Anticoncentrated $n$-bit distribution from $\log(n)$ qubits

TL;DR

This paper introduces holographic random circuit sampling (HRCS), a protocol that achieves anticoncentration with significantly fewer qubits, challenging the assumed link between anticoncentration and quantum advantage.

Contribution

The authors demonstrate that anticoncentration can be achieved with only logarithmic qubits, providing a new space-time trade-off and potential for classical simulation.

Findings

HRCS can generate anticoncentrated distributions with O(log n) qubits.

Experimental validation achieved sampling of 200 bits with only 20 qubits.

The association between anticoncentration and quantum advantage is not fundamental.

Abstract

Random circuit sampling (RCS) is a leading approach to demonstrate quantum advantage, with its believed classical hardness rooted in anticoncentration of output distributions and average-case hardness of probability estimation. Here we show that this association is not fundamental. We introduce holographic random circuit sampling (HRCS), a spatiotemporal protocol that interleaves random unitary evolution with mid-circuit measurements. We prove that $n$ classical bits exhibiting $\epsilon$-approximate anticoncentration of Haar random states can be generated using only $\mathcal{O}(\log n)$ physical qubits and linear depth, establishing a precise space-time trade-off and indicating efficient classical simulation. Our analyses is built upon exact formulas for collision probability and higher-order power sums. Our experimental validation on IBM Quantum devices demonstrates sampling up to 200 classical bits using only 20 qubits.