If one modifies the laws of Quantum Mechanics to allow nonlinear evolution of quantum states, this paper shows that NP-complete problems would be efficiently solvable in polynomial time with bounded probability (NP in BQP). With that (admittedly very unlikely) assumption, this is demonstrated by describing a polynomially large network of quantum gates that solves the 3SAT problem with bounded probability in polynomial time. As in a previous paper by Abrams and Lloyd (but by a somewhat simpler argument), allowing nonlinearity in the laws of Quantum Mechanics would prove the "weak Church-Turing thesis" to be false. General Relativity is suggested as a possible mechanism to supply the necessary nonlinearity.
The aim of the paper is to answer a long-standing open problem on the relationship between NP and BQP. The paper shows that BQP contains NP by proposing a BQP quantum algorithm for the MAX-E3-SAT problem which is a fundamental NP-hard problem. Given an E3-CNF Boolean formula, the aim of the MAX-E3-SAT problem is to find the variable assignment that maximizes the number of satisfied clauses. The proposed algorithm runs in
O(m2) for an E3-CNF Boolean formula with
m clauses and in the worst case runs in
O(n6) for an E3-CNF Boolean formula with
n inputs. The proposed algorithm maximizes the set of satisfied clauses using a novel iterative partial negation and partial measurement technique. The algorithm is shown to achieve an arbitrary high probability of success of
1−ε for small
ε>0 using a polynomial resources. In addition to solving the MAX-E3-SAT problem, the proposed algorithm can also be used to decide if an E3-CNF Boolean formula is satisfiable or not, which is an NP-complete problem, based on the maximum number of satisfied clauses.
Quantum computers are widely believed have an advantage over classical computers, and some have even published some empirical evidence that this is the case. However, these publications do not include a rigorous proof of this advantage, which would have to minimally state that the class of problems decidable by a quantum computer in polynomial time, BQP, contains problems that are not in the class of problems decidable by a classical computer with similar time bounds, P. Here, I provide the proof of a stronger result that implies this result: BQP contains problems that lie beyond the much larger classical computing class NP. This proves that quantum computation is able to efficiently solve problems which are far beyond the capabilities of classical computers.
What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires
Θ(n) queries to an NP oracle to compute a witness to a given SAT formula, quantumly
Θ(logn) queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires
Ω(logn) queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires
Ω(logn) queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making
o(logn) NP queries, then
BPPNP[o(n)] contains a
PNP-complete problem if the algorithm is classical and
FBQPNP[o(n)] contains an
FPNP-complete problem if the algorithm is quantum.
The complexity class
PSPACE includes all computational problems that can be solved by a classical computer with polynomial memory. All
PSPACE problems are known to be solvable by a quantum computer too with polynomial memory and are, thus, known to be in
BQPSPACE. Here, we present a polynomial time quantum algorithm for a
PSPACE-complete problem, implying that
PSPACE is equal to the class
BQP of all problems solvable by a quantum computer in polynomial time. In particular, we outline a
BQP algorithm for the
PSPACE-complete problem of evaluating a full binary
NAND tree. An existing best of quadratic speedup is achieved using quantum walks for this problem, so that the complexity is still exponential in the problem size. By contrast, we achieve an exponential speedup for the problem, allowing for solving it in polynomial time. There are many real-world applications of our result, such as strategy games like chess or Go. As an example, in quantum sensing, the problem of quantum illumination, that is treated as that of channel discrimination, is
PSPACE-complete. Our work implies that quantum channel discrimination, and so, quantum illumination, can be performed efficiently by a quantum computer.
Abstract. In this paper we study quantum computation from a complexity theoretic viewpoint. Our rst result is the existence of an ecient universal quantum Turing machine in Deutsch’s model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97{117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specied. We prove that O(log T) bits of precision suce to support a T step computation. This justies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the rst formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church{Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class BPP. The class BQP of languages that are eciently decidable (with small error-probability) on a quantum Turing machine satises BPP ⊆ BQP ⊆ P]P. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.