Introduction to Complexity Theory by Henry Yuen
Introducing Complexity Theory
Computer science studies inputs and outputs. Using a pocket calculator to multiply two enormous numbers or deciphering a sophisticated code, you take a string of numerical data, generally represented by 0s and 1s, and solve it. For almost 30 years, computational complexity theory academics have utilized this approach to explain why some transformations are harder to implement. They discovered that quantum computers tackle classical issues like prime factorization much faster than conventional ones.
But Columbia University professor Henry Yuen says this standard perspective is insufficient now. Complexity theorists have studied how quantum computers handle classical data, but they have only just begun to study many scenarios with genuinely quantum inputs and outputs. Yuen says standard complexity theory is “silent” on these professions. He is leading an ambitious effort to establish a “fully quantum” complexity theory, a new mathematical language to close this gap.
Video Games to Quantum Frontiers
Yuen began his research of the most obscure mathematics at a Southern California family restaurant. Rural Cambodian refugees from the 1970s Khmer Rouge slaughter gave birth to Yuen in 1989. Before attending university, Yuen and his brother worked in the family company, but his mother's family fled across land mine-filled fields from labor camps.
Early computer science interest stemmed from his desire to make video games, but a critical undergraduate incident changed his path. The innovative research methods of his mentor, Len Adleman, and theorist Scott Aaronson inspired Yuen to study quantum computing theory. He established his pioneer status in 2020 by demonstrating a substantial quantum entanglement strength discovery. He is studying “atypical inputs” that defy convention.
The Limits of Classical Thinking
Yuen's investigation starts with the idea that quantum power is not always parallel to traditional processing capacity. Yuen shows this using bit commitment cryptography. In classical times, this was like sealing a message until it was revealed. Despite being the core of modern security, these approaches assume some mathematical issues are insoluble.
If you imagine a “quantum envelope,” the laws change. Someone with unlimited classical computing power may not be able to crack a quantum bit commitment scheme. This suggests that classical and quantum computer activity may differ fundamentally. Yuen calls this the “unitary synthesis problem”: can a device that solves classical problems fast also transform quantum states? Negative answers place quantum-input issues in a different logical category.
Yuen and colleagues have begun exploring this new area, discovering that many seemingly unrelated quantum problems are essentially comparable in complexity. Uhlmann's theorem, a fundamental quantum information theory discovery, centers this map.
The theorem describes the best way to change quantum states when you can only act on one of two entangled particles. Yuen's team observed that this physical theorem is a “hub” from which many other jobs branch out after treating it as a computer problem. Deciphering black hole Hawking radiation is interesting. As a quantum-input problem, the analysis of black hole-emitted entangled particles is the Uhlmann transformation issue “in disguise”.
Yuen said this research aims to find the right word for the quantum age, not just prove technical theorems. Lack of terminology prevents scientists from thinking effectively about the cosmos, he claims. By taking a “left turn” while others went a right, he is constructing a framework that may explain quantum secrecy and black hole secrets.
“I get to enjoy this privilege of thinking about mathematics and quantum physics,” Yuen says, reflecting on his life away from his parents' labor camps. His research shows that the “fully quantum” unknown is where the future's complexity lives, even though we've mastered the past's 0s and 1s.
