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Quantum supremacy conjecture for random circuit sampling

From papers

Let A{\cal A} be a circuit architecture, and let HA{\cal H}_{\cal A} be the distribution of circuits in A{\cal A} whose gates are independently drawn from Haar measure. Random circuit sampling (RCS) is the task of receiving a circuit CC sampled from HA{\cal H}_{\cal A} and an error parameter ϵ>0\epsilon>0, then sampling from the distribution induced by CC, namely

Pr⁡(y)=∣⟨y∣C∣0⟩∣2\Pr(y)=|\langle y|C|0\rangle|^2

up to total variation distance ϵ\epsilon in time poly⁡(n,1/ϵ)\operatorname{poly}(n,1/\epsilon). Here nn is the number of qubits.

Quantum supremacy conjecture. There is no classical randomized algorithm that performs RCS to inverse-polynomial total variation-distance error.

This conjecture asserts the classical computational hardness of sampling from random quantum-circuit output distributions and is a central formulation of quantum computational supremacy. The supplied text gives no resolution evidence, so its status remains open.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The quantum supremacy conjecture for random circuit sampling

    Let A\mathcal{A} be an architecture over circuits, let HA\mathcal{H}_{\mathcal{A}} be the distribution over circuits in A\mathcal{A} whose local gates are independently drawn from the Haar measure, and let RCS be the task of, given C∈HAC\in\mathcal{H}_{\mathcal{A}} and parameters ϵ>0\epsilon>0 and δ>0\delta>0, sampling from the output distribution induced by CC with probability 1−δ1-\delta over the choice of CC, up to total variation distance ϵ\epsilon in time poly⁡(n,ϵ−1)\operatorname{poly}(n,\epsilon^{-1}). Quantum supremacy conjecture. There is no classical randomized algorithm that performs RCS in time poly⁡(n,ϵ−1)\operatorname{poly}(n,\epsilon^{-1}), where ϵ\epsilon is the total variation distance error. This conjecture is the average-case hardness assumption underlying random circuit sampling proposals for demonstrating quantum supremacy; the source gives no resolution of it.

    source: Ramis Movassagh, “Quantum supremacy and random circuits”, arXiv:1909.06210 (2020).

Sources & referencesView supporting material

Primary source

Ramis Movassagh, “Efficient unitary paths and quantum computational supremacy: A proof of average-case hardness of Random Circuit Sampling”, arXiv:1810.04681 (2018).

computational-complexityquantum-complexityquantum-computingrandom-circuitsrandom-circuit-sampling

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