Random unitaries in extremely low depth (2407.07754v1)

Abstract: We prove that random quantum circuits on any geometry, including a 1D line, can form approximate unitary designs over $n$ qubits in $\log n$ depth. In a similar manner, we construct pseudorandom unitaries (PRUs) in 1D circuits in $\text{poly} \log n $ depth, and in all-to-all-connected circuits in $\text{poly} \log \log n $ depth. In all three cases, the $n$ dependence is optimal and improves exponentially over known results. These shallow quantum circuits have low complexity and create only short-range entanglement, yet are indistinguishable from unitaries with exponential complexity. Our construction glues local random unitaries on $\log n$-sized or $\text{poly} \log n$-sized patches of qubits to form a global random unitary on all $n$ qubits. In the case of designs, the local unitaries are drawn from existing constructions of approximate unitary $k$-designs, and hence also inherit an optimal scaling in $k$. In the case of PRUs, the local unitaries are drawn from existing unitary ensembles conjectured to form PRUs. Applications of our results include proving that classical shadows with 1D log-depth Clifford circuits are as powerful as those with deep circuits, demonstrating superpolynomial quantum advantage in learning low-complexity physical systems, and establishing quantum hardness for recognizing phases of matter with topological order.

• The paper shows that logarithmic-depth circuits can form approximate unitary designs, reducing depth requirements from polynomial to logarithmic scales.
• It constructs pseudorandom unitaries in poly log n depth for 1D and poly log log n depth for fully connected circuits, outperforming earlier methods.
• The authors back their findings with rigorous proofs using advanced quantum information theory tools, paving the way for efficient quantum measurements and simulations.

Overview of "Random unitaries in extremely low depth"

This paper by Schuster, Haferkamp, and Huang establishes compelling advancements in the generation of random quantum unitaries within unprecedentedly low circuit depths. Through theoretical analysis and construction, the authors demonstrate that approximate unitary designs can be formed using logarithmic-depth quantum circuits for $n$ qubits. Such significant reduction in depth stems from effectively gluing together local random unitary designs across various geometries, including one-dimensional (1D) and all-to-all connected circuits.

Key Contributions and Methodology

1. Approximate Unitary Designs: Central to the paper is showing that quantum circuits of logarithmic depth ($\log n$) can approximate unitary designs. Consequently, they significantly reduce the depth required compared to existing methods, which typically relied on extensive circuit depths proportional to $n$ or higher.
2. Pseudorandom Unitaries: The authors also show how pseudorandom unitaries can be constructed within $poly \log n$ depth for 1D architectures and $poly \log \log n$ for highly connected circuits. This dramatically improves on the previously known constructions that necessitated polynomial-depth circuits in $n$.
3. Theoretical Framework and Proofs: Leveraging advanced tools in quantum information theory—such as permutation operators and Weingarten calculus—the authors elegantly prove that local quantum circuits can facilitate exponential reductions in the required depth for forming unitary designs. They employ robust lemmas for gluing approximations of unitary designs and analyzing residual terms in the Haar twirl.

Implications and Applications

The findings present profound implications in various quantum computing applications. The constructions pave the way for classical shadow protocols using low-depth circuits, potentially allowing for implementations under realistic experimental constraints marked by noise. This sets a stage for more feasible and efficient quantum measurements, such as fidelity estimations and non-local observable measurements, in contemporary quantum devices.

Additionally, through their construction of pseudorandom unitaries, the authors provide rigorous theoretical support for the quantum hardness of tasks like recognizing topological order—a central challenge in quantum many-body physics. They demonstrate that such recognitions are computationally hard even at depths where previously it was thought feasible.

Further, the paper highlights conceivable quantum advantages in learning scenarios extending to lower-complexity physical systems, aligning with advancements in quantum machine learning and information processing.

Theoretical and Practical Future Directions

This paper not only enhances the theoretical landscape of quantum complexity but holds significant potential for impacting practical quantum computing. The profound depth reductions achieved may influence future hardware development geared towards efficient implementations of quantum protocols.

Theoretical future work might attempt to refine the constants underlying these logarithmic depth constructions or examine whether such unitary designs hold up under even broader classes of random processes. Furthermore, the exploration of these low-depth constructions in non-standard geometries or more complex Hamiltonian models could yield essential insights into intricate quantum systems.

In summary, by showing exponential improvements on the requisite depths for constructing unitary designs and pseudorandom unitaries, the researchers offer a powerful toolkit for shaping future explorations and applications across quantum computing disciplines.