CoChat
Sign Up
Cited by 35Theoretical and Computational PhysicsQuantum many-body systemsQuantum Computing Algorithms and Architecture

The DOI matches its Crossref record; 5 of the 5 references it cites were found in a registry or an index.

The quantum transition of the two-dimensional Ising spin glass

Open access
Bernaschi, Massimo, González-Adalid Pemartín, Isidoro, Martín-Mayor, Víctor, Parisi, Giorgio·2024·Nature
DOI 10.1038/s41586-024-07647-yPMC PMC11269196
Open this paper in CoChat — free account

Abstract

Quantum annealers are commercial devices that aim to solve very hard computational problems 1, typically those involving spin glasses 2,3. Just as in metallurgic annealing, in which a ferrous metal is slowly cooled 4, quantum annealers seek good solutions by slowly removing the transverse magnetic field at the lowest possible temperature. Removing the field diminishes the quantum fluctuations but forces the system to traverse the critical point that separates the disordered phase (at large fields) from the spin-glass phase (at small fields). A full understanding of this phase transition is still missing. A debated, crucial question regards the closing of the energy gap separating the ground state from the first excited state. All hopes of achieving an exponential speed-up, compared to classical computers, rest on the assumption that the gap will close algebraically with the number of spins 5–9. However, renormalization group calculations predict instead that there is an infinite-randomness fixed point 10. Here we solve this debate through extreme-scale numerical simulations, finding that both parties have grasped parts of the truth. Although the closing of the gap at the critical point is indeed super-algebraic, it remains algebraic if one restricts the symmetry of possible excitations. As this symmetry restriction is experimentally achievable (at least nominally), there is still hope for the quantum annealing paradigm 11–13.

Overview

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.1038/s41586-024-07647-y

Cites

5 of 5 references verified0 citation errors · 0 not checkable

Statistics

Figures

Fig. 1 Phase diagram and critical scaling for the two-dimensional quantum spin glass. a, Phase diagram for a two-dimensional Ising spin glass in terms of temperature T and transverse field Γ. For all T > 0, the system is disordered when studied at large length scales, so that it is in the paramagnetic phase (PM). At T = 0, the ground state seems disordered for Γ > Γc (from the point of view of the computational basis). For Γ < Γc, we encounter the spin-glass phase (SG), which is different for every disorder realization (equation (1)). b, Our finite-size scaling analysis (see, for example, refs. 48,60) of the critical point at T = 0 and Γ = Γc, in terms of the parameter k that represents Γ in the Trotter–Suzuki formulation (‘The Trotter–Suzuki formula’ in Methods; k grows as Γ decreases). Left, correlation length ξ(3) in units of the lattice size L versus k. The curves for the different L’s intersect at the critical point kc ≈ 0.29. Right, data in the left-hand panel of b, when represented as a function of the scaling variable L1/ν(k − kc) with 1/ν = 0.7, converge to a limiting curve as L grows. Points in b are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source Data
Fig. 1 Phase diagram and critical scaling for the two-dimensional quantum spin glass. a, Phase diagram for a two-dimensional Ising spin glass in terms of temperature T and transverse field Γ. For all T > 0, the system is disordered when studied at large length scales, so that it is in the paramagnetic phase (PM). At T = 0, the ground state seems disordered for Γ > Γc (from the point of view of the computational basis). For Γ < Γc, we encounter the spin-glass phase (SG), which is different for every disorder realization (equation (1)). b, Our finite-size scaling analysis (see, for example, refs. 48,60) of the critical point at T = 0 and Γ = Γc, in terms of the parameter k that represents Γ in the Trotter–Suzuki formulation (‘The Trotter–Suzuki formula’ in Methods; k grows as Γ decreases). Left, correlation length ξ(3) in units of the lattice size L versus k. The curves for the different L’s intersect at the critical point kc ≈ 0.29. Right, data in the left-hand panel of b, when represented as a function of the scaling variable L1/ν(k − kc) with 1/ν = 0.7, converge to a limiting curve as L grows. Points in b are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source DataCC BY · PubMed Central
Fig. 2 Ensuring that the zero-temperature limit has been reached by comparing PBCs and APBCs over Euclidean time. a, Correlation length ξ(3) (‘One-time observables’ in Methods) versus k, as computed for our largest systems with L = 24 and Lτ = 2,048 and with both PBCs and APBCs for the same set of 1,280 samples. The statistical agreement for PBCs and APBCs indicates that the T → 0 limit has been effectively reached for this quantity. b, As a for the Binder cumulant (‘One-time observables’ in Methods). The dashed line represents the critical point, kc ≈ 0.29. c, The even correlation functions Q2(τ) (‘Two-times observables’ in Methods), as computed for a single sample of L = 20 at k = 0.29, rather quickly reach their large-τ plateau. The functions depend on both Lτ and the boundary conditions. The PBC plateau decreases upon increasing Lτ, whereas the APBC plateau notably increases. The reason behind the stronger sensitivity of Lτ for APBCs is understood (‘The limit of zero temperature’ in Methods). Points in a, b, and c are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source Data
Fig. 2 Ensuring that the zero-temperature limit has been reached by comparing PBCs and APBCs over Euclidean time. a, Correlation length ξ(3) (‘One-time observables’ in Methods) versus k, as computed for our largest systems with L = 24 and Lτ = 2,048 and with both PBCs and APBCs for the same set of 1,280 samples. The statistical agreement for PBCs and APBCs indicates that the T → 0 limit has been effectively reached for this quantity. b, As a for the Binder cumulant (‘One-time observables’ in Methods). The dashed line represents the critical point, kc ≈ 0.29. c, The even correlation functions Q2(τ) (‘Two-times observables’ in Methods), as computed for a single sample of L = 20 at k = 0.29, rather quickly reach their large-τ plateau. The functions depend on both Lτ and the boundary conditions. The PBC plateau decreases upon increasing Lτ, whereas the APBC plateau notably increases. The reason behind the stronger sensitivity of Lτ for APBCs is understood (‘The limit of zero temperature’ in Methods). Points in a, b, and c are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source DataCC BY · PubMed Central
Fig. 3 Studying the spectra of even excitations at the critical point. a, Sample-averaged subtracted correlation function Q2,s(τ) (‘Fitting process and estimating the Euclidean correlation length’ in Methods) becomes compatible with zero for moderate values of τ, for all our system sizes. b, Left, after computing the Euclidean correlation length ηe^(s) for each sample, we computed for each L the empirical distribution function F(ηe), namely the probability F of finding a sample with ηe^(s)<ηe (note the horizontal error bars). Right, the data in the left-hand panel of b, when plotted as a function of the scaling variable u (equation (4)) do not show any L residual L dependence other than for our smallest sizes L = 8 and 12. Points in a and b are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source Data
Fig. 3 Studying the spectra of even excitations at the critical point. a, Sample-averaged subtracted correlation function Q2,s(τ) (‘Fitting process and estimating the Euclidean correlation length’ in Methods) becomes compatible with zero for moderate values of τ, for all our system sizes. b, Left, after computing the Euclidean correlation length ηe^(s) for each sample, we computed for each L the empirical distribution function F(ηe), namely the probability F of finding a sample with ηe^(s)<ηe (note the horizontal error bars). Right, the data in the left-hand panel of b, when plotted as a function of the scaling variable u (equation (4)) do not show any L residual L dependence other than for our smallest sizes L = 8 and 12. Points in a and b are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source DataCC BY · PubMed Central
Fig. 4 Studying the spectra of odd operators at the critical point. a, The decay of the sample-averaged correlation function C(τ) (equation (14)) approaches a power law as L increases. The dashed line is a guide for the eyes. Indeed, we needed to represent C(τ) in terms of τ~=(Lτ/π)sin(πτ/Lτ) to avoid distortions due to the PBCs (τ~ and τ are almost identical for small τ/Lτ). b, Empirical distribution function F(η) as a function of logη for all our system sizes. Note that we can compute only up to some L-dependent F because our largest Lτ is not large enough to allow for a safe determination of η in some samples. c, For large η, the asymptotic behaviour F(η)=1−B/η^b is evinced by the linear behaviour (in logarithmic scale) of 1 − F as a function of η. We fond b ≈ 0.8. The dashed line is a guide for the eyes. Points in a, b, and c are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source Data
Fig. 4 Studying the spectra of odd operators at the critical point. a, The decay of the sample-averaged correlation function C(τ) (equation (14)) approaches a power law as L increases. The dashed line is a guide for the eyes. Indeed, we needed to represent C(τ) in terms of τ~=(Lτ/π)sin(πτ/Lτ) to avoid distortions due to the PBCs (τ~ and τ are almost identical for small τ/Lτ). b, Empirical distribution function F(η) as a function of logη for all our system sizes. Note that we can compute only up to some L-dependent F because our largest Lτ is not large enough to allow for a safe determination of η in some samples. c, For large η, the asymptotic behaviour F(η)=1−B/η^b is evinced by the linear behaviour (in logarithmic scale) of 1 − F as a function of η. We fond b ≈ 0.8. The dashed line is a guide for the eyes. Points in a, b, and c are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source DataCC BY · PubMed Central
Extended Data Fig. 1 Schematic representation of the energy spectrum. As it is explained in the main text, the parity symmetry splits the spectra into even and odd sectors according to the parity of states. We shall name the even eigenvectors of the transfer matrix (8) as ∣0e⟩, ∣1e⟩, …, with corresponding eigenvalues e^(−kEGS), and e^(−k(EGS+Δn,e)) for n = 1, 2, 3, … [we use the shorthand Δe = Δ1,e]. For the odd sector, we have ∣0o⟩, ∣1o⟩, … with eigenvalues e^(−k(EGS+Δ)), and e^(−k(EGS+Δ+Δn,o)) for n = 1, 2, 3, … [we use the shorthands Δ = E0,o − EGS, and Δo = Δ1,o]. Notice that expectation values at T = 0 are determined solely by ∣0e⟩.
Extended Data Fig. 1 Schematic representation of the energy spectrum. As it is explained in the main text, the parity symmetry splits the spectra into even and odd sectors according to the parity of states. We shall name the even eigenvectors of the transfer matrix (8) as ∣0e⟩, ∣1e⟩, …, with corresponding eigenvalues e^(−kEGS), and e^(−k(EGS+Δn,e)) for n = 1, 2, 3, … [we use the shorthand Δe = Δ1,e]. For the odd sector, we have ∣0o⟩, ∣1o⟩, … with eigenvalues e^(−k(EGS+Δ)), and e^(−k(EGS+Δ+Δn,o)) for n = 1, 2, 3, … [we use the shorthands Δ = E0,o − EGS, and Δo = Δ1,o]. Notice that expectation values at T = 0 are determined solely by ∣0e⟩.CC BY · PubMed Central
Extended Data Fig. 2 Quantifying the finite-temperature effects. Even correlation functions Q2(τ) defined in Eq. (14), as computed for a single sample of L = 20 at k = 0.29 ≈ kc. The corresponding Q2 value calculated from TrM²/L^2D is represented by a complementary colored horizontal line. Points are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source Data
Extended Data Fig. 2 Quantifying the finite-temperature effects. Even correlation functions Q2(τ) defined in Eq. (14), as computed for a single sample of L = 20 at k = 0.29 ≈ kc. The corresponding Q2 value calculated from TrM²/L^2D is represented by a complementary colored horizontal line. Points are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source DataCC BY · PubMed Central
Extended Data Fig. 3 Sample dependence of the Euclidean correlation lengths. Empirical distribution function of the different Euclidean correlation lengths presented in the system, for different values of k. Data from the exact diagonalization of a L = 6 system. Data from k = 0.295, and k = 0.3 are calculated over 320 samples, instead of 1280. Source Data
Extended Data Fig. 3 Sample dependence of the Euclidean correlation lengths. Empirical distribution function of the different Euclidean correlation lengths presented in the system, for different values of k. Data from the exact diagonalization of a L = 6 system. Data from k = 0.295, and k = 0.3 are calculated over 320 samples, instead of 1280. Source DataCC BY · PubMed Central
Extended Data Fig. 4 Dependence on k of the odd correlation length η. The figure shows that the logarithm of the ratio of η(k = 0.31) and η(k = 0.295) (computed for the same L = 6 sample through exact diagonalization) is very approximately a linear function of logη(k=0.295), with a positive slope. The figure shows data for the 350 samples that we have studied at both k = 0.295 and k = 0.31. Source Data
Extended Data Fig. 4 Dependence on k of the odd correlation length η. The figure shows that the logarithm of the ratio of η(k = 0.31) and η(k = 0.295) (computed for the same L = 6 sample through exact diagonalization) is very approximately a linear function of logη(k=0.295), with a positive slope. The figure shows data for the 350 samples that we have studied at both k = 0.295 and k = 0.31. Source DataCC BY · PubMed Central
Extended Data Fig. 5 Determination of the critical point. When studied as a function of k on two system sizes La < Lb, the curves for dimensionless quantities cross at a point k*(La, Lb), see Fig. 1-b. The figure shows k* (as computed for B, ξ(2)/L and ξ(3)/L) versus F(La, Lb) (18). We set 1/ν = 0.7 and ω = 1 to compute F(La, Lb). The curves should extrapolate linearly to kc as F(La, Lb) tends to zero. The shaded area encompass our uncertainty in the estimation of kc. Points are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source Data
Extended Data Fig. 5 Determination of the critical point. When studied as a function of k on two system sizes La < Lb, the curves for dimensionless quantities cross at a point k*(La, Lb), see Fig. 1-b. The figure shows k* (as computed for B, ξ(2)/L and ξ(3)/L) versus F(La, Lb) (18). We set 1/ν = 0.7 and ω = 1 to compute F(La, Lb). The curves should extrapolate linearly to kc as F(La, Lb) tends to zero. The shaded area encompass our uncertainty in the estimation of kc. Points are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source DataCC BY · PubMed Central
Extended Data Fig. 6 The power-law decay with Euclidean time of the odd correlation functions. Sample-averaged Euclidean correlation function as a function of the Euclidean distance τ (left panel), τ~=Lτ/πsin(πτ/Lτ) to avoid distortions due to the periodic boundary conditions (center and right panels). Left and center panels show the system size dependence for k = 0.29, despite the right panel shows the k-dependence for the bigger system, L = 24 and Lτ = 211. The dashed line in the right panel is a guide to the eye to show the critical exponent b~=1 encountered for k = 0.285 (see Sect. 5.6). Points are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source Data
Extended Data Fig. 6 The power-law decay with Euclidean time of the odd correlation functions. Sample-averaged Euclidean correlation function as a function of the Euclidean distance τ (left panel), τ~=Lτ/πsin(πτ/Lτ) to avoid distortions due to the periodic boundary conditions (center and right panels). Left and center panels show the system size dependence for k = 0.29, despite the right panel shows the k-dependence for the bigger system, L = 24 and Lτ = 211. The dashed line in the right panel is a guide to the eye to show the critical exponent b~=1 encountered for k = 0.285 (see Sect. 5.6). Points are statistical averages, and errors are one standard deviation. Our data set is fully described in Extended Data Table 1. Source DataCC BY · PubMed Central

Verification

Claims the paper makes

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Citations

Cites

Show all 5 references

Research works printed but not linked: 72

  1. Charbonneau, P. et al. (eds) Spin Glass Theory and Far Beyond (World Scientific, 2023). · DOI
  2. Binder, K. & Young, A. P. Spin glasses: experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58, 801–976 (1986). · DOI
  3. Kirkpatrick, S., Gelatt, C. D. & Vecchi, M. P. Optimization by simulated annealing. Science 220, 671–680 (1983). · DOI · Paper page
  4. Rieger, H. & Young, A. P. Zero-temperature quantum phase transition of a two-dimensional Ising spin glass. Phys. Rev. Lett. 72, 4141–4144 (1994). · DOI · Paper page
  5. Guo, M., Bhatt, R. N. & Huse, D. A. Quantum critical behavior of a three-dimensional Ising spin glass in a transverse magnetic field. Phys. Rev. Lett. 72, 4137–4140 (1994). · DOI · Paper page
  6. Guo, M., Bhatt, R. N. & Huse, D. A. Quantum Griffiths singularities in the transverse-field Ising spin glass. Phys. Rev. B 54, 3336–3342 (1996). · DOI
  7. Rieger, H. & Young, A. P. Griffiths singularities in the disordered phase of a quantum Ising spin glass. Phys. Rev. B 54, 3328–3335 (1996). · DOI
  8. Singh, R. R. P. & Young, A. P. Critical and Griffiths–McCoy singularities in quantum Ising spin glasses on d-dimensional hypercubic lattices: a series expansion study. Phys. Rev. E 96, 022139 (2017). · DOI
  9. Miyazaki, R. & Nishimori, H. Real-space renormalization-group approach to the random transverse-field Ising model in finite dimensions. Phys. Rev. E 87, 032154 (2013). · DOI
  10. Brooke, J., Bitko, D., Rosenbaum, & Aeppli, G. Quantum annealing of a disordered magnet. Science 284, 779–781 (1999). · DOI · Paper page
  11. Farhi, E. et al. A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472–475 (2001). · DOI · Paper page
  12. Papadimitriou, C. Computational Complexity (Addison-Wesley, 1994).
  13. Barahona, F. On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen. 15, 3241 (1982). · DOI
  14. Istrail, S. Statistical mechanics, three-dimensionality and NP-completeness. I. Universality of intracatability for the partition function of the Ising model across non-planar surfaces (extended abstract). In Proc. 32nd Annual ACM Symposium on Theory of Computing (eds Yao, F. & Luks, E.) 87–96 (ACM, 2000). · DOI
  15. Goto, H., Tatsumura, K. & Dixon, A. R. Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems. Sci. Adv. 5, eaav2372 (2019). · DOI
  16. Matsubara, S. et al. Digital annealer for high-speed solving of combinatorial optimization problems and its applications. In Proc. 2020 25th Asia and South Pacific Design Automation Conference (ASP-DAC) (eds Cheng, K.-T. T. & Yang, H.) 667–672 (IEEE, 2020). · DOI
  17. Takemoto, T., Hayashi, M., Yoshimura, C. & Yamaoka, M. A 2 × 30k-spin multi-chip scalable CMOS annealing processor based on a processing-in-memory approach for solving large-scale combinatorial optimization problems. IEEE J. Solid-State Circuits 55, 145–156 (2020). · DOI
  18. McMahon, P. L. et al. A fully programmable 100-spin coherent Ising machine with all-to-all connections. Science 354, 614–617 (2016). · DOI
  19. Baity-Jesi, M. et al. Janus II: a new generation application-driven computer for spin-system simulations. Comput. Phys. Commun. 185, 550–559 (2014). · DOI
  20. King, A. D. et al. Quantum critical dynamics in a 5,000-qubit programmable spin glass. Nature 617, 61–66 (2023). · DOI · Paper page
  21. Parisi, G. Field Theory, Disorder and Simulations (World Scientific, 1994).
  22. Grinstein, G. & Luther, A. Application of the renormalization group to phase transitions in disordered systems. Phys. Rev. B 13, 1329–1343 (1976). · DOI
  23. Parisi, G. Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43, 1754–1756 (1979). · DOI · Paper page
  24. Parisi, G. & Sourlas, N. Random magnetic fields, supersymmetry, and negative dimensions. Phys. Rev. Lett. 43, 744–745 (1979). · DOI
  25. Cardy, J. Scaling and Renormalization in Statistical Field Theory, Vol. 5 of Lecture Notes in Physics (eds Goddard, P. & Yeomans, J.) (Cambridge Univ. Press, 1996).
  26. Ballesteros, H. G. et al. Critical exponents of the three-dimensional diluted Ising model. Phys. Rev. B 58, 2740–2747 (1998). · DOI
  27. Hasenbusch, M., Pelissetto, A. & Vicari, E. Critical behavior of three-dimensional Ising spin glass models. Phys. Rev. B 78, 214205 (2008). · DOI
  28. Fytas, N. G. & Martín-Mayor, V. Universality in the three-dimensional random-field Ising model. Phys. Rev. Lett. 110, 227201 (2013). · DOI
  29. Fytas, N. G., Martín-Mayor, V., Picco, M. & Sourlas, N. Phase transitions in disordered systems: the example of the random-field Ising model in four dimensions. Phys. Rev. Lett. 116, 227201 (2016). · DOI
  30. Fytas, N. G., Martín-Mayor, V., Parisi, G., Picco, M. & Sourlas, N. Evidence for supersymmetry in the random-field Ising model at d = 5. Phys. Rev. Lett. 122, 240603 (2019). · DOI
  31. Fernandez, L. A., Marinari, E., Martin-Mayor, V., Parisi, G. & Ruiz-Lorenzo, J. J. Universal critical behavior of the two-dimensional Ising spin glass. Phys. Rev. B 94, 024402 (2016). · DOI
  32. McCoy, B. M. & Wu, T. Theory of a two-dimensional Ising model with random impurities. I. Thermodynamics. Phys. Rev. 176, 631–643 (1968). · DOI
  33. McCoy, B. M. & Wu, T. Theory of a two-dimensional Ising model with random impurities. II. Spin correlation functions. Phys. Rev. 188, 982–1013 (1969). · DOI
  34. McCoy, B. M. Theory of a two-dimensional Ising model with random impurities. III. Boundary effects. Phys. Rev. 188, 1014–1031 (1969). · DOI
  35. Fisher, D. S. Random transverse field Ising spin chains. Phys. Rev. Lett. 69, 534–537 (1992). · DOI
  36. Albash, T. & Lidar, D. A. Adiabatic quantum computation. Rev. Mod. Phys. 90, 015002 (2018). · DOI · Paper page
  37. Thill, M. J. & Huse, D. A. Equilibrium behaviour of quantum Ising spin glass. Phys. A: Stat. Mech. Appl. 214, 321–355 (1995). · DOI
  38. Matoz-Fernandez, D. A. & Romá, F. Unconventional critical activated scaling of two-dimensional quantum spin glasses. Phys. Rev. B 94, 024201 (2016). · DOI
  39. Griffiths, R. B. Nonanalytic behavior above the critical point in a random Ising ferromagnet. Phys. Rev. Lett. 23, 17–19 (1969). · DOI
  40. Farhi, E. et al. Performance of the quantum adiabatic algorithm on random instances of two optimization problems on regular hypergraphs. Phys. Rev. A 86, 052334 (2012). · DOI
  41. Yang, C. N. Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors. Rev. Mod. Phys. 34, 694–704 (1962). · DOI
  42. Sinova, J., Canright, G. & MacDonald, A. H. Nature of ergodicity breaking in Ising spin glasses as revealed by correlation function spectral properties. Phys. Rev. Lett. 85, 2609–2612 (2000). · DOI
  43. Correale, L., Marinari, E. & Martín-Mayor, V. Eigenvalue analysis of the density matrix of four-dimensional spin glasses supports replica symmetry breaking. Phys. Rev. B 66, 174406 (2002). · DOI
  44. Amit, D. J. & Martín-Mayor, V. Field Theory, the Renormalization Group and Critical Phenomena 3rd edn (World Scientific, 2005). · DOI
  45. Chayes, J. T., Chayes, L., Fisher, D. S. & Spencer, T. Finite-size scaling and correlation lengths for disordered systems. Phys. Rev. Lett. 57, 2999–3002 (1986). · DOI
  46. King, A. D. et al. Computational supremacy in quantum simulation. Preprint at https://arxiv.org/abs/2403.00910 (2024). · arXiv
  47. Hen, I. & Young, A. P. Exponential complexity of the quantum adiabatic algorithm for certain satisfiability problems. Phys. Rev. E 84, 061152 (2011). · DOI
  48. Knysh, S. Zero-temperature quantum annealing bottlenecks in the spin-glass phase. Nat. Commun. 7, 12370 (2016). · DOI
  49. Khoshbakht, H. & Weigel, M. Domain-wall excitations in the two-dimensional Ising spin glass. Phys. Rev. B 97, 064410 (2018). · DOI
  50. Fernandez, L. A., Martín-Mayor, V., Parisi, G. & Seoane, B. Temperature chaos in 3D Ising spin glasses is driven by rare events. Europhys. Lett. 103, 67003 (2013). · DOI
  51. Marshall, J., Martin-Mayor, V. & Hen, I. Practical engineering of hard spin-glass instances. Phys. Rev. A 94, 012320 (2016). · DOI
  52. Billoire, A. et al. Dynamic variational study of chaos: spin glasses in three dimensions. J. Stat. Mech.: Theory Exp. 2018, 033302 (2018). · DOI
  53. Martín-Mayor, V. & Hen, I. Unraveling quantum annealers using classical hardness. Sci. Rep. 5, 15324 (2015). · DOI
  54. Albash, T., Martin-Mayor, V. & Hen, I. Temperature scaling law for quantum annealing optimizers. Phys. Rev. Lett. 119, 110502 (2017). · DOI
  55. Albash, T., Martin-Mayor, V. & Hen, I. Analog errors in Ising machines. Quantum Sci. Technol. 4, 02LT03 (2019). · DOI
  56. Nightingale, M. P. Scaling theory and finite systems. Phys. A: Stat. Mech. Appl. 83, 561–572 (1976). · DOI · Paper page
  57. Toulouse, G. Theory of the frustration effect in spin glasses. Commun. Phys. 2, 115–119 (1977).
  58. Suzuki, M. Relationship between d-dimensional quantal spin systems and (d + 1)-dimensional Ising systems: equivalence, critical exponents and systematic approximants of the partition function and spin correlations. Prog. Theor. Phys. 56, 1454–1469 (1976). · DOI
  59. Parisi, G. Statistical Field Theory (Addison-Wesley, 1988). · DOI
  60. Palassini, M. & Caracciolo, S. Universal finite-size scaling functions in the 3D Ising spin glass. Phys. Rev. Lett. 82, 5128–5131 (1999). · DOI
  61. Ballesteros, H. G. et al. Critical behavior of the three-dimensional Ising spin glass. Phys. Rev. B 62, 14237–14245 (2000). · DOI
  62. Bernaschi, M., Pemartín, I. G.-A., Martín-Mayor, V. & Parisi, G. The QISG suite: high-performance codes for studying quantum Ising spin glasses. Comput. Phys. Commun. 298, 109101 (2024). · DOI
  63. Hukushima, K. & Nemoto, K. Exchange Monte Carlo method and application to spin glass simulations. J. Phys. Soc. Jpn 65, 1604 (1996). · DOI
  64. Efron, B. & Tibshirani, R. J. An Introduction to Bootstrap (Chapman & Hall/CRC, 1994). · DOI · Paper page
  65. Fisher, M. E. & Ferdinand, A. E. Interfacial, boundary, and size effects at critical points. Phys. Rev. Lett. 19, 169–172 (1967). · DOI
  66. Fisher, M. E. & Barber, M. N. Scaling theory for finite-size effects in the critical region. Phys. Rev. Lett. 28, 1516–1519 (1972). · DOI
  67. Barber, M. N. Finite-size Scaling, Vol. 8 of Phase Transitions and Critical Phenomena (Academic Press, 1983).
  68. Cardy, J. Finite-size Scaling (Elsevier, 2012).
  69. Ballesteros, H. G., Fernandez, L. A., Martín-Mayor, V. & Muñoz Sudupe, A. New universality class in three dimensions?: The antiferromagnetic RP2 model. Phys. Lett. B 378, 207 (1996). · DOI
  70. Yllanes, D. Rugged Free-Energy Landscapes in Disordered Spin Systems. PhD thesis, Complutense Univ. of Madrid (2011).
  71. Binder, K. Finite size scaling analysis of Ising model block distribution functions. Z. Phys. B: Condens. Matter 43, 119–140 (1981). · DOI
  72. Belletti, F. et al. Nonequilibrium spin-glass dynamics from picoseconds to one tenth of a second. Phys. Rev. Lett. 101, 157201 (2008). · DOI

Cited by

Sorted by citations

Metadata

Authors
Bernaschi, M., González-Adalid Pemartín, I., Martín-Mayor, V., & Parisi, G.
Affiliations
  • Istituto per le Applicazioni del Calcolo Mauro Picone (IT)
  • Universidad Complutense de Madrid (ES)
ORCID
Venue
Nature
Publisher
Springer Science and Business Media LLC
Published
2024-07-10
Volume
631
Issue
8022
Pages
749-754
ISSN
  • 0028-0836 (ISSN-L)
  • 1476-4687
DOI
10.1038/s41586-024-07647-y
PMC
PMC11269196
OpenAlex
W4400491633
Cited by
35
References
5
Citations by year
2026: 14 · 2025: 19 · 2024: 2
FWCI
5.05
Type
journal article
Language
en
Open access
Licence
Locations
  • Nature (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Ministerio de Ciencia, Innovación y Universidades (10.13039.501100011033, PID2022-136374NB-C21, FPU18/02665, 10.13039, .13039/501100011033, 501100011033)
  • European Commission (694925, 13039/501100011033, 501100011033, HORIZON2020)
  • European Regional Development Fund (10.13039/501100011033, 13039/501100011033, 501100011033, PID2022-136374NB-C21)
  • Agencia Estatal de Investigación (501100011033, 10.13039/501100011033, 13039, PID2022-136374NB-C21, .13039/501100011033, 10.13039)
Topics
  • Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
Keywords
quantum annealing, quantum phase transitions, infinite randomness fixed point, transverse magnetic field, quantum fluctuations, renormalization group
MeSH terms
Computer Simulation, Glass, Quantum Theory, Temperature, Phase Transition, Magnetic Fields

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review