Homomorphic Aggregation of Continuous-Variable GKP States

Homomorphic Aggregation of Continuous-Variable GKP States Nilesh Vyas Airbus Central R&T, Taufkirchen, 82024 Germany (Dated: August 14, 2026) Aggregating logical quantum information encoded in continuous-variable phase space is essential for distributed quantum computing. However, passive linear optics fail for non-Gaussian GottesmanKitaev-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.

arXiv:2608.13227v1 [quant-ph] 13 Aug 2026

I.

INTRODUCTION

Continuous-variable (CV) quantum cryptography and distributed quantum computation require robust encoding schemes to protect discrete logical information within infinite-dimensional Hilbert spaces. The GottesmanKitaev-Preskill (GKP) code [1] is a primary candidate for fault-tolerant encoding, utilizing non-local superpositions of position and momentum eigenstates to form a stabilizing phase-space lattice. This translational symmetry intrinsically protects quantum data against lowweight displacement noise, making it highly attractive for near-term scalable quantum hardware [2–4]. A significant challenge in scaling multi-node quantum networks is the homomorphic aggregation of spatially distributed encrypted states within continuousvariable architectures [5]. In the classical continuousvariable paradigm, n encrypted inputs can be aggregated via a passive linear optical network. A 50:50 beamsplitter unitary ÛBS acts linearly √ √ on coherent states: ÛBS (|α1 ⟩ ⊗ |α2 ⟩) = |(α1 + α2 )/ 2⟩ ⊗ |(α1 − α2 )/ 2⟩. Because coherent states do not entangle under this interaction, applying a partial trace over the difference mode isolates a pure, aggregated coherent state. This passive approach fails for discrete logical qubits encoded in grid states [6]. A 50:50 √ beam splitter scales the phase-space lattice vectors by 1/ 2, compressing √ the standard square-lattice GKP spacing from ∆ = 2 π to √ 2π and pushing the state outside the stabilized subspace. Furthermore, non-Gaussian GKP states become highly entangled under beam-splitter dynamics; tracing out auxiliary spatial modes injects entropy and irreversibly decoheres the logical payload. While continuous-variable cluster states and MBQC teleportation protocols are well-established for universal computation within centralized fault-tolerant quantum processors [5, 7], extending these techniques to spatially distributed networks introduces unique topological and cryptographic challenges. Standard MBQC literature assumes localized generation and processing of GKP states; however, the problem of routing and aggregating independently encrypted spatial modes across a network

requires a dedicated, secure continuous-variable routing architecture. In this paper, we elevate CV-MBQC from a local computational tool to a secure, fault-tolerant network routing primitive. We establish 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. Our specific contributions are fourfold: First, we formally prove the failure modes of passive spatial GKP aggregation, driven by deterministic symplectic lattice compression and entanglementinduced 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 (CVOTP), 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.

II.

CONTEXT AND RELATED WORK

To situate our contribution, it is necessary to distinguish the specific quantum networking primitive proposed here from existing paradigms in both classical computer science and continuous-variable quantum information. a. Classical State Aggregation We define quantum homomorphic state aggregation as the completely positive trace-preserving (CPTP) mapping of spatially distributed, encrypted quantum states into a single logical payload (e.g., |ψ1 ⊕ · · · ⊕ ψn ⟩) without intermediate decryption. Unlike classical state space reduction or quantum multi-signature compression [8], this framework ad-

2 TABLE I. Impact of Passive Beam-Splitting on Lattice Geometries Lattice Square (Z2 ) Hexagonal (A2 ) Scale-Invariant (Asymptotic)

Inter-Codeword Distance (dmin ) √ π √ ≈ 1.07 π 0 (Dense)

dresses the physical fusion of continuous-variable states across a distributed network. b. Connection to Lattice-Based Homomorphic Frameworks Structurally, our continuous-variable aggregation framework operates as a quantum-physical analog to classical lattice-based homomorphic commitment schemes, such as the BDLOP framework [9]. Classical lattice schemes encode discrete message vectors onto algebraic module lattices, utilizing additive, bounded noise to mask the payload while relying on homomorphic properties for linear operations. Similarly, our protocol embeds discrete logical information into a continuous phase-space lattice (Z2 ), applying a continuous-variable one-time pad (CV-OTP) sampled uniformly over the Voronoi cell. A central challenge in both paradigms is noise propagation under homomorphic addition: just as classical lattice schemes require strict noise bounds to prevent overflow beyond the algebraic modulus, physical GKP state fusion accumulates additive Gaussian displacement variance scaling as O(n), necessitating intermediate syndrome extraction and soft-decision decoding at network vertices to reset the physical noise floor. We establish a formal structural mapping between classical gadget matrix decompositions and quantum homodyne syndrome extraction in Sec. VIII. c. Quantum Homomorphic Encryption (QHE) There is significant existing research on Quantum Homomorphic Encryption (QHE), including CV-QHE [10, 11]. Recent advancements have established rigorous bounds on QHE, notably proving the impossibility of information-theoretically secure fully homomorphic encryption without exponential communication resources, thus restricting standard schemes to polynomial-sized circuits or computational security models [12]. While these QHE frameworks excel at evaluating generic quantum circuits on centralized, unified quantum registers [10], the problem of distributed spatial aggregation—routing n independently encrypted modes into a single logical mode across a physical quantum network—introduces unique symplectic challenges that standard single-server QHE frameworks do not address. Furthermore, we distinguish the security of our spatial aggregation protocol from standard continuous-variable quantum homomorphic encryption (CV-QHE). Frameworks such as Marshall et al. [11] evaluate encrypted continuous variables, but uncorrected noise accumulation strictly limits their computational circuit depth. In contrast, our protocol guarantees ϵ-information-theoretic se-

Post-Merge Integrity Sub-threshold distance Packing compression Unphysical infinite-energy state

curity. By sampling the physical one-time pad mask uniformly over the GKP Voronoi cell, cryptographic leakage is mathematically bounded by the complementary error function of the optical squeezing (Theorem 1). This achieves unconditional statistical security without relying on computational hardness d. Distributed State Aggregation GKP states and CV cluster states are established tools for MeasurementBased Quantum Computing (MBQC) [5, 7, 13]. The specific synthesis of these tools can be directly applied to solve the linear optical scaling limit of GKP states during routing. Specifically, we address how a network can combine n spatially distributed GKP-encrypted logical states without destroying the precise CV lattice geometry. Rather than introducing fundamentally new physics, this paper optimizes existing MBQC teleportation frameworks specifically for non-destructive, spatially distributed homomorphic aggregation, formally identifying the passive optical failure modes and bounding the cryptographic leakage of the active MBQC solution.

III.

LIMITATIONS OF PASSIVE LINEAR OPTICS

We formulate the GKP code space to formally analyze the failure modes of passive state aggregation. The ideal single-mode GKP code√is stabilized by the displace√ ment operators Sq = exp[i2 π p̂] and Sp = exp[−i2 π q̂], where the quadrature operators satisfy [q̂, p̂] = i. The ideal logical basis states, |0⟩L and |1⟩L , are Dirac combs over position eigenstates |q⟩q : |0⟩L =

X

√ |2s π⟩q ,

s∈Z

|1⟩L =

X

√ |(2s + 1) π⟩q .

(1)

s∈Z

√ The inter-codeword distance is dmin = π, establishing a strict √ correctable displacement error threshold of qthresh = π/2.

A.

Symplectic Scaling Limits

Passive linear optical operations correspond to orthogonal symplectic transformations S ∈ Sp(2n, R) ∩ O(2n) [14]. For a two-mode 50:50 beam splitter, the symplectic transformation on the joint quadrature vector

3 x̂ = (q̂1 , p̂1 , q̂2 , p̂2 )T is defined by: 1 x̂out = SBS x̂ = √ 2

The resulting von Neumann entropy is strictly positive: I2 I2 I2 −I2

! x̂.

(2)

S(ρout,1 ) = −tr(ρout,1 ln ρout,1 ) = −

d X

λk ln λk > 0.

k=1

While the orthogonal matrix SBS globally preserves the 4-dimensional Euclidean norm of the phase-space vectors, extracting the aggregated quantum information requires a spatial reduction to a single mode. 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 postp (1) π/2. This in turn interaction distance to dmin = compresses√the maximum correctable noise threshold to (1) (1) qthresh = 2√π2 ≈ 0.6266. Because qthresh < qthresh = √ π/2 ≈ 0.8862, this strictly degrades the noise tolerance and violates the fault-tolerant subspace preservation condition [1, 2]. As summarized in Table I, this geometric failure extends beyond the standard square lattice. While the hexagonal (A2 ) GKP code offers optimal phase-space packing and a strictly √ inter-codeword dis√ larger initial tance (dmin ≈ 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. Thus, maintaining discrete finite-distance scale invariance under passive linear optics is mathematically impossible for local subsystems across any viable encoding geometry. B.

(5) This entropy injection bounds the logical purity to tr(ρ2out,1 ) < 1, physically corrupting the lattice encoding and necessitating an active, measurement-based aggregation protocol.

IV.

MEASUREMENT-BASED STATE AGGREGATION

To achieve homomorphic state aggregation, we define a completely positive trace-preserving (CPTP) map ⊗n M : D(HCV ) → D(HCV ). This map bypasses symplectic limitations via continuous-variable measurementbased quantum computing (MBQC) and non-Gaussian resource injection.

A.

Non-Gaussian Resources

Universal CV computation strictly requires nonGaussian elements [6]. For this, we utilize two-mode GKP Bell states |Φ+ ⟩L . Although the deterministic generation of multi-mode GKP states remains experimentally demanding [3], we adopt pre-shared GKP Bell states as foundational network resources. These states are generated by mixing two independent GKP qunaught states [1], denoted |∅⟩L , which contain no logical information but establish the grid topology: |∅⟩L ∝

Inter-Mode Entanglement and Entropy

X √ |s 2π⟩q .

(6)

s∈Z

While spatial reduction via partial tracing preserves state purity for tensor products of Gaussian coherent states, this structural preservation fails for non-Gaussian GKP codewords. Because GKP states are highly nonclassical, applying the beam-splitter unitary ÛBS = exp[ π4 (â†1 â2 − â1 â†2 )] to the separable input state |Ψin ⟩ = |ψ1 ⟩L ⊗ |ψ2 ⟩L yields a highly entangled output state |Ψ12 ⟩ [15]. We express this using the Schmidt decomposition: |Ψ12 ⟩ = ÛBS (|ψ1 ⟩L ⊗ |ψ2 ⟩L ) =

d p X

λk |ϕk ⟩1 ⊗ |φk ⟩2 ,

Applying the 50:50 beam-splitter unitary ÛBS = exp[ π4 (â†1 â2 − â1 â†2 )] to the separable state yields the entangled resource: |Φ+ ⟩L,AB = ÛBS (|∅⟩L,A ⊗ |∅⟩L,B ).

(7)

√ This operation geometrically maps the 2π qunaught √ spacing into the π spacing required for the logical GKP Bell state, yielding continuous-variable entanglement precisely aligned with the target logical lattice.

k=1

(3) P where λk > 0, k λk = 1, and the Schmidt rank d > 1. Tracing out the auxiliary mode produces a strictly mixed reduced density matrix for the primary mode: ρout,1 = tr2 (|Ψ12 ⟩⟨Ψ12 |) =

d X k=1

λk |ϕk ⟩⟨ϕk |1 .

(4)

B.

Aggregation Circuit

homomorphic aggregation of n distributed states NThe n |ψ j ⟩L relies on a sequence of Gaussian operations j=1 and homodyne measurements. We define the primary non-Gaussian resource, the two-mode GKP Bell state

4 1. Resource Generation

4. Classical Control & Output

2 & 3. Cluster Formation & CV-BSM q̂1 BS m1

|ψ1 ⟩L BS SWAP

|∅⟩L,A

|ψ1 ⊕ ψ2 ⟩L

D̂B (−mΣ ) |Φ+ ⟩L,AB

BS

p̂A mA

|∅⟩L,B

Classical Control mΣ = f (m1 , mA , m2 )

q̂2 m2

|ψ2 ⟩L

FIG. 1. Continuous-variable measurement-based quantum state aggregation protocol for n = 2 logical inputs. Stage 1 generates a Gottesman-Kitaev-Preskill (GKP) Bell pair |Φ+ ⟩L,AB from continuous-variable vacuum qunaught states |∅⟩L,A and |∅⟩L,B . In Stages 2 and 3, Data Mode 1 (|ψ1 ⟩L ) and Data Mode 2 (|ψ2 ⟩L ) couple to Ancilla Mode A via sequential beam splitters (BS) and a physical waveguide SWAP gate (×—×), followed by dual homodyne detection of quadratures q̂1 , p̂A , and q̂2 . Stage 4 processes the extracted classical eigenvalues m1 , mA , and m2 within a classical control unit to evaluate the net displacement vector mΣ = f (m1 , mA , m2 ), executing the active feed-forward displacement D̂B (−mΣ ) on unmeasured Ancilla Mode B to yield the aggregated logical state |ψ1 ⊕ ψ2 ⟩L .

|Φ+ ⟩L,AB , as the +1 eigenstate of the joint Pauli operators X̄ ⊗ X̄ and Z̄ ⊗ Z̄ −1 : 1 |Φ+ ⟩L,AB = √ (|0⟩L,A ⊗ |0⟩L,B + |1⟩L,A ⊗ |1⟩L,B ) . 2 (8) Expanding into the continuous position basis reveals the entangled grid structure: X X √ √ |Φ+ ⟩L,AB ∝ |(2sA +k) π⟩qA |(2sB +k) π⟩qB , k∈{0,1} sA ,sB ∈Z

(9) which satisfies continuous-variable√ Einstein-PodolskyRosen (EPR) correlations modulo π. The aggregation protocol proceeds as follows: 1. Resource Generation: Prepare n − 1 GKP Bell pairs |Φ+ ⟩L,AB . For finite-energy hardware, the Dirac peaks broaden into Gaussian envelopes with intrinsic position and momentum variance σ 2 ≈ e−2r /2, where r is the optical squeezing parameter. 2. Continuous Bell-State Measurement (BSM): Following standard CV teleportation formalism [16], we project the joint subsystem Hj ⊗ HA onto the continuous Bell basis |m⟩jA . Physically, this corresponds to coupling input mode j to ancilla (jA) mode A via a balanced beam splitter (ÛBS ) and executing dual homodyne detection to extract the continuous syndrome m = qm + ipm . Note that the physical waveguide SWAP gate depicted in the circuit architecture (Fig. 1) is strictly a spatial routing operation; it physically steers the unmeasured aggregate mode (B) to the output transmission line

while directing the entangled ancilla mode (A) to the detectors. It carries no logical weight and does not alter the mathematical derivation of the Kraus operators. 3. Classical Feed-Forward: The BSM yields the continuous outcome m ∈ C. The active aggregation step operates as a CPTP map M defined by the CV gate teleportation integral [7]. The Kraus operators mapping Hj → HB are defined as: (jA)

K̂(m) = D̂B (−m)⟨m|jA ÛBS

yielding the output R 2 † d m K̂(m)ρj K̂ (m). C

Ij ⊗ |Φ+ ⟩L,AB , (10) state

ρout,B

=

If the input is encrypted with a continuous displacement mask α = qenc + ipenc , the measured quadrature decomposes algebraically as qm = qstate + qnoise + qenc . The feed-forward displacement operator D̂B (−m) = exp[iqm p̂B − ipm q̂B ] actively translates the unmeasured mode B back into the logical code space, preserving the underlying quantum information without decrypting the physical payload.

C.

Cryptographic Security Bounds

Treating the classical encryption mask α as a continuous quantum one-time pad requires careful analysis. Continuous uniform distributions defined strictly over a finite boundary (the Voronoi cell of the GKP lattice) introduce modulo-wrapping artifacts for finiteenergy states.

5 Theorem 1 (Cryptographic Hiding in Finite-Energy CV-OTP). For a finite-energy GKP state with intrinsic variance σ 2 , homodyne syndrome extraction under √ a uniformly sampled displacement mask qenc ∼ √ 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. Proof. Let the continuous measurement outcome be m = √ (qstate +qnoise +qenc ) (mod π). For finite squeezing, the intrinsic state noise follows a normal distribution qnoise ∼ N (0, σ 2 ). The mask is uniformly distributed √ encryption √ qenc ∼ U[− π/2, π/2). The statistical distinguishability between the conditional probability distributions P0 (m) ≡ P (m | |0⟩L ) and P1 (m) ≡ P (m | |1⟩L ) is quantified by the total variation distance: Z √π/2 1 DT V (P0 , P1 ) = |P0 (m) − P1 (m)|dm. (11) 2 −√π/2 Because the encryption mask qenc uniformly √ is sampled √ over the primary Voronoi cell [− π/2, π/2), the √ modulo addition m = (qstate + qnoise + qenc ) (mod π) acts as a perfect one-time pad provided the physical noise remains confined within √ the cell. If the noise is bounded such that |qnoise | ≤ π/2, the resulting distribution is perfectly uniform, yielding P0 (m) = P1 (m) exactly. The statistical difference |P0 (m) − P1 (m)| becomes non-zero exclusively due to the probability mass of the Gaussian noise tails that exceed the cell boundaries and wrap around the modulus. By the properties of total variation distance under uniform masking, DT V is upperbounded by the total probability that the physical noise escapes this bounding cell:  √  π DT V (P0 , P1 ) ≤ P |qnoise | > 2 Z ∞ x2 1 √ =2 √ e− 2σ2 dx. (12) 2πσ π/2 Evaluating this tail integral yields the standard complementary error function [2, 17]:  √  π √ . (13) DT V (P0 , P1 ) ≤ erfc 2 2σ As squeezing r → ∞ (implying σ → 0), this bound decays exponentially. To contextualize this leakage for near-term hardware, standard quantum cryptographic protocols demand a statistical security parameter of ϵ ≤ 10−9 √. Substituting the physical GKP variance σ = e−r / 2, achieving this strict cryptographic threshold requires an optical squeezing of r ≈ 14.3 dB. Because this exactly mirrors the established baseline squeezing required for fault-tolerant GKP error correction [5, 18], the measurement channel operates as a secure, continuous quantum one-time pad for any hardware capable of sustaining the logical encoding.

V.

PROOF OF QUANTUM NON-DEMOLITION

The map M must compute the logical sum without collapsing the quantum superposition. Theorem 2 (Approximate QND Projection for Finite-Energy GKP Codes). For finite-energy GKP resources with squeezing parameter r, the measurementbased aggregation map constitutes an approximate quantum non-demolition (QND) measurement. The commuˆ tator of the finite-energy Kraus operator K̃(m) and the ideal logical Pauli operators is bounded by O(e−r ) in the operator norm. Proof. The ideal aggregation channel is defined by Kraus operators K̂(m) corresponding to the Bell-state measurement outcome m and feed-forward displacement D̂(−m):  
(jA) K̂(m) = D̂(−m) ⟨m|jA ÛBS Ij ⊗ |Φ+ ⟩L,AB . (14) In the asymptotic limit (r → ∞), the Weyl-Heisenberg commutation relations guarantee that K̂(m) commutes √ exactly with the logical Paulis X̄GKP = D̂( π) and √ Z̄GKP = D̂(i π). However, physical states require a non-unitary Gaussian envelope Ê = exp(−ϵn̂) where ϵ ≈ e−2r /2. ˆ = The finite-energy Kraus operator becomes K̃(m) K̂(m)(I ⊗ Ê). Because displacement operators do not strictly commute with Ê, the commutator expands as: √ ˆ X̄GKP ] = K̂(m)[Ê, D̂( π)]. [K̃(m), (15) Using the exact identity Ê D̂(α)Ê −1 = D̂(αe−ϵ ), the non-commutativity scales directly with the envelope parameter ϵ. Because displacement operators are unbounded in infinite-dimensional Hilbert space, a global operator-norm bound is undefined. However, evaluated strictly over the restricted low-energy subspace spanned by the finite-energy GKP logical basis, the mean photon number is bounded (⟨n̂⟩ ∝ 1/ϵ). Within this computational subspace, higher-order terms in the BakerCampbell-Hausdorff expansion scale as O(ϵ2 ) and are rendered negligible. Bounding the first-order truncation in the operator norm yields: ˆ ∥[K̃(m), X̄GKP ]∥subspace ≤ O(e−r ).

(16)

Thus, the operation is QND up to an exponentially suppressed envelope deformation within the physical code space. This non-commutativity manifests physically as an additive variance σ 2 ≈ e−2r /2 in the output mode. VI. PROOF OF HOMOMORPHIC STATE ADDITION UNDER FINITE SQUEEZING

We prove homomorphic state addition by accounting for finite squeezing, continuous noise propagation, and modular lattice arithmetic [2, 17].

6 For finite squeezing r, a physical GKP basis state |µ̃⟩L for logical µ ∈ {0, 1} is modeled as an envelope-broadened superposition of displaced Gaussian wavepackets [1]: Z √ X γ − 12 [q−(2s+µ) π]2 2 |µ̃⟩L ∝ e− 2 (2s+µ) π e 2σ0 |q⟩, dq R

s∈Z

(17) where σ02 ≈ e−2r /2 governs the local peak variance and γ ≈ e−2r bounds the total state energy. The position support of any single-shot wavepacket in input mode j ∈ {1, . . . , n} is expressed as: √ (18) qj = (2sj + µj ) π + ξj , where sj ∈ Z indexes the lattice site, µj ∈ {0, 1} is the logical value, and ξj ∼ N (0, σ02 ) is the local physical displacement noise. Similarly, the finite-energy GKP Bell pairs do not satisfy exact continuous equality, but obey the EPR relation up to a Gaussian error term [16]. For the k-th resource state |Φ̃+ ⟩L,Ak Bk , we have: q̂Ak − q̂Bk = ξEPR,k ,

ξEPR,k ∼ N (0, 2σ02 ).

(19)

n X   √ (2sj + µj ) π + ξj + ξanc ,

(20)

j=1

Pn−1 where ξanc = k=1 ξEPR,k collects the net continuous noise from the n − 1 intermediate Bell pairs. The output quadrature simplifies to:   n n X X √ q̂out = 2 (21) sj + µj  π + ξout . j=1

As optical squeezing r → ∞, σout → 0 and Psucc → 1. Consequently, the physical CPTP map converges to the exact logical XOR on the density matrices:   +* n n n M M O µj µj . (24) lim M  ρ̃j  = r→∞

During the continuous Bell-state measurements, dual homodyne detection extracts the relative position syndrome for each fusion step. Applying the total classical feedP forward displacement D̂( m) to the final unmeasured aggregate mode shifts its output position quadrature q̂out to: q̂out =

The operation succeeds without logical bit flips if and only if the total displacement √ √ error remains strictly within the Voronoi cell [− π/2, π/2) for both quadratures simultaneously. Because the sequential beamsplitter operations and homodyne local oscillators are strictly phase-aligned to the logical coordinate axes (0 and π/2), the global symplectic transformation does not induce rotational mixing between the position and momentum operators. Consequently, the covariance matrix remains diagonal throughout the protocol, preserving independent Gaussian noise profiles in both the q̂ and p̂ quadratures. The joint logical success probability Psucc is therefore strictly given by the squared one-dimensional noise tail integral: " #2 Z √π/2 2 − x2 1 Psucc = √ e 2σout dx . (23) 2πσout −√π/2

j=1

Pn Here, ξout = ξanc + j=1 ξj represents the total accu2 mulated displacement noise, distributed as N (0, σout ), 2 −2r with σout ≈ (2n − 1)e matching the cascade variance bound. Decomposing the bit sum into an integer quotient c∈ Pn Z and a binary modulo-2 remainder yields µj = j=1 L  Pn n 2c + j=1 µj . Letting S = j=1 sj + c ∈ Z, the output position support evaluates precisely to:   n M √ q̂out = 2S + µj  π + ξout . (22) j=1

This demonstrates that the output position quadrature lies Ln on the target GKP grid encoding the logical sum j=1 µj , shifted by the noise term ξout .

j=1

j=1

j=1

L

This completes the proof that teleportative fusion executes homomorphic state addition on finite-energy GKP codes, provided the finite-squeezing bounds satisfy the Voronoi threshold. VII. FIDELITY BOUNDS AND FINITE-ENERGY CONSTRAINTS

Physical implementations require finite-energy GKP states characterized by an intrinsic variance σ 2 ≈ e−2r /2, where r is the optical squeezing factor [7]. During the active state-merging protocol, this variance propagates linearly through the beam-splitter network and homodyne measurements. In a single teleportative fusion step between a data mode and an ideal ancilla, the output variance updates according to the continuous-variable teleportation covariance transformation. A joint homodyne BSM on two 2 2 modes with initial variances σA and σB yields a post2 2 2 feed-forward variance of σout = σA + σB . The recursive teleportation of the running aggregate through n − 1 intermediate GKP Bell states causes the final effective variance to accumulate additively. Accounting for the dual-homodyne noise penalties, the fusion of n inputs yields a total intrinsic variance of O(n) regardless of the routing geometry: 2 σout =

n X j=1

σj2 +

n−1 X

2 σancilla,k ≈ (2n − 1)e−2r .

(25)

k=1

Theorem 3 (Logical Fidelity Lower Bound). The logical fidelity F (ρout , ρtarget ) of the aggregated state is bounded

7 by the integral of the probability density remaining within the correctable Voronoi cell. Because a logical fault occurs if the physical noise ex√ ceeds the correctable threshold qthresh = π/2 in either the position or momentum quadrature, the total logical fidelity is bounded by the joint 2D survival probability [2]: r F (ρout , ρtarget ) ≥

1−

2 π

!2

Z ∞ √ π 2σout

e

−t2 /2

.

dt

(26)

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 −

πe2r 8(2n − 1)

 .

(27)

As demonstrated in Fig. 2, this O(n) variance accumulation places physically prohibitive demands on optical squeezing (r > 20 dB for n = 16). To circumvent nearterm 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). This exponential reduction in delay-line latency suppresses time-dependent photon loss. 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. By passing the analog syndrome m as soft-decision prior information to a maximumlikelihood decoder [18], intermediate syndrome extraction actively resets the physical variance. This restricts the noise accumulation to O(1) per tree layer, enabling scalable, fault-tolerant network aggregation.

VIII. ALGEBRAIC DUALITY WITH CLASSICAL LATTICE CRYPTOGRAPHY

To synthesize our theoretical framework for the broader quantum information and cryptography communities, we formally map the physical operations of continuous-variable GKP aggregation to classical latticebased homomorphic primitives, specifically the BDLOP commitment framework [9]. Classical lattice schemes encode discrete message vectors onto algebraic module lattices, utilizing additive, bounded noise to mask the payload while relying on homomorphic properties for linear operations. Similarly, our protocol embeds discrete logical information into a continuous phase-space lattice (Z2 ), applying a continuous-variable one-time pad (CV-OTP) sampled uniformly over the primary Voronoi cell. Homomorphic

state fusion causes continuous displacement noise to accumulate as O(n), requiring intermediate syndrome extraction and soft-decision decoding at network vertices to prevent logical faults. Remark 1 (Gadget Decomposition and Phase-Space Scale Separation). In classical lattice schemes like BDLOP [9], gadget matrices G decompose state vectors into multi-scale digits, isolating large operations into lownorm components so that accumulated noise remains confined to lower-order bits without corrupting the highorder payload. Continuous-variable GKP state aggregation implements an exact physical analog of this scale separation: 1. Scale Separation: The discrete logical payload µ ∈ {0, 1}√is embedded into coarse lattice points (2s + µ) π, whereas physical squeezing noise manifests as sub-lattice Gaussian displace2 ment √ ξ ∼ √ N (0, σ ) within the local Voronoi cell [− π/2, π/2). 2. Noise Isolation: Dual homodyne Bell-state measurements act as a physical gadget matrix decomposition G−1 . They extract the continuous error syndrome m into classical registers without reading or collapsing the coarse logical grid. 3. Slack Correction: Applying the feed-forward displacement D̂(−m) uses the isolated classical syndrome to shift the output mode back to the lattice center. This clears the continuous ”slack” before √ it can cross the Voronoi boundary qthresh √ = π/2 and induce a discrete logical fault (k π for k ∈ Z \ {0})—the exact physical equivalent of modular overflow in Module-LWE schemes.

IX.

NUMERICAL SIMULATION AND RESULTS

We evaluate the CV GKP aggregation protocol using a Heisenberg-symplectic Monte Carlo simulation. The framework tracks continuous quadrature operators through all beam-splitter fusions, homodyne measurements, and feed-forward displacements. A.

Methodology and Parameter Space

To ensure statistical convergence, we perform 30,000 independent trials per configuration. We evaluate the protocol across three physical parameters: 1. Optical Squeezing (r): Swept from 5.0 dB to 30.0 dB, setting the initial single-mode variance σ02 = e−2r /2. 2. Network Size (n): Evaluated at n ∈ {2, 4, 8, 16} to quantify noise accumulation across n − 1 fusion stages.

8

FIG. 2. Performance scaling and hardware loss degradation of measurement-based continuous-variable GKP state aggregation. Top Row (Ideal Channel Scaling, η = 1.0): (a) Accumulated network noise standard deviation p σout = (2n − 1)e−2r across participant mode counts n ∈ {2, 4, 8, 16} as a function of optical squeezing r ∈ [5, 30] dB. The √ horizontal dashed line denotes the primary GKP Voronoi cell boundary qthresh = π/2 ≈ 0.8862. (b) Operational logical fidelity F (n, r) demonstrating asymptotic convergence to 100% for arbitrary network sizes n at high squeezing. Scatter markers indicate empirical Monte Carlo results (30,000 trials per step); solid curves denote the analytical lower bound. (c) Isotropic phase-space probability cloud contraction of 1,000 single-shot realizations centered around the target state (0.55, 0.30) (black star) in a lossless environment at 8 dB (red), 15 dB (blue), and 25 dB (green) squeezing. Bottom Row (Physical Channel Loss Robustness, n = 2, ηdet = 0.98): (d) Accumulated noise variance σtotal under fiber attenuation η ∈ {1.00, 0.98, 0.95, 0.90}, showing the emergence of an irreducible vacuum noise floor at high squeezing levels. (e) Upper saturation ceilings on logical fidelity F (r, η) imposed by photon-loss-induced vacuum noise injection. (f ) Phase-space state distribution under 5% channel loss (η = 0.95), illustrating phase-cloud stabilization bounded by the loss floor.

TABLE II. Algebraic and physical structural mapping between classical lattice commitment schemes (BDLOP) and the measurement-based CV GKP aggregation protocol. Feature Payload Space Noise / Slack Isolation Mechanism Noise Correction Failure Boundary

BDLOP / Gadget Schemes

CV GKP Aggregation Protocol

High-order module lattice entries Low-norm digits / small polynomial noise Gadget matrix decomposition G−1 Rounding off low-order digits Modulus overflow (> q/2)

√ Discrete GKP grid coordinates (2s + µ) π√ √ Continuous Gaussian displacement ξ ∈ [− 2π , 2π ) Joint homodyne BSM / syndrome extraction Feed-forward displacement D̂(−m) √ Voronoi boundary overflow (> π/2)

3. Optical Efficiency (η, ηdet ): Channel transmissivity η ∈ {1.00, 0.98, 0.95, 0.90} models propagation loss, with fixed detector efficiency ηdet = 0.98. In each trial, encrypted logical payloads q1 = 0.250, p1 = 0.200 and q2 = 0.300, p2 = 0.100 are masked via continuous random phase-space displacements i = (qenc,i , penc,i )T . To satisfy the cryptographic

bounds established in Theorem 1, the mask quadratures are sampled uniformly over √ √ the primary Voronoi cell, qenc,i , penc,i ∼ U[− π/2, π/2). Following joint clusterstate homodyne measurements and feed-forward mask removal, the raw decrypted aggregate payload yields the exact expected sum Qtarget = 0.550, Ptarget = 0.300 (denoted visually as quadratures X̂ and P̂ in Fig. 2)

9 perturbed by accumulated Gaussian physical noise ξ = (ξq , ξp )T . The overall physical variance accumulated in the continuous quadratures follows:   1 − η 1 − ηdet 2 σtotal + . (28) = (2n−1)e−2r +(2n−1) 2η 2ηdet Logical error correction succeeds if the physical noise deviation remains strictly within√ the primary GKP Voronoi cell boundary qthresh √ = π/2 ≈ 0.8862, preventing discrete lattice shifts k π (k ∈ Z). B.

Ideal Network Scaling Analysis

Figures 2(a)–(c) illustrate the protocol performance in the lossless limit (η = 1.00, ηdet = 1.00). As shown in Fig. standard p 2(a), the accumulated noise √ deviation σout = (2n − 1)e−2r scales as O( n). Larger networks demand higher baseline squeezing to keep total quadrature noise below the Voronoi threshold (qthresh ≈ 0.8862). Achieving σout < qthresh requires r > 5.2 dB for n = 2, scaling up to r > 15.1 dB for n = 16. Figure 2(b) confirms the empirical fidelity matches the analytical lower bound derived from the Voronoi tail integral:   √ 2 π √ F (n, r) ≥ 1 − erfc . 2 2σout

(29)

Achieving ≥ 99.0% operational fidelity requires r ≈ 13.8 dB for n = 2, increasing to r ≈ 22.1 dB for n = 16. Figure 2(c) depicts the isotropic phase-space probability cloud of 1,000 realizations around (0.55, 0.30). At 8 dB, the physical noise spans the entire Voronoi cell. Increasing squeezing to 15 dB and 25 dB contracts the phase-space area exponentially, concentrating the recovered state at the exact homomorphic sum.

C.

Impact of Physical Channel Attenuation

vacuum loss floor: s σfloor =



(2n − 1)

 1 − η 1 − ηdet + . 2η 2ηdet

(30)

This floor establishes a strict upper bound on the maximum achievable logical fidelity, Fmax (η). For a two-node network (n = 2) with detector efficiency ηdet = 0.98, a 2% channel loss (η = 0.98) yields σfloor ≈ 0.252, capping fidelity at 99.8%. Increasing loss to 5% (η = 0.95) and 10% (η = 0.90) raises the noise floor to σfloor ≈ 0.337 and 0.446, correspondingly limiting the maximal fidelities to 98.2% and 90.5% [Fig. 2(e)]. This phenomenon is visualized in phase space [Fig. 2(f)]. Unlike the lossless limit where infinite squeezing contracts the state to a Dirac peak, channel attenuation bounds the phase-space contraction, stabilizing the aggregate into a stationary Gaussian distribution governed entirely by σfloor .

X.

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. To resolve this, we formulated an active, measurement-based CPTP map using GKP Bell states and homodyne feed-forward. We proved that this protocol functions as an approximate quantum nondemolition 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, we demonstrated that logarithmic tree-aggregation architectures coupled with softdecision decoding offer a robust, fault-tolerant pathway for scalable continuous-variable quantum networking.

DATA AVAILABILITY

Figures 2(d)–(f) evaluate performance degradation under realistic optical loss (η < 1). Channel attenuation mixes vacuum fluctuations into the quadratures. As r → ∞, the intrinsic squeezing noise vanishes, and the total standard deviation σtotal flattens into an irreducible

The numerical data generated by the Heisenbergsymplectic Monte Carlo simulations and the associated Python routines used to evaluate the phase-space fidelity bounds are available from the corresponding author upon reasonable request.

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Appendix A: Subspace Truncation and Higher-Order Error Bounds for QND Projection

To formalize the quantum non-demolition (QND) bound stated in Theorem 2, we evaluate the commutator [Ê, D̂(α)] strictly within the finite-energy Gottesman√ Kitaev-Preskill (GKP) subspace. Here, α = π corresponds to the logical Pauli operator X̄GKP , and the nonunitary envelope is defined as Ê = e−ϵn̂ with ϵ ≈ e−2r /2. Leveraging the exact operator identity e−ϵn̂ âeϵn̂ = −ϵ âe , the exact commutation relation between the envelope and the displacement operator evaluates to: Ê D̂(α) = D̂(αe−ϵ )Ê.

(A1)

Consequently, the exact commutator is isolated as: h i [Ê, D̂(α)] = D̂(αe−ϵ ) − D̂(α) Ê. (A2) Because α and αϵ are strictly real, the Weyl-Heisenberg phase factor evaluates to unity (Im[α(−αϵ)] = 0). This permits the exact factorization of the residual displacement: h i D̂(αe−ϵ ) − D̂(α) = D̂(α) D̂(−αϵ) − I . (A3) To bound the operator norm of this difference, we Taylor expand the residual displacement generator D̂(−αϵ) = exp[−αϵ(↠− â)]: D̂(−αϵ)−I = −αϵ(↠−â)+

α 2 ϵ2 † (â −â)2 −O(ϵ3 ). (A4) 2

Globally, the bosonic field operators ↠and â are unbounded, causing this expansion to diverge for arbitrary infinite-dimensional states. However, within the computational subspace spanned by the finite-energy GKP logical states, the mean photon number is intrinsically constrained by the squeezing envelope, scaling as ⟨n̂⟩ ∝ 1/ϵ. Consequently, the matrix elements of the k-th power of the field operators scale as ⟨(↠− â)k ⟩ ∼ ⟨n̂⟩k/2 ∝ ϵ−k/2 . We rigorously evaluate the scaling of the expansion terms within this restricted subspace: 1. First-Order Term (k = 1): Scales as ϵ⟨â⟩ ∼ ϵ(ϵ−1/2 ) = ϵ1/2 . Substituting ϵ ≈ e−2r /2, the firstorder truncation error scales precisely as O(e−r ). 2. Second-Order Term (k = 2): Scales as ϵ2 ⟨â2 ⟩ ∼ ϵ2 (ϵ−1 ) = ϵ. This yields an error scaling of O(e−2r ). Because O(e−2r ) decreases exponentially faster than O(e−r ) in the high-squeezing regime (r ≫ 1), all higherorder terms (k ≥ 2) are explicitly negligible. Truncating the expansion at the first order is therefore mathematically rigorous, strictly bounding the QND operator deformation to: ∥[Ê, D̂(α)]∥subspace ≤ O(e−r ).

(A5)

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