The limits of Quantum Computing: an approach via Post-Quantum Cryptography

Quantum computing (QC) is emerging as a critical technology for the future of computing. QC has been shown to provide significant - sometimes even exponential - speedups on various problems, and enable protocols that would be impossible using classical computers. On the other hand, some recent results on ''dequantized algorithms'' show that it is not always straightforward to quantify the quantum advantage on some problems.

As a result, the strengths and limitations of quantum computing are still an open problem. One of the best benchmarks to evaluate the theoretical and practical limits of computing is cryptography. Indeed, cryptography is, by definition, the science of basing problems on the limits of computation. Arguably, the maturity of (classical) cryptography reflects our deep understanding of classical

computation. In contrast, post-quantum cryptography - building cryptography based on the limits of quantum computing - is very much an emerging field due to our limited understanding of quantum computing.

The emergence of post-quantum cryptography presents a tantalizing opportunity to study the theoretical and practical limits of computing. In the near-term, it can constitute a great benchmark for noisy intermediate-scale quantum computing (NISQ), providing concrete answers to questions such as: can a quantum algorithm beat any useful classical algorithm using a NISQ device of 1,000 qbits? In the long

term, more fundamental questions about the limits of quantum computers need to be answered. Beyond the known exponential and quadratic speedups that quantum algorithms can offer, one of the most promising aspects of those algorithms is to offer comparable running times with much reduced memory usage. Memory is arguably one of the most limiting aspects of classical computers.

The exponential memory blowup of simulating quantum systems, for example, suggests that understanding the limits of quantum memories is essential. Post-quantum cryptography provides ample problems to study this aspect of quantum computing and answer questions such as: can quantum computing provide exponential memory improvements for some real-life problems?

I posit that lattices and codes, fundamental mathematical objects, will play a major role in answering the questions I have put forward. Lattices have emerged as a central object for both quantum computing and cryptography. Lattices and codes play a crucial role in post-quantum cryptography, with three problems standing out as particularly relevant: the shortest vector problem (SVP), the Learning with

error problem (LWE) and the syndrome decoding problem. These problems are fundamentally about the limit of quantum computing and suggest that lattices and codes are hard enough to be quantum hard but structured enough to provide nontrivial primitives. The SVP and LWE play not only a role in cryptography but also in quantum computing.

Important search problems such as the dihedral hidden subgroup problem involve both problems. A recent breakthrough in the classical verification of quantum computations relies on LWE. LWE even enables classical parties to participate in secure quantum computation and communications protocols.

Therefore, improvements in the understanding of SVP and LWE will benefit both the quantum computing and cryptography community. Furthermore, some recent improvements in lattice algorithms, that come from codes, show the benefit of studying lattices and codes together rather than separately.