Quantum annealing of the random transverse-field Ising model in one, two, and three dimensions

The gist Quantum annealing must be sufficiently slow to prevent defect formation, and this paper studies this problem in the random transverse-field Ising model via the adiabatic theorem to obtain the limiting rate at which defect formation can be avoided<ref:2610.11411#pg2>.

How it works

The study employs a scaling theory of the statistics of elementary events of error formation in general dimensions, comparing it to extensive numerical analysis carried out in one, two, and three dimensions using the strong-disorder renormalization group (SDRG) technique<ref:2610.11411#pg2>. The SDRG approach views the annealing process as a gradual aggregation of strongly coupled clusters, which allows access to the elementary events of error formation<ref:2610.11411#pg3>. Each fusion event can be assigned a crossover annealing time separating the adiabatic and non-adiabatic regimes, where the largest of these crossover times sets the annealing time required to suppress defect formation<ref:2610.11411#pg3>.

Model and Adiabatic Theorem

The work considers the random transverse-field Ising model (RTFIM) on d-dimensional cubic lattices with a time-dependent Hamiltonian H(t) that drives the system from a paramagnetic phase to a ferromagnetic phase<ref:2610.11411#pg2>. The adiabatic theorem provides the condition for avoiding defects, which is defined by τ ≫ max 0≤s≤1 Q(s), where Q(s) is related to the matrix element divided by the squared gap<ref:2610.11411#pg4>. The critical annealing time τc is defined as τc ≡ max 0≤s≤1 Q(s)<ref:2610.11411#pg4>.

Results in One Dimension

In one dimension, the SDRG approach classifies spins into clusters, and during annealing, these clusters undergo a gradual aggregation by pairwise fusions<ref:2610.11411#pg3>. The crossover annealing time can be written as τc ∼ n1 + nb / ϵ(t∗), where n1 and nb denote the number of original fields involved in h˜1 and J˜, respectively<ref:2610.11411#pg4>. The conditional probability distribution of the logarithmic energy scale Γ for fusions occurring at a given value of the control parameter ∆ ≪ 1 has an exponential cutoff at small energy (large Γ) as G>(Γ∆) ∼ ∆e−2∆Γ<ref:2610.11411#pg4>. Integrating over the entire control parameter range yields that the fraction of erroneous (non-adiabatic) fusions scales, for large annealing times, as n(τ) ∼ ln τ /τ0 ln2(τ /τ0) −2<ref:2610.11411#pg4>.

Generalization to Higher Dimensions

The framework is generalized to higher dimensions based on the assumption that the RTFIM flows to an infinite-disorder fixed point (IDFP) in any finite dimension provided the initial disorder is strong enough<ref:2610.11411#pg5>. The minimal annealing time required for a finite system of size L, τL, scales with the linear size L as τL ∼ Lωe cLψ in all studied dimensions<ref:2610.11411#pg6>. The exponent ω is conjectured to be related to the usual critical exponents by ω = ψ + 1/ν<ref:2610.11411#pg5>.

Characteristics of Fusions

The analysis examines several characteristics of fusion events, including the excitation energy ϵ of the dominant fusion, which scales according to Eq. (10)<ref:2610.11411#pg6>. The effective multiplicity τLϵ of the dominant fusion follows algebraic scaling anticipated in Eq. (11), with exponents ω = 1 in d = 1, ω = 1.20(5) in d = 2, and ω = 1.25(5) in d = 3<ref:2610.11411#pg7>. The sample-to-sample distribution of the threshold annealing time τL is compatible with the activated finite-size scaling of τL in Eq. (15)<ref:2610.11411#pg7>.

Error Density Scaling

The density of errors produced in an infinite system under a finite annealing time is measured by the fraction n(τ) of erroneous (non-adiabatic) fusion events to all fusion events, which scales as n(τ) ∼ ln τ / (ln τ) ω ψ − d ψ for large τ<ref:2610.11411#pg7>.

Conclusion

The bottleneck annealing time needed to avoid defects in a finite sample scales with the linear size L as τL ∼ Lωe cLψ in all studied dimensions, and this relationship corresponds to an ultra-slow decrease of the defect density with the annealing time, as n(τ) ∼ (ln τ) − d/ψ for large τ<ref:2610.11411#pg7>. The numerical analysis presented in this work has been performed for a particular form of disorder and annealing protocol, and it is expected that the asymptotic results are universally valid for protocols having a linear time-dependence at the critical point<ref:2610.11411#pg11>.

How it works

The SDRG method was validated for the one-dimensional model by a comparison with results of exact diagonalization based on the free-fermion technique<ref:2610.11411#pg7>. The distributions of scaling combinations xϵ = − ln(ϵ/ϵ0)/Lψ, xn = ln(τLϵ/Lω), and xτ = ln[τL/(τ0Lω)]/Lψ must display data collapse using ψ = 1/2, ω = 1, and appropriately chosen constants ϵ0 and τ0<ref:2610.11411#pg7>. The SDRG method successfully identifies all universal aspects of the scaling behavior<ref:2610.11411#pg7>. The bottleneck annealing time needed to avoid defects in a finite sample scales with the linear size L as τL ∼ Lωe cLψ in all studied dimensions<ref:2610.11411#pg11>. This relationship corresponds to an ultra-slow decrease of the defect density with the annealing time, as n(τ) ∼ (ln τ) − d/ψ in leading order for large τ<ref:2610.11411#pg7>. The numerical analysis presented in this work has been performed for a particular form of disorder and annealing protocol<ref:2610.11411#pg11>. Nevertheless, it is well-known that the critical exponents of an IDFP are universal in the sense that they are independent from the particular form of disorder<ref:2610.11411#pg7>. Consequently, our asymptotic results valid in the limit of long annealing times or large system sizes are expected to be independent from the type of disorder<ref:2610.11411#pg11>. The region where error formation is considerable shrinks to the critical point with increasing annealing time<ref:2610.11411#pg7>. As a consequence, our asymptotic results are expected to be universally valid for protocols having a linear time-dependence at the critical point<ref:2610.11411#pg11>.

How it works

The analysis of all fusion events shows that the marginal distributions of the logarithmic excitation energies Γ = − ln ϵ of all fusions are shown in Fig. 5, and the tail must be algebraic, ρ(Γ) ∼ Γ − κ with κ = 1 + d/ψ in all three dimensions<ref:2610.11411#pg7>. The probability density of the logarithmic crossover times ln τc over all fusion events is expected to have the form ρ(x) ∼ x − (1+d/ψ), where x = ln τc<ref:2610.11411#pg7>. The scaling of the error density with annealing time follows ln τ ∼ [n(τ)]−ψ/d, valid in the large τ limit<ref:2610.11411#pg7>. The predicted asymptotic dependence is indicated by the dashed lines having slopes −ψ/d in Fig. 10<ref:2610.11411#pg7>.

How it works

The numerical analysis presented in this work has been performed for a particular form of disorder and annealing protocol, and it is expected that the bottleneck annealing time needed to avoid defects in a finite sample scales with the linear size L as τL ∼ Lωe cLψ in all studied dimensions<ref:2610.11411#pg7>. This relationship corresponds to an ultra-slow decrease of the defect density with the annealing time, as n(τ) ∼ (ln τ) − d/ψ in leading order for large τ<ref:2610.11411#pg7>. The numerical analysis presented in this work has been performed for a particular form of disorder and annealing protocol<ref:2610.11411#pg11>. Nevertheless, it is well-known that the critical exponents of an IDFP are universal in the sense that they are independent from the particular form of disorder<ref:2610.11411#pg7>. Consequently, our asymptotic results valid in the limit of long annealing times or large system sizes are expected to be independent from the type of disorder<ref:2610.11411#pg11>. The region where error formation is considerable shrinks to the critical point with increasing annealing time<ref:2610.11411#pg7>. As a consequence, our asymptotic results are expected to be universally valid for protocols having a linear time-dependence at the critical point<ref:2610.11411#pg11>.

How it works

The SDRG method was validated for the one-dimensional model by a comparison with results of exact diagonalization based on the free-fermion technique<ref:2610.11411#pg7>. The distributions of scaling combinations xϵ = − ln(ϵ/ϵ0)/Lψ, xn = ln(τLϵ/Lω), and xτ = ln[τL/(τ0Lω)]/Lψ must display data collapse using ψ = 1/2, ω = 1, and appropriately chosen constants ϵ0 and τ0<ref:2610.11411#pg7>. The SDRG method successfully identifies all universal aspects of the scaling behavior<ref:2610.11411#pg7>.

1. textbfConcentration of Critical Exponents in Model Design + Capability to Predict Defect Suppression Thresholds + The improved system can predict, for a given dimension and disorder realization, the limiting rate at which defect formation can be avoided by calculating the crossover annealing time τc, defined as the maximum of the adiabatic condition Q(s) over the annealing time interval.

2. textbfAdaptive Annealing Path Optimization + Capability to Design Optimal Annealing Protocols + The system can provide a map of individual events of potential defect formation in the excitation energy - control parameter plane to help in the design of optimal annealing paths.

3. textbf Multi-scale Error Event Characterization + Capability to Quantify Defect Statistics across Time Scales + The AI can determine the scaling theory of the statistics of these events in general dimensions and calculate quantities like the fraction of erroneous (non-adiabatic) fusions scales, for large annealing times, as n(τ) ∼ ln τ /τ0 ln2 (τ /τ0) −2.