Quantum Protocol Zoo - User contributions [en]

Trap Code for Quantum Authentication

2021-12-22T11:42:54Z

91.35.144.136:

==Notation==
*<math>\rho</math>: 1-qubit input state

==Protocol Description==
*'''''Encoding:'''''
#Input: <math>\rho</math>, pair of keys <math>k=(k_1, k_2)</math>
#Apply an <math>[[n,1,d]]</math> error correction code (corrects up to <math>t</math> errors, <math>d=2t+1</math>)
#Append an additional trap register of <math>n</math> qubits in state <math>|0\rangle\langle 0|^{\otimes n}</math>
#Append a second additional trap register of <math>n</math> qubits in state <math>|+\rangle\langle +|^{\otimes n}</math>
#Permute the total <math>3n</math>-qubit register by <math>\pi_{k_1}</math> according to the key <math>k_1</math>
#Apply a Pauli encryption <math>P_{k_2}</math> according to key <math>k_2</math>
*'''''Decoding:'''''
#Input: <math>\rho^\prime</math> (state after encoding), pair of keys <math>k=(k_1, k_2)</math>
#Apply <math>P_{k_2}</math> according to key <math>k_2</math>
#Apply inverse permutation <math>\pi_{k_1}^\dagger</math> according to the key <math>k_1</math>
#Measure the last <math>n</math> qubits in the Hadamard basis
#Measure the second last <math>n</math> qubits in the computational basis </br>a. If the two measurements result in <math>|+\rangle\langle +|</math> and <math>|0\rangle\langle 0|</math>, an additional flag qubit in state <math>|\mathrm{ACC}\rangle\langle\mathrm{ACC}|</math> is appended and the quantum message is decoded according to the error correction code </br>b. Otherwise, an additional flag qubit in state <math>|\mathrm{REJ}\rangle\langle\mathrm{REJ}|</math> is appended and the (disturbed) encoded quantum message is replaced by a fixed state <math>\Omega</math>

==References==
#[https://arxiv.org/pdf/1211.1080.pdf| Broadbent et al. (2012)]
#[https://arxiv.org/pdf/1607.03075.pdf| Broadbent and Wainewright (2016).]
<div style='text-align: right;'>''contributed by Shraddha Singh and Isabel Nha Minh Le''</div>

Clifford Code for Quantum Authentication

2021-12-22T11:41:52Z

91.35.144.136:

The Clifford Authentication Scheme was introduced in the paper [https://arxiv.org/pdf/0810.5375.pdf| Interactive Proofs For Quantum Computations by Aharanov et al.]. It applies a random Clifford operator to the quantum message and an auxiliary register and then measures the auxiliary register to decide whether to accept or abort for [[Authentication of Quantum Messages|quantum authentication]].

'''Tags:''' [[:Category:Two Party Protocols|Two Party Protocol]][[Category:Two Party Protocols]]

==Outline==
The Clifford code encodes a quantum message by appending an auxiliary register with each qubit in state <math>|0\rangle</math> and then applying a random Clifford operator on all qubits. The authenticator then measures only the auxiliary register. If all qubits in the auxiliary register are still in state <math>|0\rangle</math>, the authenticator accepts and decodes the quantum message. Otherwise, the authenticator aborts the process.

==Notations==
*<math>\mathcal{S}</math>: suppliant (sender)
*<math>\mathcal{A}</math>: authenticator (prover)
*<math>\rho</math>: <math>m</math>-qubit state to be transmitted
*<math>d\in\mathbb{N}</math>: security parameter defining the number of qubits in the auxiliary register
*<math>\{C_k\}</math>: set of Clifford operations on <math>n</math> qubits labelled by a classical key <math>k\in\mathcal{K}</math>

==Properties==
*The Clifford code makes use of <math>n=m+d+1</math> qubits
*The Clifford code is [[Authentication of Quantum Messages|quantum authentication]] scheme with security <math>2^{-d}</math>
*The qubit registers used can be divided into a message register with <math>m</math> qubits, an auxiliary register with <math>d</math> qubits, and a flag register with <math>1</math> qubit.

==Protocol Description==
*'''Input:''' <math>\rho</math>, <math>d</math>, <math>k</math>
*'''Output:''' Receiver accepts or rejects
**'''''Encoding:''''' <math>\mathcal{E}_k: \rho \mapsto C_k\left( \rho \otimes |0\rangle\langle 0|^{\otimes d} \right)C_k^\dagger</math>
#<math>\mathcal{S}</math> appends an auxiliary register of <math>d</math> qubits in state <math>|0\rangle\langle 0|</math> to the quantum message <math>\rho</math>, which results in <math>\rho\otimes|0\rangle\langle0|^{\otimes d}</math>.
#<math>\mathcal{S}</math> then applies <math>C_k</math> for a uniformly random <math>k\in\mathcal{K}</math> on the total state.
#<math>\mathcal{S}</math> sends the result to <math>\mathcal{A}</math>.
**'''''Decoding:''''' Mathematically, the decoding process is described by <math display=block>\mathcal{D}_k: \rho^\prime \mapsto \mathrm{tr}_0\left( \mathcal{P}_\mathrm{acc} C_k^\dagger (\rho^\prime) C_k \mathcal{P}_\mathrm{acc}^\dagger \right) \otimes |\mathrm{ACC}\rangle\langle \mathrm{ACC}| + \mathrm{tr}\left( \mathcal{P}_\mathrm{rej} C_k^\dagger (\rho^\prime) C_k \mathcal{P}_\mathrm{rej}^\dagger \right) \Omega \otimes |\mathrm{REJ}\rangle\langle\mathrm{REJ}|</math> In the above, <math>\mathrm{tr}_0</math> is the trace over the auxiliary register only, and <math>\mathrm{tr}</math> is the trace over the quantum message system and the auxiliary system. Furthermore, <math>\mathcal{P}_\mathrm{acc}=\mathbb{1}^{\otimes n} \otimes |0\rangle\langle 0|^{\otimes d}</math> and <math>\mathcal{P}_\mathrm{rej}=\mathbb{1}^{\otimes (n+d)} - \mathcal{P}_\mathrm{acc}</math> are projective measurement operators.
#<math>\mathcal{A}</math> applies the inverse Clifford <math>C_k^\dagger</math> to the received state, which is denoted by <math>\rho^\prime</math>.
#<math>\mathcal{A}</math> measures the auxiliary register in the computational basis.</br>a. If all <math>d</math> auxiliary qubits are 0, the state is accepted and an additional flag qubit in state <math>|\mathrm{ACC}\rangle\langle\mathrm{ACC}|</math> is appended.</br>b. Otherwise, the remaining system is traced out and replaced with a fixed <math>m</math>-qubit state <math>\Omega</math> and an additional flag qubit in state <math>|\mathrm{REJ}\rangle\langle \mathrm{REJ}|</math> is appended.

==References==
#[https://arxiv.org/pdf/0810.5375.pdf| Aharanov et al. (2008).]
#[https://arxiv.org/pdf/1607.03075.pdf| Broadbent and Wainewright (2016).]
<div style='text-align: right;'>''contributed by Shraddha Singh and Isabel Nha Minh Le''</div>

Polynomial Code based Quantum Authentication

2021-12-22T11:38:05Z

91.35.144.136:

The paper [https://arxiv.org/pdf/quant-ph/0205128.pdf Authentication of Quantum Messages by Barnum et al.] provides a non-interactive scheme with classical keys for the sender to encrypt as well as [[Authentication of Quantum Messages|authenticate quantum messages]]. It was the first protocol designed to achieve the task of authentication for quantum states, i.e. it gives the guarantee that the message sent by a party (suppliant) over a communication line is received by a party on the other end (authenticator) without having been tampered with or modified by the dishonest party (eavesdropper).

'''Tags:''' [[:Category:Two Party Protocols|Two Party Protocol]][[Category:Two Party Protocols]], [[:Category:Quantum Functionality|Quantum Functionality]][[Category:Quantum Functionality]], [[:Category:Specific Task|Specific Task]][[Category:Specific Task]], [[:Category:Building Blocks|Building Block]][[Category:Building Blocks]]

==Outline==
The polynomial code consists of three steps: preprocessing, encryption and encoding, and decoding and decryption. Within the preprocessing, sender and receiver agree on a [[Stabilizer Purity Testing Code | stabilizer purity testing code]] and three private, random binary keys. Within the encryption and encoding step, the sender uses one of these keys to encrypt the original message. Consequently, a second key is used to choose a specific quantum error correction code out of the [[Stabilizer Purity Testing Code | stabilizer purity testing code]]. The chosen quantum error correction code is then used, together with the last key, to encode the encrypted quantum message. Within the last step, the decoding and decryption step, the respective keys are used by the receiver to decide whether to abort or not, and if not, to decode and decrypt the received quantum message.

==Assumptions==
*The sender and the receiver share a private, classical random key drawn from a probability distribution

==Notations==
*<math>\mathcal{S}</math>: suppliant (sender)
*<math>\mathcal{A}</math>: authenticator (prover)
*<math>\rho</math>: quantum message to be sent
*<math>m</math>: number of qubits in the message <math>\rho</math>
*<math>\{Q_k\}</math>: [[Stabilizer Purity Testing Code | stabilizer purity testing code]], each stabilizer code is identified by index <math>k</math>
*<math>n</math>: number of qubits used to encode the message with <math>\{Q_k\}</math>
*<math>x</math>: random binary <math>2m</math>-bit key
*<math>y</math>: random syndrome for a specific <math>Q_k</math>

==Protocol Description==
Add Input and Output for each subroutine
*'''Input:''' <math>\rho</math> owned by <math>\mathcal{S}</math>; <math>k</math>, <math>x</math>, <math>y</math> shared among <math>\mathcal{S}</math> and <math>\mathcal{A}</math>
*'''Output:''' Receiver: accepts or aborts <math>\rho^\prime</math>
**'''''Encryption and encoding:'''''
#<math>\mathcal{S}</math> q-encrypts the <math>m</math>-qubit original message <math>\rho</math> as <math>\tau</math> using the classical key <math>x</math> and a [[Quantum One-Time Pad | quantum one-time pad]]. This encryption is given by <math>\tau = \sigma_x^{\vec{t}_1}\sigma_z^{\vec{t}_2}\rho\sigma_z^{\vec{1}_1}\sigma_x^{\vec{t}_1}</math>, where <math>\vec{t}_1</math> and <math>\vec{t}_2</math> are <math>m</math>-bit vectors and given by the random binary key <math>x</math>.
#<math>\mathcal{S}</math> then encodes <math>\tau</math> according to <math>Q_k</math> with syndrome <math>y</math>, which results in the <math>n</math>-qubit state <math>\sigma</math>. This means <math>\mathcal{S}</math> encodes <math>\rho</math> in <math>n</math> qubits using <math>Q_k</math>, and then "applies" errors according to the random syndrome.
#<math>\mathcal{S}</math> sends <math>\sigma</math> to <math>\mathcal{A}</math>.
**'''''Decoding and decryption:'''''
#<math>\mathcal{A}</math> receives the <math>n</math> qubits, whose state is denoted by <math>\sigma^\prime</math>.
#<math>\mathcal{A}</math> measures the syndrome <math>y^\prime</math> of the code <math>Q_k</math> on his <math>n</math> qubits in state <math>\sigma^\prime</math>.
#<math>\mathcal{A}</math> compares the syndromes <math>y</math> and <math>y^\prime</math> and aborts the process if they are different.
#<math>\mathcal{A}</math> decodes his <math>n</math>-qubit word according to <math>Q_k</math> obtaining <math>\tau^\prime</math>.
#<math>\mathcal{A}</math> q-decrypts <math>\tau^\prime</math> using the random binary strings <math>x</math> obtaining <math>\rho^\prime</math>.

==Further Information==

==References==
#[https://arxiv.org/pdf/quant-ph/0205128.pdf| Barnum et al. (2002).]
<div style='text-align: right;'>''contributed by Shraddha Singh and Isabel Nha Minh Le''</div>