arXiv subjects

Kewen Wu

Publications and source records attributed to Kewen Wu.

35 records · Page 2Linked to original sources

arXiv2024

Locality Bounds for Sampling Hamming Slices

Spurred by the influential work of Viola (Journal of Computing 2012), the past decade has witnessed an active line of research into the complexity of (approximately) sampling distributions, in contrast to the traditional focus on the complexity of computing functions. We build upon and make explicit earlier implicit results of Viola to provide superconstant lower bounds on the locality of Boolean functions approximately sampling the uniform distribution over binary strings of particular Hamming weights, both exactly and modulo an integer, answering questions of Viola (Journal of Computing 2012) and Filmus, Leigh, Riazanov, and Sokolov (RANDOM 2023). Applications to data structure lower bounds and quantum-classical separations are discussed.

cs.CC↗

arXiv2023

The Power of Adaptivity in Quantum Query Algorithms

Motivated by limitations on the depth of near-term quantum devices, we study the depth-computation trade-off in the query model, where the depth corresponds to the number of adaptive query rounds and the computation per layer corresponds to the number of parallel queries per round. We achieve the strongest known separation between quantum algorithms with $r$ versus $r-1$ rounds of adaptivity. We do so by using the $k$-fold Forrelation problem introduced by Aaronson and Ambainis (SICOMP'18). For $k=2r$, this problem can be solved using an $r$ round quantum algorithm with only one query per round, yet we show that any $r-1$ round quantum algorithm needs an exponential (in the number of qubits) number of parallel queries per round. Our results are proven following the Fourier analytic machinery developed in recent works on quantum-classical separations. The key new component in our result are bounds on the Fourier weights of quantum query algorithms with bounded number of rounds of adaptivity. These may be of independent interest as they distinguish the polynomials that arise from such algorithms from arbitrary bounded polynomials of the same degree.

quant-ph↗

arXiv2023

Parameterized Inapproximability Hypothesis under ETH

The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance, parameterized by the number of variables, from one where every assignment fails to satisfy an $\varepsilon$ fraction of constraints for some absolute constant $\varepsilon > 0$. PIH plays the role of the PCP theorem in parameterized complexity. However, PIH has only been established under Gap-ETH, a very strong assumption with an inherent gap. In this work, we prove PIH under the Exponential Time Hypothesis (ETH). This is the first proof of PIH from a gap-free assumption. Our proof is self-contained and elementary. We identify an ETH-hard CSP whose variables take vector values, and constraints are either linear or of a special parallel structure. Both kinds of constraints can be checked with constant soundness via a "parallel PCP of proximity" based on the Walsh-Hadamard code.

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arXiv2023

Fourier Growth of Communication Protocols for XOR Functions

The level-$k$ $\ell_1$-Fourier weight of a Boolean function refers to the sum of absolute values of its level-$k$ Fourier coefficients. Fourier growth refers to the growth of these weights as $k$ grows. It has been extensively studied for various computational models, and bounds on the Fourier growth, even for the first few levels, have proven useful in learning theory, circuit lower bounds, pseudorandomness, and quantum-classical separations. We investigate the Fourier growth of certain functions that naturally arise from communication protocols for XOR functions (partial functions evaluated on the bitwise XOR of the inputs to Alice and Bob). If a protocol $\mathcal C$ computes an XOR function, then $\mathcal C(x,y)$ is a function of the parity $x\oplus y$. This motivates us to analyze the XOR-fiber of $\mathcal C$, defined as $h(z):=\mathbb E_{x,y}[\mathcal C(x,y)|x\oplus y=z]$. We present improved Fourier growth bounds for the XOR-fibers of protocols that communicate $d$ bits. For the first level, we show a tight $O(\sqrt d)$ bound and obtain a new coin theorem, as well as an alternative proof for the tight randomized communication lower bound for Gap-Hamming. For the second level, we show an $d^{3/2}\cdot\mathrm{polylog}(n)$ bound, which improves the previous $O(d^2)$ bound by Girish, Raz, and Tal (ITCS 2021) and implies a polynomial improvement on the randomized communication lower bound for the XOR-lift of Forrelation, extending its quantum-classical gap. Our analysis is based on a new way of adaptively partitioning a relatively large set in Gaussian space to control its moments in all directions. We achieve this via martingale arguments and allowing protocols to transmit real values. We also show a connection between Fourier growth and lifting theorems with constant-sized gadgets as a potential approach to prove optimal bounds for the second level and beyond.

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arXiv2023

Improved Bounds for Sampling Solutions of Random CNF Formulas

Let $Φ$ be a random $k$-CNF formula on $n$ variables and $m$ clauses, where each clause is a disjunction of $k$ literals chosen independently and uniformly. Our goal is to sample an approximately uniform solution of $Φ$ (or equivalently, approximate the partition function of $Φ$). Let $α=m/n$ be the density. The previous best algorithm runs in time $n^{\mathsf{poly}(k,α)}$ for any $α\lesssim2^{k/300}$ [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]. Our result significantly improves both bounds by providing an almost-linear time sampler for any $α\lesssim2^{k/3}$. The density $α$ captures the \emph{average degree} in the random formula. In the worst-case model with bounded \emph{maximum degree}, current best efficient sampler works up to degree bound $2^{k/5}$ [He, Wang, and Yin, FOCS'22 and SODA'23], which is, for the first time, superseded by its average-case counterpart due to our $2^{k/3}$ bound. Our result is the first progress towards establishing the intuition that the solvability of the average-case model (random $k$-CNF formula with bounded average degree) is better than the worst-case model (standard $k$-CNF formula with bounded maximal degree) in terms of sampling solutions.

cs.DS↗

arXiv2023

On Differentially Private Counting on Trees

We study the problem of performing counting queries at different levels in hierarchical structures while preserving individuals' privacy. Motivated by applications, we propose a new error measure for this problem by considering a combination of multiplicative and additive approximation to the query results. We examine known mechanisms in differential privacy (DP) and prove their optimality, under this measure, in the pure-DP setting. In the approximate-DP setting, we design new algorithms achieving significant improvements over known ones.

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arXiv2022

On the Learning Mechanisms in Physical Reasoning

Is dynamics prediction indispensable for physical reasoning? If so, what kind of roles do the dynamics prediction modules play during the physical reasoning process? Most studies focus on designing dynamics prediction networks and treating physical reasoning as a downstream task without investigating the questions above, taking for granted that the designed dynamics prediction would undoubtedly help the reasoning process. In this work, we take a closer look at this assumption, exploring this fundamental hypothesis by comparing two learning mechanisms: Learning from Dynamics (LfD) and Learning from Intuition (LfI). In the first experiment, we directly examine and compare these two mechanisms. Results show a surprising finding: Simple LfI is better than or on par with state-of-the-art LfD. This observation leads to the second experiment with Ground-truth Dynamics, the ideal case of LfD wherein dynamics are obtained directly from a simulator. Results show that dynamics, if directly given instead of approximated, would achieve much higher performance than LfI alone on physical reasoning; this essentially serves as the performance upper bound. Yet practically, LfD mechanism can only predict Approximate Dynamics using dynamics learning modules that mimic the physical laws, making the following downstream physical reasoning modules degenerate into the LfI paradigm; see the third experiment. We note that this issue is hard to mitigate, as dynamics prediction errors inevitably accumulate in the long horizon. Finally, in the fourth experiment, we note that LfI, the extremely simpler strategy when done right, is more effective in learning to solve physical reasoning problems. Taken together, the results on the challenging benchmark of PHYRE show that LfI is, if not better, as good as LfD for dynamics prediction. However, the potential improvement from LfD, though challenging, remains lucrative.

cs.LG↗

arXiv2022

Optimal Space-Depth Trade-Off of CNOT Circuits in Quantum Logic Synthesis

Decoherence -- in the current physical implementations of quantum computers -- makes depth reduction a vital task in quantum-circuit design. Moore and Nilsson (SIAM Journal of Computing, 2001) demonstrated that additional qubits -- known as ancillae -- can be used to provide an extended space to parallelize quantum circuits. Specifically, they proved that, with $O(n^2)$ ancillae, any $n$-qubit CNOT circuit can be transformed into an equivalent one of $O(\log n)$ depth. However, the near-term quantum technologies can only support a limited amount of qubits, making space-depth trade-off a fundamental research subject for quantum-circuit synthesis. In this work, we establish an asymptotically optimal space-depth trade-off for CNOT circuits. We prove that any $n$-qubit CNOT circuit can be parallelized to $$O\left(\max\left\{\log n, \frac{n^2}{(n+m)\log (n+m)}\right\}\right)$$ depth with $m$ ancillae. This bound is tight even if the task is expanded from exact synthesis to the approximation of CNOT circuits with arbitrary two-qubit quantum gates. Our result can be extended to stabilizer circuits via the reduction by Aaronson and Gottesman (Physical Review A, 2004). Furthermore, we provide hardness evidence for optimizing CNOT circuits in terms of size or depth. Our result has improved upon two previous papers that motivated our work. Moore-Nilsson's construction (aforementioned) for $O(\log n)$-depth CNOT circuit synthesis: We have reduced their need for ancillae by a factor of $\log^2 n$ by showing that $m= O(n^2/\log^2 n)$ additional qubits -- which is asymptotically optimal -- suffice to build equivalent $O(\log n)$-depth $O(n^2/\log n)$-size CNOT circuits. Patel-Markov-Hayes's construction (Quantum Information & Computation 2008) for $m = 0$: We have reduced their depth by a factor of $n$ and achieved the asymptotically optimal bound of $O(n/\log n)$.

quant-ph↗

arXiv2021

Improved bounds for the sunflower lemma

A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all of them. Erdős and Rado proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ sets, must contain a sunflower with $r$ petals. The famous sunflower conjecture states that the bound on the number of sets can be improved to $c^w$ for some constant $c$. In this paper, we improve the bound to about $(\log w)^w$. In fact, we prove the result for a robust notion of sunflowers, for which the bound we obtain is sharp up to lower order terms.

math.CO↗

arXiv2021

Perfect Sampling for (Atomic) Lovász Local Lemma

We give a Markov chain based perfect sampler for uniform sampling solutions of constraint satisfaction problems (CSP). Under some mild Lovász local lemma conditions where each constraint of the CSP has a small number of forbidden local configurations, our algorithm is accurate and efficient: it outputs a perfect uniform random solution and its expected running time is quasilinear in the number of variables. Prior to our work, perfect samplers are only shown to exist for CSPs under much more restrictive conditions (Guo, Jerrum, and Liu, JACM'19). Our algorithm has two components: 1. A simple perfect sampling algorithm using bounding chains (Huber, STOC'98; Haggstrom and Nelander, Scandinavian Journal of Statistics'99). This sampler is efficient if each variable domain is small. 2. A simple but powerful state tensorization trick to reduce large domains to smaller ones. This trick is a generalization of state compression (Feng, He, and Yin, STOC'21). The crux of our analysis is a simple information percolation argument which allows us to achieve bounds even beyond current best approximate samplers (Jain, Pham, and Vuong, ArXiv'21). Previous related works either use intricate algorithms or need sophisticated analysis or even both. Thus we view the simplicity of both our algorithm and analysis as a strength of our work.

cs.DS↗

arXiv2021

Fourier Growth of Parity Decision Trees

We prove that for every parity decision tree of depth $d$ on $n$ variables, the sum of absolute values of Fourier coefficients at level $\ell$ is at most $d^{\ell/2} \cdot O(\ell \cdot \log(n))^\ell$. Our result is nearly tight for small values of $\ell$ and extends a previous Fourier bound for standard decision trees by Sherstov, Storozhenko, and Wu (STOC, 2021). As an application of our Fourier bounds, using the results of Bansal and Sinha (STOC, 2021), we show that the $k$-fold Forrelation problem has (randomized) parity decision tree complexity $\tildeΩ\left(n^{1-1/k}\right)$, while having quantum query complexity $\lceil k/2\rceil$. Our proof follows a random-walk approach, analyzing the contribution of a random path in the decision tree to the level-$\ell$ Fourier expression. To carry the argument, we apply a careful cleanup procedure to the parity decision tree, ensuring that the value of the random walk is bounded with high probability. We observe that step sizes for the level-$\ell$ walks can be computed by the intermediate values of level $\le \ell-1$ walks, which calls for an inductive argument. Our approach differs from previous proofs of Tal (FOCS, 2020) and Sherstov, Storozhenko, and Wu (STOC, 2021) that relied on decompositions of the tree. In particular, for the special case of standard decision trees we view our proof as slightly simpler and more intuitive. In addition, we prove a similar bound for noisy decision trees of cost at most $d$ -- a model that was recently introduced by Ben-David and Blais (FOCS, 2020).

cs.CC↗

arXiv2021

Parameterised Counting in Logspace

In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space bounded models developed by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). They defined the operators paraW and paraBeta for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators paraW and paraBeta by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, paraW[1] and paraBeta-Tail. Then, we consider counting versions of all four operators applied to logspace and obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0,1)-matrices is #paraBeta-Tail-L-hard and can be written as the difference of two functions in #paraBetaTail-L. For example, we show that the closure of #paraBetaTail-L under parameterised logspace parsimonious reductions coincides with #paraBeta-L, that is, modulo parameterised reductions, tail-nondeterminism with read-once access is the same as read-once nondeterminism. We show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Finally, we underline the significance of this topic by providing a promising outlook showing several open problems and options for further directions of research.

cs.LO↗

arXiv2020

On the Degree of Boolean Functions as Polynomials over $\mathbb{Z}_m$

Polynomial representations of Boolean functions over various rings such as $\mathbb{Z}$ and $\mathbb{Z}_m$ have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of fields including communication complexity, circuit complexity, learning theory, coding theory and so on. For any integer $m\ge2$, each Boolean function has a unique multilinear polynomial representation over ring $\mathbb Z_m$. The degree of such polynomial is called modulo-$m$ degree, denoted as $\mathrm{deg}_m(\cdot)$. In this paper, we investigate the lower bound of modulo-$m$ degree of Boolean functions. When $m=p^k$ ($k\ge 1$) for some prime $p$, we give a tight lower bound that $\mathrm{deg}_m(f)\geq k(p-1)$ for any non-degenerated function $f:\{0,1\}^n\to\{0,1\}$, provided that $n$ is sufficient large. When $m$ contains two different prime factors $p$ and $q$, we give a nearly optimal lower bound for any symmetric function $f:\{0,1\}^n\to\{0,1\}$ that $\mathrm{deg}_m(f) \geq \frac{n}{2+\frac{1}{p-1}+\frac{1}{q-1}}$.

cs.CC↗

arXiv2020

Decision list compression by mild random restrictions

A decision list is an ordered list of rules. Each rule is specified by a term, which is a conjunction of literals, and a value. Given an input, the output of a decision list is the value corresponding to the first rule whose term is satisfied by the input. Decision lists generalize both CNFs and DNFs, and have been studied both in complexity theory and in learning theory. The size of a decision list is the number of rules, and its width is the maximal number of variables in a term. We prove that decision lists of small width can always be approximated by decision lists of small size, where we obtain sharp bounds. This in particular resolves a conjecture of Gopalan, Meka and Reingold (Computational Complexity, 2013) on DNF sparsification. An ingredient in our proof is a new random restriction lemma, which allows to analyze how DNFs (and more generally, decision lists) simplify if a small fraction of the variables are fixed. This is in contrast to the more commonly used switching lemma, which requires most of the variables to be fixed.

cs.CC↗

arXiv2019

A Note on Lower Digits Extraction Polynomial for Bootstrapping

Bootstrapping is a crucial but computationally expensive step for realizing Fully Homomorphic Encryption (FHE). Recently, Chen and Han (Eurocrypt 2018) introduced a family of low-degree polynomials to extract the lowest digit with respect to a certain congruence, which helps improve the bootstrapping for both FV and BGV schemes. In this note, we present the following relevant findings about the work of Chen and Han (referred to as CH18): 1. We provide a simpler construction of the low-degree polynomials that serve the same purpose and match the asymptotic bound achieved in CH18; 2. We show the optimality and limit of our approach by solving a minimal polynomial degree problem; 3. We consider the problem of extracting other low-order digits using polynomials, and provide negative results.

cs.CR↗

arXiv2019

Structured decomposition for reversible Boolean functions

Reversible Boolean function is a one-to-one function which maps $n$-bit input to $n$-bit output. Reversible logic synthesis has been widely studied due to its relationship with low-energy computation as well as quantum computation. In this work, we give a structured decomposition for even reversible Boolean functions (RBF). Specifically, for $n\geq 6$, any even $n$-bit RBF can be decomposed to $7$ blocks of $(n-1)$-bit RBF, where $7$ is a constant independent of $n$; and the positions of those blocks have large degree of freedom. Moreover, if the $(n-1)$-bit RBFs are required to be even as well, we show for $n\geq 10$, $n$-bit RBF can be decomposed to $10$ even $(n-1)$-bit RBFs. For simplicity, we say our decomposition has block depth $7$ and even block depth $10$. Our result improves Selinger's work in block depth model, by reducing the constant from $9$ to $7$; and from $13$ to $10$ when the blocks are limited to be even. We emphasize that our setting is a bit different from Selinger's. In Selinger's constructive proof, each block is one of two specific positions and thus the decomposition has an alternating structure. We relax this restriction and allow each block to act on arbitrary $(n-1)$ bits. This relaxation keeps the block structure and provides more candidates when choosing positions of blocks.

cs.ET↗

arXiv2019

On the Relationship between Energy Complexity and other Boolean Function Measures

In this work we investigate into energy complexity, a Boolean function measure related to circuit complexity. Given a circuit $\mathcal{C}$ over the standard basis $\{\vee_2,\wedge_2,\neg\}$, the energy complexity of $\mathcal{C}$, denoted by $\mathrm{EC}(\mathcal{C})$, is the maximum number of its activated inner gates over all inputs. The energy complexity of a Boolean function $f$, denoted by $\mathrm{EC}(f)$, is the minimum of $\mathrm{EC}(\mathcal{C})$ over all circuits $\mathcal{C}$ computing $f$. This concept has attracted lots of attention in literature. Recently, Dinesh, Otiv, and Sarma [COCOON'18] gave $\mathrm{EC}(f)$ an upper bound in terms of the decision tree complexity, $\mathrm{EC}(f)=O(\mathrm{D}(f)^3)$. They also showed that $\mathrm{EC}(f)\leq 3n-1$, where $n$ is the input size. Recall that the minimum size of circuit to compute $f$ could be as large as $2^n/n$. We improve their upper bounds by showing that $\mathrm{EC}(f)\leq\min\{\frac12\mathrm{D}(f)^2+O(\mathrm{D}(f)),n+2\mathrm{D}(f)-2\}$. For the lower bound, Dinesh, Otiv, and Sarma defined positive sensitivity, a complexity measure denoted by $\mathrm{psens}(f)$, and showed that $\mathrm{EC}(f)\ge\frac{1}{3}\mathrm{psens}(f)$. They asked whether $\mathrm{EC}(f)$ can also be lower bounded by a polynomial of $\mathrm{D}(f)$. In this paper we affirm it by proving $\mathrm{EC}(f)=Ω(\sqrt{\mathrm{D}(f)})$. For non-degenerated functions with input size $n$, we give another lower bound $\mathrm{EC}(f)=Ω(\log{n})$. All these three lower bounds are incomparable to each other. Besides, we also examine the energy complexity of $\mathtt{OR}$ functions and $\mathtt{ADDRESS}$ functions, which implies the tightness of our two lower bounds respectively. In addition, the former one answers another open question asking for a non-trivial lower bounds for the energy complexity of $\mathtt{OR}$ functions.

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