Extremal eigenvalues of local Hamiltonians

We apply classical algorithms for approximately solving constraint satisfaction problems to find bounds on extremal eigenvalues of local Hamiltonians. We consider spin Hamiltonians for which we have an upper bound on the number of terms in which each spin participates, and find extensive bounds for the operator norm and ground-state energy of such Hamiltonians under this constraint. In each case the bound is achieved by a product state which can be found efficiently using a classical algorithm.


Introduction
The eigenvalue statistics of a local Hamiltonian are related to its structure. One example is the level spacings of chaotic vs integrable systems, which can be seen as the small-scale structure of the spectrum. What about the large-scale features, such as the extremal eigenvalues? Do these scale differently for non-interacting or interacting systems? It is generally understood that interacting systems can be frustrated, meaning that all the local terms cannot simultaneously be in their ground state. This situation is generically true for local terms with entangled ground states. But how much of an effect can frustration have on the ground-state energy of a system?
Here we study the extremal eigenvalues of quantum Hamiltonians which are only weakly interacting, in the sense that they can be written as sums of terms where each term depends only on a few qubits, and each qubit is included in only a few terms. With this mild form of locality imposed, how far apart must the largest and smallest eigenvalues be? If the Hamiltonian were non-interacting, the separation should scale with the size of the system. For a more general Hamiltonian, the extremal eigenvectors may be highly entangled and interacting terms may contribute opposite signs. Nevertheless, in this paper we show lower bounds on the norms of local Hamiltonians under very general conditions. An additional argument shows specifically that the ground-state energy is low (or if desired, that the top eigenvalue is high). In the above theorem, H is the operator norm of H and λ min (H) is the lowest eigenvalue (ground-state energy) of H. (Of course a similar statement could also be made about λ max . We focus on λ min because of its relevance to physical systems and to constraint satisfaction problems.) The notation O(f (x)), Ω(f (x)) refers to functions that are ≤ cf (x) or ≥ cf (x), respectively, for some absolute constant c. We write and Ω(f (x)).
If H were a non-interacting Hamiltonian (k = = 1) the largest and smallest eigenvalues would both be Θ(m) = Θ(n). Thus Theorem 1 can be viewed as saying that interaction can reduce the norm of H by at most a O( √ ) factor and can reduce the smallest eigenvalue by at most a O( ) factor. Observe that the bound on λ min (H) is −Ω(n) for all lattice Hamiltonians. This proves that for any such system the ground-state energy is smaller than the average energy by an extensive amount. By constrast, using our information about tr H 2 alone would only show that λ min (H) ≤ −Ω( √ m), which is in general a vanishing fraction of system size.
The restriction to terms of weight Θ(1) in Theorem 1 is not essential and is only included to simplify the bounds. Further, the hidden constants are not overly large for small k; more precise statements of our results are given below. For example, for 2-local qubit Hamiltonians, the precise bound on λ min we obtain is λ min (H) ≤ − H 1 /(24 ), where H 1 is the sum of the absolute values of the coefficients in the Pauli expansion of H. As a simple instance where this bound can be applied, consider the antiferromagnetic on a regular lattice with n vertices. Then, for any such lattice, we obtain λ min (H)/n ≤ −1/48. Theorem 1 can be applied to qudit Hamiltonians with local dimension d > 2 by embedding each subsystem in log 2 d qubits at the expense of increasing the locality from k to k log 2 d .
Proof outline.
Both results that make up Theorem 1 are based on the use of a correspondence between local quantum Hamiltonians and lowdegree polynomials, which allows us to apply classical approximation algorithms for constraint satisfaction problems. This correspondence uses a qubit 2design [5,13] to convert arbitrary qubit Hamiltonians to polynomials over boolean variables.
The operator norm bound in Theorem 1 (stated more precisely as Lemma 3 below) is based on recent work of Barak et al. [2] which gives an efficient randomised algorithm for satisfying a relatively large fraction of a set of linear equations over F 2 . The bound on λ min (stated more precisely as Lemma 5 below) is based on analysing a natural greedy algorithm which is similar to a classical algorithm of Håstad [14]. Our results can be seen as generalising these two classical algorithms to the quantum regime.
Other related work. Bansal, Bravyi and Terhal [1] have previously shown that, for 2-local qubit Hamiltonians H on a planar graph with Pauli interactions of weight Θ(1), λ min (H) ≤ −Ω(m). Similarly to our result, their proof uses a mapping between quantum and classical Hamiltonians and proves the existence of a product state achieving a −Ω(m) bound. However, the two results are not comparable; ours holds for non-planar graphs and k-local Hamiltonians for k > 2, while theirs encompasses two-local Hamiltonians on planar graphs with vertices of arbitrarily high degree. The quantum-classical mapping used is also different. Finally, the constants in our results are somewhat better (for example, they obtain λ min (H)/n ≤ −1/135 for the antiferromagnetic Heisenberg model on a 2D triangular lattice).
This work was motivated by [2] (whose main result is presented in Section 3). Ref. [2] in turn was inspired by [7,8], which gives a quantum algorithm for finding low-energy states of classical Hamiltonians. The relative performance of these different algorithms (ours/ [2] vs. [7,8]) is in general unknown, and it is also open to determine the extent to which [7] can be generalised to finding low-energy states of local Hamiltonians.
One other related work is [3], which showed that when k = 2 and the degree of the interaction graph is large, then product states can provide a good approximation for any state, with respect to the metric given by averaging the trace distance over the pairs of systems acted on by the Hamiltonian. In particular this means they can approximate the minimum and maximum eigenvalues. Both our result and [3] yield similar error bounds (ours are somewhat tighter), but in this sense apply to incomparable settings: [3] show that product states nearly match the energy of some other state (e.g. the true ground state) with possibly unknown energy while our paper puts explicit bounds on the maximum and/or minimum energy.
Another way to think about our work is as showing that interacting spins must nevertheless behave in some ways like noninteracting spins. In this picture, some vaguely related work is [10,4], which show that under some conditions lattice systems have a density of states that is approximately Gaussian. These results are incomparable to ours, even aside from the different assumptions, because we put bounds on the extremal eigenvalues while they study the density of states and/or the thermal states at nonzero temperature. Theorem 4 of Ref. [11] also bounded the density of states of k-local Hamiltonians under general conditions, but in the opposite direction: i.e. putting upper bounds on how many eigenvalues could have large absolute value.
Why product states? Ground states of local Hamiltonians may be highly entangled [9]. But our bounds on H and λ min (H) are achieved only with product states. One reason for this in the case of H is that we are using random states, and product states have much larger fluctuations than generic entangled states. Indeed the variance of ψ|H|ψ for a random unit vector |ψ is only O(m/2 n ). It is an interesting open question to find a distribution over entangled states that improves the constant factors in Theorem 1 that we achieve with product states.
Fourier analysis of boolean functions. We will need some basic facts from classical Fourier analysis of boolean functions [12]. Any function f : {±1} n → R can be written as where x S := i∈S x i and [n] := {1, . . . , n}. This is known as the Fourier expansion of f . Parseval's equality implies that where the expectation is taken over the uniform distribution on {±1} n . In addition, The influence of the j'th coordinate on f is defined as

The quantum-classical correspondence
Let H be a k-local Hamiltonian which has Pauli expansion H = s∈{I,X,Y,Z} nĤ s s 1 ⊗ s 2 ⊗ · · · ⊗ s n for some weights H s that we can view as a Fourier expansion of H analogous to that in (1). Define the norms H p := ( s | H s | p ) 1/p for p ≥ 1. In order to apply classical bounds to extremal eigenvalues of H, we observe that the action of a k-local Hamiltonian on product states corresponds to a low-degree polynomial. Define the following set of states [13,16]: These four states form a qubit 2-design; equivalently, a symmetric informationally-complete quantum measurement (SIC-POVM) on one qubit [13]. This measurement was studied in detail in [16]. Geometrically, the states describe a tetrahedron within the Bloch sphere [5].
We will now proceed to show bounds on max x∈{±1} 2n f H (x) and min x∈{±1} 2n f H (x) by viewing f H (x) as a polynomial.
As H is k-local and each function χ s (s = I) is a monomial of degree at most 2, f H is a polynomial of degree at most 2k. Because the Fourier expansion of each function χ s contains only one term, each term in H corresponds to exactly one term in the Fourier expansion of f H . Indeed Inf

Operator norm bounds
We will use the following result of Barak et al. [2], which is a constructive version of a probabilistic bound previously shown by Dinur et al. [6]: Theorem 2 (Barak et al. [2]). There is a universal constant C and a randomised algorithm such that the following holds. Let f : {±1} n → R be a polynomial with degree at most k such that Var(f ) = 1.
Let t ≥ 1 and suppose that . Then with high probability the algorithm outputs x ∈ {±1} n such that |f (x)| ≥ t. The algorithm runs in time poly(m, n, exp(k)), where m is the number of nonzero monomials in f .
Recent independent work of Håstad [15] describes an alternative, randomised algorithm achieving a similar bound.
Given the quantum-classical correspondence discussed in the previous section, we can now apply Theorem 2 to f H to prove the following result, which is one half of Theorem 1.

Lemma 3.
There is a universal constant D and a randomised classical algorithm such that the following holds. Let H be a traceless k-local Hamiltonian given as a weighted sum of m Pauli terms such that, for all j, Inf j (f H ) ≤ I max . Then with high probability the algorithm outputs a product state |ψ such that The running time of the algorithm is poly(m, n, exp(k)).
Proof. First observe that if we simply pick x ∈ {±1} 2n uniformly at random and consider the corresponding product state |ψ x , (5). In addition (see e.g. [12,Theorem 9.24]), as f H is a degree-2k polynomial, Therefore, simply picking exp(O(k)) random product states of the form |ψ x achieves | ψ x |H|ψ x | ≥ H 2 /3 k/2 with high probability. Let E be a universal constant to be chosen later. Set Note that the algorithm does not need to know whether I max is large or not, since it can simply try both strategies and see which one results in the larger value of | ψ|H|ψ |.

Bounds on extremal eigenvalues
We now describe an algorithm for bounding extremal eigenvalues which is weaker, but holds for both the largest and smallest eigenvalues. Once again, the algorithm is based on applying the quantum-classical correspondence in Section 2 to a classical algorithm. We first describe the classical algorithm, which is a simple greedy approach to find large values taken by a low-degree polynomial on the boolean cube. Consider the following algorithm, based on ideas of [14] but somewhat simpler: {±1} is picked uniformly at random.