Quantum Networks

From the point of view of quantum coordination the most crucial services of a quantum network are:

• EPR-pair distribution
• multi-partite state distribution

Multi-partite state distribution

In general there’re two approaches [GHZGeneration]:

• Centralized: one node generates, then the state is distributed
• Distributed: iterative process (smaller GHZ to larger GHZ)

States

• GHZ-state: $\frac{1}{\sqrt{2}}(|0\dots 0⟩+|1\dots 1⟩)$
• W-state: $\frac{1}{\sqrt{n}}(|10\dots 0⟩+|01\dots 0⟩+|0\dots 01⟩)$

Objectives:

• expected generation time (e.g., subject to fidelity constraints)
• maximize fidelity (e.g., subject to available initial resources, say BPs)
• maximize the size of the GHZ state (e.g., subject to initial resources, again)

Applications:

• Quantum anonymous private information retrieval [QAPIR]

  Use of states: $n$-partite $d$-dimensional GHZ states are used to:

  • quantum anonymous broadcast
  • quantum anonymous entanglement generation
  • quantum anonymous entanglement verification
  • quantum private information retrieval

  Warning

  $d\ge N$ where $N$ is the number of entries in the “database”

• Quantum Secret Sharing [qss]

  Use of states:

  Alice (sender), Bob and Charlie

  1. GHZ states (say, 3-partite) are shared
  2. Parties measure in random bases, announcing the choices
  3. Bob and Charlie use measurements to figure out Alice’s $\Rightarrow$ shared key
  4. use the key to share a message

  NB multiple measurements are needed to

  • get long shared key
  • detect eavesdropper

  Note

  Extension to more parties is possible, but apparently requires all but given only for $n=4$

• Quantum Anonymous Transmission [anonymous_transmission]

  Use of states:

  • for a unique sender one GHZ state is enough:
    1. sender applies $Z$ if input is zero
    2. everyone measures in $X$ basis
    3. done
  • for multiple senders $\log n$ states aare needed for collision detection

  Sub-applications:

  • electronic auctions
  • voting protocols

• Conference Key Agreement [conference_key_agreement]

  Use of states:

  1. multipartite states (GHZ or W) states is distributed

  2. each party measures in random basis according to the protocol

     • test rounds
     • key generation rounds

  3. eavesdroppers are detected

  4. errors are corrected

  5. hash function amplifies privacy

  NB key rates are used a metric

  Several variations of the protocol using GHZ states are presented

• Controlled multi-qubit operations [ghz_dqc]

  Use of states:

  1. a GHZ state is distributed
  2. using LOCC and $\alpha |0⟩+\beta |1⟩$ we get a “cat-like” state $\alpha |0\dots 0⟩+\beta |1\dots 1⟩$
  3. any qubit can now be used as a control for $t$
  4. cat-disentangler restores the true control qubit.

• Quantum sensing [quantum_sensing]

• (Randomized) Leader Election [CompPowerWvsGHZ]

  Use of states: Take $W$-state and measure.

• Global coin and consensus

  Use of states:

  • One GHZ state is enough to toss a weak global coin

Generation and Distribution of GHZ States in Quantum Networks

[GHZGeneration]

Goal: minimize expected generation time given (indirect) fidelity constraints

Approach:

• Fusion-Retain: (the vertex in common is retained)

  1. do a CNOT
  2. measure target in Z
  3. flip the rest if needed

• Fusion-Discard (the vertex in common is discarded)

  1. do Fusion-Retain
  2. measure control in X
  3. do a correction with a random phase-flip

Distributing Graph States Across Quantum Networks

[GraphStates]

Graph state:

• $G=(V,E)$, where vertices are qubits
• Initially all qubits are in $|+⟩$ state, followed by the controlled $Z$ operation (phase flip)

Role of graph states:

• any quantum computation can be done in a “one-way” fashion:

  1. prepare a graph state
  2. perform measurements
  3. perform single qubit operations based on 2

Protocols for Creating and Distilling Multipartite GHZ States With Bell Pairs

[GHZ]

Setting:

• to correct and track errors it is necessary to perform joint measurements
• joint measurements in a distributed quantum computing are enabled by GHZ states
• quantum networks should provide this states

Goal: produce GHZ states that at fast rate and with high fidelity. I.e., the algorithm tries to find the protocol maximizing fidelity $F_{\mathrm{GHZ}}$ given the input fidelities $F_{\mathrm{Bell}}$ and the number $k$ of BPs to consume.

Warning

There’s no decoherence and/or local gate noise

Challenges:

• if GHZ states is produced by fusing Bell pairs, the fidelity degrades exponentially
• distillation or purification

Approach:

Some form of dynamic programming.

Multipartite Entanglement Distribution in the Quantum Internet: Knowing When to Stop!

[GHZStop]

Idea: sometimes one doesn’t need to build the whole GHZ state, only a (sufficiently large) subset is enough

Setting:

• a supernode generating GHZ states
• clients connected to a supernode via quantum channels that can generate EPR pairs
• Main application: distributed quantum sensing

Goal:

• average distribution time: the average number of time slots before the distribution is arrested
• average cluster size: the average number of client nodes entangled

Improved Routing of Multiparty Entanglement over Quantum Networks

[ImprovedGHZRouting]

Goal: maximize the size of GHZ state starting from some initial graph state (e.g., grid-like)