Classical and quantum satisfiability

Eprint Arxiv Quant Ph 0210077, 2002

We describe Kitaev's result from 1999, in which he defines the complexity class QMA, the quantum analog of the class NP, and shows that a natural extension of 3−SAT, namely local Hamiltonians, is QMA complete. The result builds upon the classical Cook-Levin proof of the NP completeness of SAT , but differs from it in several fundamental ways, which we highlight. This result raises a rich array of open problems related to quantum complexity, algorithms and entanglement, which we state at the end of this survey. This survey is the extension of lecture notes taken by Naveh for Aharonov's quantum computation course, held in Tel Aviv University, 2001.

Physical Review A, 2005

A quantum algorithm is proposed to solve the Satisfiability problems by the ground-state quantum computer. The scale of the energy gap of the ground-state quantum computer is analyzed for the 3bit Exact Cover problem. The time cost of this algorithm on the general SAT problems is discussed.

New Journal of Physics

We present a quantum adiabatic algorithm to differentiate between k-SAT instances, those with no solutions and those that have many solutions. The time complexity of the algorithm is a function of the energy gap between the subspace of all 0-eigenvectors (ground states) and the first excited states manifold, and scales polynomially with the number of resources. The idea of gaps between subspaces suggests a new tool to analyze time complexity in adiabatic quantum machines. Contents 1. Introduction and motivations 2 2. The quantum adiabatic machine 3 2.

Among the NP-hard problems is 3SAT [2, 4], whichasks if there exists a satisfying assignment for theBoolean variables in a conjunctive normal form formula.This paper gives a mapping of 3SAT problemsinto multi-dimensional definite integrals. These integralstake on values greater than 1-# if the Booleanformula is satisfiable and less than 1/2 when it is not.They thus serve as e#ective indicators of the satisfiabilityof Boolean formulas. The number of dimensionsof the integral is the...

Entropy

In this paper, we proposed a novel quantum algorithm for the maximum satisfiability problem. Satisfiability (SAT) is to find the set of assignment values of input variables for the given Boolean function that evaluates this function as TRUE or prove that such satisfying values do not exist. For a POS SAT problem, we proposed a novel quantum algorithm for the maximum satisfiability (MAX-SAT), which returns the maximum number of OR terms that are satisfied for the SAT-unsatisfiable function, providing us with information on how far the given Boolean function is from the SAT satisfaction. We used Grover’s algorithm with a new block called quantum counter in the oracle circuit. The proposed circuit can be adapted for various forms of satisfiability expressions and several satisfiability-like problems. Using the quantum counter and mirrors for SAT terms reduces the need for ancilla qubits and realizes a large Toffoli gate that is then not needed. Our circuit reduces the number of ancilla...

An XSAT 𝜙 is satisfiable iff there exists a satisfying assignment 𝛼 to 𝜙. This paper shows that 𝜙 is satisfiable iff 𝜙 is reducible to 𝛼, viz., 𝜙 → 𝛼, thus 𝜙 ∧ 𝛼 is true, where 𝛼 = 𝑟 1 ∧ 𝑟 2 ∧ • • • ∧ 𝑟 𝑛 for some 𝑟 𝑙 ∈ {𝑥 𝑙 , 𝑥 𝑙 }. That is, the formula 𝜙 is unsatisfiable if it reduces to a simple formula 𝜓 inconsistent, otherwise 𝜙 → 𝜓 ∧𝜙 ′ such that 𝜙 ′ → 𝛼 ′ , and that 𝛼 = 𝜓 ∧ 𝛼 ′. Also, 𝛼 ′ = 𝜓 0 ∧ • • • ∧ 𝜓 𝑛 ′ and 𝜓 0 = (𝑟 ℓ 01 ∧ • • • ∧ 𝑟 ℓ 0 ñ),. .. ,𝜓 𝑛 ′ = (𝑟 ℓ 𝑛 ′ 1 ∧ • • • ∧ 𝑟 ℓ 𝑛 ′ ñ) such that {ℓ 01 , .

Lecture Notes in Computer Science, 2015

It has been known for almost three decades that many NP-hard optimization problems can be solved in polynomial time when restricted to structures of constant treewidth. In this work we provide the first extension of such results to the quantum setting. We show that given a quantum circuit C with n uninitialized inputs, poly(n) gates, and treewidth t, one can compute in time (n δ) exp(O(t)) a classical assignment y ∈ {0, 1} n that maximizes the acceptance probability of C up to a δ additive factor. In particular, our algorithm runs in polynomial time if t is constant and 1/poly(n) < δ < 1. For unrestricted values of t, this problem is known to be complete for the complexity class QCMA, a quantum generalization of MA. In contrast, we show that the same problem is NP-complete if t = O(log n) even when δ is constant. On the other hand, we show that given a n-input quantum circuit C of treewidth t = O(log n), and a constant δ < 1/2, it is QMA-complete to determine whether there exists a quantum state |ϕ ∈ (C d) ⊗n such that the acceptance probability of C|ϕ is greater than 1 − δ, or whether for every such state |ϕ , the acceptance probability of C|ϕ is less than δ. As a consequence, under the widely believed assumption that QMA = NP, we have that quantum witnesses are strictly more powerful than classical witnesses with respect to Merlin-Arthur protocols in which the verifier is a quantum circuit of logarithmic treewidth.

Information Processing Letters, 1988

2014

We introduce a probabilistic modal (dynamic-epistemic) quantum logic PLQP for reasoning about quantum algorithms. We illustrate its expressivity by using it to encode the correctness of the well-known quantum search algorithm, as well as of a quantum protocol known to solve one of the paradigmatic tasks from classical distributed computing (the leader election problem). We also provide a general method (extending an idea employed in the decidability proof in [12]) for proving the decidability of a range of quantum logics, interpreted on finite-dimensional Hilbert spaces. We give general conditions for the applicability of this method, and in particular we apply it to prove the decidability of PLQP.

… of Computer Science, 1999. 40th Annual …, 1999

We show that non-deterministic time N T IM E(n) is not contained in deterministic time n 2−ǫ and poly-logarithmic space, for any ǫ > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n 2−ǫ ) and polylogarithmic space. A similar result is presented for uniform circuits.