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Computational Difficulty of Computing the Density of States Computational Difficulty of Computing the Density of States Brielin Brown,1, 2 Steven T. Flammia,2 and Norbert Schuch3 1University of Virginia, Departments of Physics and Computer Science, Charlottesville, Virginia 22904, USA 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 3California Institute of Technology, Institute for Quantum Information, MC 305-16, Pasadena, California 91125, USA We study the computational difficulty of computing the ground state degeneracy and the density of states for local Hamiltonians. We show that the difficulty of both problems is exactly captured by a class which we call #BQP, which is the counting version of the quantum complexity class QMA. We show that #BQP is not harder than its classical counting counterpart #P, which in turn implies that computing the ground state degeneracy or the density of states for classical Hamiltonians is just as hard as it is for quantum Hamiltonians. Understanding the physical properties of correlated quantum computer. We show that both problems, com- quantum many-body systems is a problem of central im- puting the density of states and counting the ground state portance in condensed matter physics. The density of degeneracy, are complete problems for the class #BQP, states, defined as the number of energy eigenstates per i.e., they are among the hardest problems in this class. energy interval, plays a particularly crucial role in this Having quantified the difficulty of computing the den- endeavor. It is a key ingredient when deriving many sity of states and counting the number of ground states, thermodynamic properties from microscopic models, in- we proceed to relate #BQP to known classical counting cluding specific heat capacity, thermal conductivity, band complexity classes, and show that #BQP equals #P (un- structure, and (near the Fermi energy) most electronic der weakly parsimonious reductions). Here, the complex- properties of metals. Computing the density of states ity class #P counts the number of satisfying assignments can be a daunting task however, as it in principle involves to any efficiently computable boolean function. This can diagonalizing a Hamiltonian acting on an exponentially be a very hard problem which is believed to take exponen- large space, though other more efficient approaches which tial time; in particular, it is at least as hard as deciding might take advantage of the structure of a given problem whether the function has at least one satisfying input, are not a priori ruled out. i.e., the complexity class NP. Examples for #P-complete problems (i.e., the hardest problems in that class) include In this Letter, we precisely quantify the difficulty of counting the number of colorings of a graph, or com- computing the density of states by using the powerful puting the permanent of a matrix with binary entries. tools of quantum complexity theory. Quantum complex- Phrased in terms of Hamiltonians, what we show is that ity aims at generalizing the well-established field of clas- computing the density of states and counting the ground sical complexity theory to assess the difficulty of tasks re- state degeneracy of a classical spin system is just as hard lated to quantum mechanical problems, concerning both as solving the same problem for a quantum Hamiltonian. the classical difficulty of simulating quantum systems as Quantum complexity classes.—Let us start by intro- well as the fundamental limits to the power of quan- ducing the relevant complexity classes. The central role tum computers. In particular, quantum complexity the- in the following is taken by the verifier V , which veri- ory has managed to explain the difficulty of computing fies “quantum solutions” (also called proofs) to a given ground state properties of quantum spin systems in vari- problem. More formally, a verifier checking an n-qubit ous settings, such as two-dimensional (2D) lattices [1] and quantum proof (that is, a quantum state ψ ) consists of a arXiv:1010.3060v2 [quant-ph] 20 Jul 2011 even one-dimensional (1D) chains [2], as well as fermionic | i T = poly(n) length quantum circuit U = UT U (with systems [3]. ··· 1 local gates Ut) acting on m = poly(n) qubits, which takes We will determine the computational difficulty of two the n-qubit quantum state ψ I as an input, together | i problems: First, computing the density of states of a lo- with m n initialized ancillas, 0 A 0 0 A, applies − | i ≡ | ··· i cal Hamiltonian, and second, counting the ground state U, and finally measures the first qubit in the 0 1, 1 1 degeneracy of a local gapped Hamiltonian; in both cases, basis to return 1 (“proof accepted”) or 0 (“proof{| i | i re-} the result holds even if the Hamiltonian is restricted to jected”). Then, the class QMA contains all problems of act on a 2D lattice of qubits, or on a 1D chain. To this the form: “Decide whether there exists a ψ such that | i end, we will introduce the quantum counting class #BQP pacc(V (ψ)) > a, or whether pacc(V (ψ)) < b for all ψ , (sharp BQP), which constitutes the natural counting ver- for some chosen a b> 1/poly(n), given that one is| thei sion of the class QMA (Quantum Merlin Arthur) which it- case”. Here, the acceptance− probability of a state ψ is self captures the difficulty of computing the ground state p (V (ψ)) := ψ Ω ψ , with | i acc h | | i energy of a local Hamiltonian [4, 5]. Vaguely speaking, † ½ ½ Ω = (½I 0 A)U ( 1 1 )U( I 0 A) , (1) #BQP counts the number of possible “quantum solu- ⊗ h | | ih |1 ⊗ ⊗ | i tions” to a quantum problem that can be verified using a which we illustrate in Fig. 1. 2 tance probability above a, up to a “grace interval” [b,a] in which mistakes are tolerated (i.e., if the largest eigen- value of Ω is in [b,a], the oracle can return either out- come). Correspondingly, #BQP captures the difficulty of counting the number of eigenvalues above a, where eivenvalues in the grace interval [b,a] can be miscounted. The reason why we choose to define #BQP with a gap promise rather than with a grace interval is the same as FIG. 1: A QMA verifier consists of a sequence of T local uni- for QMA, namely to have a unique outcome associated tary gates acting on the “quantum proof” |ψi and an ancillary with any input. register initialized to |0i. The final measurement on the first Similarly, the idea of the Hamiltonian formulation of qubit returns |1i or |0i to accept or reject the proof, respec- the problem which we will discuss below is to ask for the tively. Transition probabilities can be computed by doing a number of eigenstates in a certain energy interval, where “path integral” over all intermediate configurations (ik)k. states which are in some small 1/poly(n) neighborhood of this interval may be miscounted; again, for reasons of rigor we choose to consider only Hamiltonians with no The idea behind this definition is that QMA quanti- eigenstates in that interval. It should be noted, however, fies the difficulty of computing the ground state energy that all of the equivalence proofs we give equally apply E0(H) of a local Hamiltonian H up to 1/poly(n) accu- if we choose to allow for miscounting of states in those racy. Let the verifier be a circuit estimating ψ H ψ ; h | | i grace intervals instead of requiring them to be empty, as then a black box solving QMA problems can be used the proofs do not make use of the gap promise itself, but to compute E0(H) up to 1/poly(n) accuracy by binary rather show that all states outside those grace intervals search using a single QMA query. Note also that QMA is are mapped (and thus counted) correctly. Thus, while the quantum version of the class NP, where one is given the actual number returned by the grace interval formu- an efficiently computable boolean function f(x) 0, 1 ∈{ } lation of the counting problems might change under those and one must figure out if there is an x such that mappings due to different treatment of states in the grace f(x) = 1. interval, it will still be in the correct range. The class NP has a natural counting version, known The class #BQP inherits the important property from as #P. Here, the task is to determine the number rather QMA of being stable under amplification, that is, the def- than the existence of satisfying inputs, i.e., to compute inition of #BQP is not sensitive to the choice of a and x : f(x)=1 . We will now analogously define #BQP, |{ }| b. In particular, any a b > 1/poly(n) can be ampli- the counting version of QMA. Consider the verifying map fied (by building a new− poly-size Ω′ from Ω) such that Ω of Eq. (1) for a QMA problem, with the additional a′ = 1 exp( poly(n)), b′ = exp( poly(n)), and keep- promise that Ω does not have eigenvalues between a and ing the− eigenvalue− gap between a−′ and b′, by using a b, a b > 1/poly(n). Then the class #BQP consists of − construction called strong amplification, cf. Ref. [8]; as all problems of the form “compute the dimension of the shown there, strong amplification acts on all eigenval- space spanned by all eigenvectors with eigenvalues a”. ≥ ues independently and thus also applies to #BQP. The An equivalent definition for #BQP (cf. also [6, 7]) is crucial point is that strong amplification works without the following: Consider a verifier Ω with the additional changing the proof itself, compared to weak amplifica- n promise that there exist subspaces = C2 such that tion which takes multiple copies of the proof as an input.
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