Mathematics for Quantum Computing

Ryan Bennink, Olena Burkovska, Konstantin Pieper +2 more

This review is designed to introduce mathematicians and computational scientists to quantum computing (QC) through the lens of uncertainty quantification (UQ) by presenting a mathematically rigorous and accessible narrative for understanding how noise and intrinsic randomness shape quantum computational outcomes in the language of mathematics. By grounding quantum computation in statistical inference, we highlight how mathematical tools such as probabilistic modeling, stochastic analysis, Bayesian inference, and sensitivity analysis, can directly address error propagation and reliability challenges in today's quantum devices. We also connect these methods to key scientific priorities in the field, including scalable uncertainty-aware algorithms and characterization of correlated errors. The purpose is to narrow the conceptual divide between applied mathematics, scientific computing and quantum information sciences, demonstrating how mathematically rooted UQ methodologies can guide validation, error mitigation, and principled algorithm design for emerging quantum technologies, in order to address challenges and opportunities present in modern-day quantum high performance and fault-tolerant quantum computing paradigms.

Liao Heng, Bill McColl

Wolfgang Scherer

M. A. Bashar, Mahmudul Hasan, M. Syduzzaman M. A. Chowdhury

The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations on these data.Mathematics lies at the heart of quantum computation and it is needed in every aspect of the research and development of quantum computation.In this paper we have presented mathematical concepts that are closely related to quantum computation and their relation to quantum mechanics in a way which is innovative and very easy to comprehend.Most of the mathematical concepts presented in this paper are deduced from the postulates of quantum mechanics and linear algebra.

Wolfgang Scherer

Bowen Liu

Abstract Quantum computing consists of algorithm and algorithm-output extraction. There are two major philosophical bubbles of quantum computing: the output-extraction model has low mathematical coverage closing to zero; the real mathematized goal of output-extraction, the existence of process solution, is hidden. Piercing the two bubbles, we reduce the mathematical structure of quantum computing to the two barest mathematical problems: to prove the existence of the program solution for given input and output state in algorithm; to prove the existence of process solution for given initial and final probability in output-extraction. The mathematical theme of algorithm-output extraction is two probability binaries (PA(λn), PB(λn) and (PB(λn), PBorn(λn)). For (PA(λn), PB(λn)), PA(λn) is the outcome probability obtained by the researchers by means of the eigenvalue-to-outcome transmission, PB(λn) is the probability within the state space based on the average of the results obtained from a finite discrete set of trials. The researchers assert PA(λn) = PB(λn), and assume that the mathematical basis for PA(λn) = PB(λn) is the existence of process solution satisfying PA(λn) = PB(λn). For (PB(λn), PBorn(λn), PBorn(λn) ≡|Ψ|2is the theoretical probability for single-bit output specified by quantum mechanical principles. The researchers assert PB(λn) = PBorn(λn), and assume that the mathematical basis for PB(λn) = PBorn(λn) is the so-called Law (QC) of Large Numbers. Thus, there are two major math-bugs of quantum computing: the existence of process solution satisfying PA(λn) = PB(λn) is unproven, the Law (QC) of Large Numbers guaranteeing PB(λn) = PBorn(λn) is unproven. For the first math-bug, we introduce a completely new proof method, nonconstructive proof based on path classification. We reformulate the problem of the existence of process solution as the existence problem of path classes. Physicists rarely have the opportunity to use non-constructive proofs, which are very effective for the desired existence problem. For our nonconstructive proof, it is not necessary to provide an example of a path and a mathematical algorithm that produces an example of the path. By means of non-constructive proof, we have shown the nonexistence of process solution satisfying PA(λn) = PB(λn). For the second math-bug, we show that the law (QC) of large numbers is not a corollary of any known law of large numbers in probability theory. For the qubit-output probability of quantum computing experiments there is no reliable basis of probability theory. We provide the three examples to prove that all known minimized quantum factorization experiments are invalid. We conclude that Quantum computing cannot survive the two major math-bugs. quantum computing theory is wrong because the mathematics of the output-extraction theory is wrong; the output-extraction theory, which assumes PA(λn) = PB(λn) = PBorn(λn), is wrong because PA(λn) = PB(λn) is wrong and PB(λn) = PBorn(λn) is unproven. Our study shows that two major math-bugs of quantum computing are very serious and unfixable, because they challenge mathematicians' understanding of the underlying mathematical rules.