Stephen Jordan's Homepage

PhD in Physics 2008 MIT
BS in Physics 2003 Penn State
Currently visiting scientist at RIKEN

Presently I study quantum algorithms and fault-tolerance of quantum computers.

Research

It is now believed that quantum computers can solve certain problems more efficiently than classical computers. For example, quantum algorithms have been discovered which can factor large numbers in an amount of time which varies roughly as the cube of the number of digits. In contrast, the best known classical algorithm for factoring requires time which grows exponentially with the number of digits.

Such results raise many questions. The two main questions I am currently interested in are:

How can we build a quantum computer?
What could we do with a quantum computer if we had one?

The first question has led me to study fault tolerance, and the second question has led me to study quantum algorithms, as described below.

My previous research involved condensed matter experiment and simulation.

Fault Tolerance

It is widely believed that achieving robustness against noise will be one of the primary challenges in building a practical quantum computer. Through the efforts of many members of the quantum computation community, it has been shown that quantum error correcting codes can protect against noise. In fact, the threshold theorem shows that once the noise probability per operation in a quantum computer is below a certain threshold, quantum circuits of indefinite size can be successfully implemented using quantum error correcting codes.

Recently, there has been growing interest in an alternative model of quantum computation, called adiabatic quantum computation. In this model, instead of implementing a series of quantum gates by applying electromagnetic pulses to a set of spins, the computation is performed by controlling the quantum computer a slow and continuous way such that the qubits always remains in their lowest energy state. There are some arguments which suggest that adiabatic quantum computers may be easier to experimentally implement than standard quantum computers. Furthermore, it is now known that adiabatic quantum computers are universal, that is, they can efficiently simulate any quantum circuit.

The results on error correcting codes such as the threshold theorem were originally derived mainly with the quantum circuit model in mind, and they do not automatically apply to adiabatic quantum computers. In collaboration with Edward Farhi and Peter Shor, I have found that a class of quantum error correcting codes called stabilizer codes can be used to improve the fault-tolerance of adiabatic quantum computers without harming the locality of the Hamiltonian too severely.

Quantum Algorithms

It was shown in the 1980s that quantum computers can in principle efficiently simulate any classical computation. Thus current research into what quantum computers can do generally focuses on finding quantum algorithms which solve a problem faster than the best known classical algorithm, and proving that, for certain problems, quantum computers provide no significant advantage over classical computers.

My first studies of quantum algorithms focused on quantum numerical algorithms. Specifically, I found that it is possible to use a generalization of the Bernstein-Vazirani algorithm to perform numerical gradient estimation in any number of dimensions using a constant number of function evaluations, whereas classically the required number of function evaluations scales linearly with the dimension. More recently, in collaboration with Peter Shor, I've found that a certain problem of approximating the Jones polynomial for knots can be solved efficiently on quantum computer in which all but one of the qubits are uninitialized. Furthermore, in a well-defined sense, no problem solvable on such computers is harder than this one.

Publications

Stephen P. Jordan and Edward Farhi
Perturbative Gadgets at Arbitrary Orders
Phys. Rev. A 77, 062329 (2008) [arXiv:0802.1874]
I. Kassal, S. Jordan, P. Love, M. Mohseni, and A. Aspuru-Guzik
Quantum algorithms for the simulation of chemical dynamics
[arXiv:0801.2986]
Peter W. Shor and Stephen P. Jordan
Estimating Jones polynomials is a complete problem for one clean qubit.
Quantum Information and Computation
Vol 8, No 8/9, pg. 681-714 (2008)
[arXiv:0707.2831]
A. Childs, R. Cleve, S. Jordan, and D. Yeung
Discrete-query quantum algorithm for NAND trees.
[quant-ph/0702160]
Stephen P. Jordan, Edward Farhi, and Peter W. Shor
Error correcting codes for adiabatic quantum computation.
Phys. Rev. A 74, 052322 (2006) [quant-ph/0512170]
Stephen P. Jordan
Fast quantum algorithm for numerical gradient estimation.
Phys. Rev. Lett. 95, 050501 (2005) [quant-ph/0405146]
Stephen P. Jordan and Vincent H. Crespi
Mechanical manipulation of graphene nanocones: Chiral inversion of a micron-scale three-dimensional object.
Phys. Rev. Lett. 93, 255504 (2004)
R. Garcia, S. Jordan, J. Lazzaretti, and M. Chan
Quartz microbalance study of thick He-4 film near the superfluid transition
J. Low Temp. Phys. 134 (1-2): 527-533 Jan 2004

Teaching

I have had the privilege of working with many excellent students. Specifically, I was teaching assistant for junior lab I in fall 2004 and for junior lab II in spring 2005. In addition, I was a referee for term papers in quantum mechanics III during spring 2005, 2006, 2007, and 2008.