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arXiv:quant-ph/0102001 · Physical Review Letters

Quantum Fingerprinting

Harry Buhrman, Richard Cleve, John Watrous, Ronald de Wolf

Published Sep 26, 2001• 1,188 citations
arXivHTMLPDF

Abstract

Classical fingerprinting associates with each string a shorter string (its fingerprint), such that any two distinct strings can be distinguished with small error by comparing their fingerprints alone. The fingerprints cannot be made exponentially smaller than the original strings unless the parties preparing the fingerprints have access to correlated random sources. We show that fingerprints consisting of quantum information can be made exponentially smaller than the original strings without any correlations or entanglement between the parties. This implies an exponential quantum/classical gap for the equality problem in the simultaneous message passing model of communication complexity.

DOI
10.1103/physrevlett.87.167902

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Classical fingerprinting associates with each string a shorter string (its fingerprint), such that any two distinct strings can be distinguished with small error by comparing their fingerprints alone. The fingerprints cannot be made exponentially smaller than the original strings unless the parties preparing the fingerprints have access to correlated random sources. We show that fingerprints consisting of quantum information can be made exponentially smaller than the original strings without any correlations or entanglement between the parties. This implies an exponential quantum/classical gap for the equality problem in the simultaneous message passing model of communication complexity.
We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical shared-randomness SMP protocols by means of quantum SMP protocols without shared randomness (Qpar-protocols). Our first result is to extend Yao's simulation to the strongest possible model: every many-round quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by Qpar-protocols. We apply our technique to obtain an efficient Qpar-protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols
We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical shared-randomness SMP protocols by means of quantum SMP protocols without shared randomness (Q∥Q^\parallelQ∥-protocols). Our first result is to extend Yao's simulation to the strongest possible model: every many-round quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by Q∥Q^\parallelQ∥-protocols. We apply our technique to obtain an efficient Q∥Q^\parallelQ∥-protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols.
One of the standard communication complexity problems is deciding whether bit strings held by two separate parties Alice (A) and Bob (B) are equal or different. In the simultaneous message passing model, A and B send certain information to a Referee, who makes the decision. The overall goal is to minimize the amount of communication involved while not exceeding desired error probability. This is achieved by sending only fingerprints of the original n-bit strings. In the best classical protocols, these scale as O(√n) if A and B have no access to shared randomness [1]. The recently proposed [2] and demonstrated quantum fingerprinting [3], exploiting phase-modulated weak coherent pulses, provides O(log2n) scaling in the number of used qubits, but needs phase stability between A and B. Here we introduce and analyse a quantum fingerprinting protocol based on two-photon interference which does require a shared phase reference and works in scenarios when the global phase of transmitted signals becomes completely random. Interestingly, the performance of this protocol, quantified in terms of the Chernoff information [4], is enhanced through the use of non-classical single-photon states, even in the regime of high transmission loss.
Fingerprinting enables two parties to infer whether the messages they hold are the same or different when the cost of communication is high: each message is associated with a smaller fingerprint and comparisons between messages are made in terms of their fingerprints alone. In the simultaneous message passing model, it is known that fingerprints composed of quantum information can be made exponentially smaller than those composed of classical information. For small message lengths, we present constructions of optimal classical fingerprinting strategies with one-sided error, in both the one-way and simultaneous message passing models, and provide bounds on the worst-case error probability with the help of extremal set theory. The performance of these protocols is then compared to that for quantum fingerprinting strategies constructed from spherical codes, equiangular tight frames and mutually unbiased bases.
When communication is expensive, the important task of testing two binary strings for equality can be achieved by generating relatively short fingerprints of each string and comparing those fingerprints. Quantum fingerprinting, in which the fingerprint is encoded onto qubits (the quantum version of the bit), is significantly less expensive: exponentially better in the case of no shared randomness and potentially perfect for the case of shared entanglement. Single-qubit quantum fingerprinting is feasible, and we have demonstrated its advantages theoretically and experimentally