We develop a polynomial method on finite fields to amplify the hardness of spare sets in nondeterministic time complexity classes on a randomized streaming model. One of our results shows that if there exists a $2^{n^{o(1)}}$-sparse set in $NTIME(2^{n^{o(1)}})$ that does not have any randomized streaming algorithm with $n^{o(1)}$ updating time, and $n^{o(1)}$ space, then $NEXP\not=BPP$, where a $f(n)$-sparse set is a language that has at most $f(n)$ strings of length $n$. We also show that if MCSP is $ZPP$-hard under polynomial time truth-table reductions, then $EXP\not=ZPP$.
In this paper, we study the computational complexity of the commutative determinant polynomial computed by a class of set-multilinear circuits which we call regular set-multilinear circuits. Regular set-multilinear circuits are commutative circuits with a restriction on the order in which they can compute polynomials. A regular circuit can be seen as the commutative analogue of the ordered circuit defined by Hrubes,Wigderson and Yehudayoff [HWY10]. We show that if the commutative determinant polynomial has small representation in the sum of constantly many regular set-multilinear circuits, the…
The problem of approximating a dense matrix by a product of sparse factors is a fundamental problem for many signal processing and machine learning tasks. It can be decomposed into two subproblems: finding the position of the non-zero coefficients in the sparse factors, and determining their values. While the first step is usually seen as the most challenging one due to its combinatorial nature, this paper focuses on the second step, referred to as sparse matrix approximation with fixed support. First, we show its NP-hardness, while also presenting a nontrivial family of supports making the pr…
We consider the hardness of computing additive approximations to output probabilities of random quantum circuits. We consider three random circuit families, namely, Haar random, $p=1$ QAOA, and random IQP circuits. Our results are as follows. For Haar random circuits with $m$ gates, we improve on prior results by showing $\mathsf{coC_=P}$ hardness of average-case additive approximations to an imprecision of $2^{-O(m)}$. Efficient classical simulation of such problems would imply the collapse of the polynomial hierarchy. For constant depth circuits i.e., when $m=O(n)$, this linear scaling in th…
We show that lower bounds for explicit constant-variate polynomials over fields of characteristic $p > 0$ are sufficient to derandomize polynomial identity testing over fields of characteristic $p$. In this setting, existing work on hardness-randomness tradeoffs for polynomial identity testing requires either the characteristic to be sufficiently large or the notion of hardness to be stronger than the standard syntactic notion of hardness used in algebraic complexity. Our results make no restriction on the characteristic of the field and use standard notions of hardness. We do this by combinin…
This paper aims to present an advanced version of PP-hardness proof of Minesweeper by Bondt. The advancement includes improved Minesweeper configurations for 'logic circuits' in a hexagonal Minesweeper. To do so, I demonstrate logical uncertainty in Minesweeper, which ironically allows a possibility to make some Boolean operators. The fact that existing hexagonal logic circuits did not clearly distinguish the true and false signal needs an improved form of a hexagonal wire. I introduce new forms of logic circuits such as NOT, AND, OR gates, a curve and a splitter of wires. Moreover, these new …
We say that two given polynomials $f, g \in R[X]$, over a ring $R$, are equivalent under shifts if there exists a vector $a \in R^n$ such that $f(X+a) = g(X)$. Grigoriev and Karpinski (FOCS 1990), Lakshman and Saunders (SICOMP, 1995), and Grigoriev and Lakshman (ISSAC 1995) studied the problem of testing polynomial equivalence of a given polynomial to any $t$-sparse polynomial, over the rational numbers, and gave exponential time algorithms. In this paper, we provide hardness results for this problem. Formally, for a ring $R$, let $\mathrm{SparseShift}_R$ be the following decision problem. Giv…
A critical milestone on the path to useful quantum computers is quantum supremacy - a demonstration of a quantum computation that is prohibitively hard for classical computers. A leading near-term candidate, put forth by the Google/UCSB team, is sampling from the probability distributions of randomly chosen quantum circuits, which we call Random Circuit Sampling (RCS). In this paper we study both the hardness and verification of RCS. While RCS was defined with experimental realization in mind, we show complexity theoretic evidence of hardness that is on par with the strongest theoretical propo…
Large-scale quantum information processing requires the use of quantum error correcting codes to mitigate the effects of noise in quantum devices. Topological error-correcting codes, such as surface codes, are promising candidates as they can be implemented using only local interactions in a two-dimensional array of physical qubits. Procedures such as defect braiding and lattice surgery can then be used to realize a fault-tolerant universal set of gates on the logical space of such topological codes. However, error correction also introduces a significant overhead in computation time, the numb…
Consider the regularized sparse minimization problem, which involves empirical sums of loss functions for $n$ data points (each of dimension $d$) and a nonconvex sparsity penalty. We prove that finding an $\mathcal{O}(n^{c_1}d^{c_2})$-optimal solution to the regularized sparse optimization problem is strongly NP-hard for any $c_1, c_2\in [0,1)$ such that $c_1+c_2<1$. The result applies to a broad class of loss functions and sparse penalty functions. It suggests that one cannot even approximately solve the sparse optimization problem in polynomial time, unless P $=$ NP.
Recently, Google announced the first demonstration of quantum computational supremacy with a programmable superconducting processor. Their demonstration is based on collecting samples from the output distribution of a noisy random quantum circuit, then applying a statistical test to those samples called Linear Cross-Entropy Benchmarking (Linear XEB). This raises a theoretical question: how hard is it for a classical computer to spoof the results of the Linear XEB test? In this short note, we adapt an analysis of Aaronson and Chen [2017] to prove a conditional hardness result for Linear XEB spo…
In neuroscience, an important aspect of understanding the function of a neural circuit is to determine which, if any, of the neurons in the circuit are vital for the biological behavior governed by the neural circuit. A similar problem is to determine whether a given small set of neurons may be enough for the behavior to be displayed, even if all other neurons in the circuit are deactivated. Such a subset of neurons forms what is called a degenerate circuit for the behavior being studied. Recent advances in experimental techniques have provided researchers with tools to activate and deactivate…
The Strong Exponential Time Hypothesis (SETH) asserts that for every $\varepsilon>0$ there exists $k$ such that $k$-SAT requires time $(2-\varepsilon)^n$. The field of fine-grained complexity has leveraged SETH to prove quite tight conditional lower bounds for dozens of problems in various domains and complexity classes, including Edit Distance, Graph Diameter, Hitting Set, Independent Set, and Orthogonal Vectors. Yet, it has been repeatedly asked in the literature whether SETH-hardness results can be proven for other fundamental problems such as Hamiltonian Path, Independent Set, Chromatic Nu…
A hitting-set generator (HSG) is a polynomial map $G:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $C$ of small enough circuit size and degree, if $C$ is nonzero, then $C\circ G$ is nonzero. In this paper, we give a new construction of such an HSG assuming that we have an explicit polynomial of sufficient hardness. Formally, we prove the following over any field of characteristic zero: Let $k\in \mathbb{N}$ and $δ> 0$ be arbitrary constants. Suppose $\{P_d\}_{d\in \mathbb{N}}$ is an explicit family of $k$-variate polynomials such that $\operatorname{deg} P_d = d$ and…
Noise is the defining feature of the NISQ era, but it remains unclear if noisy quantum devices are capable of quantum speedups. Quantum supremacy experiments have been a major step forward, but gaps remain between the theory behind these experiments and their actual implementations. In this work we initiate the study of the complexity of quantum random circuit sampling experiments with realistic amounts of noise. Actual quantum supremacy experiments have high levels of uncorrected noise and exponentially decaying fidelities. It is natural to ask if there is any signal of exponential complexity…
We consider the fine-grained complexity of sparse graph problems that currently have $\tilde{O}(mn)$ time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important path problems on both directed and undirected graphs, including APSP, MWC (minimum weight cycle), and Eccentricities, which is the problem of computing, for each vertex in the graph, the length of a longest shortest path starting at that vertex. We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-ti…
Center for Theoretical Physics, MIT（麻省理工学院理论物理中心）；CWI（CWI数学与计算机科学中心）；QuSoft（QuSoft量子软件研究中心）；Joint Center for Quantum Information and Computer Science, University of Maryland（马里兰大学量子信息与计算机科学联合中心）
In this paper we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of PAC learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results. 1) Hardness of learning AC$^0$ and TC$^0$ under the uniform distribution. Our first result concerns the concept class TC$^0$ (resp. AC$^0$), the class of constant-depth and polynomial-sized circuits with unbounded fan-in majority gates (resp. AND, OR, NOT gates). We…
We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in $\text{Quasi-NP} = \text{NTIME}[n^{(\log n)^{O(1)}}]$ and $\text{NEXP}$ do not have small circuits from various circuit classes ${\cal C}$, by showing that ${\cal C}$ admits non-trivial satisfiability and/or $\#$SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of non-trivial $\#$SAT algorithm for a circuit class ${\mathcal C}$. Say a symmetr…
In the past decade, sparse principal component analysis has emerged as an archetypal problem for illustrating statistical-computational tradeoffs. This trend has largely been driven by a line of research aiming to characterize the average-case complexity of sparse PCA through reductions from the planted clique (PC) conjecture - which conjectures that there is no polynomial-time algorithm to detect a planted clique of size $K = o(N^{1/2})$ in $\mathcal{G}(N, \frac{1}{2})$. All previous reductions to sparse PCA either fail to show tight computational lower bounds matching existing algorithms or …
A hitting set construction, with application to arithmetic circuit lower bounds
多项式同一性测试算法必须确定给定的输入多项式是否完全等于0。给出了一元多项式$\sum_{j=0}^t c_j X^{α_j} (a + b X)^{β_j}$的确定性黑盒恒等式检验算法。从我们的算法中，我们推导出一个指数下界表示的多项式，如$\prod_{i=1}^{2^n} (X^i-1)$在这种形式下。已经证明，这些多项式很难用一般的算术电路来计算。我们的结果表明，“从去随机化的硬度”的方法是可行的，为一类受限制的算术电路的下界。证明是基于代数数论的技术，更确切地说，是基于代数数的高度函数的性质。
In a sequence of seminal results in the 80's, Kaltofen showed that the complexity class VP is closed under taking factors. A natural question in this context is to understand if other natural classes of multivariate polynomials, for instance, arithmetic formulas, algebraic branching programs, bounded depth arithmetic circuits or the class VNP, are closed under taking factors. In this paper, we show that all factors of degree at most $\log^a n$ of polynomials with poly(n) size depth $k$ circuits have poly(n) size circuits of depth at most $O(k + a)$. This partially answers a question of Shpilka…
The minimum circuit size problem (MCSP) is a string compression problem with a parameter $s$ in which, given the truth table of a Boolean function over inputs of length $n$, one must answer whether it can be computed by a Boolean circuit of size at most $s(n) \ge n$. Recently, McKay, Murray, and Williams (STOC, 2019) proved a hardness magnification result for MCSP involving (one-pass) streaming algorithms: For any reasonable $s$, if there is no $\mathsf{poly}(s(n))$-space streaming algorithm with $\mathsf{poly}(s(n))$ update time for $\mathsf{MCSP}[s]$, then $\mathsf{P} \neq \mathsf{NP}$. We p…
The ZX-calculus is a graphical language for reasoning about quantum computation using ZX-diagrams, a certain flexible generalisation of quantum circuits that can be used to represent linear maps from $m$ to $n$ qubits for any $m,n \geq 0$. Some applications for the ZX-calculus, such as quantum circuit optimisation and synthesis, rely on being able to efficiently translate a ZX-diagram back into a quantum circuit of comparable size. While several sufficient conditions are known for describing families of ZX-diagrams that can be efficiently transformed back into circuits, it has previously been …
We use the approximation method of Razborov to analyze the locality barrier which arose from the investigation of the hardness magnification approach to complexity lower bounds. Adapting a limitation of the approximation method obtained by Razborov, we show that in many cases it is not possible to combine the approximation method with typical (localizable) hardness magnification theorems to derive strong circuit lower bounds. In particular, one cannot use the approximation method to derive an extremely strong constant-depth circuit lower bound and then magnify it to an $NC^1$ lower bound for a…
Chinese University of Hong Kong（香港中文大学）；Microsoft Research New England（微软新英格兰研究院）；UC Berkeley（加州大学伯克利分校）；Simons Institute for the Theory of Computing（西蒙斯计算理论研究所）
We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising $p$-spin glass model, and (b) finding a large independent set in a sparse Erdős-Rényi graph. The following families of algorithms are considered: (a) low-degree polynomials of the input; (b) low-depth Boolean circuits; (c) the Langevin dynamics algorithm. We show that these families of algorithms fail to produce nearly optimal solutions with high probability. For the case of Boolean cir…
A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node $v \in G$ stores its distance to the so-called hubs $S_v \subseteq V$, chosen so that for any $u,v \in V$ there is $w \in S_u \cap S_v$ belonging to some shortest $uv$ path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme a…
We study linear equations in combinatorial Laplacians of $k$-dimensional simplicial complexes ($k$-complexes), a natural generalization of graph Laplacians. Combinatorial Laplacians play a crucial role in homology and are a central tool in topology. Beyond this, they have various applications in data analysis and physical modeling problems. It is known that nearly-linear time solvers exist for graph Laplacians. However, nearly-linear time solvers for combinatorial Laplacians are only known for restricted classes of complexes. This paper shows that linear equations in combinatorial Laplacians o…
Finding important nodes in a graph and measuring their importance is a fundamental problem in the analysis of social networks, transportation networks, biological systems, etc. Among popular such metrics are graph centrality, betweenness centrality (BC), and reach centrality (RC). These measures are also very related to classic notions like diameter and radius. Roditty and Vassilevska Williams~[STOC'13] showed that no algorithm can compute a (3/2-δ)-approximation of the diameter in sparse and unweighted graphs faster that n^{2-o(1)} time unless the widely believed strong exponential time hypot…
Hardness magnification reduces major complexity separations (such as $\mathsf{\mathsf{EXP}} \nsubseteq \mathsf{NC}^1$) to proving lower bounds for some natural problem $Q$ against weak circuit models. Several recent works [OS18, MMW19, CT19, OPS19, CMMW19, Oli19, CJW19a] have established results of this form. In the most intriguing cases, the required lower bound is known for problems that appear to be significantly easier than $Q$, while $Q$ itself is susceptible to lower bounds but these are not yet sufficient for magnification. In this work, we provide more examples of this phenomenon, and …
Charles University in Prague（布拉格查理大学）；University of Oxford（牛津大学）
We prove several results giving new and stronger connections between learning, circuit lower bounds and pseudorandomness. Among other results, we show a generic learning speedup lemma, equivalences between various learning models in the exponential time and subexponential time regimes, a dichotomy between learning and pseudorandomness, consequences of non-trivial learning for circuit lower bounds, Karp-Lipton theorems for probabilistic exponential time, and NC$^1$-hardness for the Minimum Circuit Size Problem.
