Focus beyond quadratic speedups for error-corrected quantum advantage (2011.04149v2)

Abstract: In this perspective, we discuss conditions under which it would be possible for a modest fault-tolerant quantum computer to realize a runtime advantage by executing a quantum algorithm with only a small polynomial speedup over the best classical alternative. The challenge is that the computation must finish within a reasonable amount of time while being difficult enough that the small quantum scaling advantage would compensate for the large constant factor overheads associated with error-correction. We compute several examples of such runtimes using state-of-the-art surface code constructions under a variety of assumptions. We conclude that quadratic speedups will not enable quantum advantage on early generations of such fault-tolerant devices unless there is a significant improvement in how we would realize quantum error-correction. While this conclusion persists even if we were to increase the rate of logical gates in the surface code by more than an order of magnitude, we also repeat this analysis for speedups by other polynomial degrees and find that quartic speedups look significantly more practical.

Analysis of Polynomial Speedups for Quantum Computing

The paper "Focus beyond quadratic speedups for error-corrected quantum advantage" by Ryan Babbush et al. raises pivotal questions about the viability of modest fault-tolerant quantum computers achieving computational advantages over classical systems through polynomial speedups. The authors rigorously explore the conditions under which early-generation quantum devices might surpass classical alternatives, particularly when the quantum system promises only a small polynomial runtime improvement. Given the substantial overhead associated with quantum error-correction, they argue that quadratic speedups are unlikely to deliver tangible advantages unless significant advancements in error-correction techniques occur. This analysis extends to examining higher-order polynomial speedups, such as quartic, which appear more promising.

Key Insights and Findings

• Error-Correcting Overheads: Quantum error correction introduces significant constant factor slowdowns, primarily due to the time-consuming distillation of Toffoli gates using surface codes. This process results in a spacetime volume discrepancy of roughly ten orders of magnitude when compared against classical operations, posing substantial challenges for realizing a quantum advantage.
• Quadratic Advantage Limitations: The paper concludes that quantum computers will struggle to achieve advantage using quadratic speedups unless error-correction methods are drastically improved. Current projections suggest exorbitantly long runtimes required for classical algorithms' scaling to be overpowered even with modest quantum device capabilities.
• Viability of Higher Speedups: The analysis indicates that quartic speedups could potentially provide a more feasible pathway to achieving runtime advantage. Some existing quantum algorithms demonstrate quartic speedups, suggesting practical benefit with even moderate fault-tolerant resources.

Theoretical Implications

• The paper provides a quantitative assessment, employing formulas like ${\cal T}_Q = M \, t_Q$ and ${\cal T}_C = M^d \, t_C$, to model how quantum algorithms scale in comparison to classical ones. The models suggest that while quadratic speedups are theoretically intriguing, their practical implementation remains fraught with challenges under current conditions.
• Higher polynomial degrees in speedups, such as cubic and quartic, show markedly improved feasibility for achieving quantum computational advantage with realistic constraints on classical and quantum resources.

Practical Implications and Speculative Future Developments

• Algorithm Focus: Encourages the quantum computing community to refocus efforts on algorithms offering speedups beyond quadratic, potentially concentrating on quartic or even higher orders for industrially relevant applications.
• Improved Error-Correction Techniques: Proposes that breakthrough advancements in error-correction methods are essential for enabling early fault-tolerant quantum devices to compete with powerful classical computers, especially in tasks benefiting from quadratic speedups.
• Potential Architectural Shifts: Identifies that leveraging high connectivity among quantum systems, such as 3D or nonlocal error-correcting codes, may offer pathways to realize efficient non-Clifford gates, providing more accessible routes to practical quantum advantages.

In summary, the authors deliver a compelling discourse on the limitations and potential of quantum computing with error-correction overheads. Emphasizing the need for advancements in algorithms and error-correction methodologies, Babbush et al. set the stage for future developments that might redefine computational paradigms, underscoring the importance of strategic focus on polynomial speedups beyond the quadratic field.