Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs

Sinclair, Alistair; Srivastava, Piyush; Thurley, Marc

doi:10.1137/1.9781611973099.75

Cited by 80 publications

(122 citation statements)

References 17 publications

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“…The above result indicates that almost all imaginary Ising partition functions are substantially hard to calculate even in the approximated case with a multiplicative error. This result contrasts with the existence of a FPRAS in the ferromagnetic cases with magnetic fields shown by Jerrum and Sinclair [33] and antiferromagnetic cases on a sort of lattices shown by Sinclair, Srivastava, and Thurley [88]. In these cases, an exact calculation is #P-hard but its approximation with a multiplicative error is easy.…”

Section: Hardness Of Approximating Ising Partition Functionsmentioning

confidence: 70%

Commuting quantum circuits and complexity of Ising partition functions

Fujii

Morimae

2017

New J. Phys.

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Instantaneous quantum polynomial-time (IQP) computation is a class of quantum computation consisting only of commuting two-qubit gates and is not universal. Nevertheless, it has been shown that if there is a classical algorithm that can simulate IQP efficiently, the polynomial hierarchy collapses to the third level, which is highly implausible. However, the origin of the classical intractability is still less understood. Here we establish a relationship between IQP and computational complexity of calculating the imaginary-valued partition functions of Ising models. We apply the established relationship in two opposite directions. One direction is to find subclasses of IQP that are classically efficiently simulatable by using exact solvability of certain types of Ising models. Another direction is applying quantum computational complexity of IQP to investigate (im)possibility of efficient classical approximations of Ising partition functions with imaginary coupling constants. Specifically, we show that a multiplicative approximation of Ising partition functions is #P-hard for almost all imaginary coupling constants even on planar lattices of a bounded degree.Aaronson and Arkhipov introduced BOSONSAMPLING [15], a sampling problem according to the probability distribution of n bosons scattered by linear optical unitary operations. The probability distribution is given by the permanent of a complex matrix, which is determined by the linear optical unitary operations. Calculation of the permanent of complex matrices is known to be #P-hard [16,17]. Since a polynomial-time machine with an oracle for #P can solve all problems in the PH according to Toda's theorem [18], an exact classical simulation (in the strong sense [19,20] meaning a calculation of the probability distribution of the output) of BOSONSAMPLING is highly intractable in a classical computer. They showed under assumptions of plausible conjectures that if there exists an efficient classical approximation of BOSONSAMPLING (classical simulation in the weak sense [19, 20] meaning a sampling according the probability distribution of the output), the PH collapses to the third level, which is unlikely to occur. (The detailed notions of classical simulation are provided in section 3.) This result brings a novel perspective on linear optical quantum computation and drives many researchers into the recent proof-of-principle experiments [21][22][23][24][25][26][27][28].Another subclass of quantum computation of this kind is instantaneous quantum polynomial-time computation (IQP) proposed by Shepherd and Bremner [29]. IQP consists only of commuting unitary gates, such as q  Î [ ] Z exp i k S k . Here q p Î [ ) 0, 2 is a rotational angle, Z k indicates the Pauli operator on the kth qubit, and S indicates a set of qubits on which the commuting gate acts. (A detailed definition will be provided in the next section.) The input is given by +ñ Ä | n with +ñ º ñ + ñ | (| | ) 0 1 2, and the output qubits are measured in the X-basis. Since all unitary operations are commutabl...

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Section: Hardness Of Approximating Ising Partition Functionsmentioning

confidence: 70%

Commuting quantum circuits and complexity of Ising partition functions

Fujii

Morimae

2017

New J. Phys.

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“…Our general setup is two-spin systems on an input graph G = (V , E) with maximum degree Δ. We follow the setup of several related previous works [12,26,18]. Configurations of the system are assignments σ : V → {−1, +1}.…”

Section: Introductionmentioning

confidence: 99%

“…For the hard-core model, Kelly [17] showed that non-uniqueness holds if and only if λ > λ c ( (see e.g. [26]). For general antiferromagnetic two-spin models with soft constraints (i.e., B 1 B 2 > 0), non-uniqueness holds if and only if √ B 1 B 2 < (Δ − 2)/Δ and λ ∈ (λ 1 , λ 2 ) for some critical values λ 1 (Δ, B 1 , B 2 ), λ 2 (Δ, B 1 , B 2 ) (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

Galanis

Štefankovič

Vigoda

2016

Combinator. Probab. Comp.

113

127

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Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006) establish a beautiful picture for the computational complexity of approximating the partition function of the hard-core model. Let λc(T∆) denote the critical activity for the hard-model on the infinite ∆-regular tree. Weitz presented an FPTAS for the partition function when λ < λc(T∆) for graphs with constant maximum degree ∆. In contrast, Sly showed that for all ∆ ≥ 3, there exists ε∆ > 0 such that (unless RP = N P ) there is no FPRAS for approximating the partition function on graphs of maximum degree ∆ for activities λ satisfying λc(T∆) < λ < λc(T∆) + ε∆.We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Sinclair, Srivastava, and Thurley (2014) extended Weitz's approach to the antiferromagnetic Ising model, yielding an FPTAS for the partition function for all graphs of constant maximum degree ∆ when the parameters of the model lie in the uniqueness region of the infinite ∆-regular tree. We prove the complementary result for the antiferrogmanetic Ising model without external field, namely, that unless RP = N P , for all ∆ ≥ 3, there is no FPRAS for approximating the partition function on graphs of maximum degree ∆ when the inverse temperature lies in the non-uniqueness region of the infinite tree T∆. Our proof works by relating certain second moment calculations for random ∆-regular bipartite graphs to the tree recursions used to establish the critical points on the infinite tree.

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“…However, this algorithm, as well as the follow-up algorithms (e.g. [19,21,22]) require particularly strong spatial mixing conditions, that is, they require what we call the strong spatial mixing condition. On the other hand, the weaker, in terms of approximation guarantees, algorithms in [3,17] require a condition called weak spatial mixing and locally tree-like structure 1 for the underlying graphs.…”

Section: Introductionmentioning

confidence: 99%

Deterministic counting of graph colourings using sequences of subgraphs

Efthymiou

2020

Combinator. Probab. Comp.

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In this paper we propose a polynomial-time deterministic algorithm for approximately counting the k-colourings of the random graph G(n, d/n), for constant d>0. In particular, our algorithm computes in polynomial time a $(1\pm n^{-\Omega(1)})$ -approximation of the so-called ‘free energy’ of the k-colourings of G(n, d/n), for $k\geq (1+\varepsilon) d$ with probability $1-o(1)$ over the graph instances. Our algorithm uses spatial correlation decay to compute numerically estimates of marginals of the Gibbs distribution. Spatial correlation decay has been used in different counting schemes for deterministic counting. So far algorithms have exploited a certain kind of set-to-point correlation decay, e.g. the so-called Gibbs uniqueness. Here we deviate from this setting and exploit a point-to-point correlation decay. The spatial mixing requirement is that for a pair of vertices the correlation between their corresponding configurations becomes weaker with their distance. Furthermore, our approach generalizes in that it allows us to compute the Gibbs marginals for small sets of nearby vertices. Also, we establish a connection between the fluctuations of the number of colourings of G(n, d/n) and the fluctuations of the number of short cycles and edges in the graph.

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Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs

Cited by 80 publications

References 17 publications

Commuting quantum circuits and complexity of Ising partition functions

Commuting quantum circuits and complexity of Ising partition functions

Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

Deterministic counting of graph colourings using sequences of subgraphs

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