A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. Here we show that such a result does not generalize to quartic polynomials and 2-query algorithms, even when we allow for additive approximations. We also show that the additive approximation implied by their result is tight for bounded bilinear forms, which gives a new characterization of the Grothendieck constant in terms of 1-query quantum algorithms. Along the way we provide reformulations of the completely bounded norm of a form, and its dual norm.
A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. Here we show that such a result does not generalize to quartic polynomials and 2-query algorithms, even when we allow for additive approximations. We also show that the additive approximation implied by their result is tight for bounded bilinear forms, which gives a new characterization of the Grothendieck constant in terms of 1-query quantum algorithms. Along the way we provide reformulations of the completely bounded norm of a form, and its dual norm.
We prove a characterization of t-query quantum algorithms in terms of the unit ball of a space of degree-(2t) polynomials. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. [``Polynomials, Quantum Query Complexity, and Grothendieck's Inequality,” in Proceedings of the 31st Conference on Computational Complexity, CCC 2016, Schloss Dagstuh, 2016, pp. 25:1--25:19]. Our proof is based on a fundamental result of Christensen and Sinclair [ J. Funct. Anal., 72 (1987), pp. 151--181] that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. Using our characterization, we show that many polynomials of degree four are far from those coming from two-query quantum algorithms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.
We consider the problem of establishing limitations on the power of quantum query algorithms via the completely bounded polynomial method. In particular, we prove several optimal functional inequalities involving different notions of completely bounded polynomials. These inequalities lead to limiting theorems for the power of quantum query algorithms that improve on prior works.
1. An optimal root-influence bound for block-multilinear polynomials. Prior work showed that block-multilinear polynomials p of degree t satisfy a root-influence bound, ∥p∥cb≥∑iInfi[p]/t2, which is stronger than the bound appearing in the Aaronson-Ambainis conjecture. We find the optimal constant in that inequality: ∥p∥cb≥∑iInfi[p]/t. Since the amplitudes of quantum algorithms that query disjoint blocks of inputs-such as t-fold forrelation- are block-multilinear polynomials with ∥p∥cb≤1, our inequality shows that they satisfy t≥∑iInfi[p]. We prove that this inequality yields both a more efficient classical simulation than prior results based on the Aaronson-Ambainis argument, and a qualitative improvement: all classical queries are nonadaptive.
2. Optimal Fourier growth of the highest level of quantum query algorithms. We show that for every polynomial p defined on {−1,1}n of degree 2t, the Fourier Growth at the level 2t, namely ∥p2t∥ℓ1, satisfies ∥p2t∥ℓ1≤(en/(2t−1))22t−1∥p∥cb. This is optimal up to the factor e, as witnessed by 2t-fold forrelation. As quantum query algorithms that make t queries (to the whole input) satisfy ∥p∥cb≤1, this yields a Fourier growth bound for these algorithms, partially resolving a question by Girish (STOC, 2026).
We show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials. Namely, a partial Boolean function f is computable by a 1-query quantum algorithm with error bounded by epsilon<1/2 iff f can be approximated by a degree-2 polynomial with error bounded by epsilon'<1/2. This result holds for two different notions of approximation by a polynomial: the standard definition of Nisan and Szegedy and the approximation by block-multilinear polynomials recently introduced by Aaronson and Ambainis [Aaronson/Ambainis, STOC 2015]. The proof uses Grothendieck's inequality to relate two matrix norms, with one norm corresponding to polynomial approximations and the other norm corresponding to quantum algorithms. We also show two results for polynomials of higher degree. First, there is a total Boolean function which requires ~Omega(n) quantum queries but can be represented by a block-multilinear polynomial of degree ~O(sqrt(n)). Thus, in the general case (for an arbitrary number of queries), block-multilinear polynomials are not equivalent to quantum algorithms. Second, for any constant degree k, the two notions of approximation by a polynomial (the standard and the block-multilinear) are equivalent. As a consequence, we solve an open problem from [Aaronson/Ambainis, STOC 2015], showing that one can estimate the value of any bounded degree-k polynomial p:{0,1}^n -> [-1,1] with O(n^{1-1/(2k)) queries.
We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC'16). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms. Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials.
In this thesis, Quantum computing, norms, and polynomials, we investigate the interplay between quantum mechanics, complexity theory, and functional analysis, three central areas of physics, computer science, and mathematics, respectively. The unifying theme throughout the thesis is the dynamic exchange between quantum computing and functional analysis: we explore new applications of functional inequalities in quantum computing, and, in the process, establish novel results in functional analysis itself. In the first part, we study quantum query algorithms through their correspondence with completely bounded polynomials, as established in earlier work. We begin by revisiting this correspondence, extending it, and presenting it in a new form. Building on this foundation, we draw an analogy between quantum query algorithms and the Grothendieck inequality. Finally, we conclude this part by employing completely bounded polynomials to solve a special case of one of the main open problems in quantum query complexity, the Aaronson–Ambainis conjecture. In the second part, we turn to quantum learning theory, which seeks to determine how much information must be extracted from a quantum system to fully characterize it. We begin by applying existing versions of the Bohnenblust–Hille inequalities and deriving new ones to obtain results in the learning of low-degree quantum objects. We conclude by presenting some of the first results in the emerging area of Hamiltonian testing and learning. We also include a third part, as a bonus, where we gather three new proofs, that we find elegant and concise, of known results related to the analysis of Boolean functions.
The Grothendieck constant KG is the smallest constant such that for every d ∈ N and every matrix A = (aij), sup u i,v j ∈B (d) X aij〈ui, vj 〉 � KG · ij sup x i,y j ∈[−1,1] X ij aijxiyj, where B (d) is the unit ball in R d. Despite several efforts [15, 23], the value of the constant KG remains unknown. The Grothendieck constant KG is precisely the integrality gap of a natural SDP relaxation for the KM,N-Quadratic Programming problem. The input to this problem is a matrix A = (aij) and the objective is to maximize the quadratic form P ij aijxiyj over xi, yj ∈ [−1, 1]. In this work, we apply techniques from [22] to the KM,N-Quadratic Programming problem. Using some standard but non-trivial modifications, the reduction in [22] yields the following hardness result: Assuming the Unique Games Conjecture [9], it is NP-hard to approximate the KM,N-Quadratic Programming problem to any factor better than the Grothendieck constant KG. By adapting a “bootstrapping ” argument used in a proof of Grothendieck inequality [5], we are able to perform a tighter analysis than [22]. Through this careful analysis, we obtain the following new results: ◦ An approximation algorithm for KM,N-Quadratic Programming that is guaranteed to achieve an approximation ratio arbitrarily close to the Grothendieck constant KG (optimal approximation ratio assuming the Unique Games Conjecture). ◦ We show that the Grothendieck constant KG can be computed within an error η, in time depending only on η. Specifically, for each η, we formulate an explicit finite linear program, whose optimum is η-close to the Grothendieck constant. We also exhibit a simple family of operators on the Gaussian Hilbert space that is guaranteed to contain tight examples for the Grothendieck inequality.