Abstract
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 standard set { X, Y, Z, H, S, T, text{ 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 varepsilon =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.
Highlights
Quantum computing promises to efficiently solve certain hard computational problems for which it is believed no efficient classical algorithms exist [1]
Unlike with quantum error correction (QEC) protocols where the distinction between “data qubits” and “ancilla qubits” is clear, here this distinction is somewhat ambiguous; all qubits involved in the algorithm are initially prepared in state |0, and some qubits that we called ancilla qubits exist from the start to the end of a full quantum-computation part
What our results suggest is that the large constant factors arise as a consequence of the desired precision forcing us into choosing large sizes for the registers, whereas the logical resource estimate (LRE) is not notably impacted by a change in problem size N
Summary
Quantum computing promises to efficiently solve certain hard computational problems for which it is believed no efficient classical algorithms exist [1]. The quantum linear-system algorithm (QLSA), first proposed by Harrow et al [3], afterward improved by Ambainis [4], and recently generalized by Clader et al [5], is appealing because of its great practical relevance to modern science and engineering. This quantum algorithm solves a large system of linear equations under certain conditions exponentially faster than any current classical method.
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