Unitary designs from statistical mechanics in random quantum circuits.

By Nicholas Hunter-Jones. 2019

147 citations recorded by the source corpus.

Random quantum circuits are proficient information scramblers and efficient generators of randomness, rapidly approximating moments of the unitary group. We study the convergence of local random quantum circuits to unitary $k$-designs. Employing a statistical mechanical mapping, we give an exact expression of the distance to forming an approximate design as a lattice partition function. In the statistical mechanics model, the approach to randomness has a simple interpretation in terms of domain walls extending through the circuit. We analytically compute the second moment, showing that random circuits acting on $n$ qudits form approximate 2-designs in $O(n)$ depth, as is known. Furthermore, we argue that random circuits form approximate unitary $k$-designs in $O(nk)$ depth and are thus essentially optimal in both $n$ and $k$. We can show this in the limit of large local dimension, but more generally rely on a conjecture about the dominance of certain domain wall configurations.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6721d9e81a77b23f4d0cc8114cccfa0f9b65e2d2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. A. Nahum
2. Matthew Fisher
3. B. Skinner
4. V. Khemani
5. J. Ruhman
6. M. Cerezo
7. S. Vijay
8. Tianci Zhou

In Atomic and Molecular Physics, and Optics

1. A. Nahum
2. Matthew Fisher
3. V. Khemani
4. B. Skinner
5. J. Ruhman
6. Tianci Zhou
7. S. Vijay
8. Sthitadhi Roy

In Statistics and Probability

1. V. Khemani
2. A. Nahum
3. Matthew Fisher
4. Tianci Zhou
5. J. Ruhman
6. B. Skinner
7. S. Vijay
8. M. Cerezo

Citations in the corpus

• Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory (2020, methodology): “There the appropriate effective “spin model” has hard constraints on the allowed spin configurations, which for example enforce causality [33, 41, 43] (these are relaxed when projection operators are included [13, 14]).”
• Random Quantum Circuits (2022, background): “A key quantity in this setting is an “order parameter” labelling pairing between Feynman trajectories (49, 50, 113, 95, 51, 135).”
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus (2021, background): “[58] Nicholas Hunter-Jones, “Unitary designs from statistical mechanics in random quantum circuits,” arXiv preprint arXiv:1905.”
• Diagnosing Barren Plateaus with Tools from Quantum Optimal Control (2021, background): “[96] Nicholas Hunter-Jones, “Unitary designs from statistical mechanics in random quantum circuits,” arXiv preprint arXiv:1905.”
• Entanglement Membrane in Chaotic Many-Body Systems (2019, background): “The approach could also be used to study questions motivated by quantum information such as the emergence of the “design” property in random circuits with non-Haar–distributed gates [5, 17, 80–82].”
• Linear growth of quantum circuit complexity (2021, background): “[9] addresses bounds on convergence to unitary designs [8, 24, 25, 28, 36, 48], translating these bounds into results about circuit complexity.”