The quantum transition of the two-dimensional Ising spin glass

Abstract

Quantum annealers are commercial devices that aim to solve very hard computational problems¹, typically those involving spin glasses^2,3. Just as in metallurgic annealing, in which a ferrous metal is slowly cooled⁴, 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 point¹⁰. 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.

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 > Γ_c (from the point of view of the computational basis). For Γ < Γ_c, 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. ^,) of the critical point at T = 0 and Γ = Γ_c, 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 k_c ≈ 0.29. Right, data in the left-hand panel of b, when represented as a function of the scaling variable L^1/ν(k − k_c) 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. 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 L_τ = 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, k_c ≈ 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 L_τ and the boundary conditions. The PBC plateau decreases upon increasing L_τ, whereas the APBC plateau notably increases. The reason behind the stronger sensitivity of L_τ 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. 3. Studying the spectra of even excitations at the critical point.

a, Sample-averaged subtracted correlation function Q_2,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 ${\eta }_{\mathrm{e}}^{\left(s\right)}$ for each sample, we computed for each L the empirical distribution function F(η_e), namely the probability F of finding a sample with ${\eta }_{\mathrm{e}}^{\left(s\right)}<{\eta }_{\mathrm{e}}$ (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. 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 $\tilde{\tau }=\left(L_{\tau }/\mathrm{\pi }\right)\sin \left(\mathrm{\pi }\tau /L_{\tau }\right)$ to avoid distortions due to the PBCs ($\tilde{\tau }$ and τ are almost identical for small τ/L_τ). b, Empirical distribution function F(η) as a function of $\log \eta$ for all our system sizes. Note that we can compute only up to some L-dependent F because our largest L_τ is not large enough to allow for a safe determination of η in some samples. c, For large η, the asymptotic behaviour $F\left(\eta \right)=1-B/{\eta }^b$ 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

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 $∣0_{\mathrm{e}}⟩$, $∣1_{\mathrm{e}}⟩$, …, with corresponding eigenvalues ${\mathrm{e}}^{-k{E}_{\mathrm{GS}}}$, and ${\mathrm{e}}^{-k\left(E_{\mathrm{GS}}+{\mathrm{\Delta }}_{n,\mathrm{e}}\right)}$ for n = 1, 2, 3, … [we use the shorthand Δ_e = Δ_1,e]. For the odd sector, we have $∣0_{\mathrm{o}}⟩$, $∣1_{\mathrm{o}}⟩$, … with eigenvalues ${\mathrm{e}}^{-k\left(E_{\mathrm{GS}}+\mathrm{\Delta }\right)}$, and ${\mathrm{e}}^{-k\left(E_{\mathrm{GS}}+\mathrm{\Delta }+{\mathrm{\Delta }}_{n,\mathrm{o}}\right)}$ for n = 1, 2, 3, … [we use the shorthands Δ = E_0,o − E_GS, and Δ_o = Δ_1,o]. Notice that expectation values at T  =  0 are determined solely by $∣0_{\mathrm{e}}⟩$.

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 ≈ k_c. The corresponding Q₂ value calculated from $\mathrm{Tr}{M}^2/L^{2D}$ 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. 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. 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 $\log \eta \left(k=0.295\right)$, 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. 5. Determination of the critical point.

When studied as a function of k on two system sizes L_a < L_b, the curves for dimensionless quantities cross at a point k*(L_a, L_b), see Fig. 1-b. The figure shows k* (as computed for B, ξ⁽²⁾/L and ξ⁽³⁾/L) versus F(L_a, L_b) (18). We set 1/ν = 0.7 and ω = 1 to compute F(L_a, L_b). The curves should extrapolate linearly to k_c as F(L_a, L_b) tends to zero. The shaded area encompass our uncertainty in the estimation of k_c. 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), $\tilde{\tau }=\frac{L_{\tau }}{\mathrm{\pi }}\sin \left(\mathrm{\pi }\tau /L_{\tau }\right)$ 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 L_τ = 2¹¹. The dashed line in the right panel is a guide to the eye to show the critical exponent $\tilde{b}=1$ 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