Ground states of local Hamiltonians can be generally highly entangled: any quantum circuit that generates them, even approximately, must be sufficiently deep to allow coupling (entanglement) between any pair of qubits. Until now this property was not known to be "robust" -the marginals of such states to a subset of the qubits containing all but a small constant fraction of them may be only locally entangled, and hence approximable by shallow quantum circuits. In this work we construct a family of 16-local Hamiltonians for which any marginal of a ground state to a fraction at least 1 − 10 −9 of the qubits must be globally entangled.This provides evidence that quantum entanglement is not very fragile, and perhaps our intuition about its instability is an artifact of considering local Hamiltonians which are not only local but spatially local. Formally, it provides positive evidence for two wide-open conjectures in condensed-matter physics and quantum complexity theory which are the qLDPC conjecture, positing the existence of "good" quantum LDPC codes, and the NLTS conjecture [19] positing the existence of local Hamiltonians in which any low-energy state is highly entangled.Our Hamiltonian is based on applying the hypergraph product by Tillich-Zémor [43] to a classical locally testable code. A key tool in our proof is a new lower bound on the vertex expansion of the output of low-depth quantum circuits, which may be of independent interest. * Center for Theoretical Physics, MIT arXiv:1510.02082v3 [quant-ph] 21 Nov 2016 2. We show that low-depth circuits not only are unable to produce ground states of residual Hamiltonians, they cannot even produce states that approximately match the classical probability distributions resulting from measuring these states in the X and Z bases. 3 3. We prove a depth lower bound that is not merely ω(1) but is Ω(log(n)). A circuit of depth Ω(log(n)) can potentially generate a non-zero correlation between every pair of qubits, hence it "saturates" all light-cone type arguments. Since the naive algorithm for estimating expectation values runs in time doubly exponential in d, our results imply that this algorithm will require time 2 n Ω(1) . 4.We not only show that d-trivial states cannot be ε-impostors, but we show that these sets of quantum states are separated by a trace distance of n −Ω(1) . 5.We use a relatively simple form of Hamiltonian, consisting only of commuting 16-local Pauli terms.NLETS. While previously known constructions of local Hamiltonians are not NLETS, this may in part be because they are either embedded on a regular grid in low dimensions, or depart from this in ways that allow for efficient classical description. Thus, our theorem suggests that the apparent fragility of many-body entanglement from these examples may be simply a sign of not considering a wide enough range of examples.

The local Hamiltonian problem plays the equivalent role of SAT in quantum complexity theory. Understanding the complexity of the intermediate case in which the constraints are quantum but all local terms in the Hamiltonian commute, is of importance for conceptual, physical and computational complexity reasons. Bravyi and Vyalyi showed in 2003 [8], using clever applications of the representation theory of C*-algebras, that if the terms in the Hamiltonian are all two-local, the problem is in NP, and the entanglement in the ground states is local. The general case remained open since then. In this paper we extend the results of Bravyi and Vyalyi beyond the two-local case, to the case of three-qubit interactions. We then extend our results even further, and show that NP verification is possible for three-wise interaction between qutrits as well, as long as the interaction graph is embedded on a planar lattice, or more generally, "Nearly Euclidean" (NE). The proofs imply that in all such systems, the entanglement in the ground states is local. These extensions imply an intriguing sharp transition phenomenon in commuting Hamiltonian systems: 3-local NE systems based on qubits and qutrits cannot be used to construct Topological order, as their entanglement is local, whereas for higher dimensional qudits, or for interactions of at least 4 qudits, Topological Order is already possible, via Kitaev's Toric Code construction. We thus conclude that Kitaev's Toric Code construction is optimal for deriving topological order based on commuting Hamiltonians.

We initiate the study of quantum Locally Testable Codes (qLTCs). We provide a definition together with a simplification, denoted sLTCs, for the special case of stabilizer codes, and provide some basic results using those definitions. The most crucial parameter of such codes is their soundness, R(δ), namely, the probability that a randomly chosen constraint is violated as a function of the distance of a word from the code (δ, the relative distance from the code, is called the proximity). We then proceed to study limitations on qLTCs. In our first main result we prove a surprising, inherently quantum, property of sLTCs: for small values of proximity, the better the small-set expansion of the interaction graph of the constraints, the less sound the qLTC becomes. This stands in sharp contrast to the classical setting. The complementary, more intuitive, result also holds: an upper bound on the soundness when the code is defined on bad small-set expanders (a bound which turns out to be far more difficult to show in the quantum case). Together we arrive at a quantum upper-bound on the soundness of stabilizer qLTCs set on any graph, which does not hold in the classical case. Many open questions are raised regarding what possible parameters are achievable for qLTCs. In the appendix we also define a quantum analogue of PCPs of proximity (PCPPs) and point out that the result of [15] by which PCPPs imply LTCs with related parameters, carries over to the sLTCs. This creates a first link between qLTCs and quantum PCPs [6].

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The permanent is #P -hard to compute exactly on average for natural random matrices including matrices over finite fields or Gaussian ensembles. Should we expect that it remains #P -hard to compute on average if we only care about approximation instead of exact computation?In this work we take a first step towards resolving this question: We present a quasipolynomial time deterministic algorithm for approximating the permanent of a typical n × n random matrix with unit variance and vanishing mean µ = O(ln ln n) −1/8 to within inverse polynomial multiplicative error. Alternatively, one can achieve permanent approximation for matrices with mean µ = 1/polylog(n) in time 2 O(n ε ) , for any ε > 0.The proposed algorithm significantly extends the regime of matrices for which efficient approximation of the permanent is known. This is because unlike previous algorithms which require a stringent correlation between the signs of the entries of the matrix [1, 2] it can tolerate random ensembles in which this correlation is negligible (albeit non-zero). Among important special cases we note:1. Biased Gaussian: each entry is a complex Gaussian with unit variance 1 and mean µ.2. Biased Bernoulli: each entry is −1 + µ with probability 1/2, and 1 with probability 1/2.