Fourier Spectrum of Noisy Quantum Algorithms

• URL: http://arxiv.org/abs/2510.06385v1
• Date: Tue, 07 Oct 2025 19:07:59 GMT
• Title: Fourier Spectrum of Noisy Quantum Algorithms
• Authors: Uma Girish,
• Abstract summary: We show that noise acting on a quantum algorithm dampens its Fourier growth in ways intricately linked to the type of noise.<n>In particular, we show that 2-Forrelation and 3-Forrelation require $NOmega(1)$ queries in the $mathsfDQC_k$ and $frac12mathsfBQP$ models respectively.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum computing promises exponential speedups for certain problems, yet fully universal quantum computers remain out of reach and near-term devices are inherently noisy. Motivated by this, we study noisy quantum algorithms and the landscape between $\mathsf{BQP}$ and $\mathsf{BPP}$. We build on a powerful technique to differentiate quantum and classical algorithms called the level-$\ell$ Fourier growth (the sum of absolute values of Fourier coefficients of sets of size $\ell$) and show that it can also be used to differentiate quantum algorithms based on the types of resources used. We show that noise acting on a quantum algorithm dampens its Fourier growth in ways intricately linked to the type of noise. Concretely, we study noisy models of quantum computation where highly mixed states are prevalent, namely: $\mathsf{DQC}_k$ algorithms, where $k$ qubits are clean and the rest are maximally mixed, and $\frac{1}{2}\mathsf {BQP}$ algorithms, where the initial state is maximally mixed, but the algorithm is given knowledge of the initial state at the end of the computation. We establish upper bounds on the Fourier growth of $\mathsf{DQC}_k$, $\frac{1}{2}\mathsf{BQP}$ and $\mathsf{BQP}$ algorithms and leverage the differences between these bounds to derive oracle separations between these models. In particular, we show that 2-Forrelation and 3-Forrelation require $N^{\Omega(1)}$ queries in the $\mathsf{DQC}_1$ and $\frac{1}{2}\mathsf{BQP}$ models respectively. Our results are proved using a new matrix decomposition lemma that might be of independent interest.

Related papers

• Distributed quantum algorithm for divergence estimation and beyond [12.925989807145301]
  We propose a distributed quantum algorithm framework to compute $rm Tr(f(A)g(B))$ within an additive error $varepsilon$.<n>This framework holds broad applicability across a range of distributed quantum computing tasks.
  arXiv  Detail & Related papers  (2025-03-12T14:28:22Z)
• Slow Mixing of Quantum Gibbs Samplers [47.373245682678515]
  We present a quantum generalization of these tools through a generic bottleneck lemma.<n>This lemma focuses on quantum measures of distance, analogous to the classical Hamming distance but rooted in uniquely quantum principles.<n>We show how to lift classical slow mixing results in the presence of a transverse field using Poisson Feynman-Kac techniques.
  arXiv  Detail & Related papers  (2024-11-06T22:51:27Z)
• Quantum State Learning Implies Circuit Lower Bounds [1.5420873135976756]
  We establish connections between state tomography, pseudorandomness, quantum state, circuit lower bounds.<n>We show that even slightly non-trivial quantum state tomography algorithms would lead to new statements about quantum state synthesis.
  arXiv  Detail & Related papers  (2024-05-16T16:46:27Z)
• Quantum algorithms for Hopcroft's problem [45.45456673484445]
  We study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry. The classical complexity of this problem is well-studied, with the best known algorithm running in $O(n4/3)$ time. Our results are two different quantum algorithms with time complexity $widetilde O(n5/6)$.
  arXiv  Detail & Related papers  (2024-05-02T10:29:06Z)
• Towards large-scale quantum optimization solvers with few qubits [59.63282173947468]
  We introduce a variational quantum solver for optimizations over $m=mathcalO(nk)$ binary variables using only $n$ qubits, with tunable $k>1$. We analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature.
  arXiv  Detail & Related papers  (2024-01-17T18:59:38Z)
• Do you know what q-means? [42.96240569413475]
  We present a classical $varepsilon$-$k$-means algorithm that performs an approximate version of one iteration of Lloyd's algorithm with time complexity.<n>We also propose an improved $q$-means quantum algorithm with time complexity.
  arXiv  Detail & Related papers  (2023-08-18T17:52:12Z)
• Quantum Depth in the Random Oracle Model [57.663890114335736]
  We give a comprehensive characterization of the computational power of shallow quantum circuits combined with classical computation. For some problems, the ability to perform adaptive measurements in a single shallow quantum circuit is more useful than the ability to perform many shallow quantum circuits without adaptive measurements.
  arXiv  Detail & Related papers  (2022-10-12T17:54:02Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• A lower bound on the space overhead of fault-tolerant quantum computation [51.723084600243716]
  The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation. We prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude noise.
  arXiv  Detail & Related papers  (2022-01-31T22:19:49Z)
• How to simulate quantum measurement without computing marginals [3.222802562733787]
  We describe and analyze algorithms for classically computation measurement of an $n$-qubit quantum state $psi$ in the standard basis. Our algorithms reduce the sampling task to computing poly(n)$ amplitudes of $n$-qubit states.
  arXiv  Detail & Related papers  (2021-12-15T21:44:05Z)
• Quantum Algorithm for Fidelity Estimation [8.270684567157987]
  For two unknown mixed quantum states $rho$ and $sigma$ in an $N$-dimensional space, computing their fidelity $F(rho,sigma)$ is a basic problem. We propose a quantum algorithm that solves this problem in $namepoly(log (N), r, 1/varepsilon)$ time.
  arXiv  Detail & Related papers  (2021-03-16T13:57:01Z)
• Quantum learning algorithms imply circuit lower bounds [7.970954821067043]
  We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity.
  arXiv  Detail & Related papers  (2020-12-03T14:03:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.