Linear-optical hyperentanglement-assisted quantum error-correcting code

We propose a linear-optical implementation of a hyperentanglement-assisted quantum error-correcting code. The code is hyperentanglement assisted because the shared entanglement resource is a photonic state hyperentangled in polarization and orbital angular momentum. It is possible to encode, decode, and diagnose channel errors using linear-optical techniques. The code corrects for polarization “flip” errors and is thus suitable only for a proof-of-principle experiment. The encoding and decoding circuits use a Knill-Laflamme-Milburn-like scheme for transforming polarization and orbital angular momentum photonic qubits. A numerical optimization algorithm finds a unit-fidelity encoding circuit that requires only three ancilla modes and has success probability equal to 0.0097.

Figure 1

(Color online) The operation of the hyperentanglement-assisted quantum code. The qubits labeled as $A$ and $A_1$ (red in the online version) belong to the sender Alice and the qubit labeled as $B_1$ (blue in the online version) belongs to the receiver Bob though all qubits belong to Bob after the noisy channel. Alice and Bob share a hyperentangled state ${\mid {\Phi }^{+}⟩}^{A_1{B}_1}$ prior to quantum communication. Alice uses the hyperentangled state to aid in encoding an information photon in the state ${\mid \psi ⟩}^A$. Her encoding circuit consists of one controlled-phase gate. She sends her photons over a noisy polarization-error quantum channel. Bob receives the photons, performs a decoding circuit, and performs two single-photon polarization-OAM analyses in the basis ${\varphi }^{\pm },{\psi }^{\pm }$ on the systems $A_1$ and $B_1$ to determine the error syndrome. Bob finally performs a recovery operation to obtain the information photon ${\mid \psi ⟩}^A$ that Alice first sent.

Figure 2

(Color online) The figure displays the results of the numerical gate optimization algorithm for our case of six computational modes and three ancilla modes. The algorithm starts an optimization at a randomly selected point in the space of $6\times 6$ matrices and maximizes the gate fidelity. The second stage of the optimization algorithm is a constrained optimization of success probability within the unit fidelity subspace. The figure displays ordered success probability for the best 500 optimization cycles of the total amount of 2000 optimizations for fidelity $>1\text{–}{10}^{-7}$. The algorithm finds an encoding circuit with a success probability almost 2 times that of the Knill combination scheme.