doi: 10.1098/rsta.2011.0352, 4620-4635370 2012 Phil. Trans. R. Soc. A

 Ben Criger, Gina Passante, Daniel Park and Raymond Laflamme quantum information processingRecent advances in nuclear magnetic resonance  

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Phil. Trans. R. Soc. A (2012) 370, 4620–4635doi:10.1098/rsta.2011.0352

REVIEW

Recent advances in nuclear magnetic resonancequantum information processing

BY BEN CRIGER1,2, GINA PASSANTE1,2, DANIEL PARK1,2

AND RAYMOND LAFLAMME1,2,3,*1Institute for Quantum Computing, and 2Department of Physics and

Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G13Perimeter Institute for Theoretical Physics, Waterloo, Ontario,

Canada N2J 2W9

Quantum information processors have the potential to drastically change the way wecommunicate and process information. Nuclear magnetic resonance (NMR) has beenone of the first experimental implementations of quantum information processing (QIP)and continues to be an excellent testbed to develop new QIP techniques. We review therecent progress made in NMR QIP, focusing on decoupling, pulse engineering and indirectnuclear control. These advances have enhanced the capabilities of NMR QIP, and haveuseful applications in both traditional NMR and other QIP architectures.

Keywords: nuclear magnetic resonance; quantum control; quantum computing

1. Introduction

For the last 60 years, nuclear magnetic resonance (NMR) has had a profoundimpact on many areas of science and technology [1]. Its influence extends fromour study of the basic elements of nature to everyday medical applications. Inthe last 10 years, this influence has included quantum information processing(QIP). The accurate and precise control over the magnetic moments of nucleithat scientists and engineers have developed has paved the way to use NMR tomanipulate information, using the rules of quantum mechanics.

The last decade has seen the impressive development of quantum informationscience, both in theory and in experiment. There are many measures that can beused to assess the achievements in the field: new algorithms, new applications andlarger quantum processors, to name a few. The discovery of quantum algorithmshas demonstrated the potential power of quantum information. To realize thispotential requires the ability to overcome the imprecision and imperfection

*Author for correspondence (laflamme@iqc.ca).

One contribution of 14 to a Theme Issue ‘Quantum information processing in NMR: theory andexperiment’.

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Review. Recent advances in NMR QIP 4621

inherent in physical systems. Quantum error correction (QEC) has provideda solution, showing that errors can be corrected with a reasonable amountof resources as long as their rate is sufficiently small [2]. Implementing QECprotocols remains one of the most important challenges in QIP.

In the experimental arena, the quest to build quantum processors that couldoutperform their classical counterparts has led to many blueprint proposals forpotential devices based on NMR, electron spin resonance (ESR), ion traps,atom traps, optics, superconducting devices and nitrogen-vacancy (NV) centres,among others [3]. Many have demonstrated not only the possibility of controllingquantum bits, but also the ability to do so in practice, showing the progressionof quantum information science from the blackboard to the laboratory.

The most common way to think about building devices is through what is calledthe quantum circuit model [4] based on the DiVincenzo criteria, as explained later.An important criterion is the ability to control quantum processors. This is thearena where NMR QIP has made a substantial amount of progress. Although itis not yet possible to implement full QEC protocols, an important contributionhas been the design of a variety of methods to reduce the error rate, whether bycompensating for unwanted internal interactions, or by reducing the impact of theenvironment. In particular, much work has been done to use shaped pulses eitherto correct pulse imprecision or to render the pulses robust against environmentalimperfections. Techniques such as strongly modulated pulses and the GRadientAscent Pulse Engineering (GRAPE) algorithm have allowed a reduction in theerror per gate, one of the merit figures of quantum control, to as little as 10−4 inan ideal one-qubit system. In slightly larger processors, the error remains larger,but has been definitely improved. The techniques first pioneered in NMR havebeen adopted in other technologies and applications.

The development of NMR QIP has not been without obstacles. The difficulty infinding larger and larger suitable molecules has restricted the number of qubitsto 12, thus far. The relatively slow coupling between nuclear spins, comparedwith the decoherence time, restricts the number of accurate gates that canbe implemented. The inability to perform fast projective measurement is adifficulty that must be overcome to quickly extract entropy for QEC. Despitethese impediments, NMR has proved to possess a sound foundation on which tobuild our intuition of QIP.

In §2, we discuss control of the NMR system in the context of the DiVincenzocriteria. Section 3 is devoted to discussing improvements in pulse design andimplementation, with the goal of achieving extreme accuracy. Pulse sequencesdesigned to decouple the NMR quantum processor from its environment aredescribed in §4. Section 5 is devoted to how control in NMR QIP can be extendedand improved by using electron spin systems. Finally, concluding remarks aremade in §6.

2. Quantum control

There are many ways in which we can imagine quantum processors. They can bebased on the quantum circuit model, one-way quantum computing, the adiabaticmodel or a combination of these architectures. In this review, we will focuson the quantum circuit model, where quantum processors must satisfy the five

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well-accepted requirements for physical realization known as the DiVincenzocriteria [5]:

— a scalable physical system with well-characterized qubits;— the ability to initialize the register to a simple fiducial state, such as |0〉⊗n ;— a universal set of quantum gates;— a qubit-specific measurement capability; and— decoherence times much longer than the average gate operation time.

Most of these criteria are satisfied by implementing algorithms on nuclear spinsin molecules subjected to a large, constant external magnetic field and radio-frequency (RF) pulses, known as NMR QIP [6]. Later, we describe the relationbetween the desired properties of a quantum information processor and theparticular features of the NMR system.

(a) Scalability with well-characterized qubits: spin-12 nuclei

In order for an implementation of QIP to be scalable, the required resourcesmust grow polynomially with the size of the system. Any implementation ofQIP using k two-level subsystems (referred to as qubits) is scalable in the totalenergy of the system or, equivalently, the precision with which the energy ismeasured. NMR QIP employs such an architecture, as each spin-1

2 nucleus hastwo well-defined energy levels in a magnetic field, described by a two-dimensionalHamiltonian,

H = 12(hgB0 + d)Z = 1

2(hgB0 + d)

[1 00 −1

], (2.1)

where B0 is the external magnetic field, g is the nuclear gyromagnetic ratio andd is the chemical shift term, imposed on each nucleus by its local molecularenvironment. When nuclei have distinct d, they provide qubits that can easilybe individually addressed through RF pulses. If this is not the case, more subtletechniques are required to address the qubits.

NMR was widely used in other disciplines before quantum information researchbegan, resulting in a large body of literature dedicated to modelling andmeasuring the NMR Hamiltonian for a given molecule [7,8]. The techniquesdeveloped in conventional NMR can be used to characterize qubits to a sufficientlevel of precision for quantum information. However, it is difficult to scaleliquid-state NMR QIP systems because the chemical shifts do not grow withthe number of qubits, limiting the available frequency space for addressing thenuclear qubits. In the solid state, it is theoretically possible to use a localgradient magnetic field to render the qubits distinguishable; however, this is hardin practice.

Liquid-state NMR experiments have successfully demonstrated universalcontrol over NMR systems ranging from a few qubits in the late 1990s and early2000s [9,10] up to 12 qubits more recently [11]. NMR QIP has also demonstratedthe creation of a 12-coherence state, but without universal control [11]. Thisnumber is an order-of-magnitude improvement on what could be done 15 yearsago (figure 1).

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two-levelspin statesobserved

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Figure 1. Number of qubits as a function of time for a variety of technologies. The red text showsexperiments that prepare a special state and do not necessarily have universal control. (Adaptedfrom M. Mandelberg 2009, personal communication.) (Online version in colour.)

(b) Initialization: the pseudopure state

To perform a quantum algorithm, we need the register to begin in a knownstate, typically |0〉⊗n . In NMR, the Boltzmann distribution for a single spin hasa bias towards the ground state of a ∼ tanh(hgB0/kBT ). In ideal conditions, itwould be possible to extract a pure state in the liquid state from a multi-qubitBoltzmann distribution. However, this is difficult in practice. Instead, it is possibleto produce a multi-qubit pseudopure initial state,

rpseudopure = 1 − a

2nI + a(|0〉〈0|)⊗n . (2.2)

For typical values of g and B0, a is of the order of 10−5. Any unitary operationwill alter the (|0〉〈0|)⊗n term, leaving the identity term unchanged. As long as ais high enough to produce an acceptable experimental signal-to-noise ratio, thefinal pseudopure state can be measured.

As the size of the system increases, the methods that have been used to producepseudopure initial states will result in values of a that decay exponentiallyin the system size, precluding scalability in liquid-state NMR QIP. However,

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experimental procedures such as dynamic nuclear polarization and algorithmiccooling, to be discussed in §5, could increase nuclear polarizations to values nearunity in solid-state systems that include electrons.

(c) A universal set of quantum gates: radio-frequency pulses and coupling

It is possible to implement an arbitrary n-qubit unitary gate using only one-and two-qubit gates [12]. Single-qubit gates are realized in NMR QIP by exertinga temporary magnetic field over the sample in the X–Y plane (the Z directionbeing determined by the background field), which implements the rotationoperators U x(a) and U y(b), where a and b are arbitrary angles determined bythe strength and duration of the RF pulses. Composing these unitaries producesan arbitrary one-qubit operation. Conditional logic in NMR QIP is achievedby allowing the nuclear magnetic state to evolve under coupling mechanismspresent in the molecule. These coupling mechanisms are direct dipolar couplingand electron-mediated dipolar coupling, known as J -coupling.

Since all of the spin-12 nuclei in a molecule possess magnetic moments, they

will interact, producing a pairwise direct dipolar coupling term in the NMRHamiltonian,

H DDjk = − m0

4p

hgjgk

r3jk

(3(sj · ejk)(sk · ejk) − sj · sk), (2.3)

where rjk is the distance between the two nuclei, s = [X , Y , Z ] and ejk is a unitvector along the line connecting the two nuclei. In the liquid state, Qjk (the anglebetween ejk and the external B-field axis) is, to a good approximation, uniformlydistributed by fast molecular tumbling. This results in zero interaction strength.Therefore, direct dipolar coupling is useful only in systems in which the Qjk arenot uniformly distributed, such as solid-state NMR (where all of the Qjk are fixedby crystal orientation) and liquid-crystal NMR (where the Qjk are distributednon-uniformly). In addition to the direct dipolar coupling, there exists an indirectcoupling, known as J -coupling. This is a low-strength coupling mediated by theelectron cloud in the molecule, according to

H Jjk = 2psj · J jk · s

ᵀk , (2.4)

where J jk is a 3 × 3 real matrix. This coupling term is non-zero, even under rapidtumbling. In isotropic liquids, where the two spins have a chemical shift differencemuch larger than the value of J , it is proportional to Z j Z k , generating a two-qubit gate. The variety of natural coupling terms renders QIP possible on manydifferent molecules, in the liquid, solid or liquid-crystal states.

(d) Measurement: free induction decay

The apparatus used to implement rotations about axes in the X–Y planecan be used to detect signals from the sample in the X–Y plane. In orderto translate the logical state of an NMR QIP register into an X observablevalue, a readout pulse of angle p/2 is applied about the X axis on each nuclearfrequency. This results in a signal proportional to the pseudopure parameter a

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Table 1. A comparison between characteristic time scales for control (chemical shifts and directdipolar coupling strengths) and decoherence in solid-state NMR QIP, using malonic acid. Notethe description of decoherence in terms of T ∗

2 , which combines decoherence from multiple sources.(Adapted from [14].)

C1 (Hz) C2 (Hz) Cm (Hz) T ∗2 (ms) T1 (s)

C1 5693 237 828 2.4 162C2 1748 1020 2.6 326Cm −3358 3.1 314

(see §2b), from which the logical state can be derived. This process does notresult in projective measurement, but still allows universal computation, includingfull state tomography [13]. The lack of projective measurement does not prohibitprotocols such as QEC, but does increase the required control of the system.

(e) Noise and decoherence: T1/T2 versus coupling strength

In order to perform an algorithm, the time used to implement theappropriate pulses must be much shorter than the characteristic time scale ofthe noise that would overwhelm the quantum processor. In addition to thenoise related to the imperfect implementation of the gates, the interactionwith the surrounding environment can lead to decoherence. In NMR, thereare two main mechanisms for this decoherence: thermal equilibration anddephasing, characterized by T1 and T ∗

2 , respectively. Period T ∗2 can be

improved to T2 by applying decoupling pulses as described in §4. In NMRQIP, the coupling terms between qubits are usually the weakest in theHamiltonian, leading to relatively slow two-qubit gates. The solid-state malonicacid system provides an example (shown in table 1); the carbon–carboncoupling ranges from 200 Hz to 1 kHz. This indicates that the associatedcontrolled Z has a gate time of 1

2J ∼ 0.5–2.5 ms, allowing the implemen-tation of multiple two-qubit gates before a single decay period T ∗

2 has elapsed.This suffices for simple algorithms, but faster gates and longer decoherence timesare both required in order to implement more complex algorithms.

If the noise is below a threshold [15], it is possible to use fault-tolerant methodsto counteract its effects. If the noise exceeds this threshold, control must beimproved until the threshold is attained. It is possible to assess the noise levelusing benchmarking [16,17]. In a system where control was optimized, it has beenpossible to reach an error-per-gate rate of 10−4 for single-qubit gates and 10−3

for two-qubit gates [18].In order to improve the implementation of one- and two-qubit gates using the

naturally occurring Hamiltonian, it is necessary to develop better techniques for:

— optimization of RF pulses to produce high-fidelity unitary operations;— isolating the system of interest from unwanted degrees of freedom; and— taking advantage of the unique properties of the electron spin.

In the following sections, we describe recent progress in these directions.

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3. Advances in pulse engineering

In NMR QIP, pulse engineering is the practice of developing control techniquesfor either generating coherent transfer from an initial spin state to a desiredspin state (state-to-state transfer) or producing an effective Hamiltonian thatimplements a desired unitary gate U (unitary propagator) by manipulating theexternal RF field. This can be achieved by augmenting the internal Hamiltonianwith a control Hamiltonian,

Hc(t) =∑

k

uk(t)2

[cos(uRFt + fk(t))X + sin(uRFt + fk(t))Y ],

where uk(t) and fk(t) denote the amplitude and the phase applied at atransmitter frequency uRF. The time evolution is typically divided into a setof N steps. At each step j , the evolution is given by the unitary propagatorUj = exp{−iDtHj(t)}, where Hj(t) is the total Hamiltonian with a control set{uj

k , fjk}. The final density operator is given by r(T ) = UN · · · U1r0U

†1 · · · U †

N . Thetask of pulse engineering is to find a set of {uj

k , fjk} so that the resulting dynamics

are sufficiently close to those desired.Pulse engineering techniques must address experimental limitations for

practical applications: qubit selectivity due to finite frequency bandwidth,instrumental errors (such as static and RF field inhomogeneity), miscalibrationof pulse power or duration and frequency offset.

Pulse design must take these artefacts into account in order to achieve highcontrol fidelity. This causes a substantial increase in the complexity of the pulsedesign problem. For example, errors caused by chemical shift dispersion, RFinhomogeneity and RF miscalibration can be suppressed if the parameters aresampled over a range of discrete values determined by the uncertainty. Theoptimization problem then becomes more difficult, because the fidelity functionhas more degrees of freedom in the parameter space.

Over the last 30 years, numerous techniques have been developed for controlin NMR QIP. Traditionally, average Hamiltonian theory (AHT) has been apowerful tool that provides intuitive guidelines for constructing pulse sequencesfor simple cases [19]. In particular, composite pulses [20], adiabatic pulses [21–24]and shaped pulses [25] were introduced during the earlier development of NMRQIP to better compensate for static and RF field inhomogeneity by increasingthe number of degrees of freedom in the pulse shape. However, the long pulsetimes produced by these techniques lead to greater decoherence and relaxationeffects, and interference of selective pulses simultaneously applied to differentspins. Also, full knowledge of the system parameters is required to implementthese pulses, and not all errors can be corrected using these techniques [26].Strongly modulated pulses [26] average out unwanted evolution by using strongcontrol fields, so that no additional corrections are required, while a desired spin-selective unitary operation is performed. This method uses high-power pulses thatdecrease the required pulse duration, and hence reduce the effect of decoherenceand relaxation.

Recently, optimal control theory (OCT), originally developed for problemsin engineering [27–29], has been used for systematic optimization of pulsedesigns in NMR QIP. Analytical solutions to time-optimal realization of unitary

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operations can be obtained by formulating a variational principle to reducethe problem to a set of first-order ordinary differential equations [30,31]. Formore than two qubits, there are no analytical solutions, and numerical methodsare required.

For larger systems, the majority of control methods for optimizing theefficiency of coherent transfer are based on a gradient approach, such as theGRAPE algorithm [32]. GRAPE starts with an initial estimate, then iterativelyimproves the control field parameters by calculating the gradient. This methodis an improvement, as it calculates the propagators using parameters alreadyestimated, decreasing the required computational resources. A number of recentNMR QIP experiments [33–39] applied GRAPE pulses and achieved high controlfidelity. However, the GRAPE algorithm searches for the nearest local optimum;so it is difficult to find extremely high-fidelity pulses.

Another well-developed numerical method for quantum control is amonotonically convergent algorithm [40–42] based on Krotov’s numericalmethod [43]. The Krotov-based method allows large changes in control parametersfrom one iteration to the next, and immediately exploits all available informationat each time step, monotonically improving the objective fidelity functional ateach iteration. Maximov et al. [29] analysed the Krotov-based algorithm in thecontext of NMR spectroscopy and compared it with GRAPE, concluding thatthe Krotov-based algorithm consumes less computational resources per iterationand is much better than GRAPE for making an initial guess towards a globalmaximum. However, the larger step size in the Krotov-based method limitsefficiency as the solution gets closer to the desired fidelity. There is an openquestion as to whether high efficiency can be obtained by combining the twomethods, using the Krotov algorithm to quickly prepare a good initial pulsesequence to load into GRAPE for final refinement [29].

One drawback to these methods is the high computational cost, one that growsquickly with increased system size. As quantum processors become larger, newtechniques will need to be developed that might allow reduced pulse imperfectionwithin blocks of nearby qubits (in frequency space) and scalable pulse designtechniques between blocks [44].

4. Advances in decoupling

In order to achieve high-fidelity control in NMR QIP, one must address system–environment interaction, as well as unintended evolution under internal couplingsbetween spins. The coupling interaction is ‘always on’ and therefore must besuppressed to perform single-qubit gates. This process is called refocusing andis relatively simple in liquid-state NMR, since the only coupling term is of theform Z j Z k . Solid-state NMR Hamiltonians have more complex coupling terms,and are therefore more difficult to refocus. Techniques based on AHT, such asmagic angle spinning [45] and Lee–Goldburg decoupling [46], have proven usefulin refocusing these terms. Furthermore, recent developments in optimal controltheory (see §3) allow the refocusing of unwanted internal interactions.

Hereinafter, we narrow the discussion to system–environment or dynamicaldecoupling (DD). A DD scheme is a sequence of control fields applied toa system for some time with the objective of increasing coherence time by

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attenuating system–environment interaction. Starting from Hahn’s discoveryof spin-echo in 1950 [47], many methods to accomplish this have beenactively studied, one seminal example being the Carr–Purcell–Meiboom–Gill(CPMG) sequence [48]. The performance of realistic DD sequences is limitedby experimental imperfections (see §3); so it is important to design a DD schemethat is more robust to such errors.

In traditional DD, a pulse sequence is periodically applied to reduce undesiredterms of the system–bath interaction Hamiltonian, known as periodic dynamicaldecoupling (PDD). These sequences were improved in 2005 by Khodjasteh &Lidar [49], who introduced concatenated DD pulse sequences (CDD), such aspn+1 = pnXpnZpnXpnZ , where pn = t0 is a delay between pulses and n is aconcatenation level. While CDD is more efficient at decoupling and is more robustto random and systematic control errors than PDD, the total number of necessarypulses grows exponentially (4n) in the concatenation level.

In 2007, Uhrig introduced an optimized DD sequence based on gradientmoment nulling in NMR [50], called Uhrig dynamical decoupling (UDD) [51–53]. What distinguishes UDD from conventional DD schemes is that the timedelay blocks between pulses are not equal. Uhrig showed that, for p pulses,splitting the total time interval t into smaller intervals 0, d1t, . . . , dnt (where dj =sin2[pj/(2n + 2)]) can suppress decoherence up to order tn+1 without exponentialcost. Moreover, Uhrig’s simulation showed that the performance of decouplingfrom the environment becomes independent of system–bath coupling strengthsfor long sequences. The drawback of UDD is that it works for only one type oferror, T1 or T2.

In an attempt to suppress both T1 and T2, Uhrig concatenated the UDDsequence (CUDD) [54]. The total number of pulses to suppress decoherenceand relaxation to order tn is proportional to 2n , decreasing the resources by2n from UDD sequences. West et al. further improved on this by creating anew UDD-based sequence that suppresses decoherence and relaxation to ordern using (n + 1)2 pulse intervals [55]. Their scheme integrates two sequences,T1-correcting UDD and T2-correcting UDD, and is known as quadratic dynamicaldecoupling (QDD).

UDD-based methods are optimal when the noise has a sharp high-frequencycutoff [52,56–59]. However, in a low-frequency-dominated noise environment,conventional Carr–Purcell (or CPMG) is preferred [57–59]. Borneman et al. [60]pointed out that, although the CPMG sequence is inherently tolerant of fieldinhomogeneities and pulse calibration errors, the robustness of the sequence islimited by the quality of the RF pulses used. They adapted the GRAPE algorithm(see §3) to design general refocusing pulses (universal p rotation around theY axis) with high fidelity over a wide range of resonance offset frequencies andRF amplitudes. The authors were able to find a pulse that refocuses 99 per cent ofthe initial magnetization over a range of frequency offsets of ±10 kHz (four timesgreater than the maximum RF amplitude) for uniform RF. For RF inhomogeneityof ±10%, they were able to find a pulse that refocuses 98 per cent of the initialmagnetization over a frequency range of ±8 kHz (3.2 times the maximum RFamplitude), an improvement over any previously published refocusing pulses ofsimilar duration and maximum RF amplitude when applied in a CPMG sequence.

Souza et al. [61] compared the performance of Knill pulses [62] with that ofsimple pulses in several pulse sequences in order to find a decoupling scheme that

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is robust against pulse imperfections or control errors. For small duty cycles (theduty cycle being the fraction of time for which a pulse is being applied), sequenceswith non-robust pulses are superior, owing to the shorter cycle time of the non-robust pulse sequence when constant duty cycles are compared. However, theperformance of these sequences saturates or decreases with increasing duty cycle.Comparatively, robust pulses continue to increase in performance for large dutycycles. Knill dynamical decoupling (KDD), a concatenation of the Knill pulse, isfound to have the best performance for large duty cycles, comparable to that ofsequences without robust pulses for small duty cycles.

5. Advances in indirect control using the electro–nuclear system

Many of the techniques used to control spin-12 nuclei in NMR can also be applied

to electrons, comprising ESR. As the g of an electron is approximately 103 timesgreater than that of a proton, this leads to a much larger Larmor frequency andhigher polarization. Electron magnetic states also experience faster decoherenceand relaxation than nuclear states. The coupling between electron and nuclearspins is governed by the hyperfine interaction,

H hf = sE · A · sᵀN, (5.1)

where E denotes the electron, N denotes the nucleus and A denotes the hyperfinecoupling tensor.

There are three quantum information techniques that exploit the properties ofcombined NMR/ESR systems:

— dynamic nuclear polarization and algorithmic cooling, which increase theinitial polarization on nuclear spins;

— hyperfine control, which produces faster control over nuclear spins at lowmagnetic field strengths; and

— spin-bus implementations and parallel information transfer, which producetwo-qubit gates between uncoupled nuclear spins.

These three techniques, described subsequently, represent the potential ofelectron control to better satisfy the DiVincenzo criteria in NMR QIP.

(a) Dynamic nuclear polarization and algorithmic cooling

A nuclear spin-12 particle in a magnetic field at room temperature has an

equilibrium density operator described by a Boltzmann distribution, with atypical bias of approximately 10−5. Although this is sufficient to manipulatequantum information and to demonstrate the principles of QIP, it does notallow for resetting ancilla qubits, as needed, for example, after rounds ofQEC, a substantial hurdle for scalable quantum information. This cannot beeffectively resolved by physical cooling or increasing the magnetic field. Forexample, reducing the temperature to 3 K increases the polarization to theorder of 10−3, and doubling the external magnetic field at most doubles thepolarization. Therefore, it is necessary to develop novel techniques to provideinitial polarization that approaches unity.

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Because electrons provide a natural source for spin polarization, a SWAP gatebetween a nuclear spin and an electron, each at equilibrium, will increase thenuclear polarization greatly. This is called dynamic nuclear polarization (DNP).In the context of QIP, such polarization transfers have been implemented in15N@C60 [63], achieving a single-spin nuclear polarization of 62 per cent at4.2 K, in an 8.6 T field. Also, DNP has been used in conjunction with naturallyoccurring spin diffusion in silicon microparticles to produce large numbers of spinswith 5 per cent polarization at 1.5 K and 2.35 T [64]. The nuclear T1 for thesesystems was shown to be dependent on the particle size, making the polarizationdecay controllable in principle. These two results indicate that control of a singleelectron spin coupled to an NMR QIP register can assist in initialization.

It is possible to obtain nuclear qubits with still greater polarizationalgorithmically, shifting entropy to the environment using an electron spin asa thermal contact. This technique, known as algorithmic cooling, while beingimplementation-independent, lends itself to NMR QIP owing to the propertiesof the electron. Indeed, this technique has recently been implemented in liquid-and solid-state NMR [34,65,66]. It also has the potential to be used in othertechnologies, such as quantum dots or NV centres.

(b) Indirect control via the hyperfine interaction

If the hyperfine interaction has a higher frequency than the nuclearLarmor precession, faster nuclear control can be obtained by manipulatingthe electron [67], producing two effective magnetic fields seen by the nucleus,depending on the electron spin state. The effective nuclear field is

Beff =(

B0 ± A2gN

)z ± B

2gNx , (5.2)

where A is the zz component of the hyperfine tensor and B is√

A2zx + A2

zy , the

component of the hyperfine tensor in the x direction on the nucleus, in a speciallychosen frame. The ± sign takes the value plus (+) when the electron spin isparallel to the external field, and minus (−) when they are antiparallel. Anyrotation of the Bloch sphere can be generated by repeated rotations about thesetwo distinct axes. One-qubit control of the nuclear spin can then be exercisedindirectly, because the free precession of the nucleus about these axes does notrequire active control.

(c) Spin-buses and parallel information transfer

Because a single electron can couple to multiple nuclei, it can be used totransfer information between them, creating an effective coupling. This indirectcoupling is the basis of the spin-bus [68], useful for performing multi-qubit gateswhen the electron–nuclear and electron–electron coupling are much strongerthan the nuclear–nuclear coupling. This concept was used to perform Deutsch’salgorithm [68], using a system of two nuclear qubits coupled by an electron inCaF2:Ce3+.

The indirect coupling of nuclei using information transfer of the type describedabove has two flaws, which must be overcome to confirm its utility as a nuclearcontrol method. The first is that the state being transferred is subject to strong

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decoherence and relaxation while it is stored on the electron. The second is thatonly two nuclei can be coupled through the bus at the same time; two-qubit gatescannot be performed in parallel. These problems were recently resolved [69], wherethe assumed architecture consists of two local nodes (taken to be sets of n nuclei,each coupled to an electron), with the only inter-node coupling being betweenelectrons. By using the interaction frame, Borneman, Granade and Cory showedthat states of the multi-nucleus nodes can be swapped in parallel, effectivelyperforming n two-qubit gates simultaneously. Also, the effect of decoherence onthe electron is mitigated by ensuring that no computational state is stored onthe electron.

The electron spin, when used as a component of the NMR QIP system,possesses properties that complement those of the nuclear spins. While the nuclearspin has a longer coherence time and low initial polarization, the electron has ashort coherence time and high initial polarization. The electron–nuclear couplingalso permits an array of techniques, making the electron a valuable asset toNMR QIP.

6. Discussion and conclusion

The most important recent advance in NMR QIP has been in the precision withwhich nuclear spins can be controlled. The pulse shaping and DD techniques,mentioned in §§3 and 4, have decreased the typical control error to 10−3 for two-qubit gates and 10−4 for one-qubit gates [18], with 1–2 orders of magnitude ofimprovement in the last decade. While these advances have allowed only a modestincrease in the number of qubits used in NMR QIP experiments, they have foundapplication in other areas.

For example, the progress in NMR techniques for quantum information has hada considerable impact in the broader QIP community. The techniques developedto obtain control over systems with natural Hamiltonians have led to greatercontrol over those systems where the Hamiltonian is designed. Composite andoptimized pulses have been studied and applied in the context of trapped-ionQIP [70–73]. Also, the application of the GRAPE technique has been studied inJosephson-array superconducting circuits [74–77]. The applicability of GRAPEto other architectures for quantum computing enables a wider tolerance fordesigned Hamiltonians.

NMR QIP techniques have also seen application outside of quantuminformation. In NMR medical imaging [78], hyperpolarized nanoparticles [64]can be used to produce greater image quality at lower concentrations. Since theCPMG sequence has applications in oil well logging [79], its optimization [80] canincrease the accuracy with which this can be accomplished. These applicationsof NMR QIP techniques outside of the field itself further motivate the study anduse of NMR systems as quantum information processors.

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