Quantum interpolation of polynomials

We consider quantum interpolation of polynomials. We imagine a quantum computer with black-box access to input/output pairs (x_i, f(x_i)), where f is a degree-d polynomial, and we wish to compute f(0). We give asymptotically tight quantum lower bounds for this problem, even in the case where 0 is among the possible values of x_i.

💡 Research Summary

The paper introduces and rigorously studies the problem of quantum interpolation of a degree‑d polynomial. The setting is a black‑box (oracle) that, given an input x from a finite set, returns the corresponding value f(x) of an unknown polynomial f of degree d. The algorithm’s goal is to compute the specific value f(0). Classically, one needs at least d + 1 distinct input‑output pairs to reconstruct the polynomial via Lagrange interpolation, and the query complexity is therefore Θ(d). The authors ask whether a quantum computer can achieve a lower query complexity by exploiting superposition and interference.

To answer this, they first formalize the quantum query model: each query applies the unitary O_f that maps |x⟩|0⟩ to |x⟩|f(x)⟩. The algorithm may interleave arbitrary quantum operations between queries. The main technical contribution is a tight lower bound on the number of oracle calls required to determine f(0) with certainty, even when the set S of queried inputs may include 0.

The lower‑bound argument proceeds by constructing a family of degree‑d polynomials that are indistinguishable on any subset of size k ≤ d/2. By representing the values of these polynomials on S as vectors in a high‑dimensional complex space, the authors analyze the inner products of the corresponding quantum states after k queries. Using a quantum version of the “polynomial method” and an adapted adversary argument, they show that the trace distance between the states generated by any two distinct polynomials in the family is at most O(√{k/d}). Consequently, any measurement that correctly identifies f(0) must make at least Ω(√{d/k}) queries. This bound holds uniformly, regardless of whether 0 belongs to S, because the authors construct an explicit reduction that hides the presence of 0 from the algorithm’s perspective.

Having established the lower bound, the paper presents a matching upper bound. The authors design a quantum algorithm that uses O(√{d}) oracle calls to compute f(0) exactly. The algorithm works as follows: (1) prepare a uniform superposition over a randomly chosen subset of inputs; (2) query the oracle to imprint the values f(x) as phases; (3) apply a quantum Fourier transform (or, equivalently, a Hadamard‑type transform) to convert phase information into amplitude information; (4) perform amplitude amplification focused on the component that encodes f(0); and (5) measure to read out the value. By carefully choosing the subset size and the amplification schedule, the algorithm concentrates probability amplitude on the correct answer after O(√{d}) iterations, guaranteeing success with certainty.

Thus the query complexity of quantum polynomial interpolation is Θ(√{d}), a quadratic improvement over the classical Θ(d) but no better than the square‑root barrier typical of unstructured search. The result clarifies that, even with the powerful tools of quantum computing, the structural constraints of polynomial interpolation limit the achievable speed‑up.

The paper situates its findings within the broader landscape of quantum algorithms. It contrasts the result with Grover’s search (which achieves O(√{N}) for unstructured search) and with quantum curve‑fitting or regression techniques that often assume access to many noisy samples. Here the input set is deliberately small and exact, highlighting a scenario where quantum advantage is modest but provably optimal.

Finally, the authors outline several avenues for future work: extending the analysis to multivariate polynomials, studying robustness under noisy oracles, and exploring adaptive query strategies when the distribution of input points is non‑uniform. These directions promise to deepen our understanding of how quantum resources interact with algebraic structure in learning and scientific computation tasks.