Quantum Supremacy Lower Bounds by Entanglement Scaling
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A contemporary technological milestone is to build a quantum device performing a computational task beyond the capability of any classical computer, an achievement known as quantum supremacy. The minimal requirements to realize quantum supremacy have so far been based on building a device to surpass extrapolated numerical simulation times. Here, we derive a formula for the minimal number of nearest-neighbor gates on a 2D lattice required to possibly generate a quantum state possessing volumetric entanglement scaling in the number of qubits. Hence, we predict the minimum random circuit depth needed to generate the maximal bipartite entanglement correlations between all problem variables (qubits). This derivation leads immediately to an interval which is constant on a coarse-grained entanglement upper bound---all existing numerical simulations fall inside this interval. These bounds make explicit a nuance implicit in other proposals with critical physical consequence. The hardware itself must be able to support super-logarithmic ebits of entanglement across some poly(n) number of qubit-bipartitions, otherwise the quantum state itself will not possess volumetric entanglement growth and full-lattice-range correlations. In conclusion we forecast that quantum supremacy will not be possible for gate-depths of less than 100 on systems of 80 to 150 qubits. These findings are just above the entanglement interval lower-bound that we derived in the study.
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