Homomorphic Aggregation of Continuous-Variable GKP States

Homomorphic aggregation of logical quantum information encoded in continuous-variable phase space is essential for distributed quantum computing. However, passive linear optics fail for non-Gaussian Gottesman-Kitaev-Preskill (GKP) codes due to symplectic lattice compression and entanglement-induced decoherence. We present an active, measurement-based framework for the homomorphic aggregation of multi-node GKP states. Utilizing GKP Bell states and homodyne feed-forward, we construct a completely positive trace-preserving map that computes the logical sum of distributed states while preserving the logical code space geometry up to correctable finite-squeezing deformations. We prove this protocol operates as an approximate quantum non-demolition measurement, bound its cryptographic leakage for continuous one-time pads, and derive analytical logical fidelity limits under finite-squeezing constraints.

The paper establishes a theoretical framework for continuous-variable quantum state aggregation that combines n independent GKP inputs while bypassing the mathematical impossibility of passive linear optical mixing. The specific contributions are fourfold: First, we formally prove the failure modes of passive spatial GKP aggregation, driven by deterministic symplectic lattice compression and entanglement-induced entropy injection. Second, we construct an active, measurement-based homomorphic router using GKP Bell states, preserving the logical code space geometry up to correctable finite-squeezing deformations via approximate quantum non-demolition (QND) measurements. Third, we prove that this measurement channel acts as a secure continuous quantum one-time pad (CV-OTP), bounding cryptographic leakage strictly as a function of the finite-energy Gaussian overlap. Finally, we derive analytical logical fidelity limits, demonstrating that scalable spatial aggregation physically mandates a logarithmic binary-tree routing topology with intermediate maximum-likelihood decoding to suppress the intrinsic O(n) MBQC variance accumulation.

Regarding the limitations of passive linear optics, passive linear optical operations correspond to orthogonal symplectic transformations S ∈ Sp(2n, R) ∩ O(2n). For a two-mode 50:50 beam splitter, projecting the output onto the primary spatial mode scales the local quadratures as q̂out,1 = (q̂1 + q̂2)/√2. Consequently, the effective local inter-codeword distance contracts by a factor of 1/√2, reducing the post-interaction distance to d min(1) = √(π/2). This in turn compresses the maximum correctable noise threshold to q thresh(1) = √(2π)/2 ≈ 0.6266. Because q thresh(1) < q thresh = √π/2 ≈ 0.8862, this strictly degrades the noise tolerance and violates the fault-tolerant subspace preservation condition. This geometric failure extends beyond the standard square lattice; while the hexagonal (A2) GKP code offers optimal phase-space packing and a strictly larger initial inter-codeword distance (d min ≈ 1.07√π), the 1/√2 symplectic scaling inevitably compresses its distance to ≈ 0.76√π, again falling fatally below the fault-tolerant threshold. Furthermore, attempting to bypass discrete lattice compression by utilizing a perfectly scale-invariant (dense) state is physically impossible; such states demand infinite energy and possess zero tolerance to physical displacement noise.

Additionally, because GKP states are highly nonclassical, applying the beam-splitter unitary to the separable input state yields a highly entangled output state. Tracing out the auxiliary mode produces a strictly mixed reduced density matrix for the primary mode, with strictly positive von Neumann entropy. This entropy injection bounds the logical purity to tr(ρ2 out,1) < 1, physically corrupting the lattice encoding and necessitating an active, measurement-based aggregation protocol.

The aggregation protocol uses two-mode GKP Bell states Φ+⟩ L,AB as foundational network resources, generated by mixing two independent GKP qunaught states. The protocol proceeds as follows: (1) Resource Generation: Prepare n−1 GKP Bell pairs; (2) Continuous Bell-State Measurement (BSM): project the joint subsystem onto the continuous Bell basis via balanced beam splitter coupling and dual homodyne detection to extract the continuous syndrome m; (3) Classical Feed-Forward: The BSM yields the continuous outcome m, and the active aggregation step operates as a CPTP map M defined by the CV gate teleportation integral, with Kraus operators K̂(m) = D̂ B(−m)⟨m jA Û BS(jA) (I j ⊗ Φ+⟩ L,AB). If the input is encrypted with a continuous displacement mask, the measured quadrature decomposes algebraically as q m = q state + q noise + q enc, and the feed-forward displacement operator actively translates the unmeasured mode B back into the logical code space.

For cryptographic security, Theorem 1 states that for a finite-energy GKP state with intrinsic variance σ2, homodyne syndrome extraction under a uniformly sampled displacement mask q enc ∼ U[−√π/2, √π/2) preserves information-theoretic security. The total variation distance between the conditional syndrome distributions for logical states 0⟩ L and 1⟩ L is bounded by a negligible function of the squeezing parameter r. The statistical difference becomes non-zero exclusively due to the probability mass of the Gaussian noise tails that exceed the cell boundaries and wrap around the modulus. The bound is D TV(P0, P1) ≤ erfc(√π/(2√2σ)). As squeezing r → ∞ (implying σ → 0), this bound decays exponentially. Achieving a strict cryptographic threshold of ϵ ≤ 10−9 requires an optical squeezing of r ≈ 14.3 dB.

Theorem 2 proves that the measurement-based aggregation map constitutes an approximate quantum non-demolition (QND) measurement. The commutator of the finite-energy Kraus operator K̃(m) and the ideal logical Pauli operators is bounded by O(e−ʳ) in the operator norm within the restricted low-energy subspace. The non-commutativity manifests physically as an additive variance σ2 ≈ e−2ʳ/2 in the output mode.

Theorem 3 provides the logical fidelity lower bound. Because a logical fault occurs if the physical noise exceeds the correctable threshold q thresh = √π/2 in either the position or momentum quadrature, the total logical fidelity is bounded by the joint 2D survival probability: F(n, r) ≥ [1 − (2/π)(∫√π/(2σ out)∞ e−t2/2 dt)2]2. Applying asymptotic approximations to the complementary error function, the operational fidelity is bounded strictly as a function of squeezing r and mode count n: F(n, r) ≥ 1 − O(exp(−πe 2r/(8(2n−1)))).

The O(n) variance accumulation places physically prohibitive demands on optical squeezing (r > 20 dB for n = 16). To circumvent near-term hardware limits, the spatial routing must be structured as a balanced logarithmic binary tree rather than a linear cascade. While the tree topology strictly preserves the intrinsic O(n) MBQC variance addition, it reduces the temporal circuit depth from O(n) to O(log2 n). To prevent the intrinsic O(n) squeezing variance from corrupting massive-scale networks (n ≫ 16), the binary tree must integrate intermediate GKP error correction at each routing vertex, using soft-decision prior information to a maximum-likelihood decoder to reset the physical variance, restricting noise accumulation to O(1) per tree layer.

The paper also establishes an algebraic duality with classical lattice cryptography (BDLOP framework). The physical operations of continuous-variable GKP aggregation map to classical lattice-based homomorphic primitives: the discrete logical payload is embedded into coarse lattice points (2s+µ)√π, physical squeezing noise manifests as sub-lattice Gaussian displacement within the local Voronoi cell, dual homodyne Bell-state measurements act as a physical gadget matrix decomposition extracting the continuous error syndrome without reading the coarse logical grid, and feed-forward displacement clears the continuous slack before it can cross the Voronoi boundary and induce a discrete logical fault.

Numerical simulations using Heisenberg-symplectic Monte Carlo with 30,000 trials per configuration evaluate the protocol across optical squeezing r from 5.0 to 30.0 dB, network sizes n ∈ 2, 4, 8, 16, and optical efficiencies η ∈ 1.00, 0.98, 0.95, 0.90 with fixed detector efficiency η det = 0.98. In the lossless limit, achieving σ out 5.2 dB for n = 2, scaling up to r > 15.1 dB for n = 16. Achieving ≥ 99.0% operational fidelity requires r ≈ 13.8 dB for n = 2, increasing to r ≈ 22.1 dB for n = 16. Under physical channel attenuation, the total standard deviation σ total flattens into an irreducible vacuum loss floor: σ floor = √((2n−1)((1−η)/(2η) + (1−η det)/(2η det))). For a two-node network with η det = 0.98, a 2% channel loss (η = 0.98) yields σ floor ≈ 0.252, capping fidelity at 99.8%. Increasing loss to 5% (η = 0.95) and 10% (η = 0.90) raises the noise floor to σ floor ≈ 0.337 and 0.446, correspondingly limiting the maximal fidelities to 98.2% and 90.5%.

In conclusion, homomorphic aggregation of discrete logical quantum data in continuous-variable networks fails under passive linear optics due to symplectic lattice compression and inter-mode entanglement. The paper resolves this by formulating an active, measurement-based CPTP map using GKP Bell states and homodyne feed-forward, proving that this protocol functions as an approximate quantum non-demolition measurement, preserving logical code spaces up to finite-energy envelope deformations while maintaining information-theoretic cryptographic security. By establishing strict logical fidelity bounds for finite-energy variance accumulation, the paper demonstrates that logarithmic tree-aggregation architectures coupled with soft-decision decoding offer a robust, fault-tolerant pathway for scalable continuous-variable quantum networking.

Improvements to AI Systems:

1. Quantum-Aware Error Correction for CV Systems:

• Improvement: Integrate the derived logical fidelity bounds and O(n) variance accumulation models into AI-based decoders for continuous-variable (CV) quantum networks.

• Capability: An AI system can now predict and mitigate finite-squeezing and photon-loss errors in real time, optimizing feed-forward displacement and intermediate maximum-likelihood decoding to maintain fault tolerance without requiring >20 dB squeezing.

2. Topology-Aware Resource Allocation:

• Improvement: Use the proven logarithmic binary-tree routing topology (vs. linear cascades) as a constraint in reinforcement learning or optimization algorithms for distributed quantum computing.

• Capability: AI planners can automatically design scalable multi-node GKP aggregation networks that minimize temporal depth and noise accumulation, reducing required squeezing from >20 dB to <15 dB for n=16, while preserving fidelity.

3. Secure Quantum Communication Protocol Design:

• Improvement: Incorporate the CV-OTP security bound (Theorem 1) into AI-driven cryptographic protocol generators.

• Capability: AI can autonomously tune squeezing levels (e.g., r≈14.3 dB for ε≤10−9) and displacement mask sampling to guarantee information-theoretic security against eavesdropping, even under finite-energy constraints.

4. Adaptive Noise-Aware Quantum Compilation:

• Improvement: Embed the analytical fidelity limits and QND commutator bounds into AI compilers for quantum circuits.

• Capability: The compiler can automatically insert intermediate error-correction steps at optimal tree vertices, balancing gate depth and noise reset, to achieve ≥99% fidelity with realistic hardware (e.g., η=0.98, r=13.8 dB for n=2).

5. Lattice-Based Homomorphic Mapping for Classical AI:

• Improvement: Translate the BDLOP duality (GKP aggregation ↔ classical lattice cryptography) into a new AI framework for homomorphic encryption.

• Capability: AI systems can perform secure computations on encrypted data using lattice-based primitives that mimic the CV protocol, with provable leakage bounds and noise management, enabling privacy-preserving machine learning on quantum-inspired hardware.

6. Predictive Performance Modeling for Quantum Hardware:

• Improvement: Train neural networks on the Monte Carlo simulation data (30,000 trials, r=5–30 dB, η=0.90–1.00) to predict fidelity and noise floors.

• Capability: AI can forecast hardware requirements (squeezing, efficiency) for arbitrary network sizes and loss profiles, enabling automated design of near-term quantum repeaters or distributed quantum processors without exhaustive simulation.

7. Autonomous Quantum Network Routing:

• Improvement: Use the protocol’s CPTP map and syndrome extraction as a differentiable layer in AI-based network controllers.

• Capability: AI can dynamically reroute quantum states through tree topologies, adjusting feed-forward operations and Bell-state resources in response to real-time noise measurements, maximizing end-to-end logical fidelity under changing channel conditions.

8. Robustness Certification for Quantum AI Models:

• Improvement: Apply the QND measurement bounds to certify the robustness of quantum machine learning models that use GKP-encoded data.

• Capability: AI can certify that classification or regression outputs remain within correctable error thresholds, even with finite squeezing, providing formal guarantees against adversarial displacement noise.

Aggregating logical quantum information encoded in continuous-variable phase space is essential for distributed quantum computing. However, passive linear optics fail for non-Gaussian Gottesman-Kitaev-Preskill (GKP) codes due to symplectic lattice compression and entanglement-induced decoherence. We present an active, measurement-based framework for the homomorphic aggregation of multi-node GKP states. Utilizing GKP Bell states and homodyne feed-forward, we construct a completely positive trace-preserving map that computes the logical sum of distributed states while preserving the logical code space geometry up to correctable finite-squeezing deformations. We prove this protocol operates as an approximate quantum non-demolition measurement, bound its cryptographic leakage for continuous one-time pads, and derive analytical logical fidelity limits under finite-squeezing constraints.