Multiplexed Quantum Communication with Surface and Hypergraph Product Codes

TL;DR

The paper investigates quantum multiplexing as a route to scalable interconnects for distributed quantum processing by encoding multiple qubits into a single photon via photonic DOFs such as polarization and time-bin. It develops and tests three error-corrected multiplexed communication strategies, analyzes their impact on logical error rates for surface codes and hypergraph product (HGP) codes, and introduces code-aware qubit-to-photon assignment strategies to mitigate correlated loss errors. For surface codes, random+threshold and distance-maximizing strategies often improve throughput while maintaining fault-tolerance, though benefits shrink with larger multiplexing unless code size grows. For HGP codes, a pruned-peeling + VH decoder enables practical decoding with multiplexing, and several assignment strategies (notably sudoku and diagonal) can match or exceed the no-multiplexing performance, underscoring the potential of multiplexed interconnects and memory in scalable quantum architectures.

Abstract

Connecting multiple processors via quantum interconnect technologies could help overcome scalability issues in single-processor quantum computers. Transmission via these interconnects can be performed more efficiently using quantum multiplexing, where information is encoded in high-dimensional photonic degrees of freedom. We explore the effects of multiplexing on logical error rates in surface codes and hypergraph product codes. We show that, although multiplexing makes loss errors more damaging, assigning qubits to photons in an intelligent manner can minimize these effects, and the ability to encode higher-distance codes in a smaller number of photons can result in overall lower logical error rates. This multiplexing technique can also be adapted to quantum communication and multimode quantum memory with high-dimensional qudit systems.

Figures (21)

• Figure 1: An example of an optical circuit encoding $2^2$-dimensional quantum information into a single photon. A horizontal (H) polarized photon is sent through a half-quarter-half wave plate appropriately rotated to generate a general polarized qubit. Then a polarizing beam splitter (PBS) divides the components of such a qubit into two spatial modes. Specifically, the horizontal component is transmitted whereas the vertical component is reflected. A delay is added to these components, and two optical switches and another PBS will recombine all the components into a single spatial mode. (B) A heralding method to transfer a general state of a multiplexed photon into two matter qubits $m_1$ and $m_2$ (red dots into boxes) by using a switching setup (Sw) module, which has been used in piparo2019quantum. (C) A heralding method to transfer a general state of two atoms into a multiplexed photon. Here a polarized photon interacts with an atom $m_1$ and then goes through a Sw module to add time-bin modes. Then this photon interacts with another atom $m_2$. We use $\alpha\ket{00} + \beta\ket{01}+\gamma\ket{10}+\delta\ket{11}$ for the input state of two-qubit memory. Upon a successful measurement in the $X$-basis of the two atoms, their state will be transferred into the multiplexed photon. (D) The equivalent circuit of the setups in (B) ((C)), respectively. The interaction of a polarized photon with matter qubit $m_1$ ($m_2$) and then with matter qubit $m_2$ ($m_1$) corresponds to a CNOT gate in which the atom is the control qubit and the polarization DOF of the photon is the target qubit. Although not explicitly shown, at the end of the circuit the photon and matter qubits are measured for the setup in (B) and (C), respectively.
• Figure 2: Flow of quantum communication with the surface code using multiplexed photons. In the first step, a quantum state is encoded into a surface code. Each circle with a number inside denotes the physical data qubits, and the grey circles without any numbers are auxiliary qubits used for stabilizer measurements. For the second step in a quantum multiplexing scenario, one assigns each physical qubit of the codeword to single photons using an assignment strategy. For instance, in this figure, two components of the time-bin DOF in each photon are used so that each photon can encode two qubits. There is freedom in which qubit is assigned to which photon, so it is necessary to define a function mapping qubits to photons. We call this function the assignment strategy. Here, the colors of the qubits indicate the photon to which it is assigned based on the assignment strategy. Then, the encoded photons pass through a lossy channel. Here, we assume that we know which photons have been lost during the transmission (erasure channel). If a photon has been lost, all the qubits assigned to this photon have been lost. Finally, we demultiplex the received photons and decode it to a codeword of the surface code using the peeling decoder delfosse2020linear and a correction method for erasure errors as described in Sec. \ref{['sec_ER']}.
• Figure 3: Performance of $[\![2d^2,2,d]\!]$ surface codes in scenario (B) with about $100$ photons. Each curve shows the case with different code sizes and the number of qubits encoded in each photon. The logical error rate can be reduced by increasing the number of qubits per photon $m$ and the code distance $d$.
• Figure 4: Scenario (C) multiplexing performance for the $[\![200,2,10]\!]$ surface code with multiplexing using different values of $m$ (the number of qubits per photon). The assignment of qubits to photons is uniformly random. Increasing $m$ allows code words to be transmitted with fewer photons, but the logical error rate increases because multiple qubits in the same photon have strongly correlated errors.
• Figure 5: Examples showing possible assignments of qubits to photons. Each colored circle denotes a qubit, and the color indicates the photon to which the qubit is assigned. Each non-colored (gray) small circle denotes auxiliary qubits for stabilizer measurement. Strategy i (shown in (A)) minimizes the distance between qubits in the same photon, while strategy ii (shown in (B)) maximizes this distance. Note that this code is defined on the torus represented as a lattice with periodic boundary conditions.