We establish a polynomial-time approximation algorithm for partition functions of quantum spin models at high temperature. Our algorithm is based on the quantum cluster expansion of Netočný and Redig and the cluster expansion approach to designing algorithms due to Helmuth, Perkins, and Regts. Similar results have previously been obtained by related methods, and our main contribution is a simple and slightly sharper analysis for the case of pairwise interactions on bounded-degree graphs.
I. INTRODUCTION
Classical algorithms for approximating partition functions of quantum models that make use of cluster expansions have occurred in two recent papers.1,2 In this paper, we provide a simple and concise exposition of how to construct such algorithms, with the intent of making the technique accessible to a wide audience.
A quantum spin system is modelled by a hypergraph G = (X, E). At each vertex x of G, there is a d-dimensional Hilbert space with d < ∞. The Hilbert space on the hypergraph is given by . An interaction Φ assigns a self-adjoint operator Φ(e) on to each hyperedge e of G. The Hamiltonian on G is defined by HG ≔ ∑e∈E(G)Φ(e). We are interested in the quantum partition function ZG(β) at inverse temperature β, defined by .
In what follows, we shall focus our attention on quantum spin systems modeled by bounded-degree graphs; however, generalizations to bounded-degree bounded-rank hypergraphs are also possible. We shall assume that for every e ∈ E, where denotes the operator norm. Note that this is always possible by a rescaling of β. To state our main result, recall that a fully polynomial-time approximation scheme for a sequence of complex numbers is a deterministic algorithm that, for any n and ϵ > 0, produces a complex number such that in time polynomial in n and 1/ϵ.
Fix . There is a fully polynomial-time approximation scheme for the partition function ZG(β) for all graphs G of maximum degree at most Δ and all complex numbers β such that .
Our algorithm is based on combining the abstract cluster expansion for quantum spin systems of Netočný and Redig3 with the algorithmic framework of Helmuth, Perkins, and Regts.4 The condition is optimal under the complexity-theoretic assumption that RP (randomized polynomial time) is not equal to NP (non-deterministic polynomial time) due to the results on the hardness of approximate counting.5,6 We remark that these results concern real values of β; however, similar computational complexity transitions from P (polynomial time) to BQP-hard (bounded-error quantum polynomial time) and P to #P-hard can also be observed for complex values of β by using the methods of Refs. 7–9. For a formal definition of these complexity-theoretic notions, we refer the reader to Ref. 10.
Previous work on polynomial-time approximate counting algorithms for classical models have typically followed one of the three approaches: the correlation decay method,11,12 Markov-chain Monte Carlo,13 or interpolation-type methods.4,14 The latter two of these methods have also been used to design classical algorithms for quantum models.1,2,9,15–17 The goal of this paper is to convey the simplicity and flexibility of the third method that results from using the cluster expansion formalism. We emphasize that ideas of this type have previously been used to establish similar algorithms: for with quasi-polynomial runtime1 and for with polynomial runtime.2 Both the runtime of our algorithm and that of Ref. 2 are polynomials of a relatively high degree; examining our proof gives an upper bound of O(log(dΔ)) for the degree. While our results represent a modest improvement in the bound for , we view our main contribution as being the simplicity of our analysis.
We note also that a priori information on the location of zeros of the partition function can be combined with the methods of this paper to develop polynomial-time algorithms. As noted in Ref. 4, this is an alternate route to the results of Patel and Regts18 using Barvinok’s method.14 For quasi-polynomial time results of this flavour in the quantum setting, see Ref. 1.
This paper is structured as follows: In Sec. II, we introduce the abstract cluster expansion. Then, in Sec. III, we show how the partition function of quantum spins systems admits such a cluster expansion. In Sec. IV, we use this framework to establish our approximation algorithm for the quantum partition function at high temperature. Finally, we conclude in Sec. V with some remarks and open problems.
II. THE ABSTRACT CLUSTER EXPANSION
The cluster expansion is a powerful tool from mathematical physics that allows one to express, via power series expansions, perturbations of a well-understood reference model. When the perturbations are sufficiently small, the power series expansions are convergent and allow one to draw many conclusions regarding correlation decay, zero-freeness, and other related properties. This method was originally introduced by Mayer in the study of imperfect gases19 but has since been greatly abstracted and simplified. The formulation we use is due to Kotecký and Preiss.20
An abstract polymer model is a triple , where is a countable set whose objects are called polymers, is a function that assigns to each polymer a complex number wγ called the weight of the polymer, and ∼ is a symmetric compatibility relation such that each polymer is incompatible with itself. Equivalently, the incompatibility relation ≁ is a symmetric and reflexive relation. A set of polymers is called admissible if all the polymers in the set are all pairwise compatible. Note that the empty set is admissible. Let denote the collection of all admissible sets of polymers from . Then, the abstract polymer partition function is defined by
Our algorithm is based on reformulating the partition function of a quantum spin system in the abstract polymer model language (see Sec. III). The utility of this is due to the following fact about .
Let Γ be a non-empty ordered tuple of polymers. The incompatibility graph HΓ of Γ is the graph with vertex set Γ and edges between any two polymers if and only if they are incompatible. Γ is called a cluster if its incompatibility graph HΓ is connected. Let denote the set of all clusters of polymers from . The abstract cluster expansion20,21 is a formal power series for in the variables wγ, defined by
where φ(H) denotes the Ursell function of a graph H,
III. THE QUANTUM CLUSTER EXPANSION
In this section, we shall show how the partition function of a quantum spin system admits an abstract polymer representation and hence an abstract cluster expansion. We return to the more general setting of hypergraphs for the remainder of this section.
Consider a quantum spin system modeled by the hypergraph G = (X, E) with interaction Φ, where at each vertex x of G, there is a d-dimensional Hilbert space with d < ∞. Recall that Φ assigns a self-adjoint operator Φ(e) on to each hyperedge e of G. Define a polymer γ in this model to be a multiset (Eγ, mγ) of hyperedges Eγ ⊆ E with the multiplicity function whose support Eγ induces a connected subgraph. Say that two polymers are compatible if and only if their supporting subgraphs are vertex disjoint. For a polymer γ, let denote its size and let denote the cardinality of its support; by a slight abuse of notation, we will write . With these definitions, the partition function ZG(β) admits an abstract polymer model representation3,22,23 as formalized by the following lemma:
Note that is the symmetric group of degree , and this assigns an ordering to the product of interactions. We prove Lemma 9 in Appendix A. Note that the abstract polymer model representation holds as a formal power series in β. As an immediate corollary, we obtain a cluster expansion for log(ZG(β)).
For algorithms, an important quantity is the truncated cluster expansion for log(ZG(β)),
where .
IV. APPROXIMATION ALGORITHM
We now establish our approximation algorithm. First, we show that the truncated cluster expansion provides a good approximation to log(ZG(β)). Netočný and Redig3 provided a sufficient condition for the convergence of the quantum cluster expansion based on the formalism of Kotecký and Preiss.20 In the following lemma, we follow their analysis in the setting of bounded-degree graphs. In particular, we obtain convergence criteria based on the maximum degree alone.
We prove Lemma 10 in Appendix B. This lemma implies that to obtain an multiplicative ϵ-approximation to ZG(β), it is sufficient to compute Tm(ZG(β)) to order . We shall proceed by establishing an algorithm for computing Tm(ZG(β)) in time . Helmuth, Perkins, and Regts4 showed that such an algorithm exists, given the following three lemmas:
Fix , and let G = (V, E) be a graph of maximum degree at most Δ. The clusters of size at most m can be listed in time .
Our proof follows that of Ref. 4, Theorem 6. First, we enumerate all connected subgraphs in G of size at most m. This can be achieved in time by Ref. 18, Lemma 3.6. Then, for each subgraph H, we enumerate all polymers (multisets) of size at most m whose corresponding subgraph in G is H. If H has size n, then there are precisely of these, and they can be enumerated in time exp(O(m)). The enumeration of clusters in the claimed time then follows as in the proof of Ref. 4, Theorem 6.■
The Ursell function φ(H) can be computed in time .
The weight wγ of a polymer γ can be computed in time .
We prove Lemma 11 in Appendix B.
Fix , and let G = (V, E) be a graph of maximum degree at most Δ. The truncated cluster expansion Tm(ZG(β)) can be computed in time .
We can list all clusters in G of size at most m in time by Lemma 5. For each of these clusters, we can compute the Ursell function in time exp(O(m)) by Lemma 6, and the polymer weights in time exp(O(m)) by Lemma 11. Hence, the truncated cluster expansion for log(ZG(β)) can be computed in time .■
Combining Lemmas 10 and 8 gives a fully polynomial-time approximation scheme for the partition function ZG(β) when G has maximum degree at most Δ and is at most . This proves Theorem 1.
V. CONCLUSION & OUTLOOK
We have discussed how classical algorithms based on cluster expansion methods apply to quantum spin systems at high temperature. Our focus has been on conveying the simplicity of the method, which has appeared previously in other forms.1,2 We note that it may be possible to use the Markov chain polymer approach of Ref. 25 to obtain an algorithm with an improved runtime.
For discrete classical spin systems, expansion methods have also been used at low temperatures, i.e., when β ≫ 1.4,26–29 It would be interesting to adapt these methods to quantum systems, e.g., by developing algorithms based on Pirogov–Sinai methods for quantum perturbations of classical systems.30 We remark, however, that it seems difficult to use this approach for low-temperature quantum systems with an infinite degeneracy of ground states, for example, when the set of ground states possesses a continuous symmetry.
ACKNOWLEDGMENTS
We thank Michael Bremner, Adrian Chapman, Ashley Montanaro, and Will Perkins for helpful discussions. R.L.M. was supported by the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme (QuantAlgo project) and EPSRC Grant Nos. EP/L021005/1 and EP/R043957/1. T.H. was supported by EPSRC Grant No. EP/P003656/1.
DATA AVAILABILITY
No new data were created during this study.
APPENDIX A: PROOF OF LEMMA 2
APPENDIX B: PROOF OF LEMMA 4 AND LEMMA 7
(restatement). The weight wγ of a polymer γ can be computed in time .