We construct a family of distributions $\{\mathcal{D}_n\}_n$ with $\mathcal{D}_n$ over $\{0, 1\}^n$ and a family of depth-$7$ quantum circuits $\{C_n\}_n$ such that $\mathcal{D}_n$ is produced exactly by $C_n$ with the all zeros state as input, yet any constant-depth classical circuit with bounded fan-in gates evaluated on any binary product distribution has total variation distance $1 - e^{-Ω(n)}$ from $\mathcal{D}_n$. Moreover, the quantum circuits we construct are geometrically local and use a relatively standard gate set: Hadamard, controlled-phase, CNOT, and Toffoli gates. All previous separations of this type suffer from some undesirable constraint on the classical circuit model or the quantum circuits witnessing the separation. Our family of distributions is inspired by the Parity Halving Problem of Watts, Kothari, Schaeffer, and Tal (STOC, 2019), which built on the work of Bravyi, Gosset, and König (Science, 2018) to separate shallow quantum and classical circuits for relational problems.

The Parameterized Inapproximability Hypothesis (PIH), which is an analog of the PCP theorem in parameterized complexity, asserts the following: there is a constant ϵ> 0 such that for any computable function f:λ.,•→λ.,•, no f(k)· nO(1)-time algorithm can, on input a k-variable CSP instance with domain size n, find an assignment satisfying 1-ϵ fraction of the constraints. A recent work by Guruswami, Lin, Ren, Sun, and Wu (STOC'24) established PIH under the Exponential Time Hypothesis (ETH). In this work, we improve the quantitative aspects of PIH and prove (under ETH) that approximating sparse parameterized CSPs within a constant factor requires nk1-o(1) time. This immediately implies, for example, that finding a (k/2)-clique in an n-vertex graph with a k-clique requires nk1-o(1) time (assuming ETH). We also prove almost optimal time lower bounds for approximating k-ExactCover and Max k-Coverage. Our proof follows the blueprint of the previous work to identify a "vector-structured"ETH-hard CSP whose satisfiability can be checked via an appropriate form of "parallel"PCP. Using further ideas in the reduction, we guarantee additional structures for constraints in the CSP. We then leverage this to design a parallel PCP of almost linear size based on Reed-Muller codes and derandomized low degree testing.

We characterize the power of constant-depth Boolean circuits in generating uniform symmetric distributions. Let fλ¶{0,1}m→{0,1}n be a Boolean function where each output bit of f depends only on O(1) input bits. Assume the output distribution of f on uniform input bits is close to a uniform distribution D with a symmetric support. We show that D is essentially one of the following six possibilities: (1) point distribution on 0n, (2) point distribution on 1n, (3) uniform over {0n,1n}, (4) uniform over strings with even Hamming weights, (5) uniform over strings with odd Hamming weights, and (6) uniform over all strings. This confirms a conjecture of Filmus, Leigh, Riazanov, and Sokolov (RANDOM 2023). This is an extended abstract. The full paper can be found at https://arxiv.org/abs/2411.08183v1. An updated version with a stronger result can be found at https://arxiv.org/abs/2411.08183.

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 ɛ fraction of constraints for some absolute constant ɛ > 0. PIH plays the role of the PCP theorem in parameterized complexity. However, PIH has only been established under the Gap Exponential Time Hypothesis (ETH), a very strong assumption with an inherent gap. In this work, we prove PIH under the 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.

Motivated by limitations on the depth of near-term quantum devices, we study the depth-computation trade-off in the query model, where 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.

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 ε fraction of constraints for some absolute constant ε > 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.

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.

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.In this work, 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 x and y 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 the communication protocol $\mathcal{C}$, defined as $h(z):=\mathbb{E}_{\boldsymbol{x}, \boldsymbol{y}}[\mathcal{C}(\boldsymbol{x}, \boldsymbol{y}) \mid \boldsymbol{x} \oplus \boldsymbol{y}=z]$.We present improved Fourier growth bounds for the XOR-fibers of randomized 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 the Gap-Hamming problem. For the second level, we show an $d^{3 / 2} \cdot \operatorname{polylog}(n)$ bound, which improves the previous $O\left(d^{2}\right)$ bound by Girish, Raz, and Tal (ITCS 2021) and implies a polynomial improvement on the randomized communication lower bound for the XOR-lift of the Forrelation problem, which extends the quantum-classical gap for this problem.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.

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.

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, for most Φ, to (approximately) uniformly sample from its solution space. Let α = m/n be the density. The previous best algorithm runs in time npoly(k,α) for any α ≲ 2k/300 [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]. In contrast, our algorithm runs in almost-linear time for any α ≲ 2k/3.

We prove that for every parity decision tree of depth d on n variables, the sum of absolute values of Fourier coefficients at level 𝓁 is at most d^{𝓁/2} ⋅ O(𝓁 ⋅ log(n))^𝓁. Our result is nearly tight for small values of 𝓁 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 Ω̃(n^{1-1/k}), while having quantum query complexity ⌈ k/2⌉. Our proof follows a random-walk approach, analyzing the contribution of a random path in the decision tree to the level-𝓁 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-𝓁 walks can be computed by the intermediate values of level ≤ 𝓁-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).

Due to the decoherence of the state-of-the-art physical implementations of quantum computers, it is essential to parallelize the quantum circuits to reduce their depth. Two decades ago, Moore and Nilsson [1] demonstrated that additional qubits (or ancillae) could be used to design “shallow” parallel circuits for quantum operators. They proved that any n-qubit CNOT circuit could be parallelized to O(log n) depth, with O(n2) ancillae. However, the near-term quantum technologies can only support 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 the design of CNOT circuits. We prove that for any m ≥ 0, any n-qubit CNOT circuit can be parallelized to depth, with m ancillae. We show that this bound is tight by a counting argument, and further show that even with arbitrary two-qubit quantum gates to approximate CNOT circuits, the depth lower bound still meets our construction, illustrating the robustness of our result. Our work improves upon two previous results, one by Moore and Nilsson [1] for O(log n)-depth quantum synthesis, and one by Patel, Markov, and Hayes [2] for m =0: for the former, we reduce the need for ancillae by a factor of log2 n by showing that m = O(n2 / log2 n) additional qubits — which is asymptotically optimal — suffice to build O(log n)-depth, O(n2 / log n)-size CNOT circuits; for the later, we reduce the depth by a factor of n to the asymptotically optimal bound . Our results can be directly extended to stabilizer circuits using an earlier result by Aaronson and Gottesman [3]. In addition, we provide relevant hardness evidence for synthesis optimization of CNOT circuits in term of both size and depth.

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. 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. The famous sunflower conjecture is 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 (logw) w . In fact, we prove the result for a robust notion of sunflowers, for which the bound we obtain is tight up to lower order terms.

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 for such approximation. This also 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.

Reversible Boolean function (RBF) 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 connection with low-energy computation as well as quantum computation. In this paper, we give a structured decomposition for even RBFs. Specifically, for n ≥ 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 these blocks have a large degree of freedom. Moreover, if the (n-1)-bit RBFs are required to be even as well, we show for n ≥ 10, even n-bit RBF can be decomposed to 10 even (n - 1)-bit RBFs. In short, 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 work. In Selinger's constructive proof, each block is placed in 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 the positions of blocks.