Anticoncentration is (almost) all you need

Abstract

Until very recently, it was generally believed that the (approximate) 2-design property is strictly stronger than anticoncentration of random quantum circuits, mainly because it was shown that the latter anticoncentrate in logarithmic depth, while the former generally need linear depth circuits. This belief was disproven by recent results which show that so-called relative-error approximate unitary designs can, in fact, be generated in logarithmic depth, implying anticoncentration. Their result does however not apply to ordinary local random circuits, a gap which we close in this letter, at least for 2-designs. More precisely, we show that anticoncentration of local random quantum circuits already implies that they form relative-error approximate state 2-designs, making them equivalent properties for these ensembles. Our result holds more generally for any random circuit which is invariant under local (single-qubit) unitaries, independent of the architecture.

4+2 pages. v4: close to published version, fixed title, no major changes

References in corpus (15)

• Randomized Benchmarking of Quantum Gates
• Quantum advantage in learning from experiments
• Foundation of Statistical Mechanics under experimentally realistic conditions
• Random Quantum Circuits
• Evenly distributed unitaries: on the structure of unitary designs
• Chaos, Complexity, and Random Matrices
• Computational advantage of quantum random sampling
• A general framework for randomized benchmarking
• Random quantum circuits are approximate unitary $t$-designs in depth $O\left(n{t}^{5+o(1)}\right)$
• Improved spectral gaps for random quantum circuits: large local dimensions and all-to-all interactions
• Anticoncentration and state design of random tensor networks
• Optimal Conversion from Classical to Quantum Randomness via Quantum Chaos
• Anticoncentration and State Design of Doped Real Clifford Circuits and Tensor Networks
• Non-Clifford Cost of Random Unitaries
• Efficient approximate unitary designs from random Pauli rotations