Experimental Approximation of the Jones Polynomial with One Quantum Bit

Abstract

We present experimental results approximating the Jones polynomial using 4 qubits in a liquid state nuclear magnetic resonance quantum information processor. This is the first experimental implementation of a complete problem for the deterministic quantum computation with one quantum bit model of quantum computation, which uses a single qubit accompanied by a register of completely random states. The Jones polynomial is a knot invariant that is important not only to knot theory, but also to statistical mechanics and quantum field theory. The implemented algorithm is a modification of the algorithm developed by Shor and Jordan suitable for implementation in NMR. These experimental results show that for the restricted case of knots whose braid representations have four strands and exactly three crossings, identifying distinct knots is possible 91% of the time.

NMR Quantum Information Processing

“…To show that a processor has the computing power to solve problems in this class, it is sufficient to show that it can solve a single problem which is complete for this class 5 . The first experimental implementation of a complete problem for DQC1 was an algorithm for approximating the Jones polynomial at the fifth root of unity [76,77], which is a problem derived from knot theory. Details of the problem are given in Ref.…”

Control aspects of quantum computing using pure and mixed states

“…This readily discriminates the threestranded knots or links by two qubits, while in Passante et al [15] only single values of q were used. Note that the experimental data nicely follow the theoretical prediction and the functional dependence is so different that the predictive power of distinguishing knots or links is higher than by mere evaluation of single points.…”

“…In Passante et al [15], only links that contain disjoint circles were evaluated. As already mentioned, a much simpler quantum calculation using fewer qubits (here 2 qubits for a 2-strand braid representation) can calculate the Jones polynomials of the given links equally well.…”

“…Already the four-carbon architecture of trans-crotonic acid used in Passante et al [15] required the optimal control algorithm GRAPE to be efficiently implemented experimentally. We therefore anticipate that control algorithms will play a major role even for algorithms inspired by topological quantum computation.…”

“…While classically it is NP-hard to distinguish two (classes of) knots in terms of their characteristic Jones polynomials, the quantum AJL algorithm, or its predecessor by Kauffman & Lomonaco [12,13], can do so more efficiently with quantum resources. Moreover, as has been experimentally demonstrated by NMR [14,15], these algorithms can be implemented using thermal mixtures of quantum states. Moreover, it suffices to approximate the trace of a controlled unitary encapsulating the information of the Jones polynomial.…”