[Paper Review] Resource analysis of the quantum linear system algorithm

Artur Scherer, Benoît Valiron+4 more|arXiv (Cornell University)|May 25, 2015

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper provides a detailed logical resource analysis of the quantum linear system algorithm (QLSA), using Quipper and manual methods to estimate gate counts, qubit requirements, circuit width, and depth for N=332,020,680. It reveals that even without oracle costs, QLSA requires circuit depth on the order of $10^{25}$, and up to $10^{29}$ when oracles are included, highlighting that oracle resources are substantial and that reductions of many orders of magnitude are needed for practicality.

ABSTRACT

We provide a detailed estimate for the logical resource requirements of the quantum linear system algorithm (QLSA) [Phys. Rev. Lett. 103, 150502 (2009)] including the recently described generalization [Phys. Rev. Lett. 110, 250504 (2013)]. Our resource estimates are based on the standard quantum-circuit model of quantum computation; they comprise circuit width, circuit depth, 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, CNOT}. 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 example problem size N=332,020,680 beyond which, according to a crude big-O complexity comparison, QLSA is expected to run faster than the best known classical linear-system solving algorithm. For this problem size, a desired calculation accuracy 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 depth of order $10^8$ and $10^{29}$, respectively, if oracle costs 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 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 resource requirements, implying that a reduction by many orders of magnitude is necessary for the algorithm to become practical.

Motivation & Objective

• To provide precise logical resource estimates for the quantum linear system algorithm (QLSA) in the fault-tolerant quantum circuit model.
• To quantify the impact of oracle resource costs on overall resource scaling, which are often neglected in prior analyses.
• To establish a baseline for future research by demonstrating the magnitude of resource requirements needed for QLSA to outperform classical algorithms.
• To illustrate how such large resource demands arise through a concrete circuit implementation, using Quipper and manual analysis.

Proposed method

• Employed the Quipper quantum programming language and compiler to generate automated resource estimates for QLSA circuits.
• Combined automated Quipper estimates with manual analysis to refine gate counts and circuit parameters.
• Focused on standard fault-tolerant gate set: {X, Y, Z, H, S, T, CNOT}, and reported counts for each.
• Analyzed circuit width, depth, total gate operations, and qubit/ancilla qubit requirements for problem size N=332,020,680.
• Distinguished between estimates excluding and including oracle costs to isolate their contribution to resource overhead.
• Used a big-O complexity comparison to justify the chosen problem size as the threshold where QLSA could outperform classical solvers.

Experimental results

Research questions

• RQ1 What are the precise logical resource requirements (qubits, gates, depth, width) for implementing the QLSA for a large-scale problem size?
• RQ2 How do oracle resource costs affect the overall resource scaling of QLSA, and why are they often overlooked?
• RQ3 To what extent do the required circuit depth and width make QLSA impractical with current fault-tolerant quantum hardware?
• RQ4 Can a concrete circuit implementation demonstrate the origin of the algorithm’s massive resource demands?
• RQ5 What magnitude of resource reduction is necessary for QLSA to achieve practical quantum advantage over classical linear system solvers?

Key findings

• For problem size N=332,020,680, QLSA requires a circuit width of approximately 340 logical qubits when oracle costs are excluded.
• Without oracle costs, the circuit depth is estimated at $10^{25}$ operations, indicating an extremely high temporal resource demand.
• When oracle costs are included, the circuit width scales to order $10^8$, and the depth increases to $10^{29}$, demonstrating that oracles dominate resource overhead.
• The study reveals that oracle resources are not negligible and significantly contribute to the overall complexity, challenging assumptions in prior analyses.
• The results provide a concrete baseline showing that QLSA’s resource requirements are currently far beyond the reach of fault-tolerant quantum hardware, requiring reductions of many orders of magnitude.
• The analysis confirms that even with optimal gate synthesis, QLSA remains impractical for near-term fault-tolerant quantum computers due to its astronomical depth and width demands.