Quantum Working Group Seminar: Apparent Universal Behavior in Second Moments of Random Quantum Circuits
- Event Type
- Seminar/Symposium
- Sponsor
- Quantum Working Group
- Location
- 1022 Lincoln Hall
- Date
- Dec 10, 2025 4:00 - 4:50 pm
- Speaker
- Daniel Belkin
- Contact
- Felix Leditzky
- leditzky@illinois.edu
- Views
- 30
- Originating Calendar
- Quantum Working Group Calendar
Speaker: Daniel Belkin
Title: Apparent Universal Behavior in Second Moments of Random Quantum CircuitsAbstract: Just how fast does the brickwork circuit form an approximate 2-design? Is there any difference between anticoncentration and being a 2-design? Does geometry matter? How deep a circuit will I need in practice?
We tell you everything you always wanted to know about second moments of random quantum circuits, but were too afraid to compute. Our answers generally take the form of numerical results for up to 50 qubits. Our first contribution is a strategy to determine explicitly the optimal experiment which distinguishes any given ensemble from the Haar measure. With this formula and some computational tricks, we are able to compute t=2 multiplicative errors exactly out to modest system sizes. As expected, we see that most families of circuits form ϵ-approximate 2-designs in depth proportional to logn. For the 1D brickwork, we work out the leading-order constants explicitly. For graphs, we find some exceptions which are much slower, proving that they require at least Ω(n2) gates. This answers a question asked by ref. 1 in the negative. We explain these exceptional architectures in terms of connectedness. Based on this intuition we conjecture universal upper and lower bounds for graph-sampled circuit ensembles. For many architectures, the optimal experiment which determines the multiplicative error corresponds exactly to the collision probability (i.e. anticoncentration). However, we find that the star graph anticoncentrates much faster than it forms an ϵ-approximate 2-design. Finally, we show that one needs only ten to twenty layers to construct an approximate 2-design for realistic parameter ranges. This is a large constant-factor improvement over previous constructions. The parallel complete-graph architecture is not quite the fastest scrambler, partially resolving a question raised by ref. 2.About: Weekly seminar of the Quantum Working Group, covering topics in quantum information theory and operator algebras. Contact Marius Junge (mjunge@illinois.edu), Felix Leditzky (leditzky@illinois.edu), or Amanda Young (ayoung86@illinois.edu) with questions or inquiries.