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.
The level-k ℓ1-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 C computes an XOR function, then C(x,y) is a function of the parity x⊕y. This motivates us to analyze the XOR-fiber of the communication protocol C, defined as h(z):=Ex,y[C(x,y)∣x⊕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(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 d3/2⋅polylog(n) bound, which improves the previous O(d2) 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.
The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models. To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations: Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most õ(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021]. Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].
The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models.
To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations:
Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most o(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021].
Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].
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 dℓ/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 Ω~(n1−1/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).
The Forrelation problem, introduced by Aaronson [A10] and Aaronson and Ambainis [AA15], is a well studied problem in the context of separating quantum and classical models. Variants of this problem were used to give exponential separations between quantum and classical query complexity [A10, AA15]; quantum query complexity and bounded-depth circuits [RT19]; and quantum and classical communication complexity [GRT19]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈1/N, that is, the success probability is larger than ≈1/2+1/N. To achieve separations when the classical protocol has smaller advantage, we study in this work the XOR of k independent copies of the Forrelation function (where k≪N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by αk, cannot compute the XOR of k independent copies of the Forrelation function with advantage better than O(Nk/2αk). This is a strengthening of a result of [CHLT19], that gave a similar result for k=1, using the technique of [RT19]. As an application of our result, we give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol of cost $\mbox{polylog}(N)$ (when players share $\mbox{polylog}(N)$ EPR pairs), however, any classical interactive randomized protocol of cost at most o~(N1/4), has quasipolynomially small advantage over a random guess. We also give the first example of a partial Boolean function that has a quantum query algorithm of cost $\mbox{polylog}(N)$, and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess.
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).
Alice and Bob hold complementary halves of the columns of an n×n matrix of k-bit integers and decide whether it is singular over Q. Up to constant-factor dimension growth, this partitioned problem is equivalent to additive singularity, where Alice and Bob hold n×n matrices A,B of k-bit integers and test the singularity of A+B over Q. We prove a bounded-error quantum lower bound Ω(min{n3/2klogn,n2log(nk)}) and a randomized upper bound O(n2min{k,log(k+logn)}). The bounds match at Θ(n2log(nk)) for k≥nlogn. The lower bound holds even under the promise of rank n−1 versus n. Its proof combines an integer witness with a local digit encoding, while the upper-bound protocol lifts modular kernel relations using short messages.
Bitwise sharing changes how the communication complexity depends on the entry bit length k. If Alice and Bob instead hold n×n matrices A,B of k-bit integers and test whether their entrywise bitwise XOR A⊕B is singular over Q, we prove a quantum lower bound Ω(n2k), matching the trivial upper bound. This lower bound holds even under the promise of rank n−1 versus n. Our proof uses a dense weighted equation and a signed Fourier witness whose exceptional dyadic frequencies can be removed at small variation cost.