Out-of-distribution generalization for learning quantum dynamics

By Caro, Huang, Ezzell, Gibbs, Sornborger, Cincio, Coles, and Holmes

Outline

• Overview of main result
• Locally scrambling ensembles
• Framework
• Numerical results
• Open Questions

Applications of Quantum Dynamics Learning.

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Main result

Out-of-distribution generalization for unitary learning is possible for a wide variety of training and testing distributions, if both distributions are locally scrambled

• A QNN capturing the action of a target unitary on only a polynomial number of random product states generalizes to test data composed of fully random states
• Possible to learn a unitary on a broad spread of highly entangled states having only studied its action on a limited number of product states
• Product states can be prepared using only single qubit gates, bringing the possibility of using QML to learn unitary processes nearer term
• A low-entangling unitary can be compiled using results from only low-entangled training states, potentially indicating a new quantum-inspired classical approach to unitary compilation.

Generalization for learning a target unitary U

Locally scrambling ensembles

• Ensembles of states that are at least locally random
• Invariant under pre-processing by tensor products of arbitrary local unitaries, i.e. of the form

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for some locally scrambled unitary ensemble

• Out-of-distribution learning results hold for a slightly broader class of ensembles with the requirement that the ensemble agrees with a locally scrambled one up to and including its complex second moments, denoted by
• divides naturally into sub-classes of ensembles that give rise to training sets and testing sets

Examples of ensembles in :

• - Products of Haar-random single-qubit states.
• - Products of random single-qubit stabilizer states.
• - Products of Haar-random k-qubit states.
• - Haar-random n-qubit states.
• - A 2-design on n-qubit states.
• - The output states of random quantum circuits,

where denotes the k-local n-qubit quantum

circuit architecture from which the random circuit is

constructed

Locally Scrambling Ensembles

Framework

• For learning an unknown n-qubit unitary , optimize classical parameters α of V(α), an n-qubit unitary QNN, so for α_opt, V(α_opt) predicts the action of U on previously unseen test states, i.e. minimize

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where testing distribution P is a probability distribution over (pure) n-qubit states and the factor of 1/4 ensures 0 ≤ R_P(α) ≤ 1.

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• A learner so cannot evaluate the above cost function (generally does not have access to the full testing ensemble P!)
• Evaluate the training cost on a training data set consisting of input-output pairs of pure n-qubit states, where N input states are drawn independently from a training distribution Q.

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• Rewrite in terms of the average fidelity as

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which can be efficiently computed using a Loschmidt echo or swap test circuit

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• Goal is to achieve small risk R_P(α) by training the parameters α of the QNN to minimize the training cost C_DQ(N)(α)

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Numerical results

• Compare the exact evolution of product states under an unknown target Hamiltonian to an L layered Trotterized ansatz.
• In-distribution risk is average prediction error over random product states
• Out-of-distribution testing risk

is over the global Haar distribution

• Linear correlation between the cost function and both and demonstrates that both in-distribution and out-of-distribution generalization have been successfully achieved
• Increasing the number of layers in the ansatz corresponds to a decreased obtainable cost function value

Out-of-distribution Generalization for Hamiltonian Learning

Open Questions

• Is out-of-distribution generalization is possible for other QML tasks such as learning quantum channels, or for performing classification tasks?
• Could out-of distribution generalization be viable for other classes of distributions?
• Is it worth investigating possible connections to Mitiq?
• Paper mentions learning pulse sequences for noise resilient gate implementations

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