Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix

Abstract

Learning about physical systems from quantum-enhanced experiments, relying on a quantum memory and quantum processing, can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum advantages have been established for a variety of state learning tasks, quantum process learning allows for comparable advantages only with a careful problem formulation and is less understood. We establish an exponential quantum advantage for learning an unknown $n$-qubit quantum process $\mathcal{N}$. We show that a quantum memory allows to efficiently solve the following tasks: (a) learning the Pauli transfer matrix of an arbitrary $\mathcal{N}$, (b) predicting expectation values of bounded Pauli-sparse observables measured on the output of an arbitrary $\mathcal{N}$ upon input of a Pauli-sparse state, and (c) predicting expectation values of arbitrary bounded observables measured on the output of an unknown $\mathcal{N}$ with sparse Pauli transfer matrix upon input of an arbitrary state. With quantum memory, these tasks can be solved using linearly-in-$n$ many copies of the Choi state of $\mathcal{N}$, and even time-efficiently in the case of (b). In contrast, any learner without quantum memory requires exponentially-in-$n$ many queries, even when querying $\mathcal{N}$ on subsystems of adaptively chosen states and performing adaptively chosen measurements. In proving this separation, we extend existing shadow tomography upper and lower bounds from states to channels via the Choi-Jamiolkowski isomorphism. Moreover, we combine Pauli transfer matrix learning with polynomial interpolation techniques to develop a procedure for learning arbitrary Hamiltonians, which may have non-local all-to-all interactions, from short-time dynamics. Our results highlight the power of quantum-enhanced experiments for learning highly complex quantum dynamics.

30+31 pages, 2+1 figures; V2 includes small corrections to Remark 4.3 and Lemma 6.1 as well improvements to the presentation

References in corpus (12)

• Learning many-body Hamiltonians with Heisenberg-limited scaling
• Scalable and Flexible Classical Shadow Tomography with Tensor Networks
• Classical shadow tomography for continuous variables quantum systems
• An approximate description of quantum states
• Practical Black Box Hamiltonian Learning
• Closed-form analytic expressions for shadow estimation with brickwork circuits
• The Learnability of Quantum States
• Scalably learning quantum many-body Hamiltonians from dynamical data
• Foundations for learning from noisy quantum experiments
• Rademacher complexity of noisy quantum circuits
• A Hierarchy for Replica Quantum Advantage
• Learning quantum processes without input control

Cited by in corpus (14)

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• Quantum learning advantage on a scalable photonic platform
• The power and limitations of learning quantum dynamics incoherently
• Structure learning of Hamiltonians from real-time evolution
• Quantum Tensor Product Decomposition from Choi State Tomography
• Ansatz-free Hamiltonian learning with Heisenberg-limited scaling
• Hamiltonian Learning via Shadow Tomography of Pseudo-Choi States
• Learning unitaries with quantum statistical queries
• Single-copy stabilizer testing
• Learning Quantum Processes with Quantum Statistical Queries
• Fast generation of Pauli transfer matrices utilizing tensor product structure
• Nearly query-optimal classical shadow estimation of unitary channels
• Learning the structure of any Hamiltonian from minimal assumptions
• Quantifying Irreversibility via Bayesian Subjectivity for Classical & Quantum Linear Maps