Why Philosophers Should Care About Computational Complexity

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory---the field that studies the resources (such as time, space, and randomness) needed to solve computational problems---leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume's problem of induction, Goodman's grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and beyond," MIT Press, 2012. Some minor clarifications and corrections; new references adde

Testing product states, quantum Merlin-Arthur games and tensor optimisation

We give a test that can distinguish efficiently between product states of n quantum systems and states which are far from product. If applied to a state psi whose maximum overlap with a product state is 1-epsilon, the test passes with probability 1-Theta(epsilon), regardless of n or the local dimensions of the individual systems. The test uses two copies of psi. We prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarising channel. A key application of the test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain several structural results that had been previously conjectured, including the fact that efficient soundness amplification is possible and that two Merlins can simulate many Merlins: QMA(k)=QMA(2) for k>=2. Building on a previous result of Aaronson et al, this implies that there is an efficient quantum algorithm to verify 3-SAT with constant soundness, given two unentangled proofs of O(sqrt(n) polylog(n)) qubits. We also show how QMA(2) with log-sized proofs is equivalent to a large number of problems, some related to quantum information (such as testing separability of mixed states) as well as problems without any apparent connection to quantum mechanics (such as computing injective tensor norms of 3-index tensors). As a consequence, we obtain many hardness-of-approximation results, as well as potential algorithmic applications of methods for approximating QMA(2) acceptance probabilities. Finally, our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalisation of classical linearity testing.Comment: 44 pages, 1 figure, 7 appendices; v6: added references, rearranged sections, added discussion of connections to classical CS. Final version to appear in J of the AC

Impossibility of Succinct Quantum Proofs for Collision-Freeness

We show that any quantum algorithm to decide whether a function f:\left[n\right] \rightarrow\left[ n\right] is a permutation or far from a permutation\ must make \Omega\left( n^{1/3}/w\right) queries to f, even if the algorithm is given a w-qubit quantum witness in support of f being a permutation. This implies that there exists an oracle A such that \mathsfSZKA\mathsfQMAA , answering an eight-year-old open question of the author. Indeed, we show that relative to some oracle, \mathsfSZK is not in the counting class \mathsfA\mathsf0\mathsfPP defined by Vyalyi. The proof is a fairly simple extension of the quantum lower bound for the collision problem..National Science Foundation (U.S.) (grant 0844626)United States. Defense Advanced Research Projects Agency (YFA grant

Computational Complexity and Graph Isomorphism

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic, that is, structurally the same. The complexity of graph isomorphism is an open problem and it is one of the few problems in NP which is neither known to be solvable in polynomial time nor NP-complete. It is one of the most researched open problems in theoretical computer science. The foundations of computability theory are in recursion theory and in recursive functions which are an older model of computation than Turing machines. In this master’s thesis we discuss the basics of the recursion theory and the main theorems starting from the axioms. The aim of the second chapter is to define the most important T- and m-reductions and the implication hierarchy between reductions. Different variations of Turing machines include the nondeterministic and oracle Turing machines. They are discussed in the third chapter. A hierarchy of different complexity classes can be created by reducing the available computational resources of recursive functions. The members of this hierarchy include for instance P and NP. There are hundreds of known complexity classes and in this work the most important ones regarding graph isomorphism are introduced. Boolean circuits are a different method for approaching computability. Some main results and complexity classes of circuit complexity are discussed in the fourth chapter. The aim is to show that graph isomorphism is hard for the class DET. Graph isomorphism is known to belong to the classes coAM and SPP. These classes are introduced in the fifth chapter by using theory of probabilistic classes, polynomial hierarchy, interactive proof systems and Arthur-Merlin games. Polynomial hierarchy collapses to its second level if GI is NP-complete

Quantum Proofs

Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class \class{NP}, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class \class{QSZK}, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.Comment: Survey published by NOW publisher