Barren plateaus preclude learning scramblers

• Scrambling is linked to late-time dynamics and 2-design statistics; chaotic systems before scrambling time provide an approximate-scrambling regime.Source: Preliminaries · p. 2
• The cost framework learns a target unitary by optimizing a parameterized circuit, and can include general channels with dissipative effects.Source: Preliminaries · p. 2
• The theorem applies irrespective of ansatz choice or parameter initialization when the target ensemble forms at least a 2-design.Source: Preliminaries · p. 3
• For a target drawn from a unitary 2-design, gradient variance generally vanishes exponentially with qubit number under practical cost and ansatz conditions.Source: Preliminaries · p. 3
• In the numerical model, increasing scrambling strength narrows the cost valley around the minimum, leaving a broad plateau for highly scrambling unitaries.Source: Preliminaries · p. 4
• For sufficiently large scrambling time and entangling rate, gradient variance scales as 2^-2n and saturates; weaker scramblers show slower size-dependent decay.The target-ensemble and ansatz-ensemble variances show the same behavior.Source: Preliminaries · p. 4
• Analytically, exponential gradient-variance suppression persists when the target ensemble is sufficiently close to a unitary 2-design.Source: Preliminaries · p. 5
• The result constrains learning random or pseudo-random unitaries, while short-time chaotic evolution may remain learnable and prior information may alter the task.Source: Preliminaries · p. 5

Appendix A: Preliminaries

• The appendix recalls Haar-measure integration formulas for the first two moments of a unitary.Source: Appendix A · p. 8
• It states useful integration identities for operators, including one involving a subsystem swap and partial trace.Source: Appendix A · p. 8
• The proof expands unitaries in the computational basis and applies Eq. (A1) to evaluate the resulting expression.Source: Appendix A · pp. 8–9

1. Proof of Proposition 1

• The proof begins by switching the order of the average and the derivative.Source: §1 · p. 9
• It represents the input state with ρ = |ψ⟩⟨ψ| before evaluating the Haar integral over V.Source: §1 · p. 9
• The Haar integral is evaluated using Eq. (A2) to obtain the remaining expression.Source: §1 · p. 9

2. Proof of Theorem 2

• The proof starts from the variance of the cost function’s partial derivative.Source: §2 · p. 9
• Proposition 1 lets the derivation neglect the squared-mean term before evaluating the integral with Eq. (A3).Source: §2 · p. 9
• The resulting right-hand-side variance is evaluated with respect to the ansatz-evolved state |χ⟩ = U|ψ⟩.Source: §2 · p. 9

3. Proof of Corollary 3

• The proof considers a layered circuit with fixed unitaries and parameterized gates generated by Hermitian operators, and makes a bipartite cut at layer k.The generator variance is considered in the state at that cut.Source: §3 · p. 10
• The generator variance is bounded using the operator infinity norm, defined here as the largest eigenvalue.The proof assumes the squared generator norm is O(1).Source: §3 · p. 10
• Assuming Tr[H^2] is O(2^n), the proof uses the generator-variance bound to obtain the variance in the cost.The supplied passage does not state the resulting scaling.Source: §3 · p. 10

Appendix C: Extension of main results to the generalized cost Cgen

• The generalized cost uses an ansatz channel and the inverse-target channel, allowing the ansatz to incorporate nonunitary effects such as dissipation.The cost can be written using a unitary dilation on a larger system, with system, dissipation, and reference registers.Source: Appendix C · pp. 10–11
• The generalized objective accommodates multiple training states and measurements, and permits training data entangled with a reference system.The training data may act on the scrambling system and reference system together.Source: Appendix C · p. 10
• Proposition 4 gives the average partial derivative of general cost Cgen(θ,V) with respect to any parameter θk for an ensemble of target unitaries V forming a 2-design.Source: Appendix C · p. 11

1. Proof of Proposition 4

• The proof switches the order of averaging over target unitaries and differentiation to express the averaged gradient of Eq. (C3).The supplied passage does not include the resulting expression.Source: §1 · p. 11
• The calculation identifies each input-state density operator as ρ_i = |ψ_i⟩⟨ψ_i|.This notation is used in evaluating the integral terms.Source: §1 · p. 11
• The proof evaluates each integral term using Eq. (A4).The passage does not state the evaluated terms.Source: §1 · p. 11

2. Proof of Theorem 5

• The proof decomposes each measurement operator and recasts the generalized cost using operators on the system and reference registers.It then decomposes the input states and factorizes the cost into terms indexed by the resulting components.Source: §2 · pp. 11–12
• The gradient is expressed in terms of a variance, with J_k defined as −iU∂θ_kU†.The proof evaluates terms involving χ_lm and notes that the l = m case reduces to expression (B5).Source: §2 · p. 12
• The proof evaluates the asymptotic scaling of the variance expression using a layered, parameterized circuit structure.The circuit consists of fixed unitaries and gates generated by Hermitian operators.Source: §2 · p. 13
• Under the stated trace conditions, the proof bounds X_max by O(2^n) and Y_min by O(2^(2n)).The passage also states conditions on the measurement weights and generator norm, but does not provide the final variance scaling.Source: §2 · p. 13

Appendix D: Expected Scaling for Typical Ans¨atze

• Typical state overlaps scale as 2^-n, and a typical random ansatz can make the gradient variance scale as 2^-2n.The overlap scale enters the estimate of typical terms in the variance.Source: Appendix D · p. 13

Appendix E: Cost function to learn an unknown unitary

• Ref. demonstrated that the Hilbert-Schmidt cost is faithful, relates to average gate fidelity, and is efficiently computable on a quantum computer.Source: Appendix E · pp. 13–14
• The Hilbert-Schmidt Test measures a cost from a maximally entangled input, applying UV† before a Bell-state measurement.The ricochet property permits U and V to be applied in parallel, reducing circuit depth.Source: Appendix E · p. 14
• The local Hilbert-Schmidt cost uses local Bell-state projectors because the global cost has barren plateaus even for short-depth ansätze.It inherits faithfulness, and the global cost is bounded above by n times the local cost.Source: Appendix E · p. 14

Appendix F: Average and Variance of Gradient for Approximate Designs

• For an epsilon-approximate 2-design under the stated strong definition, the average gradient remains zero and the variance differs from the ideal case by a factor of 1 + epsilon.Source: Appendix F · p. 15