Dynamic Complexity of Planar 3-connected Graph Isomorphism

Dynamic Complexity (as introduced by Patnaik and Immerman) tries to express how hard it is to update the solution to a problem when the input is changed slightly. It considers the changes required to some stored data structure (possibly a massive database) as small quantities of data (or a tuple) are inserted or deleted from the database (or a structure over some vocabulary). The main difference from previous notions of dynamic complexity is that instead of treating the update quantitatively by finding the the time/space trade-offs, it tries to consider the update qualitatively, by finding the complexity class in which the update can be expressed (or made). In this setting, DynFO, or Dynamic First-Order, is one of the smallest and the most natural complexity class (since SQL queries can be expressed in First-Order Logic), and contains those problems whose solutions (or the stored data structure from which the solution can be found) can be updated in First-Order Logic when the data structure undergoes small changes. Etessami considered the problem of isomorphism in the dynamic setting, and showed that Tree Isomorphism can be decided in DynFO. In this work, we show that isomorphism of Planar 3-connected graphs can be decided in DynFO+ (which is DynFO with some polynomial precomputation). We maintain a canonical description of 3-connected Planar graphs by maintaining a database which is accessed and modified by First-Order queries when edges are added to or deleted from the graph. We specifically exploit the ideas of Breadth-First Search and Canonical Breadth-First Search to prove the results. We also introduce a novel method for canonizing a 3-connected planar graph in First-Order Logic from Canonical Breadth-First Search Trees

Tree Tribes and Lower Bounds for Switching Lemmas

We show tight upper and lower bounds for switching lemmas obtained by the action of random $p$-restrictions on boolean functions that can be expressed as decision trees in which every vertex is at a distance of at most $t$ from some leaf, also called $t$-clipped decision trees. More specifically, we show the following: $\bullet$ If a boolean function $f$ can be expressed as a $t$-clipped decision tree, then under the action of a random $p$-restriction $\rho$, the probability that the smallest depth decision tree for $f|_{\rho}$ has depth greater than $d$ is upper bounded by $(4p2^{t})^{d}$. $\bullet$ For every $t$, there exists a function $g_{t}$ that can be expressed as a $t$-clipped decision tree, such that under the action of a random $p$-restriction $\rho$, the probability that the smallest depth decision tree for $g_{t}|_{\rho}$ has depth greater than $d$ is lower bounded by $(c_{0}p2^{t})^{d}$, for $0\leq p\leq c_{p}2^{-t}$ and $0\leq d\leq c_{d}\frac{\log n}{2^{t}\log t}$, where $c_{0},c_{p},c_{d}$ are universal constants

Edge Expansion and Spectral Gap of Nonnegative Matrices

The classic graphical Cheeger inequalities state that if $M$ is an $n\times n$ symmetric doubly stochastic matrix, then $$\frac{1-\lambda_{2}(M)}{2}\leq\phi(M)\leq\sqrt{2\cdot(1-\lambda_{2}(M))}$$ where $\phi(M)=\min_{S\subseteq[n],|S|\leq n/2}\left(\frac{1}{|S|}\sum_{i\in S,j\not\in S}M_{i,j}\right)$ is the edge expansion of $M$, and $\lambda_{2}(M)$ is the second largest eigenvalue of $M$. We study the relationship between $\phi(A)$ and the spectral gap $1-\text{Re}\lambda_{2}(A)$ for any doubly stochastic matrix $A$ (not necessarily symmetric), where $\lambda_{2}(A)$ is a nontrivial eigenvalue of $A$ with maximum real part. Fiedler showed that the upper bound on $\phi(A)$ is unaffected, i.e., $\phi(A)\leq\sqrt{2\cdot(1-\text{Re}\lambda_{2}(A))}$. With regards to the lower bound on $\phi(A)$, there are known constructions with $$\phi(A)\in\Theta\left(\frac{1-\text{Re}\lambda_{2}(A)}{\log n}\right),$$ indicating that at least a mild dependence on $n$ is necessary to lower bound $\phi(A)$. In our first result, we provide an exponentially better construction of $n\times n$ doubly stochastic matrices $A_{n}$, for which $$\phi(A_{n})\leq\frac{1-\text{Re}\lambda_{2}(A_{n})}{\sqrt{n}}.$$ In fact, all nontrivial eigenvalues of our matrices are $0$, even though the matrices are highly nonexpanding. We further show that this bound is in the correct range (up to the exponent of $n$), by showing that for any doubly stochastic matrix $A$, $$\phi(A)\geq\frac{1-\text{Re}\lambda_{2}(A)}{35\cdot n}.$$ Our second result extends these bounds to general nonnegative matrices $R$, obtaining a two-sided quantitative refinement of the Perron-Frobenius theorem in which the edge expansion $\phi(R)$ (appropriately defined), a quantitative measure of the irreducibility of $R$, controls the gap between the Perron-Frobenius eigenvalue and the next-largest real part of any eigenvalue

Edge Expansion and Spectral Gap of Nonnegative Matrices

The classic graphical Cheeger inequalities state that if M is an n × n symmetric doubly stochastic matrix, then 1−λ₂(M)/2 ≤ ϕ(M) ≤ √2⋅(1−λ₂(M)) where ϕ(M) = min_(S⊆[n],|S|≤n/2)(1|S|∑_(i∈S,j∉S)M_(i,j)) is the edge expansion of M, and λ₂(M) is the second largest eigenvalue of M. We study the relationship between φ(A) and the spectral gap 1 – Re λ₂(A) for any doubly stochastic matrix A (not necessarily symmetric), where λ₂(A) is a nontrivial eigenvalue of A with maximum real part. Fiedler showed that the upper bound on φ(A) is unaffected, i.e., ϕ(A) ≤ √2⋅(1−Reλ₂(A)). With regards to the lower bound on φ(A), there are known constructions with ϕ(A) ∈ Θ(1−Reλ₂(A)/log n) indicating that at least a mild dependence on n is necessary to lower bound φ(A). In our first result, we provide an exponentially better construction of n × n doubly stochastic matrices A_n, for which ϕ(An) ≤ 1−Reλ₂(A_n)/√n. In fact, all nontrivial eigenvalues of our matrices are 0, even though the matrices are highly nonexpanding. We further show that this bound is in the correct range (up to the exponent of n), by showing that for any doubly stochastic matrix A, ϕ(A) ≥ 1−Reλ₂(A)/35⋅n As a consequence, unlike the symmetric case, there is a (necessary) loss of a factor of n^α for ½ ≤ α ≤ 1 in lower bounding φ by the spectral gap in the nonsymmetric setting. Our second result extends these bounds to general matrices R with nonnegative entries, to obtain a two-sided gapped refinement of the Perron-Frobenius theorem. Recall from the Perron-Frobenius theorem that for such R, there is a nonnegative eigenvalue r such that all eigenvalues of R lie within the closed disk of radius r about 0. Further, if R is irreducible, which means φ(R) > 0 (for suitably defined φ), then r is positive and all other eigenvalues lie within the open disk, so (with eigenvalues sorted by real part), Re λ₂(R) < r. An extension of Fiedler's result provides an upper bound and our result provides the corresponding lower bound on φ(R) in terms of r – Re λ₂(R), obtaining a two-sided quantitative version of the Perron-Frobenius theorem

Behavior of O(log n) Local Commuting Hamiltonians

We study the variant of the k-local hamiltonian problem which is a natural generalization of k-CSPs, in which the hamiltonian terms all commute. More specifically, we consider a hamiltonian H over n qubits, where H is a sum of k-local terms acting non-trivially on O(log n) qubits, and all the k-local terms commute, and show the following - 1. We show that a specific case of O(log n) local commuting hamiltonians over the hypercube is in NP using the Bravyi-Vyalyi Structure theorem. 2. We give a simple proof of a generalized area law for commuting hamiltonians (which seems to be a folklore result) in all dimensions, and deduce the case for O(log n) local commuting hamiltonians. 3. We show that traversing the ground space of O(log n) local commuting hamiltonians is QCMA complete. The first two behaviours seem to indicate that deciding whether the ground space energy of O(log n)-local commuting hamiltonians is low or high might be in NP or possibly QCMA, though the last behaviour seems to indicate that it may indeed be the case that O(log n)-local commuting hamiltonians are QMA complete. </p

QCMA hardness of ground space connectivity for commuting Hamiltonians

In this work we consider the ground space connectivity problem for commuting local Hamiltonians. The ground space connectivity problem asks whether it is possible to go from one (efficiently preparable) state to another by applying a polynomial length sequence of 2-qubit unitaries while remaining at all times in a state with low energy for a given Hamiltonian $H$. It was shown in [Gharibian and Sikora, ICALP15] that this problem is QCMA-complete for general local Hamiltonians, where QCMA is defined as QMA with a classical witness and BQP verifier. Here we show that the commuting version of the problem is also QCMA-complete. This provides one of the first examples where commuting local Hamiltonians exhibit complexity theoretic hardness equivalent to general local Hamiltonians