On the average-case complexity of learning output distributions of quantum circuits
1Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Germany
2IBM Quantum, Almaden Research Center, San Jose, CA 95120, USA
3Institute for Integrated Circuits, Department of Computer Science, Johannes Kepler University Linz, Austria
4School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02318, USA
5Department of Mathematics, Saarland University, Saarbrücken, Germany
| Published: | 2025-10-13, volume 9, page 1883 |
| Editor: | Yu Tong |
| Eprint: | arXiv:2305.05765v2 |
| Doi: | https://doi.org/10.22331/q-2025-10-13-1883 |
| Citation: | Quantum 9, 1883 (2025). |
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Abstract
In this work, we show that learning the output distributions of brickwork random quantum circuits is average-case hard in the statistical query model. This learning model is widely used as an abstract computational model for most generic learning algorithms. In particular, for brickwork random quantum circuits on $n$ qubits of depth $d$, we show three main results:
– At super logarithmic circuit depth $d=\omega(\log(n))$, any learning algorithm requires super polynomially many queries to achieve a constant probability of success over the randomly drawn instance.
– There exists a $d=O(n)$, such that any learning algorithm requires $\Omega(2^n)$ queries to achieve a $O(2^{-n})$ probability of success over the randomly drawn instance.
– At infinite circuit depth $d\to\infty$, any learning algorithm requires $2^{2^{\Omega(n)}}$ many queries to achieve a $2^{-2^{\Omega(n)}}$ probability of success over the randomly drawn instance.
As an auxiliary result of independent interest, we show that the output distribution of a brickwork random quantum circuit is constantly far from any fixed distribution in total variation distance with probability $1-O(2^{-n})$, which confirms a variant of a conjecture by Aaronson and Chen.
Featured image: The statistical query complexity of learning the output distributions of local random quantum circuits of depth $d$.
Popular summary
model for the physical world. Understanding the intrinsic properties of quantum circuits is thus
of fundamental interest. One such property is quantum circuit complexity, which corresponds
to the minimum number of elementary gates necessary to implement a given quantum circuit.
Another such property is the complexity of simulating quantum circuits, which is the basis for
many quantum advantage proposals. There, one asks what computational resources are required
for sampling from the output distribution of a given quantum circuit when applied to a fixed
input product state and when measured in the computational basis. In this work, we take the
perspective of learning theory, and study the complexity of learning the output distributions of
quantum circuits. At a high level, this amounts to the resources required to reproduce samples
according to the output distribution of a quantum circuit when given black-box access to the
corresponding output distribution.
More specifically, we study the average case complexity of learning the output distributions of
quantum circuits. This amounts to the cost of learning when the quantum circuit is drawn ran-
domly according to some measure. In particular, we ask the following question:
What is the complexity of learning generic quantum circuit output distributions?
The main contribution of our work is to provide a variety of rigorous answers to this question, within the "statistical query" framework, which models the vast majority of learning algorithms. In particular, we provide a variety of lower bounds on the query complexity necessary to learn certain fraction of randomly drawn local quantum circuits, of a specific depth.
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