Abstract
There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a non-chiral E8 theory, which is based on the root lattice of E8 understood as an even self-dual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I.B. Frenkel, J. Lepowsky and A. Meurman, A natural representation of the fischer-griess monster with the modular function j as character, Proc. Natl. Acad. Sci. 81 (1984) 3256.
I. Frenkel, J. Lepowsky and A. Meurman, Vertex operator algebras and the Monster, Academic Press, New York U.S.A. (1989).
L. Dolan, P. Goddard and P. Montague, Conformal field theories, representations and lattice constructions, Commun. Math. Phys. 179 (1996) 61.
C. Dong, R. L. Griess Jr and G. Hohn, Framed vertex operator algebras, codes and the moonshine module, Commun. Math. Phys. 193 (1998) 407.
D. Gaiotto and T. Johnson-Freyd, Holomorphic SCFTs with small index, arXiv:1811.00589 [INSPIRE].
N.D. Elkies, Lattices, linear codes, and invariants. Part I, Not. AMS 47 (2000) 1238.
N. D. Elkies, Lattices, linear codes, and invariants. Part II, Not. AMS 47 (2000) 1382.
T. Hartman, D. Mazáč and L. Rastelli, Sphere packing and quantum gravity, JHEP 12 (2019) 048 [arXiv:1905.01319] [INSPIRE].
N. Afkhami-Jeddi, H. Cohn, T. Hartman, D. de Laat and A. Tajdini, High-dimensional sphere packing and the modular bootstrap, JHEP 12 (2020) 066 [arXiv:2006.02560] [INSPIRE].
N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, JHEP 01 (2021) 130 [arXiv:2006.04839] [INSPIRE].
E. Witten, Three-dimensional gravity revisited, arXiv:0706.3359 [INSPIRE].
M. R. Gaberdiel, H. R. Hampapura and S. Mukhi, Cosets of meromorphic CFTs and modular differential equations, JHEP 04 (2016) 156 [arXiv:1602.01022] [INSPIRE].
A. R. Chandra and S. Mukhi, Towards a classification of two-character rational conformal field theories, JHEP 04 (2019) 153 [arXiv:1810.09472] [INSPIRE].
A.R. Chandra and S. Mukhi, Curiosities above c = 24, SciPost Phys. 6 (2019) 53.
A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
A. Maloney and E. Witten, Averaging over Narain moduli space, JHEP 10 (2020) 187 [arXiv:2006.04855] [INSPIRE].
P. Abel, IBM PC Assembly language and programming, Prentice-Hall Inc., U.S.A. (1995).
J. Leech and N. Sloane, Sphere packings and error-correcting codes, Can. J. Math. 23 (1971) 718.
J.H. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, Springer, Germany (2013).
L.E. Danielsen and M.G. Parker, Edge local complementation and equivalence of binary linear codes, Designs Codes Crypt. 49 (2008) 161.
P.R. Östergård, Classifying subspaces of Hamming spaces, Designs Codes Crypt. 27 (2002) 297.
V.D. Tonchev, Error-correcting codes from graphs, Discr. Math. 257 (2002) 549.
M.S. Viazovska, The sphere packing problem in dimension 8, Ann. Math. 185 (2017) 991 [arXiv:1603.04246].
V.P. Nair, A.D. Shapere, A. Strominger and F. Wilczek, Compactification of the twisted heterotic string, Nucl. Phys. B 287 (1987) 402 [INSPIRE].
V. Pless, A classification of self-orthogonal codes over GF(2), Discr. Math. 3 (1972) 209.
V. Pless and N.J. Sloane, On the classification and enumeration of self-dual codes, J. Comb. Theory A 18 (1975) 313.
J.H. Conway, The sensual (quadratic) form, American Mathematical Society, U.S.A. (1997).
P.H. Ginsparg, Comment on toroidal compactification of heterotic superstrings, Phys. Rev. D 35 (1987) 648 [INSPIRE].
K. Narain, M. Sarmadi and E. Witten, A note on toroidal compactification of heterotic string theory, Nucl. Phys. B 279 (1987) 369.
K. Narain, New heterotic string theories in uncompactified dimensions < 10, Phys. Lett. B 169 (1986) 41.
M. Miyamoto, Binary codes and vertex operator (super)algebras, J. Algebra 181 (1996) 207.
C.H. Lam, Codes and vertex operator algebras (algebraic combinatorics), Not. Res. Inst. Math. Anal. 1109 (1999) 117.
G. Höhn, Self-dual codes over the kleinian four group, Math. Ann. 327 (2003) 227.
C.H. Lam and H. Yamauchi, A characterization of the Moonshine vertex operator algebra by means of virasoro frames, Int. Math. Res. Not. 2007 (2007) rnm003.
G. Höhn, Conformal designs based on vertex operator algebras, Adv. Math. 217 (2008) 2301.
C.L. Mallows and N.J. Sloane, An upper bound for self-dual codes, Inf. Control 22 (1973) 188.
C. Mallows, A. Odlyzko and N. Sloane, Upper bounds for modular forms, lattices, and codes, J. Algebra 36 (1975) 68.
E. Rains and N.J.A. Sloane, The shadow theory of modular and unimodular lattices, J. Numb. Theor. 73 (1998) 359.
I. Krasikov and S. Litsyn, Linear programming bounds for doubly-even self-dual codes, IEEE Trans. Inf. Theor. 43 (1997) 1238.
E. M. Rains, New asymptotic bounds for self-dual codes and lattices, IEEE Trans. Inf. Theor. 49 (2003) 1261.
H. Cohn and N. Elkies, New upper bounds on sphere packings I, Ann. Math. (2003) 689
H. Cohn, New upper bounds on sphere packings II, Geom. Topol. 6 (2002) 329.
H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Ann. Math. 185 (2017) 1017 [arXiv:1603.06518].
H. Cohn, A conceptual breakthrough in sphere packing, arXiv:1611.01685.
V. Pless and J. N. Pierce, Self-dual codes over GF(q) satisfy a modified Varshamov-Gilbert bound, Inf. Control 23 (1973) 35.
E.M. Rains and N.J. Sloane, Self-dual codes, math/0208001.
G. Nebe, E.M. Rains and N.J.A. Sloane, Self-dual codes and invariant theory, Springer, Germany (2006).
F.J. MacWilliams, A.M. Odlyzko, N.J. Sloane and H.N. Ward, Self-dual codes over GF(4), J. Comb. Theor. A 25 (1978) 288.
A.R. Calderbank, E.M. Rains, P. Shor and N.J. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inf. Theor. 44 (1998) 1369.
E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55 (1997) 900.
M.A. Nielsen and I. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2002).
A. Ekert and C. Macchiavello, Quantum error correction for communication, Phys. Rev. Lett. 77 (1996) 2585.
R. Laflamme, C. Miquel, J.P. Paz and W.H. Zurek, Perfect quantum error correcting code, Phys. Rev. Lett. 77 (1996) 198.
S.J. Devitt, W.J. Munro and K. Nemoto, Quantum error correction for beginners, Rept. Prog. Phys. 76 (2013) 076001.
A.R. Calderbank, E.M. Rains, N.J.A. Sloane and P.W. Shor, Quantum error correction and orthogonal geometry, Phys. Rev. Lett. 78 (1997) 405 [quant-ph/9605005] [INSPIRE].
P. Shor and R. Laflamme, Quantum analog of the Macwilliams identities for classical coding theory, Phys. Rev. Lett. 78 (1997) 1600.
E.M. Rains, Quantum weight enumerators, IEEE Trans. Inf. Theor. 44 (1998) 1388.
E.M. Rains, Quantum shadow enumerators, IEEE Trans. Inf. Theor. 45 (1999) 2361.
J. Polchinski, String theory. Volume 1. An introduction to the bosonic string, Cambridge University Press, Cambridge U.K. (1998).
K. Becker, M. Becker and J.H. Schwarz, String theory and M-theory: a modern introduction, Cambridge University Press, Cambridge U.K. (2006).
L.J. Dixon and J.A. Harvey, String theories in ten-dimensions without space-time supersymmetry, Nucl. Phys. B 274 (1986) 93 [INSPIRE].
M. Van den Nest, J. Dehaene and B. De Moor, Graphical description of the action of local Clifford transformations on graph states, Phys. Rev. A 69 (2004) 022316 [quant-ph/0308151].
D. Schlingemann and R. F. Werner, Quantum error-correcting codes associated with graphs, Phys. Rev. A 65 (2001) 012308 [quant-ph/0012111].
D.J. MacKay, G. Mitchison and P.L. McFadden, Sparse-graph codes for quantum error correction, IEEE Trans. Inf. Theor. 50 (2004) 2315.
D. Schlingemann, Stabilizer codes can be realized as graph codes, quant-ph/0111080.
W. Dür, H. Aschauer and H.-J. Briegel, Multiparticle entanglement purification for graph states, Phys. Rev. Lett. 91 (2003) 107903 [quant-ph/0303087].
G. Casati, D. L. Shepelyansky and P. Zoller, Quantum computers, algorithms and chaos, IOS press, Amsterdam The Netherlands (2006).
M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Nest and H.J. Briegel, Entanglement in graph states and its applications, quant-ph/0602096.
C. Riera, L.E. Danielsen and M.G. Parker, On pivot orbits of boolean functions, math/0604396.
M. Van den Nest and B. De Moor, Edge-local equivalence of graphs, math/0510246.
M. Aigner and H. Van der Holst, Interlace polynomials, Lin. Alg. Appl. 377 (2004) 11.
D.G. Glynn, T.A. Gulliver, J.G. Maks and M.K. Gupta, The geometry of additive quantum codes, submitted to Springer (2004).
A. Bouchet, Recognizing locally equivalent graphs, Discr. math. 114 (1993) 75.
L.E. Danielsen and M.G. Parker, On the classification of all self-dual additive codes over GF(4) of length up to 12, J. Comb. Theor. A 113 (2006) 1351.
J. Harvey and G. Moore, Moonshine, superconformal symmetry, and quantum error correction, JHEP 05 (2020) 146 [arXiv:2003.13700] [INSPIRE].
M.R. Gaberdiel, A. Taormina, R. Volpato and K. Wendland, A k3 sigma model with \( {\mathrm{\mathbb{Z}}}_2^8 \): M20 symmetry, JHEP 02 (2014) 022 [arXiv:1309.4127] [INSPIRE].
S. Hellerman, A universal inequality for CFT and quantum gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
B. Bates and F. Denef, Exact solutions for supersymmetric stationary black hole composites, JHEP 11 (2011) 127 [hep-th/0304094] [INSPIRE].
C.A. Keller and H. Ooguri, Modular constraints on Calabi-Yau compactifications, Commun. Math. Phys. 324 (2013) 107 [arXiv:1209.4649].
D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562].
J.D. Qualls and A.D. Shapere, Bounds on operator dimensions in 2d conformal field theories, JHEP 05 (2014) 091 [arXiv:1312.0038] [INSPIRE].
T. Hartman, C.A. Keller and B. Stoica, Universal spectrum of 2d conformal field theory in the large c limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
J.D. Qualls, Universal bounds on operator dimensions in general 2D conformal field theories, arXiv:1508.00548 [INSPIRE].
M. Ashrafi and F. Loran, Non-chiral 2d CFT with integer energy levels, JHEP 09 (2016) 121 [arXiv:1607.08516] [INSPIRE].
H. Kim, P. Kravchuk and H. Ooguri, Reflections on conformal spectra, JHEP 04 (2016) 184 [arXiv:1510.08772] [INSPIRE].
Y.H. Lin, S.H. Shao, Y. Wang and X. Yin, (2, 2) superconformal bootstrap in two dimensions, JHEP 05 (2017) 112 [arXiv:1610.05371] [INSPIRE].
T. Anous, R. Mahajan and E. Shaghoulian, Parity and the modular bootstrap, SciPost Phys. 5 (2018) 022 [arXiv:1803.04938] [INSPIRE].
S. Collier, Y.H. Lin and X. Yin, Modular bootstrap revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman and A. Tajdini, Fast conformal bootstrap and constraints on 3d gravity, JHEP 05 (2019) 087 [arXiv:1903.06272].
M. Cho, S. Collier and X. Yin, Genus two modular bootstrap, JHEP 04 (2019) 022 [arXiv:1705.05865] [INSPIRE].
F. Gliozzi, Modular bootstrap, elliptic points, and quantum gravity, Phys. Rev. Res. 2 (2020) 013327 [arXiv:1908.00029] [INSPIRE].
M. R. Gaberdiel, Constraints on extremal self-dual CFTs, JHEP 11 (2007) 087 [arXiv:0707.4073] [INSPIRE].
D. Gaiotto, Monster symmetry and extremal CFTs, JHEP 11 (2012) 149 [arXiv:0801.0988] [INSPIRE].
S.D. Mathur, S. Mukhi and A. Sen, On the classification of rational conformal field theories, Phys. Lett. B 213 (1988) 303.
S.G. Naculich, Differential equations for rational conformal characters, Nucl. Phys. B 323 (1989) 423 [INSPIRE].
D. Tong and C. Turner, Notes on 8 Majorana fermions, SciPost Phys. Lect. Notes 14 (2020) 1 [arXiv:1906.07199] [INSPIRE].
R.E. Borcherds, Monstrous Moonshine and monstrous Lie superalgebras, Inv. Math. 109 (1992) 405.
A. Milekhin, Quantum error correction and large N, arXiv:2008.12869 [INSPIRE].
C.D. White, C. Cao and B. Swingle, Conformal field theories are magical, Phys. Rev. B 103 (2021) 075145 [arXiv:2007.01303] [INSPIRE].
E. Witten, D = 10 superstring theory, in Fourth Workshop on grand unification, H.A. Weldon et al. eds., Springer, Germany (1983).
F. Englert and A. Neveu, Nonabelian compactification of the interacting bosonic string, Phys. Lett. B 163 (1985) 349 [INSPIRE].
S. Elitzur, E. Gross, E. Rabinovici and N. Seiberg, Aspects of Bosonization in string theory, Nucl. Phys. B 283 (1987) 413 [INSPIRE].
W. Lerche, A.N. Schellekens and N.P. Warner, Lattices and strings, Phys. Rept. 177 (1989) 1 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to the memory of John Horton Conway
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2009.01244
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Dymarsky, A., Shapere, A. Quantum stabilizer codes, lattices, and CFTs. J. High Energ. Phys. 2021, 160 (2021). https://doi.org/10.1007/JHEP03(2021)160
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2021)160