In this theoretical paper, we consider spin systems in three spatial dimensions and consider the computational complexity of estimating the ground state energy, known as the local Hamiltonian problem, for translationally invariant Hamiltonians. We prove that the local Hamiltonian problem for 3D lattices with face-centered cubic unit cells and 4-local translationally invariant interactions between spin-3/2 particles and open boundary conditions is QMAEXP-complete, where QMAEXP is the class of problems which can be verified in exponential time on a quantum computer. We go beyond a mere embedding of past hard 1D history state constructions, for which the local spin dimension is enormous: even state-of-the-art constructions have local dimension 42. We avoid such a large local dimension by combining some different techniques in a novel way. For the verifier circuit which we embed into the ground space of the local Hamiltonian, we utilize a recently developed computational model, called a quantum ring machine, which is especially well suited for translationally invariant history state constructions. This is encoded with a new and particularly simple universal gate set, which consists of a single 2-qubit gate applied only to nearest-neighbour qubits. The Hamiltonian construction involves a classical Wang tiling problem as a binary counter which translates one cube side length into a binary description for the encoded verifier input and a carefully engineered history state construction that implements the ring machine on the cubic lattice faces. These novel techniques allow us to significantly lower the local spin dimension, surpassing the best translationally invariant result to date by two orders of magnitude (in the number of degrees of freedom per coupling). This brings our models on par with the best non-translationally invariant construction.
The calculation of ground-state energies of physical systems can be formalized as the
k-local Hamiltonian problem, which is a natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set
S and scaling them by arbitrary weights. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterize the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset
S, the problem falls into one of the following categories: in
P;
NP-complete; polynomial-time equivalent to the Ising model with transverse magnetic fields; or
QMA-complete. The third of these classes has been shown to be
StoqMA-complete by Bravyi and Hastings. The characterization holds even if
S does not contain any 1-local terms; for example, we prove for the first time
QMA-completeness of the Heisenberg and XY interactions in this setting. If
S is assumed to contain all 1-local terms, which is the setting considered by previous work, we have a characterization that goes beyond 2-local interactions: for any constant
k, all
k-local qubit Hamiltonians whose terms are picked from a fixed set
S correspond to problems either in
P; polynomial-time equivalent to the Ising model with transverse magnetic fields; or
QMA-complete. These results are a quantum analogue of the maximization variant of Schaefer's dichotomy theorem for Boolean constraint satisfaction problems.
Classical simulation of quantum many-body systems is in general a challenging problem for the simple reason that the dimension of the Hilbert space grows exponentially with the system size. In particular, merely encoding a generic quantum many-body state requires an exponential number of bits. However, condensed matter physicists are mostly interested in local Hamiltonians and especially their ground states, which are highly non-generic. Thus, we might hope that at least some physical systems allow efficient classical simulation.Starting with one-dimensional (1D) quantum systems (i.e., the simplest nontrivial case), the first basic question is: Which classes of states have efficient classical representations? It turns out that this question is quantitatively related to the amount of entanglement in the state, for states with ``little entanglement'' are well approximated by matrix product states (a data structure that can be manipulated efficiently on a classical computer). At a technical level, the mathematical notion for ``little entanglement'' is area law, which has been proved for unique ground states in 1D gapped systems. We establish an area law for constant-fold degenerate ground states in 1D gapped systems and thus explain the effectiveness of matrix-product-state methods in (e.g.) symmetry breaking phases. This result might not be intuitively trivial as degenerate ground states in gapped systems can be long-range correlated.Suppose an efficient classical representation exists. How can one find it efficiently? The density matrix renormalization group is the leading numerical method for computing ground states in 1D quantum systems. However, it is a heuristic algorithm and the possibility that it may fail in some cases cannot be completely ruled out. Recently, a provably efficient variant of the density matrix renormalization group has been developed for frustration-free 1D gapped systems. We generalize this algorithm to all (i.e., possibly frustrated) 1D gapped systems. Note that the ground-state energy of 1D gapless Hamiltonians is computationally intractable even in the presence of translational invariance.It is tempting to extend methods and tools in 1D to two and higher dimensions (2+D), e.g., matrix product states are generalized to tensor network states. Since an area law for entanglement (if formulated properly) implies efficient matrix product state representations in 1D, an interesting question is whether a similar implication holds in 2+D. Roughly speaking, we show that an area law for entanglement (in any reasonable formulation) does not always imply efficient tensor network representations of the ground states of 2+D local Hamiltonians even in the presence of translational invariance. It should be emphasized that this result does not contradict with the common sense that in practice quantum states with more entanglement usually require more space to be stored classically; rather, it demonstrates that the relationship between entanglement and efficient classical representations is still far from being well understood.Excited eigenstates participate in the dynamics of quantum systems and are particularly relevant to the phenomenon of many-body localization (absence of transport at finite temperature in strongly correlated systems). We study the entanglement of excited eigenstates in random spin chains and expect that its singularities coincide with dynamical quantum phase transitions. This expectation is confirmed in the disordered quantum Ising chain using both analytical and numerical methods.Finally, we study the problem of generating ground states (possibly with topological order) in 1D gapped systems using quantum circuits. This is an interesting problem both in theory and in practice. It not only characterizes the essential difference between the entanglement patterns that give rise to trivial and nontrivial topological order, but also quantifies the difficulty of preparing quantum states with a quantum computer (in experiments).
The calculation of ground-state energies of physical systems can be formalised as the k-local Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterise the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the following categories: in P; NP-complete; polynomial-time equivalent to the Ising model with transverse magnetic fields; or QMA-complete. The third of these classes has been shown to be StoqMA-complete by Bravyi and Hastings. The characterisation holds even if S does not contain any 1-local terms; for example, we prove for the first time QMA-completeness of the Heisenberg and XY interactions in this setting. If S is assumed to contain all 1-local terms, which is the setting considered by previous work, we have a characterisation that goes beyond 2-local interactions: for any constant k, all k-local qubit Hamiltonians whose terms are picked from a fixed set S correspond to problems either in P; polynomial-time equivalent to the Ising model with transverse magnetic fields; or QMA-complete. These results are a quantum analogue of Schaefer's dichotomy theorem for boolean constraint satisfaction problems.
Given a renormalization scheme, we show how to formulate a tractable convex relaxation of the set of feasible local density matrices of a many-body quantum system. The relaxation is obtained by introducing a hierarchy of constraints between the reduced states of ever-growing sets of lattice sites. The coarse-graining maps of the underlying renormalization procedure serve to eliminate a vast number of those constraints, such that the remaining ones can be enforced with reasonable computational means. This process can be used to obtain rigorous lower bounds on the ground-state energy of arbitrary local Hamiltonians by performing a linear optimization over the resulting convex relaxation of reduced quantum states. The quality of the bounds crucially depends on the particular renormalization scheme, which must be tailored to the target Hamiltonian. We apply our method to 1D translation-invariant spin models, obtaining energy bounds comparable to those attained by optimizing over locally translation-invariant states of n≳100 spins. Beyond this demonstration, the general method can be applied to a wide range of other problems, such as spin systems in higher spatial dimensions, electronic structure problems, and various other many-body optimization problems, such as entanglement and nonlocality detection. Published by the American Physical Society 2024
We prove that estimating the ground state energy of a translationally invariant, nearest-neighbour Hamiltonian on a 1D spin chain is
QMAEXP -complete, even for systems of low local dimension (
≈40 ). This is an improvement over the best previously known result by several orders of magnitude, and it shows that spin-glass-like frustration can occur in translationally invariant quantum systems with a local dimension comparable to the smallest-known non-translationally invariant systems with similar behaviour. While previous constructions of such systems rely on standard models of quantum computation, we construct a new model that is particularly well-suited for encoding quantum computation into the ground state of a translationally invariant system. This allows us to shift the proof burden from optimizing the Hamiltonian encoding a standard computational model, to proving universality of a simple model. Previous techniques for encoding quantum computation into the ground state of a local Hamiltonian allow only a linear sequence of gates, hence only a linear (or nearly linear) path in the graph of all computational states. We extend these techniques by allowing significantly more general paths, including branching and cycles, thus enabling a highly efficient encoding of our computational model. However, this requires more sophisticated techniques for analysing the spectrum of the resulting Hamiltonian. To address this, we introduce a framework of graphs with unitary edge labels. After relating our Hamiltonian to the Laplacian of such a unitary labelled graph, we analyse its spectrum by combining matrix analysis and spectral graph theory techniques.