J.A. Jones/Progress in Nuclear Magnetic Resonance Spectroscopy 38(2001)325-360 States of this kind are often called pseudo-pure states,or effective pure states [8-10]. Unfortunately the simple approach outlined above does not work for larger spin systems, as the pattern of population devia- tions is more complicated, and does not have the desired form. Several different techniques have Fig. 7. The structure of partially deuterated cytosine obtained by however, been developed to tackle this problem. issolving cytosine in D2O; the three protons bound to nitrogen uclei exchange with solvent deuterons, leaving two H nuclei as 54. Readout an isolated two spin system(all other nuclei can be ignored) The last stage in any quantum computation is to When, as for NMR, the computational basis coin- characterise the final state of the system, so that the cides with the natural basis of the quantum system it result of the computation may be read out. Just as for initialisation, a range of different approaches have should in principle be easy to implement CLEAR as it been used, but all these approaches combine two takes the quantum computer to its energetic ground state, and this can be achieved by some cooling process major elements. For simplicity I will assume that the Unfortunately this approach is not practical in NMR as computation ends with the result qubits in eigenstates the Zeeman energy gap is small compared with the thus it is only necessary to determine whether a given Boltzman energy at any reasonable temperature: thus qubit is in(the pseudo-pure)state lo)or[1) temperature the population of all the states The simplest approach is to apply a 90 pulse to the will be almost equal, with only small deviations(around corresponding spin, and observe the NMR spectrum one part in 10")from the average. Techniques for enhan- [11]. Since 0)corresponds to the ground state, a qubit cing spin polarisation [38], such as optical pumping in o)will give rise to an absorption line; correspond [39-41], and the use of para-hydrogen [42-44) allow ingly a qubit in state 1)will give an emissive signal this deviation to be increased, but with the exception of Is, of course, necessary to acquire some sort of refer- optically pumped noble gases it has so far proved impos- ence signal, in order to distinguish between these two sible to even approach a pure ground state system. extremes, but this can be easily achieved by acquiring This apparent inability to implement the CLEAr the spectrum of the pseudo-pure initial state The second major approach[12] is to determine the peration led to NMR being rejected as a practical state of one qubit by analysing the multipl let structure technology for implementing quantum computers Recently, however, it was realised [8] that this conclu- within the spectrum of a neighbouring spin. If several sion was over hasty, as with an ensemble quantum spins are coupled together, then individual lines computer it is not actually necessary to produce a within a multiplet can be assigned to specific states pure ground state; instead it suffices to produce of these neighbours. Thus, the spectrum of one spin state which behaves in the same manner as the pure can give information on the states of several differen ground state. This point can be clarified by consider ubits ing the density matrix describing a single isolated 5.5. Some two spin syster spin-half nucleus. This exhibits nearly equal popula tions for the two eigenstates, but with a slight While a number of different systems have been in the (low energy) o) state compared with the used to build small NMR quantum computers, all (slightly higher energy)1)state. No NMR signal their major features can be explored using two differ will be observed from the equal populations, as th ent two-qubit systems which were used in the earliest signals from different molecules will cancel out, but a demonstrations of NMR quantum computation small signal can be seen which arises from the devia [I1, 12]. The most important difference between tions away from the average. Thus, ignoring questions these systems is that one uses a homonuclear two- of signal intensity, for a single isolated nucleus the spin system, while the other is heteronuclear thermodynamic equilibrium state is indistinguishable The first example system uses the two H nuclei of from a pure o)state. partially deuterated cytosine in D2O(see Fig. 7). As
When, as for NMR, the computational basis coincides with the natural basis of the quantum system it should in principle be easy to implement clear as it takes the quantum computer to its energetic ground state, and this can be achieved by some cooling process. Unfortunately this approach is not practical in NMR as the Zeeman energy gap is small compared with the Boltzman energy at any reasonable temperature; thus at room temperature the population of all the states will be almost equal, with only small deviations (around one part in 104 ) from the average. Techniques for enhancing spin polarisation [38], such as optical pumping [39±41], and the use of para-hydrogen [42±44] allow this deviation to be increased, but with the exception of optically pumped noble gases it has so far provedimpossible to even approach a pure ground state system. This apparent inability to implement the CLEAR operation led to NMR being rejected as a practical technology for implementing quantum computers. Recently, however, it was realised [8] that this conclusion was over hasty, as with an ensemble quantum computer it is not actually necessary to produce a pure ground state; instead it suf®ces to produce a state which behaves in the same manner as the pure ground state. This point can be clari®ed by considering the density matrix describing a single isolated spin-half nucleus. This exhibits nearly equal populations for the two eigenstates, but with a slight excess in the (low energy) u0l state compared with the (slightly higher energy) u1l state. No NMR signal will be observed from the equal populations, as the signals from different molecules will cancel out, but a small signal can be seen which arises from the deviations away from the average. Thus, ignoring questions of signal intensity, for a single isolated nucleus the thermodynamic equilibrium state is indistinguishable from a pure u0l state. States of this kind are often called pseudo-pure states, or effective pure states [8±10]. Unfortunately the simple approach outlined above does not work for larger spin systems, as the pattern of population deviations is more complicated, and does not have the desired form. Several different techniques have, however, been developed to tackle this problem. 5.4. Readout The last stage in any quantum computation is to characterise the ®nal state of the system, so that the result of the computation may be read out. Just as for initialisation, a range of different approaches have been used, but all these approaches combine two major elements. For simplicity I will assume that the computation ends with the result qubits in eigenstates; thus it is only necessary to determine whether a given qubit is in (the pseudo-pure) state u0l or u1l. The simplest approach is to apply a 908 pulse to the corresponding spin, and observe the NMR spectrum [11]. Since u0l corresponds to the ground state, a qubit in u0l will give rise to an absorption line; correspondingly a qubit in state u1l will give an emissive signal. It is, of course, necessary to acquire some sort of reference signal, in order to distinguish between these two extremes, but this can be easily achieved by acquiring the spectrum of the pseudo-pure initial state. The second major approach [12] is to determine the state of one qubit by analysing the multiplet structure within the spectrum of a neighbouring spin. If several spins are coupled together, then individual lines within a multiplet can be assigned to speci®c states of these neighbours. Thus, the spectrum of one spin can give information on the states of several different qubits. 5.5. Some two spin systems While a number of different systems have been used to build small NMR quantum computers, all their major features can be explored using two different two-qubit systems which were used in the earliest demonstrations of NMR quantum computation [11,12]. The most important difference between these systems is that one uses a homonuclear twospin system, while the other is heteronuclear. The ®rst example system uses the two 1 H nuclei of partially deuterated cytosine in D2O (see Fig. 7). As J.A. Jones / Progress in Nuclear Magnetic Resonance Spectroscopy 38 (2001) 325±360 335 Fig. 7. The structure of partially deuterated cytosine obtained by dissolving cytosine in D2O; the three protons bound to nitrogen nuclei exchange with solvent deuterons, leaving two 1 H nuclei as an isolated two spin system (all other nuclei can be ignored)
J.A. Jones/Progress in Nuclear Magnetic Resonance Spectroscopy 38 (2001)325-360 this system is homonuclear it is possible to excite both heteronuclear spin system is unlikely to be practical nuclei with a single hard pulse, and to observe both beyond five qubit systems, as there are only five advantage is that the pattern of Boltzmann popula- lov ous" spin-half nuclei which can be used(. nuclei in the same spectrum. Another more subtle andP). In practice NMR quantum ions is simpler in homonuclear syster n in computers with more than three qubits are likely to their heteronuclear counterparts. There are, however, include two or more spins of the same nuclear species two significant disadvantages of such as system it is, therefore, essential to consider how computation Firstly the two H multiplets have relatively similar can be performed in homonuclear systems frequencies, as they lie only about 1.51 ppm apart, and thus it is necessary to use soft frequency selective 5.6. Scaling the system up pulses [45](or sequences of hard pulses and delays with equivalent effects)in order to address the spin The requirements outlined above are adequate for individually. Secondly, the J-coupling between the building small quantum computers, suitable for two spins is relatively small(about 7 Hz), and so simple demonstrations of quantum information controlled gates take a fairly long time to implement processing. If, however, one wishes to build a large It would, of course, be possible to choose a different scale quantum computer, suitable for performin molecule, in which the chemical shift difference or J- interesting computations, then it is necessary to coupling was larger, but it is difficult to improve one consider whether the approaches used are lim without making the other worse. While it is unlikely such small systems, or whether(and if so, how)they that cytosine is the absolutely optimal choice, no other can be scaled up. A fifth requirement for practical homonuclear H system would be very much better. uantum computation [37], the implementation of The heteronuclear alternative is probably the most fault-tolerant quantum error correction, is described widely used two qubit NMR system. It is based on the in Section 12 H and C nuclei in C-labelled chloroform. This has This is an important practical question, but not one which will be addressed in detail here. The problems the huge advantage that it is possible to separately of scaling up NMR quantum computers are formid- excite the two spins using hard pulses, rendering able. and have been well described elsewhere selective excitation essentially trivial. Furthermore the relatively large size of the J-coupling allows two [38, 46, 47. Most authors now agree that NMR qubit gates to be performed much more rapidly than in approaches are likely to be limited to computers homonuclear systems. In this heteronuclear system it containing 10-20 qubits: this is significantly smaller is not possible to acquire signals from both spins than estimates of the size required to perform useful simultaneously, but this is not a major problem as it computations (50-300 qubits).Furthermore the is possible to determine the states of both spins by apparent inability of NMR systems to perform effi examining either the H or theC spectrum. Simi- cient quantum error correction rules out their use for larly, the complex pattern of populations over the four many types of problem energy levels of this system does not fit with the origi- The fundamental difficulties involved in scaling up current NMR quantum computers to large sizes have however. some more modern schemes are in fact led some authors to suggest that this approach does simpler to implement in heteronuclear systems not actually implement real quantum computation at all. This is a quite subtle question which will be Considering all these issues together, it is not easy discussed further in Section 13 below to say whether it is better to use homonuclear or heteronuclear systems to implement two qubit NM quantum computers: heteronuclear systems are 6. Qubits and NMR spin states perhaps simpler to work with, but homonuclear ystems give more elegant results. With larger spin quantum computers systems the issues become even more complex, and comprise a number of two-level systems which inter a wide range of options have been explored. It is clear, act with one another and have some specific interac however, that the simplest approach of using a fully tion with the outside world, through which they can be
this system is homonuclear it is possible to excite both nuclei with a single hard pulse, and to observe both nuclei in the same spectrum. Another more subtle advantage is that the pattern of Boltzmann populations is simpler in homonuclear systems than in their heteronuclear counterparts. There are, however, two signi®cant disadvantages of such as system. Firstly the two 1 H multiplets have relatively similar frequencies, as they lie only about 1.51 ppm apart, and thus it is necessary to use soft frequency selective pulses [45] (or sequences of hard pulses and delays with equivalent effects) in order to address the spins individually. Secondly, the J-coupling between the two spins is relatively small (about 7 Hz), and so controlled gates take a fairly long time to implement. It would, of course, be possible to choose a different molecule, in which the chemical shift difference or Jcoupling was larger, but it is dif®cult to improve one without making the other worse. While it is unlikely that cytosine is the absolutely optimal choice, no other homonuclear 1 H system would be very much better. The heteronuclear alternative is probably the most widely used two qubit NMR system. It is based on the 1 H and 13C nuclei in 13C-labelled chloroform. This has the huge advantage that it is possible to separately excite the two spins using hard pulses, rendering selective excitation essentially trivial. Furthermore, the relatively large size of the J-coupling allows two qubit gates to be performed much more rapidly than in homonuclear systems. In this heteronuclear system it is not possible to acquire signals from both spins simultaneously, but this is not a major problem as it is possible to determine the states of both spins by examining either the 1 H or the 13C spectrum. Similarly, the complex pattern of populations over the four energy levels of this system does not ®t with the original scheme for generating pseudo-pure states; however, some more modern schemes are in fact simpler to implement in heteronuclear systems. Considering all these issues together, it is not easy to say whether it is better to use homonuclear or heteronuclear systems to implement two qubit NMR quantum computers: heteronuclear systems are perhaps simpler to work with, but homonuclear systems give more elegant results. With larger spin systems the issues become even more complex, and a wide range of options have been explored. It is clear, however, that the simplest approach of using a fully heteronuclear spin system is unlikely to be practical beyond ®ve qubit systems, as there are only ®ve ªobviousº spin-half nuclei which can be used (1 H, 13C, 15N, 19F and 31P). In practice NMR quantum computers with more than three qubits are likely to include two or more spins of the same nuclear species; it is, therefore, essential to consider how computation can be performed in homonuclear systems. 5.6. Scaling the system up The requirements outlined above are adequate for building small quantum computers, suitable for simple demonstrations of quantum information processing. If, however, one wishes to build a large scale quantum computer, suitable for performing interesting computations, then it is necessary to consider whether the approaches used are limited to such small systems, or whether (and if so, how) they can be scaled up. A ®fth requirement for practical quantum computation [37], the implementation of fault-tolerant quantum error correction, is described in Section 12. This is an important practical question, but not one which will be addressed in detail here. The problems of scaling up NMR quantum computers are formidable, and have been well described elsewhere [38,46,47]. Most authors now agree that NMR approaches are likely to be limited to computers containing 10±20 qubits; this is signi®cantly smaller than estimates of the size required to perform useful computations (50±300 qubits). Furthermore the apparent inability of NMR systems to perform ef®- cient quantum error correction rules out their use for many types of problem. The fundamental dif®culties involved in scaling up current NMR quantum computers to large sizes have led some authors to suggest that this approach does not actually implement real quantum computation at all. This is a quite subtle question which will be discussed further in Section 13 below. 6. Qubits and NMR spin states Traditional designs for quantum computers comprise a number of two-level systems which interact with one another and have some speci®c interaction with the outside world, through which they can be 336 J.A. Jones / Progress in Nuclear Magnetic Resonance Spectroscopy 38 (2001) 325±360