|| Quantum Gates, Circuits, and Algorithms using NMR
Bulk quantum computation is a radically different approach which, in contrast to trapped ion and cavity-QED implementations, allows quantum
computation to be performed using ordinary liquids at room temperature. Applying these methods to nuclear magnetic resonance, and using a simple
molecule in solution, a quantum controlled-NOT gate has been realized. Quantum bits (qubits) are represented as nuclear spins, which have long
coherence times due to their natural isolation from the external world, and fiducial input states are created using state labeling techniques. Cascading elementary gates will provide an exciting opportunity to experimentally explore quantum algorithms immediately.
Quantum computers, though inherently digital in nature, are notoriously susceptible to errors due to decoherence: leakage of quantum information to
the environment. This reality can be countered by application of error correction techniques, which restore the system to a set of standardized states. Methods which generalize known classical error correction codes to work on quantum systems have been discovered, but these codes are
complicated; the simplest requires five qubits to encode one. Practical quantum computers require simple, expedient codes which are short and easy
to encode and decode, and target real-world errors like those resulting from T1 and T2 processes in NMR. Such codes will be developed and their
application will be explored using high-field NMR techniques in the laboratory.
Robust Quantum Algorithms
Quantum computation can be made reliable at the expense of intermixing quantum error correction techniques with quantum algorithms. This is
currently the subject of serious theoretical investigations; however, it can also be seen that algorithms such as Shor's quantum factoring algorithm actually use decoherence as an integral part of their procedures. In fact, simulated annealing provides an excellent classical example of the usefulness of controlled loss (heating and cooling via coupling to an external reservoir). Application of such notions toward quantum computation would provide a way to perform quantum computation with decohering systems, and thus achieve a natural robustness. These algorithms would provide practical means for accomplishing quantum computation with current technology, such as NMR.
Quantum Signature Tests
Given a black box, what distinctive signature would its dynamics have which certifiably demonstrate its capacity to perform quantum (vs. classical) computation? From the theoretical point of view, an NMR quantum computer can be treated as a black box, as can other physical
implementations. For the purposes of quantum computation, what is most important is the existence of an exponential number of degrees of freedom which are manipulated by a polynomial number of "knobs." A suitability benchmark will be designed to test for this relationship, and implemented on an NMR quantum computer.
Quantum Computer Compiler
Quantum algorithms are implemented with quantum circuits, which are networks of abstract elements such as the controlled-NOT and square-root-of-NOT gates. An important application, which will complement network design and simulation tools, is the "back-end" compiler, which outputs actual control instructions optimized for a particular physical machine model, such as an ion trap or NMR spectrometer. Depending on the experimental configuration, many simplifications can be made, and due to the imperfect nature of current experiments, such optimizations may be crucial
to practical operation. For example, sequential single qubit rotations can be combined using the rules of SU(2) Lie algebra. At present, simple
quantum circuits are hand-coded into pulse sequences for NMR quantum computation. An optimizing compiler will be developed to automate this important function.
Desktop Quantum Computer
Quantum computers will not be practically useful until they are either large enough to out-perform the best classical computers, or physically small enough to be useful as integrated co-processors. A technological opportunity afforded by the invention of bulk quantum computers is the development of such a quantum co-processor. As a step in this direction, a desktop scale quantum computer will be designed and implemented, using
permanent magnet NMR techniques, and targeted multi-qubit molecules. Ultimately, the envisioned system is a solid-state quantum computer, realized by modern semiconductor materials technology using micromachining of complex quantum structures.
|Recent FY-97 Accomplishments:
This project was initiated 8/1/97 with the goal of experimentally realizing small quantum computers using nuclear magnetic resonance (NMR) techniques. The collaboration involves three groups: Stanford developing algorithms, numerical models and micromachine based NMR technology, U.C. Berkeley synthesizing molecules and implementing algorithms at their high magnetic field NMR facility, and MIT investigating scaling to 100's of quantum bits and desktop size NMRQC apparatus. From 8/1/97 to 2/1/98 the consortium achieved a major success, by demonstrating implementations of two quantum algorithms using molecules of chloroform as our computer. These results represent the first laboratory realizations of quantum algorithms, and are reported in the two publications below. From these studies, the consortium has developed a better understanding of the potential for scaling up NMR quantum computers. In the coming year, the consortium plans to synthesize and test quantum computers with 3-4 bits, and to further investigate means for scaling these systems to significant sizes. The major contributions of each group are listed below.
Stanford's three accomplishments in this period relate to quantum algorithms, microfabricated devices, and error correction. First, Stanford designed and performed experiments implementing the Deutsch-Jozsa and Deutsch-Jozsa Cleve quantum algorithms, which exhibit an exponential speedup on a quantum computer, compared with a classical computer. These results are reported in the two papers referenced below. Second, Stanford fabricated and tested a series of planar RF-microcoils to develop a basis for a future implementation of a silicon microfabricated NMR quantum computer. This project brought on board a new graduate student, who is now expert in using our Ginzton Laboratory Microfabrication Facility. Third, Stanford developed a new experiment which demonstrates quantum error correction being performed with NMR. This experiment is in progress.
U.C. Berkeley fulfilled two main goals during this period, relating to molecular synthesis and quantum algorithms. First, special high resolution samples of several two and three qubit molecules were synthesized, including chloroform and alanine. Molecules with more qubits have been identified and their custom synthesis requirements are being explored. Second, Grover's fast quantum search algorithm was implemented using chloroform as our two qubit computer. This demonstrates the loading of a quantum computer with an initial state, performing a computation requiring fewer steps than a classical computer, and reading out the final state. The focus is to obtain NMR samples of four and five qubit molecules and implementation of quantum computing algorithms including quantum error correction developed at Stanford. The molecular synthesis and development of NMR pulse sequences is underway.
MIT's primary role in this collaborative effort has been to scale bulk quantum computing from million-dollar NMR spectrometers to the table-top, both to make it more widely accessible, and to overcome the limits of an instrument designed for spectroscopy instead of computation. The first step in this effort was an analysis of the experimental scaling constraints, leading to the construction of prototypes using electromagnets and permanent magnets. These are now resolving J-couplings and hence close to reproducing the high-field experiments. In the coming year MIT will be focusing on online shimming down to parts per billion, the pre-polarization to increase the number of qubits, and writing the compiler for pulse sequences.
- Isaac L. Chuang, Neil Gershenfeld, and Mark Kubinec, "Experimental Implementation of Fast Quantum Searching," Physical Review Letters 80, 3408 (1998).
- Isaac L. Chuang, Lieven M. K. Vandersypen, Xinlan Zhou, Debbie W. Leung, and Seth Lloyd, "Experimental realization of a quantum algorithm," Nature 393, 143 (1998).
||The consortium is working very closely with technology partners including IBM, HP, Motorola, Analog Devices, and Microsoft on transferring the instrumentation (an early version went to DARPA). In addition results have been communicated in technical articles (Science, Proceedings of the Royal Society, Physical Review Letters) and the popular press (New Scientist, The Economist, Discover, Scientific American). An invited paper on NMR Quantum Computation by Chuang, Vandersypen and Harris was presented at the IEEE International Solid State Circuits Conference. Also, joint seminars have been held at Stanford on Quantum Computation and Error Correction with Hewlett-Packard.|
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