Quantum-Computing-BB84

Pauli Error Composition Determines the Multipartite Advantage in Conference Key Agreement

Ryder Carlson, Kidus Peter, and Lochlan Fitzgerald

We test whether sharing one quantum key across three or more people at once (through GHZ states) holds up better or worse than exchanging keys person to person through pairwise BB84. The answer changes depending on the type of noise on the connection, the number of people receiving the key, and how the noise is distributed across the network.

GHZ is more efficient than BB84, but entanglement is more fragile, and what we wanted to know was whether the efficiency is worth the fragility once noise is accounted for.

Both protocols are simulated exactly in Cirq under amplitude damping, dephasing, and depolarizing noise, and the secret key rate is compared. (The secret key rate is how much usable secure key survives.) Group sizes of 3, 4, and 5 are compared as noise increases, with the pairwise protocol charged N−1 network uses per conference key against the GHZ protocol's one.

Results

Noise type (main result). Raw crossover values differ by about 8× across the three channels, but Cirq parameterizes each channel independently, so those numbers are not directly comparable. Rescaled to average channel infidelity, the spread is 2.38× at N=3 (2.20× at N=4, 2.08× at N=5), and amplitude damping and depolarizing agree to within 1% at every group size despite entirely different Kraus structure. What separates the channels is their Pauli error composition, not their name: dephasing produces no bit-flip errors, so Q_Z stays at zero and the GHZ protocol pays only the phase-error term. Mixing bit-flip-type error into pure dephasing at just 5% relative strength more than halves the threshold; equal mixing collapses it by a factor of 14.

Closed form. For arbitrary independent Pauli noise on each arm, the conference key rate has a closed form (Section 2.7 of the paper). It proves that concentrating a fixed noise budget on one arm is strictly worse than spreading it evenly — answering, for all Pauli channels, a question Proietti et al. (2021) raised numerically for depolarizing noise — and yields a zero-cost operational rule: assigning the dealer role to the cleanest arm improves the rate by ~18% in our test case, and the improvement vanishes under pure dephasing.

Group size. Adding more Bobs makes GHZ worse, and the crossover moves to lower noise with each additional party; the pairwise protocol is unaffected. This confirms depolarizing-specific results of Epping et al. and Memmen et al. under a per-qubit noise model.

Crossover vs. death. At every crossover the GHZ protocol is still producing key — the crossover precedes the noise strength at which the GHZ rate reaches zero by 15–20%. The boundary is economic (set by the resource convention), not a security failure.

Validation

The simulation is checked against a closed-form expression for dephasing. Agreement is at machine precision: the largest key-rate difference across 46 noise values and three group sizes is 1.2×10⁻¹⁵, and crossover locations agree to 2.6×10⁻¹⁰ (limited by the bisection tolerance, not the simulation). This requires DensityMatrixSimulator(dtype=np.complex128); Cirq's default complex64 introduces errors of order 10⁻⁶ near zero key rate.

Files

GHZ_CKA.py — engine: circuits, error rates, key rates, crossover solvers, infidelity conversion figstyle.py — shared matplotlib style for all figures fig_validation.py — simulation vs. closed form (Fig. 1) fig_composition.py — crossover at two noise compositions (Fig. 2) fig_mix.py — crossover collapse under admixture (Fig. 3) fig_scaling.py — thresholds vs. group size, raw and normalized (Fig. 4) fig_heatmap.py — key rate vs. asymmetric arm noise, fixed-budget contour (Fig. 5) fig_table.py — crossover table across noise types and group sizes (Table 1), writes results.csv fig_budget.py — even vs. concentrated noise at fixed budget (Table 2) and the Dirichlet sampling check fig_twirl.py — exact vs. Pauli-twirled amplitude damping (Table 3) and twirled crossover errors

Scope

Thresholds are asymptotic secret fractions per network use. Transmission loss, detector efficiency, quantum memories, and finite-key corrections are outside the model; all are known to shift the advantage boundary toward the pairwise protocol, and transmission loss in particular removes the multipartite advantage beyond roughly 15 km per link (Memmen et al.).