We provide a detailed estimate for the logical resource requirements of the quantum linear-system algorithm (Harrow et al. in Phys Rev Lett 103:150502, 2009) including the recently described elaborations and application to computing the electromagnetic scattering cross section of a metallic target (Clader et al. in Phys Rev Lett 110:250504, 2013). Our resource estimates are based on the standard quantum-circuit model of quantum computation; they comprise circuit width (related to parallelism), circuit depth (total number of steps), the number of qubits and ancilla qubits employed, and the overall number of elementary quantum gate operations as well as more specific gate counts for each elementary fault-tolerant gate from the stanThis article is part of topical collection on Quantum Computer Science.Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/NBC, or the US Government. dard set {X, Y, Z , H, S, T, CNOT}. In order to perform these estimates, we used an approach that combines manual analysis with automated estimates generated via the Quipper quantum programming language and compiler. Our estimates pertain to the explicit example problem size N = 332,020,680 beyond which, according to a crude big-O complexity comparison, the quantum linear-system algorithm is expected to run faster than the best known classical linear-system solving algorithm. For this problem size, a desired calculation accuracy ε = 0.01 requires an approximate circuit width 340 and circuit depth of order 10 25 if oracle costs are excluded, and a circuit width and circuit depth of order 10 8 and 10 29 , respectively, if the resource requirements of oracles are included, indicating that the commonly ignored oracle resources are considerable. In addition to providing detailed logical resource estimates, it is also the purpose of this paper to demonstrate explicitly (using a fine-grained approach rather than relying on coarse big-O asymptotic approximations) how these impressively large numbers arise with an actual circuit implementation of a quantum algorithm. While our estimates may prove to be conservative as more efficient advanced quantum-computation techniques are developed, they nevertheless provide a valid baseline for research targeting a reduction of the algorithmic-level resource requirements, implying that a reduction by many orders of magnitude is necessary for the algorithm to become practical.
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