Quantum Fingerprinting

This example protocol allows two parties (two quantum clients) to collaboratively compute the value of a function using both of their inputs while maintaining the privacy of their own input. The protocol does not permit the two parties to interact directly with each other, hence they send their respective inputs to a trusted third party (quantum server) who computes the correct value of the function corresponding to both inputs while minimising the amount of information sent by the two parties.

Minimisation of information is done by fingerprinting, which is a technique that associates a string to an exponentially smaller string (fingerprints) such that any two strings can by distinguished solely based on their fingerprints.

Tags: Fingerprinting

Outline

The two quantum clients want to compute a function using the input of both parties. In this protocol, we focus on a specific function, which is the equality function. This function provides boolean output 1 if the input of both the parties is the same, otherwise, the output is 0.

Preparation stage involves two parties (two quantum clients) preparing the fingerprints of their individual inputs and sending their quantum states to the server (trusted third party). The two clients do not possess any shared quantum key.

Next stage involves the server performing several operations on the fingerprints to check their equality. These operations are repeated several times on the same fingerprints to reduce the error probability.

• Client's preparation: The client prepares the fingerprint of initial input sized $n$-bits. This fingerprint has a length of $\log _{}n$ bits. The client now sends this fingerprint to the server through a quantum channel. Both the clients do this process simultaneously.

• Server's preparation: The server receives the two fingerprints from both the clients and prepares the operations to distinguish them. The server independently repeats the computation process with the fingerprints several times to reduce the error probability in detecting whether the equality of the two states.

• Measurement: The server measures the final states and announces the result.

If the initial input of the two parties are equal, this would be inferred from the corresponding fingerprints with no error probability and the outcome of the protocol would be correct. However, if the two fingerprints are of different inputs, there exists a non-zero probability that the outcome of the protocol is incorrect. Therefore, there exists a one sided error in the measurement. This one sided error is reduced by repeating the server operations several times with the same fingerprints.

Requirements

• Authenticated Quantum channel capable of sending a pair of qubits.
• Quantum memory for server to store the fingerprints.
• Measurement devices for the server.

Notation

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |h_x\rangle} , Quantum fingerprint for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -bit input $x$.
• $E(x)$, Fingerprint function associated with input ${x\in \{0,1\}^{n}}$ which maps $n$-bit input to $m$-bit fingerprint.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} , Fixed constant, such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta<0} .

Properties

• The computational complexity of this protocol is ${\mathcal {O}}(\log {}n)$.
• The two quantum clients have no shared key in this protocol.
• Given an $n$-bit input, the protocol requires a quantum fingerprint of minimum $\log {}n$ bits which contains quantum information.
• The quantum fingerprint is defined as the state $|h_{x}\rangle$, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {E(x)}} is the fingerprint of the input $x$. ${E_{i}(x)}$ is the ${i^{th}}$ bit of ${E(x)}$.
  

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |h_x\rangle = \frac{1}{\sqrt{m}}\sum_{n=1}^{m} |i\rangle|E_i(x)\rangle}

• The hamming distance i.e. the number of positions between two strings of same length, at which the corresponding symbols are different, of ${E(x)}$ and ${E(y)}$ is at least ${(1+\delta )m}$.
• Any two fingerprints, $|h_{x}\rangle$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |h_y\rangle} have an inner product of at most Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\delta}} .
• The one sided error probability is at least $({\frac {1+\delta }{2}})^{k}$ if the server computation is repeated $k$ times.
• Quantum memory would be required to store the fingerprints if the server operations are performed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} times.
• This protocol requires no quantum memory for the Client

Pseudocode

Input: ${x\in \{0,1\}^{n}}$ to First Party and ${y\in \{0,1\}^{n}}$ to Second Party.
Output: One bit by server satisfying the equality function with some error probability.
Stage 1: Client's preparation

• First Party prepares the fingerprint $|h_{x}\rangle$ from input $x$.
• Second Party prepares the fingerprint $|h_{y}\rangle$ from input Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} .
• Both parties transmit their fingerprints to the server.

Stage 2: Server's preparation

• Server prepares an ancilla qubit $|0\rangle$ for final measurement purpose, and thus starts with the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0\rangle|h_x\rangle|h_y\rangle} .
• Server creates an entangled state by applying the gate $G$.

$G={(H\otimes I)(c-SWAP)(H\otimes I)}$

• The server measures the first qubit and transmits the output to both the parties.

Related Papers

*contributed by Rhea Parekh