Classical Polynomial-time Research Articles

On the Need for Large Quantum Depth

Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “ Any quantum polynomial-time algorithm can be implemented with only O (log n ) quantum depth interspersed with polynomial-time classical computations. ” This can be formalized as asserting the equivalence of BQP and “ BQNC BPP .” However, Aaronson conjectured that “ there exists an oracle separation between BQP and BPP BQNC . ” BQNC BPP and BPP BQNC are two natural and seemingly incomparable ways of hybrid classical-quantum computation. In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter d , there exists an oracle that separates quantum depth d and 2 d +1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.

Cryptanalysis of FRS obfuscation based on the CLT13 multilinear map

The authors present a classical polynomial-time attack against the branching program obfuscator of Fernando–Rasmussen–Sahai (for short FRS, Asiacrypt’17) (with one zerotest parameter), which is robust against all known classical cryptanalyses on obfuscators when instantiated with the CLT13 multilinear map. The first step is to recover a plaintext modulus of the CLT13 multilinear map. To achieve the goal, the Coron and Notarnicola (Asiacrypt’19) algorithm is applied. However, because of parameter issues, the algorithm cannot be used directly. In order to detour the issue, the authors convert an FRS obfuscator into a new programme containing a small message space. Through the conversion, the authors obtain two zerotest parameters and encodings of zero except for two non-zero slots. Then, they are used to mitigate parameter constraints of the message space recovering algorithm. Then, a cryptanalysis of the FRS obfuscation based on the recovered message space is proposed. The authors show that there exist two functionally equivalent programmes such that their obfuscated programmes are computationally distinguishable. Thus, the FRS scheme does not satisfy the desired security without any additional constraints.

Complexity-theoretic aspects of expanding cellular automata

The expanding cellular automata (XCA) variant of cellular automata is investigated and characterized from a complexity-theoretical standpoint. An XCA is a one-dimensional cellular automaton which can dynamically create new cells between existing ones. The respective polynomial-time complexity class is shown to coincide with {le _{tt}^p}(textsf {NP}), that is, the class of decision problems polynomial-time truth-table reducible to problems in textsf {NP}. An alternative characterization based on a variant of non-deterministic Turing machines is also given. In addition, corollaries on select XCA variants are proven: XCAs with multiple accept and reject states are shown to be polynomial-time equivalent to the original XCA model. Finally, XCAs with alternative acceptance conditions are considered and classified in terms of {le _{tt}^p}(textsf {NP}) and the Turing machine polynomial-time class textsf {P}.

Multi-Objective Task Scheduling Using Hybrid Genetic-Ant Colony Optimization Algorithm in Cloud Environment

Task allocation within the cloud computing environment is a nondeterministic polynomial time class problem that is laborious to get the best solution. It is an important issue in the cloud computing setting. The usage of cloud based applications and cloud users are increasing tremendously. In order to handle the massive cloud user’s requests, effective multi-objective Hybrid Genetic Algorithm–Ant Colony Optimization (HGA–ACO) based task allocation technique is proposed in this paper. Utility based scheduler identifies the task order and suitable resources to be scheduled. The proposed HGA–ACO considers the utility based scheduler output and finds the best task allocation method based on response time, completion time and throughput. The HGA–ACO algorithm combines Genetic and Ant Colony Optimization algorithms together. Genetic algorithm (GA) initializes the effective pheromone for ant colony optimization (ACO). ACO is used to enhance the GA solutions for crossover operation of GA. The experimental results show that the proposed framework has better performance in task allocation and ensuring quality of service parameters.

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.

Commuting quantum circuits with few outputs are unlikely to be classically simulatable

We study the classical simulatability of commuting quantum circuits with n input qubits and O(log n) output qubits, where a quantum circuit is classically simulatable if its output probability distribution can be sampled up to an exponentially small additive error in classical polynomial time. Our main result is that there exists a commuting quantum circuit that is not classically simulatable unless the polynomial hierarchy collapses to the third level. This is the first formal evidence that a commuting quantum circuit is not classically simulatable even when the number of output qubits is O(log n). Then, we consider a generalized version of the circuit and clarify the condition under which it is classically simulatable. Lastly, using a proof similar to that of the main result, we provide an evidence that a slightly extended Clifford circuit is not classically simulatable.

BosonSampling is far from uniform

BosonSampling, which we proposed three years ago, is a scheme for using linear-optical networks to solve sampling problems that appear to be intractable for a classical computer. \ In a recent manuscript, Gogolin et al.\ claimed that even an ideal BosonSampling device's output would be operationally indistinguishable\textquotedblright\ from a uniform random outcome, at least \textquotedblleft without detailed a priori knowledge; or at any rate, that telling the two apart might itself be a hard problem. We first answer these claims---explaining why the first is based on a definition of a priori knowledge such that, were it adopted, almost no quantum algorithm could be distinguished from a pure random-number source; while the second is neither new nor a practical obstacle to interesting BosonSampling experiments.However, we then go further, and address some interesting research questions inspired by Gogolin et al.'s arguments. We prove that, with high probability over a Haar-random matrix $A$, the BosonSampling distribution induced by $A$ is far from the uniform distribution in total variation distance. More surprisingly, and counter to Gogolin et al., we give an efficient algorithm that distinguishes these two distributions with constant bias. Finally, we offer three bonus results about BosonSampling. First, we report an observation of Fernando Brandao: that one can efficiently sample a distribution that has large entropy and that's indistinguishable from a BosonSampling distribution by any circuit of fixed polynomial size. Second, we show that BosonSampling distributions can be efficiently distinguished from uniform even with photon losses and for general initial states. Third, we offer the simplest known proof that Fermion Sampling is solvable in classical polynomial time, and we reuse techniques from our Boson Sampling analysis to characterize random FermionSampling distributions.

The Complexity of the Gapped Consecutive-Ones Property Problem for Matrices of Bounded Maximum Degree

The Gapped Consecutive-Ones Property (C1P) Problem, or the (k, δ)-C1P Problem is: given a binary matrix M and integers k and δ, decide if the columns of M can be ordered such that each row contains at most k blocks of 1's, and no two neighboring blocks of 1's are separated by a gap of more than δ 0's. This problem was introduced by Chauve et al. ( 2009b ). The classical polynomial-time solvable C1P Problem is equivalent to the (1, 0)-C1P problem. It has been shown that, for every unbounded or bounded k ≥ 2 and unbounded or bounded δ ≥ 1, except when (k, δ) = (2, 1), the (k, δ)-C1P Problem is NP-complete (Maňuch et al., 2011 ; Goldberg et al., 1995 ). In this article, we study the Gapped C1P Problem with a third parameter d, namely the bound on the maximum number of 1's in any row of M, or the bound on the maximum degree of M. This is motivated by the reconstruction of ancestral genomes (Ma et al., 2006 ; Chauve and Tannier, 2008 ), where, in binary matrices obtained from the experiments of Chauve and Tannier ( 2008 ), we have observed that the majority of the rows have low degree, while each high degree row contains many rows of low degree. The (d, k, δ)-C1P Problem has been shown to be polynomial-time solvable when all three parameters are fixed (Chauve et al., 2009b ). Since fixing d also fixes k (k ≤ d), the only case left to consider is the case when δ is unbounded, or the (d, k, ∞)-C1P Problem. Here we show that for every d > k ≥ 2, the (d, k, ∞)-C1P Problem is NP-complete.

ABS methods for continuous and integer linear equations and optimization

ABS methods are a large class of algorithms for solving continuous and integer linear algebraic equations, and nonlinear continuous algebraic equations, with applications to optimization. Recent work by Chinese researchers led by Zunquan Xia has extended these methods also to stochastic, fuzzy and infinite systems, extensions not considered here. The work on ABS methods began almost thirty years. It involved an international collaboration of mathematicians especially from Hungary, England, China and Iran, coordinated by the university of Bergamo. The ABS method are based on the rank reducing matrix update due to Egervary and can be considered as the most fruitful extension of such technique. They have led to unification of classes of methods for several problems. Moreover they have produced some special algorithms with better complexity than the standard methods. For the linear integer case they have provided the most general polynomial time class of algorithms so far known; such algorithms have been extended to other integer problems, as linear inequalities and LP problems, in over a dozen papers written by Iranian mathematicians led by Nezam Mahdavi-Amiri. ABS methods can be implemented generally in a stable way, techniques existing to enhance their accuracy. Extensive numerical experiments have shown that they can outperform standard methods in several problems. Here we provide a review of their main properties, for linear systems and optimization. We also give the results of numerical experiments on some linear systems. This paper is dedicated to Professor Egervary, developer of the rank reducing matrix update, that led to ABS methods.

On the power of randomized multicounter machines

One-way two-counter machines represent a universal model of computation. Here we consider the polynomial-time classes of multicounter machines with a constant number of reversals and separate the computational power of nondeterminism, randomization and determinism. For instance, we show that polynomial-time one-way multicounter machines, with error probability tending to zero with growing input length, can recognize languages that cannot be accepted by polynomial-time nondeterministic two-way multicounter machines with a bounded number of reversals. A similar result holds for the comparison of determinism and one-sided-error randomization, and of determinism and Las Vegas randomization.

Computing the distribution of spearman's footrule in O ( n^4 ) time

Consider the exact distribution of Spearman's "footrule", D(\\sigma, \\pi) = \\sum_{i=1}^n \\vert \\sigma (i) - \\pi (i)\\vert , where \\sigma and \\pi are chosen independently at random from S_n , the set of all permutations of the first n integers; or, equivalently, where \\sigma is chosen at random and \\pi = (1,2,\\ldots,n) . This can obviously be obtained by complete enumeration of all n ! permutations: that is, in O(n!) time. However, we show that the intrinsically different approach used by Salama and Quade (1990) reduces the computation to the polynomial-time class, specifically, to O(n^4) time.

Relating polynomial time to constant depth

Going back to the seminal paper of Furst, Saxe, and Sipser (1984), analogues between polynomial time classes and constant depth circuit classes have been considered in a number of papers. Oracles separating polynomial time classes have been obtained by diagonalization making essential use of lower bounds for circuit classes. In this note we show how separating oracles can be obtained uniformly from circuit lower bounds without the need of carrying out a particular diagonalization. Our technical tool is the leaf language approach to the definition of complexity classes.

Polynomial time–frequency distributions and time-varying higher order spectra: Application to the analysis of multicomponent FM signals and to the treatment of multiplicative noise

One of the fundamental problems of time–frequency analysis that remained unsolved until recently is the time–frequency representation of signals with an arbitrary non-linear frequency variation in time. A certain type of higher-order time–frequency distributions (TFDs), referred to as polynomial Wigner-Ville distributions (PWVDs), can achieve delta function concentration in the time–frequency plane for mono-component polynomial FM signals (Boashash and O’Shea, 1994). This paper is a sequel to Boashash and O’Shea (1994) dealing with a broader class of signals and presenting an application to the treatment of multiplicative noise. The first part of the paper presents a general design procedure for PWVDs and the main properties of PWVDs. A specific class of polynomial time–frequency distributions that deals effectively with multicomponent signals is then described. In the second part of the paper we deal with random signals and introduce time-varying higher-order spectra (TV-HOS) an ensemble averaged PWVDs. TV-HOS are shown to be natural tools for analysis of non-stationary random signals, and we demonstrate this in the context of FM signals affected by multiplicative noise. Both moment and cumulant fourth-order TV-HOS, referred to as the Wigner-Ville trispectra are shown to be superior to the second-order methods for this application.

Computational Power of Quantum Machines, Quantum Grammars and Feasible Computation

This paper studies the computational power of quantum computers to explore as to whether they can recognize properties which are in nondeterministic polynomial-time class (NP) and beyond. To study the computational power, we use the Feynman's path integral (FPI) formulation of quantum mechanics. From a computational point of view the Feynman's path integral computes a quantum dynamical analogue of the k-ary relation computed by an Alternating Turing machine (ATM) using AND-OR Parallelism. Hence, if we can find a suitable mapping function between an instance of a mathematical problem and the corresponding interference problem, using suitable potential functions for which FPI can be integrated exactly, the computational power of a quantum computer can be bounded to that of an alternating Turing machine that can solve problems in NP (e.g, factorization problem) and in polynomial space. Unfortunately, FPI is exactly integrable only for a few problems (e.g., the harmonic oscillator) involving quadratic potentials; otherwise, they may be only approximately computable or noncomputable. This means we cannot in general solve all quantum dynamical problems exactly except for those special cases of quadratic potentials, e.g., harmonic oscillator. Since there is a one to one correspondence between the quantum mechanical problems that can be analytically solved and the path integrals that can be exactly evaluated, we can say that the noncomputability of FPI implies quantum unsolvability. This is the analogue of classical unsolvability. The Feynman's path graph can be considered as a semantic parse graph for the quantum mechanical sentence. It provides a semantic valuation function of the terminal sentence based on probability amplitudes to disambiguate a given quantum description and obtain an interpretation in a linear time. In Feynman's path integral, the kernels are partially ordered over time (different alternate paths acting concurrently at the same time) and multiplied. The semantic valuation is computable only if the FPI is computable. Thus both the expressive power and complexity aspects quantum computing are mirrored by the exact and efficient integrability of FPI.

Genericity, Randomness, and Polynomial-Time Approximations

Polynomial-time safe and unsafe approximations for intractable sets were introduced by Meyer and Paterson [Technical Report TM-126, Laboratory for Computer Science, MIT, Cambridge, MA, 1979] and Yesha [SIAM J. Comput., 12 (1983), pp. 411--425], respectively. The question of which sets have optimal safe and unsafe approximations has been investigated extensively. Duris and Rolim [Lecture Notes in Comput. Sci. 841, Springer-Verlag, Berlin, New York, 1994, pp. 38--51] and Ambos-Spies [Proc. 22nd ICALP, Springer-Verlag, Berlin, New York, 1995, pp. 384--392] showed that the existence of optimal polynomial-time approximations for the safe and unsafe cases is independent. Using the law of the iterated logarithm for p-random sequences (which has been recently proven in [Proc. 11th Conf. Computational Complexity, IEEE Computer Society Press, Piscataway, NJ, 1996, pp. 180--189]), we extend this observation by showing that both the class of polynomial-time $\Delta$-levelable sets and the class of sets which have optimal polynomial-time unsafe approximations have p-measure 0. Hence typical sets in E (in the sense of p-measure) do not have optimal polynomial-time unsafe approximations. We will also establish the relationship between resource bounded genericity concepts and the polynomial-time safe and unsafe approximation concepts.

Defying Upward and Downward Separation

"Downward separation" results show that when small classes collapse, larger ones also collapse. For example, Stockmeyer proved that if P=NP, then the polynomial hierarchy collapses to P, and this result itself holds in every relativized world. In contrast, we construct a relativized world in which the exponential-time limited nondeterminism hierarchy does not display such behavior: its tower levels collapse yet its upper levels separate. "Upward separation" results typically show that polynomial-time classes differ on sparse or tally sets if and only if their exponential analogs differ. For example, Hartmanis, Immerman, and Sewelson proved that NP-P contains sparse sets if and only if E ≠ NE, and this result itself holds in every relativized world. In contrast, we construct relativized worlds in which probabilistic classes do not display upward separation, e.g., a world A in which BPP A -P A contains sparse sets even though BPE A = E A . We also construct a relativized world B in which NP B has P B -immune sparse sets yet NE B is not E B -immune. On the other hand, we provide a structural sufficient condition for upward separation.

Decomposed software pipelining: A new perspective and a new approach

Software pipelining is an efficient instruction-level loop scheduling technique, but existing software pipelining approaches have not been widely used in practical and commercial compilers. This is mainly because resource constraints and the cyclic data dependencies make software pipelining very complicated and difficult to apply. In this paper we present a new perspective on software pipelining in which it is decomposed into two subproblems—one is free from cyclic data dependencies and can be effectively solved by the list scheduling technique, and the other is free from resource constraints and can be easily solved by classical polynomial-time algorithms of graph theory. Based on this new perspective, we develop a new instruction-level loop scheduling approach, call DEcomposed Software Pipelining (DESP).

On the polynomial IO-complexity

In connection with the notion of approximation languages the IO-complexity has been defined. We extend this concept to include polynomial time and we study the structure of deterministic and non-deterministic polynomial time classes. We restrict our attention to the uniform probability distribution

The logarithmic alternation hierarchy collapses: AΣ2L = AΠ2L

We show that AΣ 2 L = AΣ k L , k ≥ 2, by proving that AΣ 2 L coincides with AΠ 2 L . Essentially this is done by reducing the AΣ 2 L -complete set (GAP¢CoGap) (ℶ) to the question whether of two vectors A and B of n components, A contains more “solvable” components, i.e., components which are contained in GAP, than B . Moreover, using a similar technique we show AΣ 2 L = L hd (NL) . Finally, we consider the relevance of our proof technique for polynomial time classes, e.g., the Boolean NP -Hierarchy.

Relativizations of Unambiguous and Random Polynomial Time Classes

We study relationships between ${\bf P}$, $NP$ , and the unambiguous and random time classes, ${\bf U}$, and ${\bf R}$. Questions concerning these relationships are motivated by complexity issues in public-key cryptosystems. We prove that there exists a recursive oracle A such that ${\bf P}^A \ne {\bf U}^A \ne {\bf NP}^A $, and such that the first inequality is “strong,” i.e., there exists a ${\bf P}^A $-immune set in ${\bf U}^A $. Further, we construct a recursive oracle B such that ${\bf U}^B $ contains an ${\bf R}^B $-immune set. As a corollary, we obtain ${\bf P}^B \ne {\bf R}^B \ne {\bf NP}^B $ and both inequalities are strong. By use of the techniques employed in the proof that ${\bf P}^A \ne {\bf U}^A \ne {\bf NP}^A $, we are also able to solve an open problem raised by Book, Long and Selman [Quantitative relativizations of complexity classes, SIAM J. Comput. 13 (1984), pp 461–487].