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Effects of quantum resources on the statistical complexity of quantum circuits
Quantum Physics · Submitted 5 Feb 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the increase in the statistical complexity of a quantum circuit when an additional quantum channel is added to it is upper bounded by the free robustness of the added channel.
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The abstract, sentence by sentence
We investigate how the addition of quantum resources changes the statistical complexity of quantum circuits by utilizing the framework of quantum resource theories. Measures of statistical complexity that we consider include the Rademacher complexity and the Gaussian complexity, which are well-known measures in computational learning theory that quantify the richness of classes of real-valued functions. We derive bounds for the statistical complexities of quantum circuits that have limited access to certain resources and apply our results to two special cases: (1) stabilizer circuits that are supplemented with a limited number of T gates and (2) instantaneous quantum polynomial-time Clifford circuits that are supplemented with a limited number of CCZ gates. We show that the increase in the statistical complexity of a quantum circuit when an additional quantum channel is added to it is upper bounded by the free robustness of the added channel. Finally, we derive bounds for the generalization error associated with learning from training data arising from quantum circuits.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate how the addition of quantum resources changes the statistical complexity of quantum circuits by utilizing the framework of quantum resource theories. | contribution 0.98 | 0.86 | 0.05 | asserted 0.69 |
| s1 | Measures of statistical complexity that we consider include the Rademacher complexity and the Gaussian complexity, which are well-known measures in computational learning theory that quantify the richness of classes of real-valued functions. | background 0.57 | 0.19 | 0.03 | asserted 0.99 |
| s2 | We derive bounds for the statistical complexities of quantum circuits that have limited access to certain resources and apply our results to two special cases: (1) stabilizer circuits that are supplemented with a limited number of T gates and (2) instantaneous quantum polynomial-time Clifford circuits that are supplemented with a limited number of CCZ gates. | contribution 0.89 | 0.96 | 0.12 | asserted 0.99 |
| s3 | We show that the increase in the statistical complexity of a quantum circuit when an additional quantum channel is added to it is upper bounded by the free robustness of the added channel. | result 1.00 | 0.94 | 0.46 | asserted 1.00 |
| s4 | Finally, we derive bounds for the generalization error associated with learning from training data arising from quantum circuits. | result 0.59 | 0.95 | 0.11 | asserted 0.99 |
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The paper
Original abstract
We investigate how the addition of quantum resources changes the statistical complexity of quantum circuits by utilizing the framework of quantum resource theories. Measures of statistical complexity that we consider include the Rademacher complexity and the Gaussian complexity, which are well-known measures in computational learning theory that quantify the richness of classes of real-valued functions. We derive bounds for the statistical complexities of quantum circuits that have limited access to certain resources and apply our results to two special cases: (1) stabilizer circuits that are supplemented with a limited number of T gates and (2) instantaneous quantum polynomial-time Clifford circuits that are supplemented with a limited number of CCZ gates. We show that the increase in the statistical complexity of a quantum circuit when an additional quantum channel is added to it is upper bounded by the free robustness of the added channel. Finally, we derive bounds for the generalization error associated with learning from training data arising from quantum circuits.
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References
How the literature uses it
7 arXiv papers cite this in 13 sentences
Papers citing it · 6 sentences without a confident label
Builds on it 1 paper
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In [ref] , we extended the above analysis by studying the dependence of the Rademacher complexity on general quantum resources by employing tools from the framework of quantum resource theories [ref] .
Rademacher complexity of noisy quantum circuits · Introduction · 0.84
Uses its method or tool 1 paper
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We follow a setup similar to that of [ref] .
Rademacher complexity of noisy quantum circuits · Methods · 0.98
Compares with it 1 paper
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However, unlike in Ref. [ref] , the Rademacher complexity bounds of Refs. [ref] are given in terms of quantities that exhibit an implicit dependence on the data-encoding strategy.
Encoding-dependent generalization bounds for parametrized quantum circuits · Encoding-dependent complexity and generalization bounds · 0.97
Cites it as background 4 papers
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Recent work has studied the training sample complexity needed for QML models to generalize [ref] .
Theory for Equivariant Quantum Neural Networks · Equivariance versus barren plateaus, local minima and data requirements · 0.87
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Some prior works have studied the generalization capabilities of quantum models, among them the classical learning-theoretic approaches of [ref] ; the more geometric perspective of [ref] ; and the information-theoretic technique of [ref] .
Generalization in quantum machine learning from few training data · Related work on generalization · 0.99
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Previous works have focused on showing how complexity measures depend on the different parts of the quantum model that implement the observable measurement, such as the quantum circuit ansatz [ref] , or the level of noise in the model [ref] .
Structural risk minimization for quantum linear classifiers · Introduction · 0.80
What it cites · 3 sentences without a confident label
Cites as background 6 papers
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These include magic [ref] , entanglement [ref] and coherence [ref] .
Quantifying Superposition · Introduction · 0.97
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Examples of variational quantum algorithms include the variational quantum eigensolver (VQE) for quantum chemistry [ref] , the quantum approximate optimization algorithm (QAOA) for optimization [ref] , and the quantum neural network (QNN) that generalizes the classical neural network [ref] .
A Quantum Approximate Optimization Algorithm · Introduction · 0.92
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Examples of variational quantum algorithms include the variational quantum eigensolver (VQE) for quantum chemistry [ref] , the quantum approximate optimization algorithm (QAOA) for optimization [ref] , and the quantum neural network (QNN) that generalizes the classical neural network [ref] .
Classification with Quantum Neural Networks on Near Term Processors · Introduction · 0.92
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