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Russian Microelectronics

Constant-Depth Algorithm for Quantum Hashing

Alexander Vasiliev

Published Dec 1, 2023• 6 citations
Publisher

Abstract

Abstract Quantum hashing is one of the techniques that allow to construct space-efficient quantum algorithms and protocols. We present a construction that allows to make it extremely time efficient. Our approach is based on the shallow implementation of the quantum fingerprinting technique but uses additional circuit identities specific to the quantum hash function for the cyclic group. The resulting algorithm is equivalent to the one we have implemented earlier using single-photon states with orbital angular momentum encoding.

DOI
10.1134/s106373972360067x

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Abstract Quantum hashing is one of the techniques that allow to construct space-efficient quantum algorithms and protocols. We present a construction that allows to make it extremely time efficient. Our approach is based on the shallow implementation of the quantum fingerprinting technique but uses additional circuit identities specific to the quantum hash function for the cyclic group. The resulting algorithm is equivalent to the one we have implemented earlier using single-photon states with orbital angular momentum encoding.
Abstract A new version of quantum hashing technique is developed wherein a quantum hash is constructed as a sequence of single-photon high-dimensional states (qudits). A proof-of-principle implementation of the high-dimensional quantum hashing protocol using orbital-angular momentum encoding of single photons is implemented. It is shown that the number of qudits decreases with increase of their dimension for an optimal ratio between collision probability and decoding probability of the hash. Thus, increasing dimension of information carriers makes quantum hashing with single photons more efficient.
Quantum hashing is a promising generalization of the cryptographic hashing concept on the quantum domain. In this paper, we construct a quantum hash via a sequence of single-photon states and perform a proof-of-principle experiment using orbital angular momentum (OAM) encoding. We experimentally verify the collision resistance of the quantum hash function depending on the number of qubits in use. Based on these results, we conclude that theoretical estimates are confirmed for different bases of OAM states and the proposed technique can be useful in computational and cryptographic scenarios. The possibility of multiplexing different OAM bases can make this approach even more efficient.
Quantum fingerprinting is a technique that maps a classical input word to a quantum state. The obtained quantum state is much shorter than the original word, and its processing uses fewer resources, making it useful in quantum algorithms, communication, and cryptography. One of the examples of quantum fingerprinting is the quantum automata algorithm for M O D p = { a i · p ∣ i ≥ 0 } languages, where p is a prime number. However, implementing such an automaton on current quantum hardware is not efficient. Quantum fingerprinting maps a word x ∈{0, 1} n of length n to a state |ψ( x )〉 of O (log n ) qubits, and uses O ( n ) unitary operations. Computing quantum fingerprint using all available qubits of the current quantum computers is infeasible due to many quantum operations. To make quantum fingerprinting practical, we should optimize the circuit for depth instead of width, in contrast to the previous works. We propose explicit methods of quantum fingerprinting based on tools from additive combinatorics, such as generalized arithmetic progressions (GAPs), and prove that these methods provide circuit depth comparable to a probabilistic method. We also compare our method to prior work on explicit quantum fingerprinting methods. We provide a series of numerical experiments with implementation of the quantum automata for MOD 17 language on noisy simulators of IBMQ quantum devices. We show that shallow implementation based on GAPs produces results with much smaller computational error compared to standard deep circuit implementation. Despite the fact that on the ideal quantum computational device, the opposite situation arises. We show that the shallow circuit for the quantum automaton is better for near-future quantum computational devices.