Linear time algorithm for quantum 2SAT

A canonical result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors $\Pi_{ij}$ on a system of $n$ qubits, and the task is to decide whether the Hamiltonian $H=\sum \Pi_{ij}$ has a 0-eigenvalue, or it is larger than $1/n^\alpha$ for some $\alpha=O(1)$. The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding the ground state of 2-local frustration-free Hamiltonians of spin $\frac{1}{2}$, a well-studied model believed to capture certain key properties in modern condensed matter physics. While Bravyi has shown that the quantum 2-SAT problem has a classical polynomial-time algorithm, the running time of his algorithm is $O(n^4)$. In this paper we give a classical algorithm with linear running time in the number of local projectors, therefore achieving the best possible complexity.Comment: 20 page

Quantum de Finetti Theorems under Local Measurements with Applications

Quantum de Finetti theorems are a useful tool in the study of correlations in quantum multipartite states. In this paper we prove two new quantum de Finetti theorems, both showing that under tests formed by local measurements one can get a much improved error dependence on the dimension of the subsystems. We also obtain similar results for non-signaling probability distributions. We give the following applications of the results: We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the exponential time hypothesis. We show that the maximum winning probability of free games can be estimated in polynomial time by linear programming. We also show that 3-SAT with m variables can be reduced to obtaining a constant error approximation of the maximum winning probability under entangled strategies of O(m^{1/2})-player one-round non-local games, in which the players communicate O(m^{1/2}) bits all together. We show that the optimization of certain polynomials over the hypersphere can be performed in quasipolynomial time in the number of variables n by considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy of semidefinite programs. As an application to entanglement theory, we find a quasipolynomial-time algorithm for deciding multipartite separability. We consider a result due to Aaronson -- showing that given an unknown n qubit state one can perform tomography that works well for most observables by measuring only O(n) independent and identically distributed (i.i.d.) copies of the state -- and relax the assumption of having i.i.d copies of the state to merely the ability to select subsystems at random from a quantum multipartite state. The proofs of the new quantum de Finetti theorems are based on information theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor improvements. v3: added some explanations, mostly about Theorem 1 and Conjecture 5. STOC version. v4, v5. small improvements and fixe

Quantum Advantage for All

We show that the algorithmic complexity of any classical algorithm written in a Turing-complete programming language polynomially bounds the number of quantum bits that are required to run and even symbolically execute the algorithm on a quantum computer. In particular, we show that any classical algorithm $A$ that runs in $\mathcal{O}(f(n))$ time and $\mathcal{O}(g(n))$ space requires no more than $\mathcal{O}(f(n)\cdot g(n))$ quantum bits to execute, even symbolically, on a quantum computer. With $\mathcal{O}(1)\leq\mathcal{O}(g(n))\leq\mathcal{O}(f(n))$ for all $n$, the quantum bits required to execute $A$ may therefore not exceed $\mathcal{O}(f(n)^2)$ and may come down to $\mathcal{O}(f(n))$ if memory consumption by $A$ is bounded by a constant. Our construction works by encoding symbolic execution of machine code in a finite state machine over the satisfiability-modulo-theory (SMT) of bitvectors, for modeling CPU registers, and arrays of bitvectors, for modeling main memory. The FSM is linear in the size of the code, independent of execution time and space, and represents the reachable machine states for any given input. The FSM may be explored by bounded model checkers using SMT and SAT solvers as backend. However, for the purpose of this paper, we focus on quantum computing by unrolling and bit-blasting the FSM into (1)~satisfiability-preserving quadratic unconstrained binary optimization (QUBO) models targeting adiabatic forms of quantum computing such as quantum annealing, and (2)~semantics-preserving quantum circuits (QCs) targeting gate-model quantum computers. With our compact QUBOs, real quantum annealers can now execute simple but real code even symbolically, yet only with potential but no guarantee for exponential speedup, and with our QCs as oracles, Grover's algorithm applies to symbolic execution of arbitrary code, guaranteeing at least in theory a quadratic speedup

A Quantum Version of Sch\"oning's Algorithm Applied to Quantum 2-SAT

We study a quantum algorithm that consists of a simple quantum Markov process, and we analyze its behavior on restricted versions of Quantum 2-SAT. We prove that the algorithm solves this decision problem with high probability for n qubits, L clauses, and promise gap c in time O(n^2 L^2 c^{-2}). If the Hamiltonian is additionally polynomially gapped, our algorithm efficiently produces a state that has high overlap with the satisfying subspace. The Markov process we study is a quantum analogue of Sch\"oning's probabilistic algorithm for k-SAT

On the complexity of probabilistic trials for hidden satisfiability problems

What is the minimum amount of information and time needed to solve 2SAT? When the instance is known, it can be solved in polynomial time, but is this also possible without knowing the instance? Bei, Chen and Zhang (STOC '13) considered a model where the input is accessed by proposing possible assignments to a special oracle. This oracle, on encountering some constraint unsatisfied by the proposal, returns only the constraint index. It turns out that, in this model, even 1SAT cannot be solved in polynomial time unless P=NP. Hence, we consider a model in which the input is accessed by proposing probability distributions over assignments to the variables. The oracle then returns the index of the constraint that is most likely to be violated by this distribution. We show that the information obtained this way is sufficient to solve 1SAT in polynomial time, even when the clauses can be repeated. For 2SAT, as long as there are no repeated clauses, in polynomial time we can even learn an equivalent formula for the hidden instance and hence also solve it. Furthermore, we extend these results to the quantum regime. We show that in this setting 1QSAT can be solved in polynomial time up to constant precision, and 2QSAT can be learnt in polynomial time up to inverse polynomial precision.Comment: 24 pages, 2 figures. To appear in the 41st International Symposium on Mathematical Foundations of Computer Scienc