Characterizing local noise in QAOA circuits

• QAOA uses a diagonal cost Hamiltonian, a transverse-field mixer, and 2d angles for a depth-d circuit.Source: §I · p. 1
• The algorithm seeks angles that optimize the expected cost for an optimization problem encoded by the cost Hamiltonian.Source: §I · p. 1
• The noise model enables analysis of how noise rate, system size, and circuit depth affect deviations in QAOA performance.Source: §I · p. 1
• The study focuses on local depolarizing noise applied after each QAOA round and analyzes its effects on expected cost.Source: §I · p. 1

• After each QAOA round, identical local noise channels act independently on all N qubits.Source: §II · pp. 1–2
• The channel has a no-error contribution proportional to (1 −p)ρ and an operator-sum contribution weighted by p.Source: §II · p. 1
• For one round, the probability that no noise acts is (1 −p)^N, becoming (1 −p)^dN over d rounds.Source: §II · p. 2
• For unitary noise operators, normalization coefficients can be omitted, simplifying the pure-state decomposition of the noisy output.Source: §II · p. 2
• The density matrix is organized by m, the total number of noise operators acting, with m ranging from zero to N for one round.Source: §II · p. 2

• Fidelity is defined through the overlap between the noisy state and the ideal QAOA output, while cost differences quantify performance degradation.Source: §III · p. 2
• For a typical N = 8 instance under depolarizing noise, the average overlap is fitted by κ^-m with α = 0.9958 and κ = 2.71.Source: §III · p. 2
• The fitted fidelity curve matches the exact fidelity as a function of p for nearly all noise rates in the example.Source: §III · p. 2

Numerical validation

• An example for a typical problem instance is shown in Fig. 2.Source: A. Fidelity · p. 2

Fidelity approximation

• The fidelity is modeled by grouping noise terms according to their number m and approximating the corresponding average overlap by a function f_m.The approximation uses f_0 = 1 and f_m ≈ 1 + α(κ^−m − 1), which leads to a binomial expression.Source: A. Fidelity · pp. 2–3

Relation to prior form

• The proposed form generalizes the polynomial fidelity F_1 = (1 − p)^(δN) previously posited in Ref. [1] for local unitary noise.The note derives a slightly different N-dependence while retaining equivalence to the earlier form at sufficiently small p.Source: A. Fidelity · pp. 2–3

Small-noise limit

• δ = 0.63 for α = 0.996 and κ = 2.71, linking the small-noise exponent to the fitted overlap-decay parameters.The relation is δ = ακ^−1 in the small-p limit.Source: A. Fidelity · p. 3

Validity across noise rates

• The formula remains accurate across nearly the full range p ∈ [0, 1], including dominant noise with p > 0.5.A simpler (1 − p)^(δN) form fails to reproduce the appropriate large-p behavior.Source: A. Fidelity · p. 3

Cost approximation

• The expected noisy cost is expressed by grouping terms with the same number m of noise operators and replacing their averaged contribution by a function of m.The m = 0 term is the ideal cost, while the m = N term is approximated by the Haar-random expectation, here zero.Source: B. Cost · p. 3

Numerical validation

• For a typical N = 8 depolarizing-noise instance, the fitted cost curve matches the exact cost for nearly all p.The fit uses α = 1.04, α̃ = −7.41, and χ = 1.32; the plotted Hamiltonian has ground-eigenenergy −14.Source: B. Cost · p. 3

Noise dependence

• η = 0.28 for the example, and the fitted cost equation remains accurate across nearly the full range p ∈ [0, 1].The inset specifically demonstrates accuracy where p > 0.5 is not a small parameter.Source: B. Cost · p. 3

C. QAOA-d

• Replacing N with Nd in the cost model matches the QAOA-d data very well across circuit depths.The extension treats Nd as the total possible number of noise terms across d rounds.Source: C. QAOA-d · pp. 3–4
• The QAOA-d density-matrix expression tracks which layer, qubit, and operator each of the m noise events acts on.The layer index records the QAOA round in which each noise operator acts.Source: C. QAOA-d · p. 4
• The fitting parameters α and χ depend on depth because different depths correspond to completely different circuits.Thus, depth-specific parameter estimates are required when applying the model.Source: C. QAOA-d · p. 4
• For large p, going beyond d = 1 provides no benefit because the output approaches the maximally mixed state.The depth advantage therefore disappears as depolarizing noise becomes dominant.Source: C. QAOA-d · p. 4
• For p > 0.25, d = 1 is always the best choice in the N = 6 example, whereas larger depths are preferable only at lower noise.Once p exceeds 2%, there is no reason to go beyond d = 3 in this example.Source: C. QAOA-d · p. 4
• Estimating α and χ for several depths allows the optimal circuit depth to be selected for a given noise rate p.This provides a practical way to balance deeper ideal optimization against additional noise.Source: C. QAOA-d · p. 4

IV. CONCLUSION

• The analysis extends prior QAOA noise work with a slightly different binomial form for fidelity and cost, supported by numerical simulations.The approach decomposes the density operator according to the number of noise operators acting.Source: §IV · p. 4
• The equations predict that sufficiently large noise rates favor shorter QAOA circuits over deeper circuits.This captures the trade-off between optimization accuracy and noise degradation.Source: §IV · p. 4
• The approach can serve as a scalable, tunable benchmark for comparing theoretical noise-model predictions with experimental quantum-device outcomes.Depth and qubit count can act as proxy variables for noise strength.Source: §IV · p. 4