Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy

We introduce a simple sub-universal quantum computing model, which we call the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and therefore it is in the second level of the Fourier hierarchy. We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level. The proof technique is different from those used for previous sub-universal models, such as IQP, Boson Sampling, and DQC1, and therefore the technique itself might be useful for finding other sub-universal models that are hard to classically simulate. We also study the classical verification of quantum computing in the second level of the Fourier hierarchy. To this end, we define a promise problem, which we call the probability distribution distinguishability with maximum norm (PDD-Max). It is a promise problem to decide whether output probability distributions of two quantum circuits are far apart or close. We show that PDD-Max is BQP-complete, but if the two circuits are restricted to some types in the second level of the Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a Merlin-Arthur system with quantum polynomial-time Merlin and classical probabilistic polynomial-time Arthur.Comment: 30 pages, 4 figure

Small Circuits Imply Efficient Arthur-Merlin Protocols

The inner product function ? x,y ? = ?_i x_i y_i mod 2 can be easily computed by a (linear-size) AC?(?) circuit: that is, a constant depth circuit with AND, OR and parity (XOR) gates. But what if we impose the restriction that the parity gates can only be on the bottom most layer (closest to the input)? Namely, can the inner product function be computed by an AC? circuit composed with a single layer of parity gates? This seemingly simple question is an important open question at the frontier of circuit lower bound research. In this work, we focus on a minimalistic version of the above question. Namely, whether the inner product function cannot be approximated by a small DNF augmented with a single layer of parity gates. Our main result shows that the existence of such a circuit would have unexpected implications for interactive proofs, or more specifically, for interactive variants of the Data Streaming and Communication Complexity models. In particular, we show that the existence of such a small (i.e., polynomial-size) circuit yields: 1) An O(d)-message protocol in the Arthur-Merlin Data Streaming model for every n-variate, degree d polynomial (over GF(2)), using only O?(d) ?log(n) communication and space complexity. In particular, this gives an AM[2] Data Streaming protocol for a variant of the well-studied triangle counting problem, with poly-logarithmic communication and space complexities. 2) A 2-message communication complexity protocol for any sparse (or low degree) polynomial, and for any function computable by an AC?(?) circuit. Specifically, for the latter, we obtain a protocol with communication complexity that is poly-logarithmic in the size of the AC?(?) circuit

A Hierarchy Theorem for Interactive Proofs of Proximity

The number of rounds, or round complexity, used in an interactive protocol is a fundamental resource. In this work we consider the significance of round complexity in the context of Interactive Proofs of Proximity (IPPs). Roughly speaking, IPPs are interactive proofs in which the verifier runs in sublinear time and is only required to reject inputs that are far from the language. Our main result is a round hierarchy theorem for IPPs, showing that the power of IPPs grows with the number of rounds. More specifically, we show that there exists a gap function g(r) = Theta(r^2) such that for every constant r geq 1 there exists a language that (1) has a g(r)-round IPP with verification time t=t(n,r) but (2) does not have an r-round IPP with verification time t (or even verification time t\u27=poly(t)). In fact, we prove a stronger result by exhibiting a single language L such that, for every constant r geq 1, there is an O(r^2)-round IPP for L with t=n^{O(1/r)} verification time, whereas the verifier in any r-round IPP for L must run in time at least t^{100}. Moreover, we show an IPP for L with a poly-logarithmic number of rounds and only poly-logarithmic erification time, yielding a sub-exponential separation between the power of constant-round IPPs versus general (unbounded round) IPPs. From our hierarchy theorem we also derive implications to standard interactive proofs (in which the verifier can run in polynomial time). Specifically, we show that the round reduction technique of Babai and Moran (JCSS, 1988) is (almost) optimal among all blackbox transformations, and we show a connection to the algebrization framework of Aaronson and Wigderson (TOCT, 2009)

The Jones polynomial: quantum algorithms and applications in quantum complexity theory

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.Comment: 34 pages. Substantial revision. Increased emphasis on HOMFLYPT, greatly simplified arguments and improved organizatio

07411 Abstracts Collection -- Algebraic Methods in Computational Complexity

From 07.10. to 12.10., the Dagstuhl Seminar 07411 ``Algebraic Methods in Computational Complexity\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Fine-grained Complexity Meets IP = PSPACE

In this paper we study the fine-grained complexity of finding exact and approximate solutions to problems in P. Our main contribution is showing reductions from exact to approximate solution for a host of such problems. As one (notable) example, we show that the Closest-LCS-Pair problem (Given two sets of strings $A$ and $B$, compute exactly the maximum $\textsf{LCS}(a, b)$ with $(a, b) \in A \times B$) is equivalent to its approximation version (under near-linear time reductions, and with a constant approximation factor). More generally, we identify a class of problems, which we call BP-Pair-Class, comprising both exact and approximate solutions, and show that they are all equivalent under near-linear time reductions. Exploring this class and its properties, we also show: $\bullet$ Under the NC-SETH assumption (a significantly more relaxed assumption than SETH), solving any of the problems in this class requires essentially quadratic time. $\bullet$ Modest improvements on the running time of known algorithms (shaving log factors) would imply that NEXP is not in non-uniform $\textsf{NC}^1$. $\bullet$ Finally, we leverage our techniques to show new barriers for deterministic approximation algorithms for LCS. At the heart of these new results is a deep connection between interactive proof systems for bounded-space computations and the fine-grained complexity of exact and approximate solutions to problems in P. In particular, our results build on the proof techniques from the classical IP = PSPACE result