Advances MBQC with Binomial Codes and Cavity-Qed for Quantum Computing

Scientists are tackling a key challenge in measurement-based quantum computation (MBQC): creating the complex quantum states needed for photonic systems. G. P. Teja and Radim Filip, from the Department of Optics, Palacký University, alongside et al., demonstrate a novel approach utilising binomial codes , a simpler alternative to complex states like GKP , and cavity-QED tools to generate the necessary cluster states and Pauli measurements. This research is significant because it proposes a practical pathway for utilising existing atom-cavity architectures, potentially accelerating the development of scalable quantum technologies and offering a more experimentally feasible route to MBQC.

Binomial Codes and Photonic Quantum Computation

The team achieved this by focusing on binomial codes, which, unlike the more complex continuous-variable encodings like GKP states, present a simpler, experimentally viable alternative for encoding quantum information in photons. This innovative approach proposes the first concrete steps for adapting existing atom-cavity architectures to facilitate their use in advanced quantum processing. The study reveals a method for creating entangled resource states, specifically cluster states, essential for MBQC, through a carefully designed interaction between atoms and optical cavities. Researchers employed a cavity-QED protocol, leveraging the strong coupling between single atoms and the electromagnetic field within an optical cavity to conditionally generate these binomial code states.
This protocol incorporates a controlled-phase-flip operation, atomic rotations, and precise atomic measurements, all meticulously accounted for within a framework that addresses scattering losses using a Kraus operator approach. Numerical simulations, performed using QuTiP, demonstrate the evolution of the density matrix throughout these operations, validating the feasibility of the proposed method. Experiments show the ability to prepare states conforming to binomial code requirements, specifically those capable of correcting photon-loss errors, a major challenge in optical quantum computing. The team demonstrated the generation of states described by the equations |0⟩ + |4⟩/√2 and |2⟩, representing the foundational building blocks for more complex error-correcting codes.

Furthermore, the protocol can be extended to generate states necessary for correcting dephasing loss, expanding the robustness of the quantum computation. State quality was rigorously quantified using the fidelity metric F(ρ1, ρ2) = trp√ρ1ρ2√ρ1, ensuring the reliability of the generated quantum states. This work establishes a crucial link between theoretical MBQC principles and practical implementation using readily available atom-cavity systems. Unlike GKP codes, which rely on translation symmetry, binomial codes exhibit rotation symmetry, offering a unique advantage in manipulating quantum information. The research highlights the potential for utilising existing superconducting platforms, where binomial codes have already been successfully demonstrated, as a benchmark for optical implementations.,.

Binomial Code Cluster State Generation via Cavity QED

This work details a method for optical cat-state generation, employing cavity QED to conditionally filter binomial code superpositions from Gaussian inputs via a controlled-phase-flip (CPF) operation, U(φ) = eiφn ⊗|g⟩⟨g|+I ⊗|s⟩⟨s|, followed by atomic rotation R and subsequent atomic measurement. The crucial CPF is realised through atom-cavity reflections, and researchers accounted for scattering losses using a Kraus operator framework, Eφ(ρ) →P j Kj(φ)ρK† j (φ), avoiding complex virtual cavity methods. The density matrix evolution incorporates noise dependent on the cooperativity C = g2/(κγ) and cavity efficiency β = κc/κ, where g, γ, κc, and κl represent atom-cavity coupling, atomic decay, cavity emission, and loss rates, respectively; numerical simulations were performed using QuTiP. The atom-light state transforms after these three operations as O(φ, R, m) ≡|m⟩⟨m| ⊗R[α, β, ζ] ⊗Eφ(ρ) and ρ(n+1) ≡O Eφ(ρ(n)) O†/Tr[ O Eφ(ρ(n)) O†], where |m⟩∈{|g⟩, |s⟩}, R is a unitary matrix, and ρ(n) is the light mode state after n atom-cavity iterations.

Scientists harnessed standard Jaynes-Cummings (JC) interactions to realise the CPF between an atomic qubit and an optical mode. Experiments began by conditionally preparing code and magic states from displaced squeezed vacuum states, defined as D(α)S(r) |0⟩= 5X n=0 cn |n⟩+ O(6) and |B⟩(θ,Φ) = cos θ 0 + eiΦ sin θ 1. To generate general superposition states, the team initially set Φ = 0 and optimised α and r to approximate binomial superpositions with arbitrary θ, ensuring negligible high Fock-state contributions, P5 n=0 |cn|2 ≈1, and matching target amplitude ratios c 1/c 0 = tan θ. While direct attainment was limited to θ = π/3.3, a second atom-cavity iteration with O(π/2, R[β, ζ], g) produced general binomial superposition states, where θ and Φ are controlled by atomic rotation, satisfying −cos ζ + eiβ sin ζ cos ζ + eiβ sin ζ tan π 3.3 = tan θ eiΦ. Atomic measurements proved profitable for generating high-fidelity target states; tracing out the atomic system after reflections yielded the |+⟩ state with fidelities of (0.50, 0.25), while projective measurements achieved higher fidelities (0.97, 0.98). The protocol successfully generated T-type and H-type magic states with fidelities exceeding 0.98, essential resources for non-Clifford gate teleportation, as demonstrated by density matrix elements showing well-preserved coherences ρmn = ⟨m|ρ|n⟩.,.

Binomial Code Cluster States via Cavity-QED

Experiments revealed the successful generation of star-shaped cluster states, with stabilizer measurements demonstrating the crucial role of low cavity losses in maintaining stabilizer values, essential for high-quality entanglement. The team measured expectation values ⟨C|Si|C⟩ for the generated cluster state, confirming that minimising cavity losses is paramount for preserving the integrity of the quantum information. Data shows that teleportation fidelities of {0.98, 0.96} were achieved for β = {0.999, 0.99} when teleporting a state using qubits 1-4 and projecting it onto qubit 5, surpassing the 2/3 threshold required for a reliable quantum channel. This breakthrough delivers a significant step towards practical quantum communication and computation.

Researchers employed photon-number-resolving detectors (PNRDs) and an ancillary state to perform conditional projective measurements |χ⟩t in the XY-plane on a cluster graph, enabling efficient manipulation of quantum states. Table 0.4(e) presents fidelities of post-measurement states with respect to ideal projected states, demonstrating consistent performance for both 3-chain and 5-star cluster states across varying cavity losses β. Measurements in the XY-plane are sufficient for MBQC, although a Z-measurement can be obtained through a single atom-cavity iteration. The ancillary state (|A⟩t) and the corresponding positive operator-valued measure (POVM) (|χ⟩t) are defined by equations (11a) and (11b) respectively, allowing for precise control over the measurement process.

To quantify the impact of noise, the team analysed measurements performed on both 3-chain and 5-star cluster states, revealing that the performance remains consistent even with varying projection parameters and measurement locations. The CZ-gate between photons is implemented via an atom-light CZ operation, naturally extending to the creation of hybrid cluster states combining atomic and photonic qubits. Furthermore, the work proposes replacing face-center qubits with atomic qubits, simplifying the circuit and requiring only atom-cavity reflections to generate hybrid cluster states. Scientists anticipate that atom-cavity and atom-waveguide systems, with coupling strengths ranging from MHz to GHz, will provide an ideal testing ground for these components, enabling future advancements in autonomous error correction and POVM implementations based on homodyne detection.,.

Binomial Code MBQC with Atom-Cavity Systems

The proposed methods demonstrate consistent performance across varying cavity losses and measurement parameters, suggesting robustness in practical implementations. This work establishes key components for MBQC, including deterministic creation of superposition states, CZ-gates, cluster states, and conditional Pauli measurements within the binomial code space, all crucial for scalable quantum computation. Atom-cavity and atom-waveguide systems, with coupling strengths ranging from MHz to GHz, are identified as ideal platforms for testing these components, particularly those already demonstrating cat-state and cluster-state generation, as well as nonlinear phase shifts. Furthermore, the approach allows for the creation of hybrid cluster states combining atomic and photonic qubits, potentially simplifying circuits and leveraging the strengths of both qubit types. The authors acknowledge limitations related to deterministic code generation and autonomous error correction under photon loss, suggesting these as areas for future investigation. They also propose exploring POVM implementations based on homodyne detection to further simplify resource requirements.