We describe a quantum algorithm for the Planted Noisy kXOR problem (also known as sparse Learning Parity with Noise) that achieves a nearly quartic (4th power) speedup over the best known classical algorithm while also only using logarithmically many qubits. Our work generalizes and simplifies prior work of Hastings [Has20], by building on his quantum algorithm for the Tensor Principal Component Analysis (PCA) problem. We achieve our quantum speedup using a general framework based on the Kikuchi Method (recovering the quartic speedup for Tensor PCA), and we anticipate it will yield similar speedups for further planted inference problems. These speedups rely on the fact that planted inference problems naturally instantiate the Guided Sparse Hamiltonian problem. Since the Planted Noisy kXOR problem has been used as a component of certain cryptographic constructions, our work suggests that some of these are susceptible to super-quadratic quantum attacks.
We describe a quantum algorithm for the Planted Noisy kXOR problem (also known as sparse Learning Parity with Noise) that achieves a nearly quartic (4th power) speedup over the best known classical algorithm while also only using logarithmically many qubits. Our work generalizes and simplifies prior work of Hastings [Has20], by building on his quantum algorithm for the Tensor Principal Component Analysis (PCA) problem. We achieve our quantum speedup using a general framework based on the Kikuchi Method (recovering the quartic speedup for Tensor PCA), and we anticipate it will yield similar speedups for further planted inference problems. These speedups rely on the fact that planted inference problems naturally instantiate the Guided Sparse Hamiltonian problem. Since the Planted Noisy kXOR problem has been used as a component of certain cryptographic constructions, our work suggests that some of these are susceptible to super-quadratic quantum attacks.
We describe a quantum algorithm for the Planted Noisy k XOR Problem (also known as Sparse Learning Parity with Noise) that achieves a nearly (fourth-power) speedup over the best known classical algorithm while using exponentially less space. Our work generalizes and simplifies prior work of Hastings [], by building on his quantum algorithm for the tensor principal component analysis (PCA) problem. We achieve our quantum speedup using a general framework based on the Kikuchi method (recovering the quartic speedup for Tensor PCA), and we anticipate it will yield similar speedups for further planted inference problems. These speedups rely on the fact that planted inference problems naturally instantiate the guided sparse Hamiltonian problem. Since the Planted Noisy k XOR Problem has been used as a component of certain cryptographic constructions, our work suggests that some of these are susceptible to superquadratic quantum attacks.
We introduce hypergraph cycle enumeration (HCE), a simple spectral framework for detecting and recovering planted signals in noisy hypergraph inference problems. We study HCE on one representative problem from this class, namely planted noisy k-XOR, also known as sparse Learning Parity with Noise (LPN). Both theoretically and empirically, HCE achieves comparable recovery behavior to Kikuchi method, matching Kikuchi hierarchy. We identify parameter regimes where HCE outperforms Kikuchi method, showing that neither algorithm is universally better. We also show a subquadratic quantum speedup for HCE by replacing one of the algorithm's steps with a quantum walk. Our classical and quantum algorithms do not refute the quartic quantum speedup of quantum Kikuchi method, which remains valid for the practically relevant regime of large size and high noise strength.
Spectral methods are a leading approach for tensor PCA with a ``spiked" Gaussian tensor. The methods use the spectrum of a linear operator in a vector space with exponentially high dimension and in Ref. 1 it was shown that quantum algorithms could then lead to an exponential space saving as well as a quartic speedup over classical. Here we show how to accelerate both classical and quantum algorithms quadratically, while maintaining the same quartic separation between them. That is, our classical algorithm here is quadratically faster than the original classical algorithm, while the quantum algorithm is eigth-power faster than the original classical algorithm. We then give a further modification of the quantum algorithm, increasing its speedup over the modified classical algorithm to the sixth power. We only prove these speedups for detection, rather than recovery, but we give a strong plausibility argument that our algorithm achieves recovery also.
Note added: After this paper was prepared, A. Schmidhuber pointed out to me Ref. 3. This improves the best existing bounds on the spectral norm of a certain random operator. Because the norm of this operator enters into the runtime, with this improvement on the norm, we no longer have a provable polynomial speedup. Our results are phrased in terms of certain properties of the spectrum of this operator (not merely the largest eigenvalue but also the density of states). So, if these properties still hold, the speedup still holds. Rather than modify the paper, I have left it unchanged but added a section at the end discussing the needed property of density of states and considering for which problems (there are several problems for which this kind of quartic quantum speedup has been used and the techniques here will likely be applicable to several of them) the property is likely to hold.
Planted noisy kXOR and the strong refutation of random kXOR are governed by a conjectured trade-off between signal strength and time: Level ℓ of the Kikuchi hierarchy should achieve the smooth curve \begin{equation*}
m\ \gtrsim\ ρ^{-2}n^{k/2}/\ell^{k/2-1}\ \text{clauses}
\quad\Longleftrightarrow\quad
\text{solvable in time }n^{O(\ell)}, \end{equation*} where ρ is the bias of the planted signal or, for refutation, the target advantage. However, every spectral analysis of sparse kXOR to date loses polylogarithmic factors against this curve, a loss that enters the exponent of the running time.
We show that a normalized variant of the Kikuchi hierarchy achieves the sharp conjectured trade-off, with no logarithmic loss, at every arity k≥3. At the scale above, our algorithms achieve strong detection, weak recovery, and strong refutation; an additional cleanup step boosts weak recovery to exact recovery, and the refutation certificates yield sum-of-squares proofs of degree Ok(ℓ). We also prove matching lower bounds in the same model. The inference and refutation upper bounds transfer to more general planting laws and predicates. Finally, we give a quantum algorithm that achieves a quartic speedup over the classical spectral algorithms for detection and weak recovery.
The proofs rest on two key ingredients: a normalization of the sparse Kikuchi matrix, and a sharp count of the closed walks in its trace expansion. We use a closely related trace-walk count to prove Feige's 2008 hypergraph Moore bound conjecture in a companion paper.
Quantum algorithms for linear algebra that access a matrix A through quantum random access memory (QRAM) prepare quantum states proportional to the solutions of linear systems and related problems. Dequantized classical algorithms, when given sampling and query access to A, return sampling and query access to an approximate solution. Both kinds of algorithms use a number of queries that is polynomial in the normalized Frobenius norm ρ=∥A∥F/∥A∥ and does not directly depend on the dimension. For linear systems at fixed condition number and accuracy, quantum algorithms use O(ρ) queries, whereas the best dequantized algorithms use O(ρ4). We prove that this quartic gap is tight in the worst case, up to a logarithmic factor. For every constant κ0>1, we construct dense linear systems with condition number at most κ0 for which sampling from the solution within a sufficiently small constant total variation distance requires Ω(ρ4/logρ) classical queries but only Θ(ρ) quantum queries. The same separation holds, for errors below smaller thresholds, for positive-definite systems, least squares, ridge regression, matrix powers and square roots, imaginary-time and real-time Hamiltonian evolution, and sampling from the top principal component. Our results therefore show that many quantum algorithms for linear algebra, achieve a near-quartic speedup in the worst case, over classical algorithms with sampling and query access. All of these classical lower bounds follow from the same framework, which is based on a lower bound for sampling from distributions that depend on a fixed set of Fourier coefficients of an unknown Boolean function.
Along with the development of quantum technology, finding useful applications of quantum computers has been a central pursuit. Despite various quantum algorithms have been developed, many of them often require strong input assumptions, which is hardware demanding. In particular, recent advances on dequantization have revealed that the quantum advantage is more of a mere artifact of strong input assumptions. In this work, we propose a variant of these algorithms, leveraging both classical and quantum resources. Provided the classical knowledge (the entries) of the matrix/vector of interest, a classical procedure is used to pre-process this information. Then they are fed into a quantum circuit which is shown to be a block encoding of the matrix of interest. From this block-encoding, we show how to use it to tackle a wide range of problems, including principal component analysis, linear equation solving, Hamiltonian simulation, preparing ground state, and data fitting. We also analyze our protocol, showing that both the classical and quantum procedure can achieve logarithmic complexity in the input dimension, thus implying its potential for near term realization. We then discuss several implications and corollaries of our result. First,, our results suggest there are certain matrices/Hamiltonians where our method can provide exponential improvement compared to the existing ones with respect to the sparsity. Regarding dense linear systems, our method achieves exponential speed-up with respect to the inverse of error tolerance, compared to the best previously known quantum algorithm for dense systems. Last, and most importantly, regarding quantum data fitting, we show how the output of our quantum algorithms can be leveraged to predict unseen data. Thus, it provides an end-to-end application, which has been an open aspect of the previous quantum data fitting algorithm.