Mildly-Interacting Fermionic Unitaries are Efficiently Learnable

Abstract: Recent work has shown that one can efficiently learn fermionic Gaussian unitaries, also commonly known as nearest-neighbor matchcircuits or non-interacting fermionic unitaries. However, one could ask a similar question about unitaries that are near Gaussian: for example, unitaries prepared with a small number of non-Gaussian circuit elements. These operators find significance in quantum chemistry and many-body physics, yet no algorithm exists to learn them. We give the first such result by devising an algorithm which makes queries to an $n$-mode fermionic unitary $U$ prepared by at most $O(t)$ non-Gaussian gates and returns a circuit approximating $U$ to diamond distance $\varepsilon$ in time $\textrm{poly}(n,2^t,1/\varepsilon)$. This resolves a central open question of Mele and Herasymenko under the strongest distance metric. In fact, our algorithm is much more general: we define a property of unitary Gaussianity known as unitary Gaussian dimension and show that our algorithm can learn $n$-mode unitaries of Gaussian dimension at least $2n - O(t)$ in time $\textrm{poly}(n,2^t,1/\varepsilon)$. Indeed, this class subsumes unitaries prepared by at most $O(t)$ non-Gaussian gates but also includes several unitaries that require up to $2^{O(t)}$ non-Gaussian gates to construct. In addition, we give a $\textrm{poly}(n,1/\varepsilon)$-time algorithm to distinguish whether an $n$-mode unitary is of Gaussian dimension at least $k$ or $\varepsilon$-far from all such unitaries in Frobenius distance, promised that one is the case. Along the way, we prove structural results about near-Gaussian fermionic unitaries that are likely to be of independent interest.

An Analysis of Efficient Learning in Mildly-Interacting Fermionic Systems

The paper "Mildly-Interacting Fermionic Unitaries are Efficiently Learnable" by Vishnu Iyer advances the foundational understanding of efficiently learning quantum systems in the context of mildly-interacting fermionic unitaries. The study is situated at the intersection of quantum mechanics and computational learning, addressing critical gaps in the recognition and implementation of quantum processes. By extending diverse systems from non-interacting to moderately interacting domains, the paper introduces a novel approach to approximating complex unitaries that underlie the core principles of quantum chemistry and many-body physics. This paper provides theoretical analysis, algorithmic contributions, and practical insights with significant implications across quantum learning theory.

Technical Contributions

1. Extension from Non-Interacting to Mildly-Interacting Systems: While previous work efficiently addressed the learning of fermionic Gaussian unitaries, commonly referred to as FLO, this paper breaks new ground in the realm of unitaries that are nearly Gaussian by leveraging new techniques to handle unitaries represented with a small number of non-Gaussian elements. These systems bridge real-world applications, where perfect Gaussian approximations aren't adequate.

2. Algorithmic Framework: The study presents the first viable algorithm for learning $n$-mode fermionic unitaries with Gaussian dimensions severely exceeding traditional FLO boundaries. The proposed algorithm queries a given fermionic unitary $U$, approximated to diamond distance, operating proficiently in time $\poly(n, 2^t, 1/\epsilon)$.

3. Gaussian Dimension and Correlation: The notion of 'unitary Gaussian dimension' is rigorously defined, quantifying Gaussian properties in given systems, tied intimately to diagonalizing unitary operations. In turn, correlation matrices are leveraged to deduce Gaussian dimensions, allowing generalized fermionic systems to be efficiently learned.

4. Property Testing and Learning Algorithm: Through precise algorithmic steps, the paper elucidates distinct property testing and learning algorithms. The properties of Gaussian dimension efficacy are examined with a focus on stability and robustness, accommodating perturbations that simulate realistic quantum circumstances.

5. Robustness and Perturbation Bounds: Singular value decomposition and perturbation analysis are deployed, showing that fermionic systems close to Gaussian can be systematically rounded to higher Gaussian dimensions. The algebraic completeness posits connections with the distance and commutative properties of Gaussian operators, broadening the tractability for learning substantial physical systems.

Implications and Future Directions

Theoretical Implications:

• The critical insight that quantum systems with even minor non-Gaussian interference can be precisely learned implies a broader range of quantum phenomena are amenable to computational simulations, beyond historic constraints of purely non-interacting systems.
• Understanding and utilizing Gaussian dimensions within fermionic systems propels the mathematization of near-Gaussian unitaries, providing fresh pathways for theoretical advancements in quantum learning models and extending them to mixed-dimensional spaces.

Practical Implications:

• The ability to learn and approximate such unitaries efficiently has immediate applications in developing simulations for quantum impurity models and quantum chemistry problems, offering low-complexity resolutions for substantial electronic interaction scenarios.
• Enhancements in terms of scaling polynomially with dimensionality and exponentially in perturbative elements herald potential advances in quantum computation hardware and software systems development.

Future Directions:

• While the study acknowledges the complexity bounds as necessary, future work could explore tightening polynomial constants or explore learning models addressing agnostic quantum processes beyond theoretical Gaussianity.
• Extending these techniques to include $t$-doped Clifford circuits or bosonic Gaussian processes should be considered, with potential heuristic optimizations offering improved decisions under uncertainty.

Overall, the paper constitutes a rigorous stride forward in quantum learning theory by addressing longstanding challenges related to learning efficiency and interaction within fermionic systems, underpinning diverse practical and theoretical opportunities in quantum science and technology.