CHALMERS UNIVERSITY OF TECHNOLOGY

MARCH 2006

NMR Quantum Computing Ihab Sinno1 1

Nanoscale Science and Technology IMP Student

Abstract Generally speaking, a computer is an apparatus having some controllable input pattern that can be transformed by means of certain rules given for such a system. When properly interpreted, the transformation process can be understood as the execution of some desired computation, resulting in a detectable output pattern. Using the rules of quantum physics, a quantum computer is definitely to make current digital computers obsolete once technological hurdles are conquered. Various proposals for building a quantum computer have been realized, with each having its advantages and drawbacks. Amongst the first phenomena to be devised in such a quantum simulation was nuclear magnetic resonance (NMR), which was already a well established field due to its long-standing application in spectroscopy (such as the analysis of complex biomolecules containing thousands of nuclear spins). Given this much of development and controllability, NMR was a favorite candidate for quantum computation, where it has quickly shown success (realization up to seven qubits in Shor’s factorization algorithm). On the other hand, being an established field has enabled researchers to envision the limits of such technology, where modern estimates suggest that NMR might be inadequate for realizing systems having more than 10-20 qubits [1]. In this paper, the characteristics of liquid-state NMR quantum computing will be discussed, starting from DiVincenzo’s criteria [2] as the backbone of discussion; thereby, analyzing the merits and defects of such a quantum system. Finally, different opinions and suggestions regarding the prospects and potentials of the technology will be presented.

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Introduction Using the rules of quantum physics, the angular momentum of a given particle can be expressed as a multiple of (ħ), where the multiplied is the particle’s “spin” value. The spin property can be thought of as a magnetic moment vector, causing the particle to behave like a tiny magnet with different orientations (poles). For a spin (½) particle, there are two possible orientations, both having equal energies. However, when a magnetic field is applied, energy levels will split. NMR is the phenomenon that occurs when non-zero net-spin nuclei are immersed in a static external magnetic field (B) for alignment, while being exposed to a secondary RF oscillating magnetic field (to control the direction of spin vectors). In such a case, a particle (nucleus) with a net magnetic moment (γ) can absorb a photon of energy (E): E=hν=hγB By such absorption, this particle can be excited to a higher vacant energy state (vice versa for emission).

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After this brief glimpse over NMR’s basics, it is time to consider DiVincenzo’s criteria and discuss their impact upon realizing a useful spin (½) NMR quantum computer.

Initialization Initialization arises from the computing requirement that registers should be initialized to a known value before the start of computation, typically the energetic ground-state I0›= I00…0›, regardless of the previous state. Clearly, this can’t be done by any unitary process, and cooling has to be used. However, the Zeeman energy splitting between different states is tiny, and for the typical field strengths used in NMR (2.3 21.1 T), this corresponds to an upper gap bound of 0.4-3.7 μeV (except for active tritium 3H). At room temperature, this means that the initial state of the spins is nearly completely random, since the Boltzmann energy will be of much larger value (25 meV), and only a tiny excess population will stay in the lower level. Luckily, starting from a pseudo-pure state (ρ) was found to be sufficient for NMR computation [3]: ρ = (1 – Є) 1/N + Є I0›‹0I Where ρ is a mixture of the desired quantum state Iψ› (with a probability Є) and the maximally mixed state 1/N, where N is the dimension of the Hilbert space. After a computation is carried out, the output in an error-free system will be: ρ' = (1 – Є) 1/N + Є Iψ›‹ψI

The resonant frequency (ν) is referred to as the Larmor or precession frequency.

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Therefore, by using a macroscopic ensemble (1025) of samples, the ensemble’s averaged output will yield the desired answer (since the maximally mixed states will cancel-out). It should be noted that in order to create a pseudo-pure state, a non-unitary process is Göteborg - Sverige

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required. As to NMR, there are two commonly used approaches in pulse sequences with non-unitary effects: magnetic field gradients and phase cycling. Both elements have been used to assemble pseudopure states in quantum computation. The first is carried out by applying variable Hamiltonians along the ensemble, thereby causing a non-unitary averaged evolution. On the other hand, phase cycling relies on combining the results of several experiments by a non-unitary post-processing [4]. It can be deduced that available methods described above, lack the ability to reinitialize individual qubits in the middle of computation since they work on the system as a whole. This will make essential error correction schemes totally impractical for implementation in NMR computing systems.

Scaling Using pseudo-pure states as a method for initialization will impose some major drawbacks on the whole system. The basic problem relies in the fact that Є (desired pure-state probability) will vary with the size of the system’s qubits (n) according to the following relation: Є = 2sinh((nhν)/(2kT))/(2cosh((hν)/(2kT)))n

MARCH 2006

system won’t be in liquid state anymore, and totally different considerations have to be considered. Moreover, at room temperature, the pseudopure state leaves the region where the lower bound implies the possibility of separation (at about 13 qubits), but never enters the region where the upper bound guarantees the existence of entangled states [5]. This has caused some authors to dismiss NMR from being a practical approach for quantum computing since no entanglement is possible so far. Nevertheless, some effective solutions for NMR problems do exist, and they greatly enhance the spin polarization in spin systems. Optical pumping for instance decreases the apparent temperature of the spin system without affecting the rest of the sample, while methods such as para-hydrogen induced polarization (PHIP) have proved to work well over spin polarization in liquidstate NMR [6]. In addition, some computational approaches have allowed the improvement of low spin polarization signals. [7, 8] It can be drawn that the major defect for NMR in quantum computing is the problem of scaling; where systems having more than 20 qubits are unlikely to develop in the foreseeable future.

In liquid-state NMR → hν << kT Hence, Є will decrease exponentially by a factor of (n/2n) with the size of the spin system, and only having an exponentially larger sample or repeating the experiment exponentially more times will compensate for such effect (non-scalability). However, some people might think that a straightforward solution can be obtained by decreasing the temperature or using high fields. Unfortunately, the critical fields and temperatures required are given (for 1H) by 150000T at room temperature (which is far beyond achievement), or 0.043K at 21.1T respectively. At such low temperatures, the 3/5

Universal Set of Quantum Gates A quantum algorithm is typically specified as a sequence of unitary transformations, where each transformation (U) acts on a small number of qubits: U = exp (iH(t)t/ ħ) The physical system should be designed so that it is possible to switch the Hamiltonian (H) or different parts of H on and off in a synchronized manner.

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MARCH 2006

Building quantum gates is relatively simple in NMR, despite the fact that it is not possible to selectively initialize qubits.

If the J-coupling was high enough, this Hamiltonian won’t apply; hence the condition:

Moreover, gates need to be modified so that larger spin systems can fit in. These modifications were found to be of a quadratic overhead.

|2π JIS| << |ωI - ωS|

One qubit gates are simply described using the Bloch vector representation [9] where a qubit is represented by a point on the surface of the unitary Bloch sphere. A single qubit gate corresponds to rotations on this sphere (where the power and length of the RF pulse determines the rotation angle, while the phase of the pulse corresponds with the rotation axis within the x-y plane). However, it should be remembered that for liquid-state NMR, a qubit is actually an ensemble of spins which are in the liquid state (thus each spin is in continual rapid motion). Also, the wavelength of RF signals is huge (1 m range) when compared with particles’ separation distances. In other words, spatial localization is impossible in such a system. Therefore, computation in such a case relies on manipulating different nuclear species by targeting their resonant frequencies (frequency selection). Remarkably, even with two or more nuclei of the same type, different environments will render different chemical shifts and thus different resonant frequencies [10]. Multi qubit gates require considering the interactions between different spins. For instance, the two spin Hamiltonian is given by: H = ωIIz + ωSSz + 2π JIS Iz Sz Where Iz and Sz are spin operators (scaled versions of Pauli matrices), while ωI and ωS are the resonance frequencies of the two spins. JIS is the scalar spin-spin coupling (Jcoupling).

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In two qubit gates, selective pulse techniques are applied so that only one spin will be excited at a time (energies are split by Jcoupling). This direct approach has been also used for building three qubit gates [11]. Another more popular approach uses the multipulse NMR techniques to sculpt the Hamiltonian into a more suitable form. Thus, spin echoes will refocus specific interactions in the Hamiltonian to either create scaled elements in the Hamiltonian, or completely deleting them. For a general n spin system the Hamiltonian is: H = ∑ ω i I zi + 2∑ π J ik I zi I zk i

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Including n Larmor frequency terms and a total of n(n-1)/2 spin-spin coupling terms. So far, it was assumed that it is possible to address individual qubits without affecting the other. However, this is only applicable when NMR transitions have well separated frequencies. Therefore, larger systems will suffer from this interaction since the limited frequency range will quickly “fill up”. Using heteronuclear spin systems (different species) will be a partial solution, since different nuclei have well separated resonant frequencies. Unfortunately, the number of suitable nuclei in application is rather small (5), where obvious candidates are: 1H, 13C, 15N, 19F and 31 P. This problem is yet another main factor that prevents further scaling of NMR quantum computers, since the largest number of spins of one nuclear type used to date is six (1H), giving a total optimistic limit of just 30 qubits.

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MARCH 2006

Decoherence

Read-out

Single qubit gates have their gate rates provided by the inverse of the time required to perform a 2π rotation. For a heteronuclear spin system, this rate is bounded to the RF power and voltage values (typically 10-100 KHz). However, homonuclear spin systems have to use selective excitations in which the gate rate is limited by the resonant frequency difference between the “to be excited” and the “to be untouched” spins (gate rate is often below 1 KHz).

Finally, the result of a computation must be read out, a thing that requires the ability to measure specific qubits.

Upon increasing the number of qubits involved in a gate, the case is similar to that of homonuclear systems, except that the frequency splitting is equal to JIS. Similarly, when using multiple pulse techniques, the relevant rate is given by 1/ 2JIS. Thus, gate rates will dramatically decrease to the 10 Hz range. Decoherence times on the other hand, are considered to be extremely good in NMR, since rough estimates suggest that it should be possible to implement about 104-106 gates before decoherence becomes a problem. However, when being more precise, one have to realize that decoherence errors, when averaged over the ensemble will largely cancel out. The effect of such decoherence will be an exponential decay in the measured signal at the output. Briefly, experiments involving hundreds of two-qubit gates have been performed reflecting a main advantage (in decoherence) for the NMR quantum computer.

References [1] J.A. Jones, NMR Quantum Computation, Fortschr. Phys. 48 (2000) [2] David P. DiVincenzo, Fortschr. Phys. 48 (2000) [3] D.G. Cory, A.F. Fahmy, and T.F. Havel, PhysComp ‘96, pp. 87-91 (1996) [4] G. Bodenhausen, H. Kogler, and R.R. Ernst, J. Magn.Reson. 58, 370 (1984) [5] S.L. Braunstein, C.M. Caves, R. Jozsa, N. Linden, S. Popescu, and R. Schack, Phys. Rev. Let. (1999)

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In NMR, read-out is done by measuring the voltage signal induced by precessing magnetic moment. This method involves quick ensemble measurements; hence, expectation values rather than traditional measurements will apply. For instance, an interesting situation occurs when the algorithm yields a superposition state. While a conventional quantum computer will end up with one of the corresponding answers at random, an ensemble quantum computer (such as NMR) will return an ensemble average over the set of all possible answers. However, in most cases, this result is not particularly useful, and it is necessary to recast the algorithm so that a single well defined result is obtained.

Conclusion In conclusion, despite its limitations, NMR technology has really been the main phenomenon used for laying out the foundation of quantum computing. The relative ease with which NMR quantum circuits can be implemented has been always a tempting advantage. Moreover, simple read-out and excellent decoherence rates all have provided the first working quantum computer, which has been a key reason behind the boost of nanotechnology in the last decade.

[6] J. Natterer and J. Bargon, Prog. NMR Spectrosc. 31, 293 (1997) [7] L.J. Schylman and U. Vazirani, LANL e-print quant-ph/9804060 [8] E. Knill and R. Laflamme, Phys. Rev. Let. 81, 5672 (1998) [9] F. Bloch, Phys. Rev. 102, 104 (1956) [10] L. Emsley and A. Pines, Lectures on Pulsed NMR in B. Maraviglia (1992) [11] N. Linden, H. Bariat, and R. Freeman, Chem. Phys. Let. 296, 61 (1998)

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