More than 80 years after Schrödinger published his famous equation, quantum physics is still, in many ways, a mystery. "I think I can safely say that nobody understands quantum theory," Richard Feynman, the celebrated Caltech physicist, once said.
EECS Professor Umesh Vazirani. (Photo by Peg Skorpinski) One of the theory's most confounding aspects is that describing the evolution of even a modest-sized system of 500 atoms requires storing more numbers than there are particles in the universe. "How and where does nature store all this information?" asks Umesh Vazirani. "It's baffling."
Vazirani is one of the founders of the field of quantum computing, a model of computation that has caught the popular imagination as a basis for building more powerful computers. But equally intriguing to quantum researchers is the promise that the computational viewpoint may shed light on the mysteries of quantum physics. "That the description of a quantum system is exponentially larger than the system itself is an essential aspect of quantum theory that has never been tested," Vazirani says. "Physicists had missed the significance of this point, but taking the computational perspective really underscores its importance."
Vazirani first became interested in quantum physics in 1992, after stumbling upon a paper written 10 years earlier by Feynman. In the paper, titled "Simulating Physics with Computers," Feynman had observed that the familiar model for a computer, in which information is manipulated and stored according to the principles of classical physics, appears incapable of simulating a several-particle quantum system efficiently. If this is the case, Vazirani reasoned, the converse opens up an exciting prospect: a computer that harnesses quantum physics might be more powerful than a classical computer.
At the time, the gospel of computer science was the Extended Church-Turing Thesis, an axiom about computation developed in the 1960s and 1970s that postulates that the set of problems that can be solved efficiently on a computer is independent of the computer’s design. A computer based on quantum physics might violate this thesis, Vazirani realized. Moreover, a "quantum computer" could have profound implications for quantum physics, providing the means for testing unproved hypotheses of the theory. For instance, demonstrating that a quantum computer is exponentially faster at some task than a classical one would prove that the description of a quantum system cannot be of less than exponential size.
"Once I realized there are deep connections between quantum physics and computation, I knew I wanted to learn about quantum physics," Vazirani says, and he did. A year later, Vazirani and Ethan Bernstein, his Ph.D. student at the time, established the theoretical viability of quantum computers by showing that such a computer would act as a digital device and could be programmed. The researchers then came up with a quantum algorithm for a version of the Fourier sampling problem. Their result demonstrated that quantum computation apparently violates the Extended Church-Turing Thesis and launched the field of quantum complexity theory.
In contrast to classical bits, which are typically represented by a voltage that is high or low, quantum bits are represented as states of elementary particles, which, according to the theory of quantum mechanics, can be not only just spin up or spin down (1 or 0 respectively) but a linear combination of both. As the size of the quantum system grows, the number of parameters needed to describe it grows exponentially, suggesting that a quantum system could potentially store a vast amount of information.
Accessing the extra information in a quantum system involves performing a measurement, which collapses the quantum state into a classical one and destroys much of the additional information. Bernstein and Vazirani's Fourier sampling algorithm featured a clever technique for lining up quantum states before measuring them to extract more information.
In 1994, Peter Shor, now a professor of mathematics at the Massachusetts Institute of Technology, built upon this technique to devise an efficient quantum algorithm for factoring integers, an achievement that could have dramatic practical consequences. A working quantum computer of sufficient size could break the cryptosystems used to secure financial transactions over the Internet.
Meanwhile, repeated failures at designing algorithms for other problems have hinted at limits to quantum computation's power. Also in 1994, Vazirani and Bernstein, who is now at Microsoft, joined forces with two other researchers to give formal evidence that the NP-complete problems are unlikely to fall to a quantum computer. With Charles Bennett of IBM's Thomas J. Watson Research Center in Yorktown Heights, N.Y. and Gilles Brassard of the University of Montreal, they showed that if finding a solution to an NP-complete problem is simply a matter of finding a needle in an exponentially large haystack, then a quantum computer could not help all that much: It would provide only quadratic gains over what a classical computer could achieve. "We were too ambitious," said Vazirani. "We were trying to solve NP-complete problems on a quantum computer, and then we realized that there was a fundamental reason why we kept failing."
One of Vazirani's current research efforts is to find a function that is easy to compute classically, but for which there is evidence that its inverse, unlike factoring, is hard to compute on a quantum computer. Such a function could be a basis for a public key cryptosystem that would be robust against quantum computation. "We are trying to find a cure to the havoc that a quantum computer would wreak," he says.
Vazirani is also exploring the power of the computational viewpoint in teaching quantum physics. In 2003, together with colleagues in physics and chemistry, he designed and taught an interdisciplinary course called "Qubits, Quantum Mechanics, and Computers," and he is currently designing an introductory course in quantum physics, viewed from the computational perspective.