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Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits
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Constructs explicit data re-uploading PQCs (no parameterized classical processing) for multivariate polynomials and Hölder-smooth functions, giving non-asymptotic approximation error bounds in qubits, depth, and parameter count, and compares favorably with ReLU networks for smooth functions.
Problem
Quantitative (non-asymptotic) approximation error bounds for parameterized quantum circuits approximating general multivariate functions, without relying on parameterized classical data pre/post-processing.
Approach
Builds PQCs from single-qubit Pauli-X-encoded quantum signal processing monomials, combines monomials via linear combination of unitaries and Hadamard test, and approximates Hölder-smooth functions piecewise using a localization PQC (sign-function polynomial) plus local Taylor expansions with trainable coefficients, with all trainable parameters in rotation gates.
Key results
- Numerical experiment: test MSE of 3.57e-4 (K=2) and 1.04e-4 (K=10) for the single-qubit localization PQC learning D(x).
- Numerical experiment: test MSE of 2.22e-4 (K=2) and 9.82e-5 (K=10) for approximating the bivariate polynomial with a nested PQC with 120 to 1272 trainable parameters.
- Theory: for Hölder-smooth functions the PQC error is d^{s+β/2}K^{-β}, and the PQC size can be exponentially smaller in d than the near-optimal ReLU FNN for the same error, under the paper's comparison.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase K from 2 to 10 | Single-qubit PQC for learning D(x) · Piecewise-constant D(x), test set | MSE: 3.57d-4 -> 1.04d-4 | Helped |
| Increase K from 2 to 10 | Nested PQC for approximating f(x, y) · Bivariate polynomial f(x, y), test set | MSE: 2.22d-4 -> 9.82d-5 | Helped |
Limitations
- Numerical experiments are small-scale (bivariate, 200 training points) and only validate approximation on a single target.
- Comparison with ReLU networks is asymptotic and only for specific smoothness index; the bounds on depth and width are large (exponential in d for the Taylor part).
- Trivial-error caveat: the trifling region of measure dK*Delta is handled only in L2 with Delta set to K^{-d}.
Experimental setup
Datasets: randomly sampled points from [0,1] and [0,1]^2 for numerical experiments
Benchmarks: bivariate polynomial f(x,y) approximation piecewise-constant localization function D(x)
Metrics: mean squared error theoretical approximation error bound
Baselines: ReLU deep neural networks (near-optimal approximation) Lipschitz PQC of Pérez-Salinas et al. tensor-train PQC
Builds on: data re-uploading PQC quantum signal processing linear combination of unitaries Bernstein polynomials
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