Shallow Quantum Hashing Achieves Equivalent Collision Resistance with Depth 1 Circuitry

Hashing forms the bedrock of many efficient algorithms and protocols, enabling substantial reductions in computational space, and researchers continually seek ways to optimise this crucial technique. Ilnar Zinnatullin from Kazan Federal University and Alexander Vasiliev, also at Kazan Federal University, now demonstrate a powerful connection between shallow hashing and single-qubit hashing for amplitude-based systems. Their work establishes that a circuit of just one layer achieves the same level of collision resistance previously requiring a more complex, two-layer circuit, representing a significant advance in quantum information processing. This simplification promises to accelerate the development of more compact and efficient quantum algorithms, potentially unlocking new possibilities in fields like data analysis and machine learning.

Current quantum machines are NISQ (Noisy Intermediate-Scale Quantum) devices, susceptible to decoherence and noise, and lack fault tolerance, making time-efficient quantum algorithms essential. The team proposes a quantum hash function that achieves equivalent collision resistance to shallow quantum hashing, but with significantly improved time efficiency, requiring no quantum entanglement and potentially simplifying practical implementation.

Efficient Quantum Hashing With Simplified Circuits

The research team significantly advanced quantum hashing techniques by developing a highly efficient circuit for the amplitude form of shallow quantum hashing, reducing its circuit depth to one. This streamlines the process for current quantum hardware, benefiting from circuits requiring only single-qubit rotations around the y-axis on the Bloch sphere. Researchers engineered a method that replaces complex multi-qubit controlled rotations with simpler two-qubit controlled rotations, substantially decreasing computational demands. The core of the method involves defining a classical-quantum function that maps elements of a set to quantum states using single-qubit operations.

For a given input, the amplitude-form quantum hash is constructed from n single qubits, each defined by a rotation angle determined by a set of parameters. The team rigorously demonstrated that for any two distinct inputs, the overlap between their quantum hashes is minimized, ensuring effective collision resistance. This is achieved through careful selection of the parameter set, guaranteeing that the resulting quantum states are nearly orthogonal. To validate the approach, the researchers mathematically proved that the overlap between quantum hashes is directly proportional to the product of cosine terms, each associated with a specific parameter and the difference between the input values, allowing for precise control over the degree of orthogonality and reliability of the hashing process.

Efficient Quantum Hashing with Minimal Collisions

Scientists have achieved a significant breakthrough in quantum hashing, developing methods to efficiently map data to quantum states while minimizing collisions. The team proposes a circuit of depth 1 for shallow hashing, a substantial improvement over previous designs. The research centers on creating quantum hashes that are “collision-resistant”, meaning that different inputs produce distinct quantum states with a high degree of separation. The team defines a function that maps inputs to n single qubits, utilizing rotations about the y-axis on the Bloch sphere. Crucially, they demonstrate that for any two distinct inputs, the overlap between their quantum hashes is minimized.

Specifically, the team proved that this overlap is equal to the product of n cosine terms, each representing the angle of rotation for a single qubit. Further investigation led to the discovery of a set, S, containing only O(log q) elements, which guarantees collision resistance. This means that by carefully selecting a small set of parameters, the team can create a hashing function that effectively distinguishes between different inputs. The team then extended this work to shallow quantum hashing, defining a function that maps inputs to quantum states using an ε-biased set. This approach utilizes a circuit to construct the quantum state. A key finding is that by expressing the elements of the ε-biased set as linear combinations of elements from the set S, the team can simplify the circuit design, reducing the computational complexity and paving the way for practical applications in quantum data processing and security.

This achievement lies in demonstrating that a complex hashing process, previously requiring multiple quantum operations, can be realized using a simpler, single-qubit approach without compromising security. This simplification represents a step towards more efficient quantum communication protocols and cryptographic systems. The authors acknowledge that further research is needed to fully explore the potential of this method and to assess its performance in various communication scenarios, potentially focusing on optimizing the function for specific hardware platforms and investigating its resilience against advanced attacks.