[Paper Review] Learning quantum processes without input control

Marco Fanizza, Yihui Quek+1 more|arXiv (Cornell University)|Nov 9, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces a statistical learning theory for quantum processes that map classical inputs to quantum outputs without requiring control over the input distribution. By adapting shadow tomography and hypothesis selection with uniform convergence guarantees, the authors provide finite-sample learning algorithms even for infinite concept classes, establishing conditions for efficient learning of quantum processes in settings like astrophysical or biological systems where inputs are uncontrollable and only single copies per input are available.

ABSTRACT

We introduce a general statistical learning theory for processes that take as input a classical random variable and output a quantum state. Our setting is motivated by the practical situation in which one desires to learn a quantum process governed by classical parameters that are out of one's control. This framework is applicable, for example, to the study of astronomical phenomena, disordered systems and biological processes not controlled by the observer. We provide an algorithm for learning with high probability in this setting with a finite amount of samples, even if the concept class is infinite. To do this, we review and adapt existing algorithms for shadow tomography and hypothesis selection, and combine their guarantees with the uniform convergence on the data of the loss functions of interest. As a by-product we obtain sufficient conditions for performing shadow tomography of classical-quantum states with a number of copies which depends on the dimension of the quantum register, but not on the dimension of the classical one. We give concrete examples of processes that can be learned in this manner, based on quantum circuits or physically motivated classes, such as systems governed by Hamiltonians with random perturbations or data-dependent phase-shifts.

Motivation & Objective

• To develop a statistical learning framework for quantum processes that map classical inputs to quantum outputs when inputs are uncontrollable and sampled from an unknown distribution.
• To overcome the limitation of prior work requiring identical copies of quantum states prepared at controlled inputs by enabling learning from single, non-identical copies per input.
• To establish finite-sample learning guarantees for infinite concept classes of quantum processes, even in the agnostic setting where the true process may not be in the hypothesis class.
• To provide sufficient conditions for shadow tomography of classical-quantum states with a copy cost independent of the classical register dimension.
• To demonstrate practical applicability through examples involving quantum circuits, random Hamiltonian perturbations, and data-dependent phase shifts.

Proposed method

• Adapt the framework of hypothesis selection and shadow tomography to the setting of uncontrolled classical inputs, using a data-dependent loss function over quantum outputs.
• Introduce a pseudometric on the concept class of quantum processes based on the L1 norm of output states across input samples, enabling uniform convergence analysis.
• Define a covering number for the concept class that depends on the quantum system dimension but not on the classical input dimension, enabling finite-sample learning.
• Use uniform convergence of loss functions over the data to bound generalization error, ensuring high-probability learning with finite samples.
• Apply operator norm approximations and perturbation bounds (e.g., using the Dynkin formula) to control the distance between quantum processes under small changes in parameters.
• Derive explicit covering number bounds for specific classes, such as unitary-evolved states and projectors onto low-energy subspaces of perturbed Hamiltonians, using operator norm inequalities and spectral perturbation theory.

Experimental results

Research questions

• RQ1 Can quantum processes be learned when the input is not under the learner's control and only a single copy of each output state is available?
• RQ2 What conditions ensure finite-sample learning of an infinite class of quantum processes in the agnostic setting, where the true process may not be in the hypothesis class?
• RQ3 Can shadow tomography be applied to classical-quantum states when the classical register has infinite dimension, and if so, under what conditions?
• RQ4 How can covering numbers for quantum process classes be bounded in the absence of input control, and what determines their sample complexity?
• RQ5 What are the sample complexity bounds for learning processes involving unitary evolution under random or data-dependent Hamiltonians?

Key findings

• The paper establishes sufficient conditions for uniform convergence of loss functions over data, enabling high-probability learning of quantum processes even with infinite concept classes and no input control.
• A finite-sample learning algorithm is constructed that succeeds with high probability, with sample complexity depending only on the quantum system dimension and not on the classical input dimension.
• For the class of processes defined by unitary evolution under bounded Hamiltonians, the covering number is bounded by a function of the system dimension d, the bound B on the Hamiltonian norm, and the input function variation b.
• The paper provides a covering number bound of order (d²) for the class of projectors onto low-energy subspaces of perturbed Hamiltonians, with explicit dependence on the perturbation strength and spectral gap.
• The shadow tomography procedure is improved for classical-quantum states: the number of copies required scales with the quantum dimension but is independent of the classical register size.
• Explicit covering number bounds are derived for three classes: (1) unitary-evolved states, (2) projectors onto low-energy subspaces of fixed Hamiltonians, and (3) projectors onto low-energy subspaces of perturbed Hamiltonians, all with finite sample complexity under the proposed framework.