.1016/S0003-4916(02)00018-0, arXiv:quant-ph/9707021]
(introducing the toric code)
Songyang Pu, Jainendra K. Jain: Composite anyons on a torus, Phys. Rev. B 104 (2021) 115135 [doi:10.1103/PhysRevB.104.115135, arXiv:2106.15705]
Steven H. Simon: Anyon Vacuum on a Torus and Quantum Memory, Section 4.3 in: Topological Quantum, Oxford University Press (2023) [ISBN:9780198886723, pdf, webpage]
Shang Liu: Anyon quantum dimensions from an arbitrary ground state wave function, Nature Communications 15 (2024) 5134 [doi:10.1038/s41467-024-47856-7, arXiv:2304.13235]
But anyonic states may alternatively be localized in more abstract spaces. Anyons localized not in position space but in “reciprocal momentum space”, namely on the Brillouin torus of quasi-momenta of electrons in a crystal, are considered in
Also proposals to classify (free or interacting) topological phases of matter by topological quantum field theory mean to consider them on all base space topologies, including tori. This idea may originate around:
Anton Kapustin: Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology [spire:1283873, arXiv:1403.1467]
“Our basic assumption is that a gapped state of matter with short-range interactions can be put on a curved space-time of arbitrary topology […] At short distances a system is usually defined on a regular lattice, with short-range interactions. However, if we allow for disorder, then dislocations in the lattice are possible, and more general triangulations also become possible”
Anton Kapustin: Bosonic Topological Insulators and Paramagnets: a view from cobordisms [spire:1292830, arXiv:1404.6659]
“SPT phases are usually defined on a spatial lattice, while time may or may not be discretized. In the effective action approach we want to allow space-time to have an arbitrary topology, thus we discretize both space and time and regard the system as being defined on a general triangulation $K$ of a $d$-dimensional manifold $X$.”
and is implicit also in the proposal of classification via invertible field theory of:
Daniel S. Freed, Michael J. Hopkins: Section 9.3 of: Reflection positivity and invertible topological phases, Geom. Topol. 25 (2021) 1165-1330 [doi:10.2140/gt.2021.25.1165, arXiv:1604.06527]
Kazuya Yonekura: On the cobordism classification of symmetry protected topological phases, Commun. Math. Phys. 368 (2019) 1121–1173 [doi:10.1007/s00220-019-03439-y, arXiv:1803.10796]
The broad idea that TQFT is the right language to speak about anyonic topological order is now often stated as if self-evident, e.g. in:
Critical commentary on the assumption of non-trivial topology in position space appears in the following (whose authors then suggest that using extended TQFT may ameliorate the problem, p. 2):
Davide Gaiotto, Theo Johnson-Freyd: Condensations in higher categories [spire:1736539, arXiv:1905.09566]
“This relationship between gapped condensed matter systems and TQFTs is perplexing, particularly so if one takes a “global” approach to TQFTs, defining them à la Atiyah 1988 in terms of partition functions attached to non-trivial Euclidean space-time manifolds and spaces of states attached to non-trivial space manifolds. From that perspective, matching a given lattice system to a TQFT would require identifying a lot of extra structure to be added to the definition of the lattice system in order to define it on discretizations of non-trivial space manifolds and to define adiabatic evolutions analogous to non-trivial space-time manifolds.”
“If one takes the information-theoretic perspective that a phase of matter is fully characterized by the local entanglement properties of the ground-state wavefunction of the system, with no reference to a time evolution, then even more work may be needed.”
“These concerns are not just abstract. Given some phase of matter, in the lab or in a computer, it is hard to extract the data which would pin down the corresponding TQFT, or even know if the TQFT exists. For example, we can hardly place a three-dimensional material on a non-trivial space-manifold. We can only try to simulate that by employing judicious collections of defects in flat space.”
Last revised on August 25, 2024 at 11:10:08. See the history of this page for a list of all contributions to it.