[Paper Review] The NISQ Complexity of Collision Finding

Yassine Hamoudi, Qipeng Liu+1 more|arXiv (Cornell University)|Nov 23, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces the hybrid compressed oracle framework, a novel technique for analyzing quantum-classical query tradeoffs in the random oracle model. It enables consistent tracking of knowledge gained from both classical and quantum queries in incompatible bases, providing optimal query complexity bounds for collision finding and simplifying proofs for search problems.

ABSTRACT

Collision-resistant hashing, a fundamental primitive in modern cryptography, ensures that there is no efficient way to find distinct inputs that produce the same hash value. This property underpins the security of various cryptographic applications, making it crucial to understand its complexity. The complexity of this problem is well-understood in the classical setting and $Θ(N^{1/2})$ queries are needed to find a collision. However, the advent of quantum computing has introduced new challenges since quantum adversaries $\unicode{x2013}$ equipped with the power of quantum queries $\unicode{x2013}$ can find collisions much more efficiently. Brassard, Höyer and Tapp and Aaronson and Shi established that full-scale quantum adversaries require $Θ(N^{1/3})$ queries to find a collision, prompting a need for longer hash outputs, which impacts efficiency in terms of the key lengths needed for security. This paper explores the implications of quantum attacks in the Noisy-Intermediate Scale Quantum (NISQ) era. In this work, we investigate three different models for NISQ algorithms and achieve tight bounds for all of them: (1) A hybrid algorithm making adaptive quantum or classical queries but with a limited quantum query budget, or (2) A quantum algorithm with access to a noisy oracle, subject to a dephasing or depolarizing channel, or (3) A hybrid algorithm with an upper bound on its maximum quantum depth; i.e., a classical algorithm aided by low-depth quantum circuits. In fact, our results handle all regimes between NISQ and full-scale quantum computers. Previously, only results for the pre-image search problem were known for these models by Sun and Zheng, Rosmanis, Chen, Cotler, Huang and Li while nothing was known about the collision finding problem.

Motivation & Objective

• To address the lack of general techniques for proving query lower bounds in hybrid quantum-classical algorithms with black-box access to a random oracle.
• To resolve the open problem of determining the optimal quantum-classical tradeoff for the collision finding problem in the random oracle model.
• To develop a unified framework that consistently records knowledge from both classical queries (standard basis) and quantum queries (Fourier basis).
• To provide a more practical model for quantum algorithm analysis by accounting for realistic constraints such as limited quantum query access and high quantum query costs.
• To support post-quantum cryptography and near-term quantum computing by enabling tighter resource estimates and security proofs under hybrid access models.

Proposed method

• Proposes the hybrid compressed oracle framework as an extension of Zhandry's compressed oracle technique, enabling consistent recording of queries in incompatible bases.
• Introduces a transcript-based approach that tracks knowledge gain from both classical and quantum queries using orthogonal projectors in standard and Fourier bases.
• Employs a hybrid query model where algorithms make a bounded number of quantum queries (in superposition) and classical queries (in standard basis), with the framework tracking their combined progress.
• Applies the framework to derive bounds on query complexity by decomposing the state into subspaces based on query history and measuring overlaps via projectors.
• Uses inequalities such as Cauchy–Schwarz and triangle inequality to bound error terms arising from overlapping query subspaces.
• Leverages structural properties of projectors (e.g., orthogonality, completeness) to derive recursive bounds on the probability of success across query layers.

Experimental results

Research questions

• RQ1 What is the optimal tradeoff between quantum and classical queries for collision finding in the random oracle model?
• RQ2 Can a unified framework be developed to analyze hybrid quantum-classical query algorithms that treat classical and quantum queries consistently?
• RQ3 How can query lower bounds be established for hybrid algorithms when existing techniques fail to account for incompatible query bases?
• RQ4 What are the implications of this framework for post-quantum cryptography and near-term quantum computing with limited quantum depth?
• RQ5 Can the framework simplify or improve upon existing lower bounds for the search problem in the hybrid query model?

Key findings

• The paper establishes the optimal quantum-classical tradeoff for the collision finding problem in the random oracle model, resolving an open question posed by Rosmanis.
• The hybrid compressed oracle framework provides a simpler and more systematic proof of the optimal tradeoff for the search problem compared to prior work.
• The framework achieves tight bounds by modeling knowledge gain from both classical and quantum queries using orthogonal projectors in incompatible bases.
• The analysis shows that the success probability of a hybrid algorithm is limited by the sum of contributions from different query subspaces, with error terms bounded using Cauchy–Schwarz and triangle inequalities.
• The framework demonstrates that quantum speedups are constrained when quantum queries are costly or limited, even with many classical queries.
• The results imply that algorithms for preimage search and collision finding cannot achieve significant speedups with only a few quantum queries, even if classical query counts are high.