[Paper Review] Learnability of the output distributions of local quantum circuits

Marcel Hinsche, Marios Ioannou+7 more|arXiv (Cornell University)|Oct 11, 2021

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper investigates the learnability of output distributions from local quantum circuits, showing that while Clifford circuits of super-logarithmic depth are not efficiently learnable via statistical queries (implying no quantum advantage in this setting), they are efficiently learnable by classical learners when given direct sample access. The work provides the first rigorous analysis of quantum circuit Born machines from a probabilistic modeling perspective.

ABSTRACT

There is currently a large interest in understanding the potential advantages quantum devices can offer for probabilistic modelling. In this work we investigate, within two different oracle models, the probably approximately correct (PAC) learnability of quantum circuit Born machines, i.e., the output distributions of local quantum circuits. We first show a negative result, namely, that the output distributions of super-logarithmic depth Clifford circuits are not sample-efficiently learnable in the statistical query model, i.e., when given query access to empirical expectation values of bounded functions over the sample space. This immediately implies the hardness, for both quantum and classical algorithms, of learning from statistical queries the output distributions of local quantum circuits using any gate set which includes the Clifford group. As many practical generative modelling algorithms use statistical queries -- including those for training quantum circuit Born machines -- our result is broadly applicable and strongly limits the possibility of a meaningful quantum advantage for learning the output distributions of local quantum circuits. As a positive result, we show that in a more powerful oracle model, namely when directly given access to samples, the output distributions of local Clifford circuits are computationally efficiently PAC learnable by a classical learner. Our results are equally applicable to the problems of learning an algorithm for generating samples from the target distribution (generative modelling) and learning an algorithm for evaluating its probabilities (density modelling). They provide the first rigorous insights into the learnability of output distributions of local quantum circuits from the probabilistic modelling perspective.

Motivation & Objective

• To determine whether output distributions of local quantum circuits can be efficiently learned using probabilistic modeling techniques.
• To assess the potential for quantum advantage in generative modeling tasks based on quantum circuit Born machines.
• To evaluate the learnability of these distributions under different oracle models—specifically, statistical query access versus direct sample access.
• To establish theoretical limits on classical and quantum learning algorithms for such distributions.
• To provide foundational insights into the feasibility of quantum machine learning for generative modeling using near-term quantum devices.

Proposed method

• Analyzes the Probably Approximately Correct (PAC) learnability of quantum circuit Born machines in two oracle models: statistical query and direct sample access.
• Proves that output distributions of super-logarithmic depth Clifford circuits are not sample-efficiently learnable in the statistical query model using bounded functions over the sample space.
• Demonstrates that in the more powerful direct sample access model, classical learners can efficiently PAC-learn the output distributions of local Clifford circuits.
• Employs tools from finite field linear algebra, including the probability of random binary matrices having full column rank, to establish probabilistic bounds on subspace recovery.
• Uses an affine subspace recovery algorithm that samples k vectors, transforms them to a vector subspace via XOR with a reference sample, and applies Gaussian elimination to recover a basis.
• Applies concentration bounds and probabilistic inequalities to show that k = n + ⌈log(1/δ)⌉ samples suffice to recover the subspace with probability at least 1−δ.

Experimental results

Research questions

• RQ1 Can the output distributions of local quantum circuits be efficiently learned using statistical query access?
• RQ2 Is there a quantum advantage in learning the output distributions of local quantum circuits via quantum or classical algorithms?
• RQ3 Under what conditions can classical learners efficiently learn the output distributions of local Clifford circuits?
• RQ4 How does the depth of a quantum circuit affect the learnability of its output distribution in the statistical query model?
• RQ5 What is the minimal number of samples required to reliably reconstruct the support of a quantum circuit's output distribution?

Key findings

• Output distributions of super-logarithmic depth Clifford circuits are not sample-efficiently learnable in the statistical query model, implying that any learning algorithm relying on empirical expectation values of bounded functions cannot efficiently learn these distributions.
• This result implies that both classical and quantum algorithms face fundamental hardness in learning such distributions when using statistical queries, which are common in practical generative modeling algorithms.
• In contrast, when direct access to samples is available, the output distributions of local Clifford circuits are computationally efficiently PAC-learnable by a classical learner.
• The sample complexity required for classical PAC learning is O(n + log(1/δ)), where n is the number of qubits and δ is the failure probability.
• The analysis establishes that with high probability (at least 1−δ), k = n + ⌈log(1/δ)⌉ samples suffice to recover the affine subspace structure of the output distribution.
• The results apply equally to both generative modeling (learning to sample) and density modeling (learning to evaluate probabilities), providing a unified theoretical foundation for quantum circuit Born machines.