Evaluating the noise resilience of variational quantum algorithms

A. Variational quantum algorithms

• VQAs combine parameterized quantum state preparation and measurement with a classical optimization loop that updates parameters to minimize a cost function.The ansatz is a parameterized unitary applied to an initial state, while the classical optimizer searches over its parameters.Source: §I · p. 1, A. Variational quantum algorithms · p. 2

I. INTRODUCTION

• The study targets the poorly understood intermediate regime between successful noisy optimization and asymptotic cost-landscape flattening.It quantitatively evaluates how specific noise models affect VQA performance and how circuit design and optimization strategy mitigate those effects.Source: §I · p. 1
• VQAs can partly mitigate state-preparation noise through parameter optimization, while redundant parameters can produce noiselessly equivalent states with different noise resilience.These degenerate parameter sets can be lifted by noise, allowing some optimized states to be more resilient than others.Source: §I · pp. 1–2
• Low-noise propagation in random-target VQAs follows a stochastic model with linear deviation, but a circuit-dependent threshold can lead optimization toward states with different physical properties.Noise levels must remain below this threshold for the linear description and practical noise mitigation to remain reliable.Source: §I · p. 2

B. Noise model

• The simulations interleave noiseless gates with identical product noise channels applied across all qubits, focusing on amplitude damping, phase damping, and symmetric depolarization.Noise strength is parameterized by γ from 0 for no noise to 1 for the channel’s fixed point; coherent errors, crosstalk, readout noise, and shot noise are excluded.Source: B. Noise model · pp. 2–3

1. VQE simulations

• VQE minimizes the Hamiltonian expectation value to identify the ground state of a given Hamiltonian.The Hamiltonian is used as the cost observable, so the cost function returns the trial state’s energy.Source: §1 · p. 3

C. Quality measures

• Fidelity measures the overlap between the prepared state and the exact ground state, providing a quality criterion alongside energy.The definition applies to general states and simplifies when one state is pure.Source: §1 · p. 3
• Concurrence assesses bipartite entanglement for noisy two-qubit mixed states and equals zero for separable mixtures and one for Bell states.For larger systems, the study uses the maximum concurrence over all qubit pairs.Source: §1 · p. 3

2. Random target state fidelity optimisation

• Random-target optimization minimizes infidelity, equivalently maximizing fidelity between a layered circuit’s output and a randomly selected pure target state.The target-state cost can be expressed within the VQA framework using O = 1 − ρT.Source: §2 · p. 3
• Performance is averaged over nT pure target states sampled from the Haar distribution over real states using average optimal infidelity.Zero average infidelity would mean the circuit exactly represents every state in the ensemble, while noise increases the value because noisy outputs are mixed.Source: §2 · pp. 3–4
• The study compares non-reoptimised circuits, evaluated with noise after noiseless optimization, against reoptimised circuits trained with noise channels present.The non-reoptimised cost is an upper bound on the reoptimised cost and equals it only when noise leaves the minimum unchanged.Source: §2 · p. 4
• Average optimal relative infidelity isolates noise-induced error by comparing noisy infidelity with the ideal noiseless infidelity.The relative measure removes contributions that arise independently of the noise channels.Source: §2 · p. 4

A. Two qubit system

• All three ansätze can prepare any real two-qubit state without noise, while the 4-parameter circuit adds one redundant parameter for studying overparameterisation.Source: A. Two qubit system · p. 4, Figure 1 · p. 4

III. VQE FOR ENERGY MINIMISATION

• The simulations use exact density matrices and BFGS optimisation across γ ∈[0, 1] for phase, amplitude damping, and symmetric depolarising noise.Source: A. Two qubit system · p. 4, §III · p. 4

A. Two qubit system

• Phase and amplitude damping are least destructive across energy, fidelity, and concurrence, whereas symmetric depolarising noise has a much more dramatic effect.Source: A. Two qubit system · pp. 4–5
• Amplitude damping produces multiple local minima, with two solutions for each 3-parameter circuit and three for the 4-parameter circuit; phase damping and symmetric depolarising noise yield one global minimum.Source: A. Two qubit system · p. 5
• Circuit placement matters under noise: shifting a rotation gate between qubits changes state quality for phase and amplitude damping, but not for symmetric depolarising noise.Source: A. Two qubit system · p. 5, Figure 2 · p. 5
• The 4-parameter overparameterised circuit yields a solution significantly better than any 3-parameter circuit across all state-quality measures.Its redundant rotation parameter is associated with improved noise mitigation and resilience.Source: A. Two qubit system · p. 5

B. Four qubit system

• Amplitude and phase damping are less destructive than symmetric depolarising noise, while the four-qubit system exhibits more irregular local minima than the two-qubit case.Multiple amplitude-damping minima appear at low noise, whereas phase damping and symmetric depolarising noise do not show them.Source: B. Four qubit system · p. 6, C. Discussion · p. 6

C. Discussion

• Concurrence falls to zero at γ ≈0.2 −0.3 in both simulations, making it more stringent than energy or fidelity for detecting degraded state quality.At maximum noise, energy and fidelity can remain within 10-20% of their exact values, while concurrence vanishes.Source: C. Discussion · p. 6
• Amplitude damping creates multiple local minima whose quality measures decrease at different rates, so some solutions are more noise-resilient than others.These minima branch from equivalent zero-noise solutions as the noise parameter increases.Source: C. Discussion · p. 6, Figure 4 · p. 6
• Damping noise distinguishes circuits that are equivalent at zero noise, and circuit placement must therefore be optimized for noise resilience.Redundant parameters can further improve the final state, especially when damping noise breaks equivalences.Source: C. Discussion · p. 7

D. Parameter degeneracies

• Parameter degeneracy means that different parameter sets produce the same state and cost in the noiseless circuit, forming discrete minima or continuous valleys.A state-dependent disturbance can split these equivalent minima into local minima with different energies.Source: D. Parameter degeneracies · p. 7
• Redundant rotations improve damping-noise resilience because noise breaks continuous degeneracies and lets optimization explore alternative, less-affected state-preparation paths.For symmetric depolarising noise, the degeneracies remain unbroken, so redundant parameters do not improve results.Source: D. Parameter degeneracies · pp. 7–8, Figure 5 · p. 7

IV. RANDOM TARGET STATES

• The ansatz repeats a layer of four rotation gates and three CX gates, using two noise channels per layer because the first two CX gates can run in parallel.The structure is chosen for expressibility and compactness.Source: §IV · p. 8
• Reoptimized parameters generally improve average fidelity substantially over non-reoptimized parameters by adapting to the noise-distorted cost landscape.The comparison directly measures how variational optimization compensates for noise.Source: §IV · p. 8
• For L ≥4, noiseless fidelity equals one for all target states, whereas noise reduces fidelity and produces a noise-dependent optimal depth without reoptimization.For phase and amplitude damping with reoptimization, fidelity continues increasing with depth even at high noise.Source: §IV · p. 8
• At γ = 10−4, relative infidelity peaks at L = 4 after reoptimization but increases monotonically without reoptimization.These results are reported for phase damping; other noise types are shown in the appendix.Source: §IV · p. 8, Figure 8 · p. 9
• At L = 4, amplitude damping spreads fidelities across degenerate parameter states, whereas phase damping and symmetric depolarising preserve the degeneracies as sharp peaks.All degenerate states have identical fidelity at zero noise.Source: §IV · p. 9, Figure 9 · p. 10
• A representative random-target example shows a noise-induced transition near γ ≈0.04, although transition behavior depends on the target state and may be absent.The post-transition state in this example remains entangled, so the transition is not universally characterized by loss of entanglement.Source: §IV · pp. 9–10

A. Discussion

• Overparameterised circuits become more consistent across target states and gain resilience to amplitude and phase damping, but not symmetric depolarising noise.The improvement is attributed to noise breaking continuous parameter degeneracies; it is not expected to continue indefinitely.Source: A. Discussion · p. 11

1. Stochastic model

• The stochastic model captures low-noise fidelity trends, especially without reoptimisation, but diverges when reoptimisation benefits from additional layers.Its agreement worsens at higher noise and for reoptimised circuits with L ≥4.Source: §1 · p. 11
• Above a circuit-dependent noise threshold, optimisation can switch to more resilient parameters and produce states that cannot be linearly extrapolated to the noiseless state.Noise-mitigation extrapolation therefore requires data collected below this threshold.Source: Figure 10 · p. 10, §1 · p. 11

V. CONCLUSIONS

• Across the studied systems, symmetric depolarising noise is most detrimental, while amplitude and phase damping can be mitigated through gate placement, overparameterisation, and noise-aware reoptimisation.Energy alone can be misleading, so the study also evaluates fidelity and entanglement.Source: §V · pp. 11–12
• Noise-aware reoptimisation improves fidelity with added layers for amplitude and phase damping, whereas symmetric depolarising noise has an optimal circuit depth.For non-reoptimised circuits, added noise eventually outweighs the accuracy benefit of more parameters.Source: A. Discussion · p. 10, §V · p. 12

APPENDIX A: Kraus operators for simulations of noise

• Kraus operators define noisy output states through channel conjugation while preserving trace through the completeness condition.Product channels are formed from tensor products of single-qubit Kraus-operator sets.Source: Appendix A · p. 13
• Phase damping suppresses density-matrix off-diagonal components without changing diagonal components.This describes dephasing without population relaxation.Source: Appendix A · p. 13
• Amplitude damping suppresses off-diagonal components and shifts diagonal populations toward the |0⟩ state.The channel represents relaxation toward the ground state.Source: Appendix A · p. 13
• Symmetric depolarising noise contracts Bloch vectors uniformly toward the completely mixed state, with γ = 1 producing that state.The implementation uses Pauli matrices and a scaling chosen for this endpoint.Source: Appendix A · p. 13
• The simulator applies product noise channels efficiently as Pauli transfer-matrix multiplication rather than exponentially many Kraus-operator terms.States are represented as vectors in the Pauli basis.Source: Appendix A · p. 13

APPENDIX B: Linear model of noise propagation

• The circuit is divided into d layers, with noiseless evolution represented as an ordered product of layer unitaries.Each Ui denotes the unitary operation of layer i.Source: Appendix B · p. 13

1. Example: global depolarising noise

• For global depolarising noise, each channel replaces the state with the completely mixed state with probability γ.The channel acts on any input state and leaves the maximally mixed state invariant under subsequent operations.Source: §1 · pp. 13–14
• With d interleaved global depolarising channels, the probability of avoiding every channel is (1 − γ)^d.Any channel event produces a maximally mixed contribution in the final state.Source: §1 · pp. 13–14
• In the low-noise regime, infidelity increases approximately linearly with both γ and the number of layers d.The prefactor approaches 1 for large systems.Source: §1 · p. 14
• The derived relation applies to global depolarising noise, not the product of single-qubit depolarising channels used in the main simulations.The global result motivates analogous models for more general noise channels.Source: §1 · p. 14

• For a general noise channel, the first-order action is represented by a map λ, with higher-order effects in γ neglected when γ ≪1.Source: §2 · p. 14
• The noisy state differs from the noiseless result by an additive contribution proportional to γ.Source: §2 · p. 14
• Relative infidelity is defined as the fidelity loss between the ideal and noisy optimized states.Source: §2 · p. 14
• The total infidelity is approximately linear in the number of noise channels d and the local noise parameter γ.Source: §2 · p. 14
• Each channel’s contribution depends on the target state, the intermediate circuit state, and its image under λ.Source: §2 · p. 14
• Different circuit paths can reach the same noiseless state yet have different noisy infidelities because their intermediate states differ.Source: §2 · p. 15

• The ensemble figure of merit is the average optimal relative infidelity over nT pure target states.Source: §3 · p. 15
• The approximation treats intermediate target and optimized states as identically distributed to the final states.Source: §3 · p. 15
• The proportionality constant α may exceed 1 because λ(ρopt) is generally not itself a normalized quantum state.Source: §3 · p. 15
• The stochastic model assumes that states encountered at successive noise channels are perfectly correlated because they are connected by short-depth unitary sequences.Source: §3 · p. 16
• Eqs. (B25) and (B29) define the stochastic model for noise propagation, with the cited passage reporting rel ≈ β γ2d.Source: §3 · p. 16

• α and β are approximated by replacing λ(ρopt) with λ(ρT), removing explicit dependence on the circuit.This assumes a sufficiently expressive circuit whose optimized output is close to the target state.Source: §4 · p. 16
• The constants can be estimated from the target-state distribution and the first-order noise-channel action.Source: §4 · p. 16
• Finite differences provide a practical estimate of the required derivative when target states can be sampled.Source: §4 · p. 16
• For the four-qubit real-state distribution, the estimated α and β values are reported in Table I from 10000 randomly chosen states.Source: §4 · p. 16, Table I · p. 16

5. Scaling of α with the number of qubits

• α scales as O(N) for independently distributed single-qubit product states, so the noise effect is expected to grow linearly with qubit number.Source: §5 · pp. 16–17

1. VQE for noise levels varying across qubits

• For unequal noise levels, amplitude damping produces substantially different circuit outcomes, whereas phase and symmetric depolarising noise show no difference between configurations.The two configurations assign ten times more noise to opposite qubits.Source: §1 · p. 18, Figure 11 · p. 18
• The amplitude-damping minima branch in distinct directions, indicating that relative noise strength can break parameter degeneracies differently across qubits.Source: §1 · p. 18

2. Relative infidelity for different noise channels

• At low noise, the stochastic model agrees well with numerical relative-infidelity results for amplitude damping and symmetric depolarising noise.Source: §2 · p. 19, Figure 12 · p. 19
• The model more closely follows non-reoptimised results, while agreement improves with circuit depth as random-intermediate-state assumptions become more appropriate.Source: §2 · p. 19

3. Noise-induced state transitions for further random states

• Critical noise thresholds occur across random targets, but transitions may be abrupt, discontinuous, smooth, or absent as circuit and target state vary.Concurrence can reveal transitions more clearly when fidelity changes are difficult to detect.Source: §3 · p. 20, Figure 13 · p. 20