Nuclear magnetic resonance quantum computer

The ideal picture of liquid state NMR (LSNMR) quantum information processing (QIP) is based on a molecule in which some of its atom's nuclei behave as spin-{{sfrac|1|2}} systems.{{Cite journal|author=Neil Gershenfeld |author2=Isaac L. Chuang |title=Quantum computing with molecules |journal=Scientific American |volume=278 |issue=6 |year=1998 |pages=66–71 |doi=10.1038/scientificamerican0698-66 |bibcode=1998SciAm.278f..66G |url=http://cba.mit.edu/docs/papers/98.06.sciqc.pdf}} Depending on which nuclei we are considering they will have different energy levels and different interaction with its neighbours and so we can treat them as distinguishable qubits. In this system we tend to consider the inter-atomic bonds as the source of interactions between qubits and exploit these spin-spin interactions to perform 2-qubit gates such as CNOTs that are necessary for universal quantum computation. In addition to the spin-spin interactions native to the molecule an external magnetic field can be applied (in NMR laboratories) and these impose single qubit gates. By exploiting the fact that different spins will experience different local fields we have control over the individual spins.

The picture described above is far from realistic since we are treating a single molecule. NMR is performed on an ensemble of molecules, usually with as many as 10^15 molecules. This introduces complications to the model, one of which is introduction of decoherence. In particular we have the problem of an open quantum system interacting with a macroscopic number of particles near thermal equilibrium (~mK to ~300 K). This has led the development of decoherence suppression techniques that have spread to other disciplines such as trapped ions. The other significant issue with regards to working close to thermal equilibrium is the mixedness of the state. This required the introduction of ensemble quantum processing, whose principal limitation is that as we introduce more logical qubits into our system we require larger samples in order to attain discernable signals during measurement.

Solid state NMR (SSNMR), unlike LSNMR uses a solid state sample, for example a nitrogen vacancy diamond lattice rather than a liquid sample.{{Cite web|url=https://www.aps.org/publications/apsnews/200805/diamond.cfm |title=Diamond Sparkles in Quantum Computing}} This has many advantages such as lack of molecular diffusion decoherence, lower temperatures can be achieved to the point of suppressing phonon decoherence and a greater variety of control operations that allow us to overcome one of the major problems of LSNMR that is initialisation. Moreover, as in a crystal structure we can localize precisely the qubits, we can measure each qubit individually, instead of having an ensemble measurement as in LSNMR.

The use of nuclear spins for quantum computing was first discussed by Seth Lloyd and by David DiVincenzo.{{cite journal | author=Seth Lloyd | author-link=Seth Lloyd | title=A Potentially Realizable Quantum Computer | volume=261 | issue=5128 | year=1993 | journal=Science| pages=1569–1571 | doi=10.1126/science.261.5128.1569 | pmid=17798117 | bibcode=1993Sci...261.1569L | s2cid=38100483 }}{{cite journal | author=David DiVincenzo | author-link=David DiVincenzo | title=A Two-bit gates are universal for quantum computation | volume=51 | issue=2 | year=1995 | journal=Phys. Rev. A| pages=1015–1022 | doi=10.1103/PhysRevA.51.1015 | pmid=9911679 | arxiv=cond-mat/9407022 | bibcode=1995PhRvA..51.1015D | s2cid=2317415 }}{{cite journal | author=David DiVincenzo | author-link=David DiVincenzo | title=Quantum computation | volume=270 | issue=5234 | year=1995 | journal=Science}}

Manipulation of nuclear spins for quantum computing using liquid state NMR was introduced independently by Cory, Fahmy and Havel{{Cite news|last1=Cory|first1=David G.|last2=Fahmy|first2=Amr F.|last3=Havel|first3=Timothy F.|date=1996|title=Nuclear Magnetic Resonance Spectroscopy: An Experimentally Accessible Paradigm for Quantum Computing | language=en | pages=87–91|publisher=Phys-Comp 96, Proceedings of the Fourth Workshop on Physics and Computation, edited by T.Toffoli, M.Biafore, and J.Leao (New England Complex Systems Institute}}{{Cite journal|last1=Cory|first1=David G.|last2=Fahmy|first2=Amr F.|last3=Havel|first3=Timothy F.|date=1997-03-04|title=Ensemble quantum computing by NMR spectroscopy|journal=Proceedings of the National Academy of Sciences|language=en|volume=94|issue=5|pages=1634–1639|issn=0027-8424|pmc=19968|pmid=9050830|bibcode = 1997PNAS...94.1634C |doi = 10.1073/pnas.94.5.1634 |doi-access=free}} and Gershenfeld and Chuang{{Cite journal|last1=Gershenfeld|first1=Neil A.|last2=Chuang|first2=Isaac L.|date=1997-01-17|title=Bulk Spin-Resonance Quantum Computation|journal=Science|language=en|volume=275|issue=5298|pages=350–356|doi=10.1126/science.275.5298.350|issn=0036-8075|pmid=8994025|citeseerx=10.1.1.28.8877|s2cid=2262147}} in 1997. Some early success was obtained in performing quantum algorithms in NMR systems due to the relative maturity of NMR technology. For instance, in 2001 researchers at IBM reported the successful implementation of Shor's algorithm in a 7-qubit NMR quantum computer.{{cite journal |vauthors=Vandersypen LM, Steffen M, Breyta G, Yannoni CS, Sherwood MH, Chuang IL |year=2001 |title=Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance |journal=Nature |volume=414 |issue=6866 |pages=883–887 |doi=10.1038/414883a |pmid=11780055|arxiv = quant-ph/0112176 |bibcode = 2001Natur.414..883V |s2cid=4400832 }} However, even from the early days, it was recognized that NMR quantum computers would never be very useful due to the poor scaling of the signal-to-noise ratio in such systems.{{cite journal |author=Warren WS |year=1997 |title=The usefulness of NMR quantum computing |journal=Science |volume=277 |issue=5332 |pages=1688–1689 |doi=10.1126/science.277.5332.1688}} More recent work, particularly by Caves and others, shows that all experiments in liquid state bulk ensemble NMR quantum computing to date do not possess quantum entanglement, thought to be required for quantum computation. Hence NMR quantum computing experiments are likely to have been only classical simulations of a quantum computer.{{cite journal |vauthors=Menicucci NC, Caves CM |year=2002 |title=Local realistic model for the dynamics of bulk-ensemble NMR information processing |journal=Physical Review Letters |volume=88 |issue=16 |doi=10.1103/PhysRevLett.88.167901|arxiv = quant-ph/0111152 |bibcode = 2002PhRvL..88p7901M |pmid=11955265 |page=167901|s2cid=14583916 }}

The ensemble is initialized to be the thermal equilibrium state (see quantum statistical mechanics). In mathematical parlance, this state is given by the density matrix:

:$\backslash rho\; =\; \backslash frac\{e^\{-\; \backslash beta\; H\}\}\{\backslash operatorname\{Tr\}(e^\{-\; \backslash beta\; H\})\},$

where H is the hamiltonian matrix of an individual molecule and

:$\backslash beta\; =\; \backslash frac\{1\}\{k\; \backslash ,\; T\}$

where $k$ is the Boltzmann constant and $T$ the temperature. That the initial state in NMR quantum computing is in thermal equilibrium is one of the main differences compared to other quantum computing techniques, where they are initialized in a pure state. Nevertheless, suitable mixed states are capable of reflecting quantum dynamics which lead to Gershenfeld and Chuang to term them "pseudo-pure states".

Operations are performed on the ensemble through radio frequency (RF) pulses applied perpendicular to a strong, static magnetic field, created by a very large magnet. See nuclear magnetic resonance.

Consider applying a magnetic field along the z axis, fixing this as the principal quantization axis, on a liquid sample. The Hamiltonian for a single spin would be given by the Zeeman or chemical shift term:

:$H\; =\; \backslash mu\; B\_z\; =\; I\_z\; \backslash omega$

where $I\_z$ is the operator for the z component of the nuclear angular momentum, and $\backslash omega$ is the resonance frequency of the spin, which is proportional to the applied magnetic field.

Considering the molecules in the liquid sample to contain two spin-{{sfrac|1|2}} nuclei, the system Hamiltonian will have two chemical shift terms and a dipole coupling term:

:$H\; =\; \backslash omega\_1\; I\_\{z1\}+\backslash omega\_2\; I\_\{z2\}+2J\_\{12\}I\_\{z1\}I\_\{z2\}$

Control of a spin system can be realized by means of selective RF pulses applied perpendicular to the quantization axis. In the case of a two spin system as described above, we can distinguish two types of pulses: "soft" or spin-selective pulses, whose frequency range encompasses one of the resonant frequencies only, and therefore affects only that spin; and "hard" or nonselective pulses whose frequency range is broad enough to contain both resonant frequencies and therefore these pulses couple to both spins. For detailed examples of the effects of pulses on such a spin system, the reader is referred to Section 2 of work by Cory et al.{{cite journal |author1=Cory D. |year=1998 |title=Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing |journal=Physica D |volume=120 |issue=1–2 |doi = 10.1016/S0167-2789(98)00046-3 |arxiv = quant-ph/9709001|bibcode = 1998PhyD..120...82C |display-authors=etal |pages=82–101|s2cid=219400 }}

• Kane quantum computer

{{Reflist}}

{{quantum computing}}

Category:Quantum information science

Category:Nuclear magnetic resonance