Information-theoretic bounds on quantum advantage in machine learning (2101.02464v2)

Abstract: We study the performance of classical and quantum ML models in predicting outcomes of physical experiments. The experiments depend on an input parameter $x$ and involve execution of a (possibly unknown) quantum process $\mathcal{E}$. Our figure of merit is the number of runs of $\mathcal{E}$ required to achieve a desired prediction performance. We consider classical ML models that perform a measurement and record the classical outcome after each run of $\mathcal{E}$, and quantum ML models that can access $\mathcal{E}$ coherently to acquire quantum data; the classical or quantum data is then used to predict outcomes of future experiments. We prove that for any input distribution $\mathcal{D}(x)$, a classical ML model can provide accurate predictions on average by accessing $\mathcal{E}$ a number of times comparable to the optimal quantum ML model. In contrast, for achieving accurate prediction on all inputs, we prove that exponential quantum advantage is possible. For example, to predict expectations of all Pauli observables in an $n$-qubit system $\rho$, classical ML models require $2^{\Omega(n)}$ copies of $\rho$, but we present a quantum ML model using only $\mathcal{O}(n)$ copies. Our results clarify where quantum advantage is possible and highlight the potential for classical ML models to address challenging quantum problems in physics and chemistry.

• The paper establishes that classical ML models can match quantum models in achieving small average-case prediction errors, highlighting theoretical bounds on query complexity.
• The study proves an exponential separation between classical and quantum models for worst-case prediction error when estimating Pauli operator expectation values.
• Analytical results and numerical experiments demonstrate that quantum advantage emerges in worst-case scenarios, guiding future hybrid algorithm research.

Information-Theoretic Bounds on Quantum Advantage in Machine Learning

The paper presented in this paper evaluates the effectiveness of classical and quantum ML models in predicting the outcomes of physical experiments that involve an unknown quantum process. Specifically, the authors focus on the ability of these ML models to predict a certain function, $f(x) = \Tr(O \cE(\lvert x \rangle \! \langle x \rvert))$, where $\cE$ represents a completely positive trace-preserving (CPTP) quantum map, $O$ a known observable, and $x$ an input parameter. The paper establishes a significant distinction between the capabilities of classical and quantum ML models under differing conditions on prediction error.

The central result is concerned with the average-case prediction error. The paper demonstrates that a classical ML model can be as effective as a quantum ML model in achieving a small average-case prediction error given a fixed input distribution $\mathcal{D}(x)$. The authors assert that, in some scenarios, a classical ML model requires a comparable number of queries to access the quantum process $\cE$, differing only by a polynomial factor compared to the optimal quantum ML model. This implies that exponential quantum advantage is absent in average-case prediction scenarios with respect to query complexity, although not ruling out potential advantages in computational complexity.

Furthermore, the result suggests that classical ML models, which process classical measurement data post-experiment, are quite efficient in average-case scenarios. Classical ML models can utilize the measurement outcomes from repeated investments in quantum experiments and adapt to challenging quantum problems, which strengthens the prospects for near-term applications in fields such as quantum chemistry and material sciences.

Contrastingly, when focusing on worst-case prediction errors, the authors identify scenarios with the potential for exponential quantum advantage. A notable example involves predicting the expectation values of Pauli operators in an $n$-qubit state $\rho$. Here, the authors prove that a quantum ML model can achieve accurate predictions with $O(n)$ samples. However, they show that classical ML models, even those with the ability to perform sophisticated adaptive measurements, inherently demand at least $2^{\Omega(n)}$ samples to ensure similar predictive accuracy of all Pauli observables. This gap underscores a stark distinction between classical and quantum processing capabilities, particularly highlighting the limits of classical techniques when each quantum process measurement must be concluded with a classical data recording.

The paper's theoretical groundwork is supported by analytical results and numerical experiments. It succinctly delineates the required sample complexities, demonstrating an exponential separation between classical and quantum learning models under worst-case scenarios. This validation of quantum advantage through rigorous theoretical insights and empirically substantiated results accentuates the potential for quantum algorithms in tackling specific quantum tasks.

In conclusion, the findings of this research clarify the domains where quantum advantage is possible, offering theoretical benchmarks for constructing effective ML models. The insights encourage further exploration into hybrid classical-quantum algorithms and their applications in the context of near-term quantum technologies. Such discernment into quantum ML models' capabilities and limitations serves as a foundation for future inquiries and developments within the AI landscape.