Entangling spins by measuring charge: A parity-gate toolbox

Radu Ionicioiu

Phys. Rev. A 75, 032339 – Published 27 March 2007

Abstract

The parity gate emerged recently as a promising resource for performing universal quantum computation with fermions using only linear interactions. Here we analyze the parity gate ( $P$ gate) from a theoretical point of view in the context of quantum networks. We present several schemes for entanglement generation with $P$ gates and show that native networks simplify considerably the resources required for producing multiqubit entanglement, such as $n$ –Greenberg-Horne-Zellinger (GHZ) states. Other applications include a Bell-state analyzer and teleportation. We also show that cluster state fusion can be performed deterministically with parity measurements. We then extend this analysis to hybrid quantum networks containing spin and mode qubits. Starting from an easy-to-prepare resource (spin-mode entanglement of single electrons) we show how to produce a spin $n$ -GHZ state with linear elements (beam splitters and local spin flips) and charge-parity detectors; this state can be used as a resource in a spin quantum computer or as a precursor for constructing cluster states. Finally, we construct an alternative spin-controlled- $Z$ gate by using the mode degrees of freedom as ancillæ.

Received 15 September 2006

DOI:https://doi.org/10.1103/PhysRevA.75.032339

Authors & Affiliations

Radu Ionicioiu^*

Quantum Information Group, Institute for Scientific Interchange (ISI), Viale Settimio Severo 65, I-10133 Torino, Italy

^*Present address: Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom.

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 75, Iss. 3 — March 2007

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(Color online) (Top) A quantum network for a spin CZ gate using two spin-parity measurements (pink boxes). (Bottom) Each spin-parity gate (a) is constructed out of a charge-parity gate (white box) via charge-to-spin conversion (the CNOT gates between spin and charge qubits) as in (b). Implementation of the previous circuit using ballistic electrons (c), see Ref. 7; the gray box is a nonabsorbing charge-parity detector measuring the parity $n$ in the lower arm of the interferometer; all sums are mod 2.Reuse & Permissions
Figure 2
Gate identities for the parity gate. The $P$ gate commutes with (a) general $Z$ rotations $e^{i φ Z}$ , (b) global spin flips $X_{1} \otimes X_{2}$ , and (d) arbitrary controlled- $U$ gates. (c) Commuting with $X$ changes the value $p$ of the parity into $p \oplus 1$ . Since the $P$ gate is symmetric in the two inputs, all the commutation relations work on both qubit lines. (e) Two $P$ gates commute when acting on three qubits. (f) A quantum network model for the $P$ gate; the ancilla is symmetrically coupled to the two qubits and then measured, giving the parity of the input states $p = x + y \mod 2$ .Reuse & Permissions
Figure 3
(a) Producing Bell states with the parity gate. After the $P$ gate, the output state is $∣ B_{p 0} ⟩ = ∣ 0 ⟩ ∣ p ⟩ + ∣ 1 ⟩ ∣ p + 1 ⟩$ , where $p$ is the result of the parity measurement. (b) The $P$ gate leaves invariant the Bell states and outputs $p = i$ . (c) $H^{\otimes 2}$ maps Bell states into Bell states. (d) A quantum network for a Bell-state analyzer; the first $P$ gate measures $i$ , then $H^{\otimes 2}$ swaps $i$ and $j$ and the final $P$ gate measures $j$ . (e) Adding to the previous network $H^{\otimes 2}$ leaves the input Bell state $∣ B_{i j} ⟩$ invariant.Reuse & Permissions
Figure 4
A teleportation circuit using two $P$ gates and two Hadamard gates; note that no ancilla and no final measurement of Alice’s two qubits is needed. After the measurement, Alice remains with a Bell state ${(- 1)}^{p_{1} p_{2}} ∣ B_{p_{2} p_{1}} ⟩$ . Bob recovers $∣ ψ ⟩$ by applying a local transformation $Z^{p_{2}} X^{p_{1}}$ .Reuse & Permissions
Figure 5
(Color online) (a) Entanglement swapping between spin (red) and charge (blue) qubits. The initial state contains only single-particle entanglement (both electrons are entangled in spin charge) ${(∣ ↑ 0 ⟩ + ∣ ↓ 1 ⟩)}^{\otimes 2}$ . By performing a Bell-state measurement on the charge qubits we obtain a hyperentangled state of two electrons (entangled in both spin and charge) ${∣ EPR ⟩}_{σ} {∣ EPR ⟩}_{k}$ . (b) Generalizing to $n$ qubits; for simplicity, only the mode qubits are depicted, as no operation on spins is performed. The initial state is ${(∣ ↑ 0 ⟩ + ∣ ↓ 1 ⟩)}^{\otimes n}$ ; the output of the quantum network is ${∣ {GHZ}_{n} ⟩}_{σ} {∣ {GHZ}_{n} ⟩}_{k}$ .Reuse & Permissions
Figure 6
(Color online) The new spin-CZ gate using the mode qubits as ancillæ.Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information