School of Computer Science Senior Thesis QCNMR: Simulation of a Nuclear Magnetic Resonance Quantum Computer

Transcription

1 School of Computer Scence Senor Thess QCNMR: Smulaton of a Nuclear Magnetc Resonance Quantum Computer Matthew W. Anderson ma@andrew.cmu.edu Carnege Mellon Unversty 3/4/4

2 Astract We present the mplementaton of a nuclear magnetc resonance (NMR) unversal quantum computer (QC) smulator we have named QCNMR. Our quantum computer smulator uses a pre-exstng open source package, GAMMA, for performng the underlyng NMR smulaton. QCNMR reads n a quantum crcut converts t nto a sequences of pulses that can e smulated on an NMR system, our system then returns the output of the smulated computaton to the user. Our smulator s unversal over the space of quantum computaton, analogous to the way a classcal computer may e called unversal. I. Introducton A. Overvew Classcal computers have een around for a large part of a century and the computatons that they can perform are entrely determnstc. There exst classes of prolems that are not known to e effcently solvale n polynomal tme (O(n k )) y classcal computers; these types of prolems are known as non-polynomal (NP) tme prolems. In 996, Grover desgned an algorthm to set for a marked tem on a quantum computer that provded a square root mprovement over the est tme a classcal computer s thought to take to e ale to solve the prolem [Grov96]. Grover s search algorthm provded a more effcent way to solve a NP-complete prolem; complete refers to the fact that prolem s canoncal to the set of NP prolems therefore an effcent soluton for ths prolem could e translated nto an effcent soluton for any other NP prolem. Grover algorthm, however, dd not provde the crucal speedup, as hs algorthm stll took NP tme n the worst case. In the later n 997, Peter Shor devsed an algorthm for fndng the prme factors of a numer n polynomal (P) tme on a quantum computer. Ths represents a drastc speedup over the wdely held elef that the prolem takes NP tme on a classcal computer [Shor97]. Factorng, however, s not a NP-complete so these results could not e used to translate the speedup to all NP prolems. There are relatvely few other examples of algorthms where there s an evdent speedup when the computer s allowed to take advantage of quantum mechancal effects. It s not known whether allowng use of quantum effects provdes n nherent ncrease n the avalale computatonal power. It s also unknown as to whether there exst other algorthms that may e more effcently performed usng a quantum computer. In ether case, t s mportant that whle one s attemptng to devse new algorthms for quantum computers that one has ready access to uld and test ther algorthm to see how and f t performs as expected. B. Goals To that end, the mplementaton of a framework to smulate quantum algorthm on readly avalale classcal hardware would e useful to help researchers study quantum algorthms. There are several possle approaches, each wth ts own level of accuracy and practcalty. In the frst and smplest approach you assume that you have an deal quantum computer and that you can perform more or less artrary operatons on the quts n your computer. The mplementaton tself s just a seres of straghtforward matrx operatons. It should e noted that the tme to perform the calculatons to smulate the deal QC s not the same asymptotc tme that a real quantum computer would take, as each matrx

3 operaton take O( n ) tme n the numer of quts. owever, to our knowledge such an deal physcal system for performng quantum computaton does not even exst, t can only e approxmated y other less accurate and more lmted physcal systems. So, whle t s mportant that the algorthm functon on an deal quantum computer, t must also functon on a realstc physcal one, otherwse the algorthm has lmted applcalty. The next level of ncreased realsm s the smulaton of a physcally ased quantum computaton system. Ths means that the smulator must calculate the computaton ased on some physcal quantum mechancal effects and uld a framework for quantum computaton lmted y these effects. Ths s the method that we have chosen to mplement n ths paper. Ths method dscards the assumpton that our system s deal, and requres that the computaton have ass n a physcally realzale system. The thrd and most tellng method s to actually mplement the computaton on a physcal quantum system. Ths method s y far the most complcated and expensve of the three. Implementng algorthms correctly and effectvely on real systems can take a long amount of tme and plannng; weeks or months. In addton, these systems are y no means wdely avalale, requre a great deal of techncal expertse and must e custom ult and talored to the specfc task. There exst a numer of possle physcal realzatons for a physcal quantum computer: nuclear magnetc resonance, quantum dots, on traps and optcal photon quantum computers and others [Cra95, Cory96, DV95, Yann99]. owever, t remans to e seen whch physcal realzaton, f any, wll e vale n the end. The purpose of our smulator, QCNMR, s two-fold. The frst purpose s to allow NMR quantum computer researchers to smulate algorthms efore they attempt to perform them on a physcal NMR system. Ths wll hopefully decrease the amount of turnaround tme to study an algorthm for NMR QCs. It allows asc mplementaton wthout the nvestment of much tme. The second purpose s purely educatonal; t wll allow students to quckly experment wth quantum algorthms, even wth very lttle knowledge of NMR tself; hopefully, gan a more ntutve understandng of the algorthm and quantum computaton as a whole. C. Specfc work Our man contruton n ths paper was to the mplement framework for quantum computaton on top of an exstng open-source NMR smulator, GAMMA [Smt94]. QCNMR exsts as a C++ lrary that has the alty to create quts, a quantum crcut and to run the crcut and return the fnal state of the system after the crcut has run ts course. QCNMR frst parses the nput crcut and then converts nto a sequence of NMR pulses, whch represents the computaton we wsh to perform. Pulses are essentally NMR nstructons. We pass these pulses on to GAMMA to smulate the NMR system and then read ack nformaton and nterpret t for the user. The man work n ths project was to mplement the generaton of pulse sequences on the NMR system so that we could selectvely address ndvdual quts and leave the rest of the system nvarant, usng a technque know as refocusng. Tme was also taken to make the graphcal nterface for the QCNMR lrary that allows the user to quckly construct and run quantum crcuts. D. Organzaton of Paper Secton two presents some ackground nformaton on NMR, quantum computaton and performng quantum computaton operatons through NMR methods. Secton three explans the structure and mplementaton of QCNMR. Secton four descres usng QCNMR to mplement a quantum teleportaton algorthm as well as some general comments on our assumptons n uldng QCNMR. Secton fve 3

4 concludes ths paper and dscusses future work, related work and summarzes the results of ths project. II. Background A. Quantum Computaton Quantum computaton s perhaps est ntroduced n as an analog to classcal computaton. In classcal computaton, the oject that stores or represents nformaton s known as the t ; t can take only two dscrete values zero () or one (). The analog for quantum computaton s known as the qut ; the nformaton the qut stores, however, s not dscrete valued. The qut s composed of a lnear comnaton of vectors n complex space. Normally, the two-dmensonal lert space s descre usng the two orthonormal ass states > and >. As a sde note, there exst n-ary quts that are the lnear comnaton of n orthonormal ass states. The value of a general -qut can e wrtten as: q a +. Where a and are complex constants and the norm of q> s. The values of a sngle qut can e thought of as a vector from the orgn Eucldean three-space to the surface of a unt sphere; ths s known as the Bloch s sphere representaton (Fgure ). It s clear that the orentaton of a qut can e parameterzed y two real values, gvng t the capacty to store Fgure : The Bloch Sphere ( ) real-valued nformaton opposed to the dscretevalued nformaton that s stored n the classcal t. owever, the nformaton that s stored n that real-valued state of the qut cannot e drectly read. The most tellng measurements that can e performed are ones that measure n orthogonal ass states, returnng the ass state that the qut s state projected nto. Ths measurement provdes a sngle t of classcal nformaton. To get a more accurate determnaton of the actual qut state more measurements are requred, each provdng a lttle more nformaton aout the qut. owever, ths s a prolem n many physcal realzatons as ther measurement operatons tend to e destructve; meanng that the measurement of a qut dsturs t from ts orgnal state creatng hstory-dependent effects due to measurement; ths often makes s dffcult to make more than one measurement on a gven physcal qut n practce. Smlar to the theory of unversal classcal computaton y logcal crcut, we can descre an analog for quantum computaton: unversal quantum crcuts. The wres of our crcut are quts and the two drectons along the wre represent the tme evoluton of the computaton. We can place gates on the wres that transform quts analogous to the transforms on classcal ts y classcal gates. The gates n oth cases can e thought of as transformaton matrces, n classcal case the only values that the transformaton matrces can contan are zeros and ones; n the quantum case the transformaton matrces are artrary untares. Classcally there s a noton of unversal computaton, whch s eng ale to perform to construct artrary functon from n to m ts, wth some suset of all vald gates. One example s XOR and AND, whch are suffcent for unversal classcal computaton [Nel]. There s the same noton n quantum computaton, the controlled-not operator, U CNOT, the adamard operator, U, and the phase shft operator, U P, represents a ass for unversal quantum computaton. 4

5 U X U U S., U CNOT, ( ) Other useful operators are can e thought of as rotatons aout the x-, y-, and z-axs, respectvely U X, U Y, U Z : An addtonal feature of quantum computaton that does not have an analog n classcal computaton s the notaton of entanglement. Entanglement s essentally the correlaton n the states of two or more quts. To make ths more tangle we wll present a ref example wth one of the smplest entangled states, a Bell par: B ( + ). ( 4) Ths represents a state n the tensor product space of two quts. We can make a measure of the state n the frst or second qut s space that wll return a > or a >, n ether of these cases the measurement of the second qut n the same ass s fully determned: M B I B M, U Y, U. X A A ( ) ( I + I ) ( 5) B a. a a ( I ) ( I + I ) ( 6) a. a B a a a a a ( 3) We see f we measure a > for the frst qut then the second qut must also measure a >, a smlar thng happens for measurng >. Most of the power of quantum computaton seems to come from the alty to entangle quts, allowng nformaton to e correlated across tme and dstance. It allows the state to exst n the superposton as the lnear comnaton of multple states. Ths allows for the parallel processng of many states through the evoluton of a couple of quts. In the end, ths parallel state processng provdes the speedup n most cases of quantum mprovement over classcal algorthms. Constructng entanglement requres at least two-qut operatons, such as the controlled-not gate, as seen aove. One note on termnology, a densty matrx s one representaton of a system s state. Suppose we have some state > the densty matrx representaton of ths state s ρ ψ ψ. ( 7) Where s the outer product of > wth ts conjugate, <. For example: ψ, 8 ρ ψ ψ +, 9 ρ. ( ) ( ) ( ) ( ) ( ) Notce that the dagonal terms of sum to one; all densty matrces of normalzed pure states have ths property. There s one restrcton on qut operatons. It s the so-called no clonng theorem, t smply states that we cannot clone the state of one qut to another qut, unless we know the nput comes from a known set of orthogonal states [Nel]. Ths theorem prevents us from allowng fan-out gates whch clones the proalty dstrutons of quts, though we are free to dstrute nformaton etween the quts y entanglng them. owever, dong so does not ncrease the amount 5

6 of avalale nformaton, t only spreads t out. Thus t s not possle to copy a state many tmes n order to extract more nformaton from a destructve measurement process. B. NMR Nuclear magnetc resonance s a technque used n a numer of felds. It can e used n analytc chemstry for chemcal analyss and s the techncal ass for magnetc resonance magng (MRI) procedures for medcal purposes. The sotopes of some elements have non-zero nuclear magnetc moments. These moments can e thought of as a unt vector centered at the orgn; not surprsngly, t s very smlar to the concept of the Bloch sphere as shown aove n Fgure. If the atom s placed n an external magnetc feld, B z; canoncally the feld s algned pontng up the z-axs. The nucle s magnetc moment wll egn to rotate aout the z-axs at a rate known as the Larmor frequency: ω γb z ˆ. ( ) Where s the gyromagnetc rato of the nucleus; ths dea s shown dagrammatcally n Fgure elow. The frequency of the rotaton s lnearly proportonal to the strength of the external feld. In the asence of other atoms and other effects the moton of the sngle atom s constant n tme. When we add other atoms n a molecule wth the orgnal atom other effects egn to appear. One effect s chemcal shft,, whch represents the relatve shft from the Larmor frequency of the atom n solaton; ths s due to sheldng effects ased on the arrangement of electrons medatng the onds etween atoms [Keel4]. The chemcal shft allows us to dfferentate etween atoms two atoms of the same sotope n the same molecule. Another effect of the onds etween atoms s the J- couplng effect, J; t s smply an nteracton that encourages the moments of the two nucle to pont n the same or opposte drectons. The J-couplng nteracton can e used to medate nformaton exchange etween two nucle. The J-couplng nteracton s ansotropc, however, ts effectve strength n the drecton of the external magnetc feld, far outweghs ts strength n the two other orthogonal drectons [Ladd3, Keel4]. Now f we allow multple copes of the same molecule to resde n the same area, usually n a lqud, there can e effects caused y nteracton etween molecules. These effects are collectvely known as decoherence. The frst decoherence effect s known as thermal relaxaton, whch s caused y nteracton etween the molecules as a whole. It s lke randomly applyng rotatons to the molecule [Grze3]. It has the effect of rngng the system of molecule to thermal equlrum; an essentally random confguraton ased on the temperature of the system. It has a characterstc onset tme known as T. The second decoherence effect s caused y J- couplng nteractons takng place etween atoms on adjacent molecules ths also dsturs the state of the system. The characterstc onset tme for the second decoherence effect s known as T [Keel4, Ladd3]. The mnmum of T and T represents the maxmum tme efore the system decoheres and loses a sustantal amount of nformaton. Nuclear magnetc resonance systems exst as a conglomeraton of molecules. Usually the molecule of nterest s put nto a lqud soluton wth some other molecule that s not actve n NMR, has only zero nuclear magnetc moments, so that the nterestng molecule s propertes can e dstngushed from the soluton s propertes. The amalgamaton s an ensemle, so that any measurement s represents the average value of the measurement over all molecules n the ensemle. owever, snce t s an ensemle, measurements that are taken are not necessarly as destructve as n other physcal realzatons. The measurement acton n an NMR system s know as the Free Inducton Decay (FID). The measurement s taken y allowng the fnal state of the system to decay to the thermal state. The 6

7 change n the orentaton of the nuclear magnetc moments s made evdent y a current nduce n the cols that perform the r.f. pulses. The tme-varyng nducton, rather the Fourer transform of the tme-varyng nducton, s used to determne the fnal average state of the molecular ensemle efore t egan to decay [Keel4]. In physcs, one way of descrng the tme evoluton of a system s through the amltonan of the system. The amltonan of the NMR system of the ensemle of molecules, n the frame of the laoratory, can e wrtten as: la n ω J z + n j + ω Jj J z J j z. ( ) Where J z s the z spn operator of th nucleus; the frst term represents Zeeman Effect, whch s essentally the Larmor precesson of the nuclear magnetc moment aout the external feld axs. The second term, whch s the summaton, represents the J-couplng nteractons etween all pars of nucle. One should notce that ths constructon of the amltonan s tme ndependent; therefore the way the system evolves s ndependent of the amount of tme that has elapsed snce t started. The tmedependent formulaton removes the consderaton of relaxaton effect ecause ther onset s tme-dependent. owever, to frstorder, the effects of relaxaton can e consdered as maxmum runtme cutoffs. Fgure : Larmor Precesson n the rotatng frame The amltonan n the rotatng frame of each sotope s smply the prevous equaton wth susttuted for [Ladd3]. In order to perform any sort of computaton we have to e ale to manpulate the NMR system. We can manpulate the system through somethng known n NMR as a pulse. A pulse s smply a rado frequency (r.f.) magnetc feld appled to the system n a drecton perpendcular to the external magnetc feld. In order to apply the pulses n a more ntutve fashon we tend to apply them n the rotatng frame of the sotope. Ths means f we have m dfferent sotopes we have m dfferent rotatng frames or channels ; these channels have a specfc carrer frequency that represents the frequency of the rotatng frame relatve the la frame. Generally the Larmor frequency of dfferent sotopes s suffcently dfferent as to e ale to address each channel separately; to e ale to apply pulses to a channel wthout t affectng the other channels. The smplest pulse s a one on a sngle qut, t uses an r.f. pulse on the carrer frequency at the same frequency as the Larmor frequency n the qut s rotatng frame (fg 3). In the rotatng frame the effectve external magnetc s many orders of magntude smaller ecause the moments are now only rotatng wth respect to the already rotatng frame, though they may not e rotatng at all n the rotatng frame f ther chemcal shft s zero. Fgure 3: Larmor Precesson n the rotatng frame of a pulse. 7

8 Therefore the domnant effect s the appled pulse whch causes the qut s magnetc moment to rotate aout the axs of the appled r.f. magnetc feld (Fgure 3 (aove)). So n order to get a nnety degree rotaton aout the x- axs, we must apply a pulse on the rotatng x- axs for a tme proportonal to the Larmor frequency n the rotatng frame aout the pulsed axs. It should e noted that the only pulses that can e appled are pulses aout an axs n the x- y plane. There exsts no z-axs pulse ecause there s already a large constant magnetc feld n the z drecton. In order for the pulse to e selectve the Fourer transform of the pulse must have neglgle value at the frequences of atoms you do not wsh to affect; ths loosely translates nto a constrant on the length of tme you may apply a pulse. Longer pulses are more hghly selectve and shorter pulse tme lengths are more ndscrmnate. One mportant thng to note s that whle a pulse s eng appled selectvely to one qut all the other quts are contnung to evolve as they normally would wthout the presence of the pulse, ths wll necesstate the use of the technque known as refocusng, whch we wll descre later on. Basc parameters for defnng a pulse are: start tme, pulse tme, central resonant frequency and r.f. strength. C. NMR for QC In order to perform quantum computaton usng an NMR system we need to defne a mappng from quts and operators to molecules and pulses. The mappng s mostly ovous. The separate quts n the NMR system are unque atoms n a gven molecule wth unquely addressale spns. If there are two atoms wth very smlar resonant frequency t wll e dffcult to use them n any way to store or transfer nformaton. They are stll part of the system and must e refocused to prevent ther nteractons from affectng the rest of the computaton state. They are more or less gnored except n refocusng when they are grouped together and refocused. We wll assume from now on that the only quts that are n the system are ndvdually addressale and wll e useful at some stage of the computaton, although the second assumpton s not really necessary. When you wsh to perform a QC on physcal NMR equpment, much thought must go nto the engneerng of the specfc molecule that you wll use. You must have suffcently dfferent Larmor frequences, whch means that the molecule wll lkely consst of dfferent sotopes or the small sotopes wth large chemcal shfts; ths s all necessary to e ale to relaly address nucle. You must have the approprate molecular structure to oth medate a strong J-couplng nteracton, for qucker nformaton exchange and a structure that ncreases relaxaton tmes, T and T, to allow more tme for computaton. Ths s only a qualtatve dscusson of the requrements of NMR propertes for roust QC; however, a real quanttatve dscusson s eyond the scope of ths paper, n the realm of chemcal engneerng. The quantum computaton operators are analogous to pulse sequences. Let us consder a smple molecule wth only one atom; ths negates the need for any refocusng ecause nothng else s evolvng. We can perform a U Z operaton y smple watng the tme requred for the magnetc moment to rotate y around the z-axs (or more lkely wat for m + ). We can apply U X and U Y y applyng r.f. pulses for the approprate tme aout the x- and y-axs respectvely. We can therefore apply any artrary rotaton of a qut on the Bloch sphere. We can generate the phase shft y performng a Z( /), watng half the tme as for the Z( ) [Nel]. The adamard operator can e expressed as the sequence sngle qut rotatons: Y( /4)X( )Y(- /4). To have the twoqut U CNOT, we ovously now need two quts, the U CNOT pulse s constructed usng several sngle qut rotatons and allowng the J- 8

9 couplng nteracton etween the quts to evolve (eqn 3). π 3π π π U CNOT > Z Z j X j J j π Y j 3 j Thus we now have a ass for of a unversal set of quantum untary operators avalale. Ths gves us unversalty of computaton n the realm of untary quantum operators, analogous to the unversal set n the classcal realm. The fnal ssue that remans s that measurement of fnal computatonal state. Measurements n NMR cannot e made of ndvdual atoms. Measurements made n NMR are ensemle measurements of the average value of some state varale, lke the projecton of the magnetc moment of the th qut onto the z-axs. Measurements are determned y the FID measurement as descred efore. In general, most of the average fnal state of the system can e determned, and we can use the FID to generate the fnal densty matrx descrng the system, though the densty matrx may not e unque. D. Refocusng ( ) ( ) matrces are oolean n-dmenson square matrces, where n s an even nteger or one. adamard matrces have the property that all pars of rows and all pars of columns dffer y n/ elements. These matrces are somorphc under swappng of rows and swappng of columns and the logcal negaton of rows and columns. It s always possle to make one row and one column of a adamard matrx all TRUE elements. Note, that n all ut the one row or column that has all TRUE elements, the numer of TRUE elements n each row or column must e exactly n/. There are several methods for generatng adamard matrces, most are farly complex, however, there s one smple constructon method for the some (eqn 6): n [ ],, 4, ( 5) n n. ( 6) n n Refocusng s the technque of stoppng the tme evoluton of certan nucle n a NMR system. Durng a pulse unless a gven nucle s eng addressed y the pulse that nucle wll contnue to evolve as t normally would. The evoluton of the entre system can e expressed, y the rotatonal frame, y the amltonan: rot n ω J z + n j + ω Jj J z J j x. ( 4) What we would deally lke to e ale to do s perform an artrary untary operaton on some suset of quts and have the rest reman unchanged. In order to perform refocusng we must determne a sequence of auxlary pulses to prevent the unwanted tme evoluton. The technque we used for generatng approprate refocusng pulse sequences s from Leung [Leun99]. Ths technque utlzes the propertes of adamard matrces. adamard adamard matrces are not guaranteed to exst for all even numers nor are all adamard matrces of a gven dmenson necessarly somorphc under smple swap and negaton operatons. In order to perform effcent refocusng we need a adamard matrx wth dmenson at least as great as the numer of nucle n our system. The dmenson of the matrx s the same as the numer of dscrete refocusng ntervals we must use. The refocusng method s as follows:. We transform the adamard matrx so that the frst row and the last column of the matrx contan only TRUE values.. Each nucle s assgned a row n the matrx. 9

10 3. Each column n the matrx represent a T/n step n the tme evoluton, where T s the total evoluton tme of spent refocusng. 4. Assume wthout loss of generalty, that all the elements outsde the matrx are TRUE. 5. A 8-degree rotaton (pulse) aout the y- axs s appled to the nucle at the egnnng of each tme step where the element n the matrx s FALSE and the prevous element was TRUE. 6. A -8-degree rotaton (pulse) aout the y- axs s appled to the nucle at the egnnng of each tme step where the element n the matrx s TRUE and the prevous element was FALSE. Because of (.) no refocusng pulses are appled to quts assgned the frst row of the matrx, ths means that Zeeman evoluton s allowed to contnue for ths nucle, through the J-couplng nteracton s refocused wth nucle assgned to other rows. Also from (.) there are no pulses appled at the end tme of the refocusng sequence, T, whch s a practcal concern snce pulses take a non-zero, fnte amount of tme to apply. If any quts share assgned rows the J- couplng nteracton etween them s not refocused. The reasons that ths scheme works are a t mathematcal and would take a t of tme to explan; we drect you to the orgnal paper for a more nsghtful and thorough explanaton [Leun99]. The ntuton, however, s to thnk that the value of the element descres the drecton n tme the nucle s evolvng. TRUE mples forward tme moton and FALSE ackward tme moton. Because adamard matrces have the property that all ut the frst row dffer y exactly n/ elements all the nucle assgned other rows than the frst evolve half the tme gong forward and half the tme gong ackward so the net evoluton s zero. Ths effectvely refocuses the Zeeman/Larmor evoluton. The J- couplng s refocused smlarly, as all pars of unque rows dffer y n/ elements meanng that half the tme the two quts are movng n the same drecton and half the tme they are movng n opposte drectons effectvely stoppng the tme-evoluton of the nteracton etween them. To make thng more concrete we wll descre how we appled ths scheme to construct our quantum gates. The Z ( ) gate pulse sequence wth refocusng s smple. Assgn the nucle we wsh to evolve the frst row of the matrx and all the other nucle to unque rows n the matrx. The total evoluton tme s gven: m + θ T ( ) z θ π, ( 7) ω where m s the numer of extra full perod z- rotatons. Ths depends on the length of the refocusng pulses and the numer of quts eng refocused. The constrant on T can e formed as T t mpl n, ( 8) where t mpl s the maxmum pulse length over all possly appled pulses on a sngle nucle for any of the refocusng ntervals. Gven these two constrants we can choose a mnmum T suject to oth those constrants. One thng to note s that f the chemcal shft s zero the evoluton tme s nfnte. It s a removale sngularty, however, snce the carrer frequency of the rotatonal frame s artrary, so we can change t to keep the refocusng tmes fnte and reasonale. The constructons for the X ( ) and Y ( ) gates are very smlar to the Z ( ) gate. There are two mportant changes, however. The total evoluton tme s now: m Tx ( θ ) Ty ( θ ) ( 9). ω The nucle are assgned rows the same way as for the Z ( ) gate, and the second constrant s stll n effect. At the egnnng of the pulsng sequence we apply the approprate pulse to generate the rotaton we want on the actve qut. The tme constrant serves to refocus the Zeeman nteracton for the actve nucle, ecause t was assgned the frst row. Strctly

11 speakng, ths s not necessary, ecause we can assgn the actve qut a dfferent row n the matrx, therey refocusng the Zeeman nteracton usng pulses nstead of lettng the evoluton repeat. Ths would remove the constrant nvolvng the chemcal shft entrely and let the evoluton tme e only dependent on the maxmum pulse length constrant. The J j ( ) gate s also very smlar. The th and j th nucle are assgned same row of the matrx, though not the TRUE-valued row, and all other nucle are assgned unque rows n the matrx, though also not n the TRUE-valued row. The tme constrant for ths gate s as follows: m + θ T ( ) J θ π. ( ) ω J j The standard constrant of maxmum pulse length (Eqn. 8) s also consdered n the choce of evoluton tme. Note t s necessary that Jj e non-zero for there to exst a fnte refocusng tme. Ths s not a removale sngularty lke the one for the Z ( ) gate, the nteracton must exst for the gate to e perform and the closer Jj s to zero t s the longer the t wll take to perform ths acton. III. Implementaton A. GAMMA capaltes GAMMA s an open-source lrary for conductng NMR Smulatons created y researchers at ET n Zurch, Swtzerland for the study of nuclear magnetc resonance technques [Smt94]. For our purposes, the GAMMA lrary provdes a means of constructng spn systems wth varous nteractons and propertes of sotopes. It provdes a means of constructng and applyng pulses to the system. At the end of a run GAMMA can produce an FID (and more mportantly ts Fourer transform) to read the output of the fnal system state or can smply just provde the fnal densty matrx. Constructng our molecule for use n GAMMA s smply a matter of tellng GAMMA the sotopes we would lke to use, ther chemcal shft values and the J-couplng nteracton strengths that exst etween the varous nucle. We can then choose to set the ntal state of the nucle to whatever ntal state we wsh y nputtng the ntal densty matrx to GAMMA. The capalty to automatcally generate ntal thermal equlrum states s provded y the lrary. GAMMA allows us several dfferent types of pulses to apply to a system of nuclear spns. The frst type of pulse s an deal pulse. The deal pulse s performed nstantaneously n smulaton tme and acts on a sngle sotope rotatng all nucle of a gven sotope y a set amount. The pulse s tme-strength functon s a delta functon. Ths s nether practcal nor realstc. Pulses must have some fnte tme duraton and some selectvty. The next type of pulse that GAMMA provdes s a hard pulse. The hard pulse pulses all nucle of a certan sotope aout a set angle. The hard pulse length s non-zero n tme, through t s usually qute short. The tme-strength functon for ths pulse type s a square functon. These sorts of pulses are approprate when there exsts only one nucleus n the molecule per sotope, f there exst more than one nucle n the channel, they wll oth e excted y the pulse. The fnal type of pulses GAMMA provdes are shaped pulses. Shaped pulses are smply pulses who s tmestrength functon s nputted y the user. These types of pulses allow the user to tune the selectvty of the pulse and excte only nucle n certan frequency regons of the channel y varyng the pulse strength over tme. The frequences that are excted y the shaped pulse are dependent on the Fourer transform of the tme-strength functon. One can tune the shaped pulse to selectvely address multple nucle on a gven channel wthout affectng other nucle on the same channel. GAMMA provdes some capalty for the constructon of the tmestrength functon and even has some ult-n

12 functons that are usually used n these types of pulses (.e. Gaussan functons). One thng that GAMMA does not provde s the alty to apply more than one pulse smultaneously n smulaton tme. Whle t represents some of a complcated ssue to even physcally apply two smultaneous pulses on a sngle channel; t s not a physcal restrcton that prevents applyng two smultaneous pulses on two dfferent channels. Applyng two pulses at the same tme on dfferent channels s a common occurrence n everyday NMR, GAMMA s lack of capacty for ths task s lkely due to the nherent complexty n makng an API that provde for that general purpose use. As a result we wll have to mplement ths capacty n QCNMR. GAMMA can return output n two ways. The frst s to smply return the densty matrx of the fnal state. The second way s to smulate the free nducton decay of the system and return an array that stores the current nduced n the r.f. cols as a functon of tme. It also provdes that capalty to take the Fourer transform of the FID. owever, GAMMA stops there; there s lttle ult n to the API that allows for analyss of the FID. There s nothng that attempts to estmate the fnal state y analyss the FID, there s nothng ult n to even fnd peaks on the FID s Fourer transform. Wth the lack of ult n functonal for analyzng FIDs, producng FIDs s only for qualtatve purposes. GAMMA has several lmtatons. The frst was already mentoned, the lack of smultaneous pulsng. The second lmtaton s the lack of FID analyss. The thrd lmtaton s that ncludng the T and T relaxaton effects n a way that would have een meanngful for QCNMR would have een very dffcult. The mplementaton of that part of the lrary was not fully featured or complete, though the authors left notes n some of the source fles for hntng at possle ways to expand the mplementaton. owever, despte ths ts lmtatons, t s very smple to construct and run pulses for smple NMR smulatons usng GAMMA. GAMMA made the work of ths paper possle, even though t was not talored exactly to our needs. B. Program Structure The core of QCNMR s ts C++ lrary kernel. The kernel provdes all the capactes for constructng quts, crcut gate, executng the crcut and marshallng the output. The lrary s meant to work along sde GAMMA provdng extra functonalty for quantum computaton tasks. Some of the datatypes that are returned from QCNMR are quantum computatonally useful mathematcal types lke matrces and operators, whch nherently provde for more power computaton n QCNMR user s programs. In addton, there s a graphcal user nterface that allows the quck constructon and testng of crcuts, wthout the need to comple a C++ program; ts computatonal power s lmted due to the complexty of makng an nterface that can fully exercse the lrary under an artrary C++ program. owever, the graphcal nterface s more meant as an educatonal tool than a power computatonal apparatus. The Kernel The kernel conssts of three man classes:. QCNMRQut. QCNMROperator 3. QCNMRProgram The respectve purpose of each of these classes s almost ovous, ut we wll explan t anyway. The nstantaton of the QCNMRQut class s the equvalent of a sngle qut. It contans all nformaton aout the qut ncludng ts sotope, chemcal shft, J-couplng nteracton strengths, ponters to all the operators that wll operate on t, ponters to all the other quts of the same sotope and nformaton aout what types of pulses can e appled to t.

13 . Fgure 4: QCNMR Desgn Layout There really s not much work eng done wthn ths class durng the course of the computaton, t s really just a glorfed struct. The nstantaton of the QCNMROperator class s the equvalent of a sngle qut gate. It stores nformaton aout whch QCNMRQuts are eng operated on, at what tme the gate s appled and the types of pulses need to occur at the top level (gnorng refocusng) to accomplsh the requested operaton. An auxlary class QCNMROperatorGen s used to actually generate nstances of QCNMROperators; ths s ecause some gloal knowledge must e stored to determne whether or not that creaton of an operator on certan quts at a certan tme s a vald thng to do. The fnal kernel class s QCNMRProgram. It s the workhorse of the mplementaton. It takes n an array of QCNMRQuts and then allows the addton of crcut elements n the form of QCNMROperators. Most of the work happens n the QCNMRProgram::Run() functon, whch actually nterfaces wth GAMMA to transform our QCNMRQuts nto ther GAMMA representaton. Then QCNMRProgram converts the nputted QCNMROperators n pulse sequences that t then runs on GAMMA, the QCNMROperators are destroyed n ths process, though the pulse sequence that they were converted nto to can e returned to the user. The dea s that user s e allowed to perform some artrary crcut, look at the output and then proceed from the fnal state y addng new elements and runnng the new crcut. QCNMRProgram has the power to return pulse sequences to user or to save them to dsk as n a human readale format. QCNMRProgram can also output a text-ased dagram of the crcut that t sends to the dsk. The man computaton n QCNMRProgram s dedcated to generatng and runnng refocusng pulses. The frst part of ths was mplementng that refocusng pulse sequence generaton as was fully descred n secton two. The second key concern was allowng GAMMA to run smultaneous pulses. To understand what was done we must frst gve a ref explanaton of how pulse sequences are mplemented n GAMMA. A pulse on a sngle channel, so, on a specfc resonant chemcal shft frequency,, s mplemented n the followng fashon (wthout loss of generalty the pulse s for radan aout the x-axs):. Shftng all nucle of that sotope from the orgnal rotatng frame nto the rotatng frame of the resonant chemcal shft frequency, we get a new amltonan: rot rot ω so J z. ( ) 3

14 . Next we add the r.f magnetc feld n the drecton of our axs (the x-axs). The strength of the feld s determned y the pulse tme length and the angle we wsh to rotaton through: eff rot + θ π t so J x. ( ) 3. We then leave the system to evolve under eff for tme t. 4. Snce we shfted to another rotatng frame we need to shft ack to the orgnal frame, ths can e done y constructng a amltonan: ω shft J z so 5. We evolve the system agan for tme t under shft, though ths tme s not real smulaton tme, t s just some mathematcal ookkeepng to put us ack n the orgnal rotatng frame. 6. The pulsng s now complete and we return to usng our orgnal rot.. ( 3) Ths method performs a hard rectangular pulse or, f the tme s long enough a soft rectangular pulse centered on the specfed frequency. In order to perform more complex pulses ths method s used as an atomc operaton. To perform shaped pulse we apply a seres of rectangular pulses wth varyng strengths. The shaped pulse s appled through the dscretzaton of ts tme-strength functon. So n order to allow pulses to fre smultaneously n a general sort of way you have to keep track of all the changes that you have made and all the changes you need to make to the effectve amltonan. The dscrete tme step method for performng shaped pulses adds even more complexty to the prolem. owever, when you get down to, t s not a techncally dffcult prolem, t s just tedous. We wll not go nto the mplementaton detals for schedulng and orderng changes to the amltonan and evolvng the state approprately as t s just a programmng task and does not add anythng nterestng to ths dscusson. QCNMRProgram s man output s the fnal densty matrx of the system. Ths s clearly not a realstc feature, an NMR machne wll not smply gve the fnal densty matrx of the system; you must compute t from the FID. There are ascally two reasons we choose not to provde the output va FID. The reason one reads an FID and converts t to a densty matrx was smply ecause the densty matrx tself s unavalale, so snce GAMMA freely gves us the densty matrx t makes lttle sense to transform the densty matrx to an FID for the purposes of transformng t ack to an FID. The second reason s that as we mentoned efore GAMMA does not have much ult n capacty for analyzng FIDs, so t would have requred sgnfcant extra work to add that on to support an admttedly duous process. We dd add the alty to take the partal trace of a densty matrx aganst some parts of the space to make t easer to examne the states of suspaces of the system.. The GUI The graphcal front-end of QCNMR s almost an afterthought. It has a smply pont-clck nterface that allows the user to quckly construct quantum crcuts and vew the fnal output of the system as a graphcally expressed densty matrx or as a dagram of pulses. At the tme of wrtng ths paper, the frst verson of the GUI s fnshed, though a few mnor ugs stll reman. It makes t very easy to just put down a few gates and experment wth some small crcuts. The output capaltes are lmt to dagrams and graphs, ecause large amounts of textual data are confusng and often not useful. In the short tme we have had to experment wth t, t seems as though t wll e useful educatonally, even f a person knows lttle 4

15 aout the nuclear magnetc resonance effects t s ased on. It s an nterestng thng to play wth f nothng else. Fgure 5: Graphcal User Interface Screenshot IV. Results - Quantum Teleportaton One nterestng quantum algorthm that we had a chance to mplement usng QCNMR, was the canoncal quantum teleportaton crcut. The crcut s shown elow n Fgure 6. All three quts start out n the state >. Operatons may e performed on the Q, state to produce a dfferent nput to the system (step one on the dagram). Next qut two and three are ntalzed nto an entangled Bell state (step two on the dagram): B oo ( + ) ( 4) In the orgnal formulaton of the prolem the qut two was gven to Alce and qut three was gven to Bo after the quts have een entangled. Alce starts y ntalzng qut one nto some state. Then Alce can make some local measurements on qut one and qut two and send the results classcal to Bo, who may e some dstance away. Bo can then use the classcal nformaton that Alce sent hm to decde whether or not to apply certan operatons to hs qut three. The end result s that the fnal state of Bo s qut three s the same quantum state as ntal state that Alce prepared. Because we have not mplemented classcal channels wth QCNMR, we have to use an equvalent crcut wth only quantum operatons. We mplemented ths crcut on a heteronuclear three-qut system. The spn parameters where more or less artrary, the chemcal shft for each nucle was z, and the J-couplng strength etween each par was set to z. Snce we have a heteronuclear system all pulses are hard pulses on a channel. The nomnal pulse length was one mcrosecond. The pulse sequence for the sectons two and three of the teleportaton crcut are shown elow n Fgure 7. As you can see the total executon tme for the crcut s on the order of.3 seconds. Notce the four symmetrc open regons n the pulse dagram. These are the regons where the J-couplng nteracton was eng allowed to evolve. Snce the J-couplng term s an order of magntude smaller than the chemcal shft, the tme requred to perform the J-couplng over some angle ncreases y an order of magntude. Smlarly, the tghtly packed regons of the dagram correspond to the sngle-qut evoluton that occur as part of the CNOT gates and the ntal adamard gate. The numercal results themselves are not too nterestng. The orgnal densty matrx after preparaton of the qut one traced down to qut one s almost dentcal to the fnal densty matrx traced down to qut three. The nterestng thng s that when we nterestng the length of the hard pulses, the fdelty of the fnal teleported state to the ntal prepared state decreases dramatcally. owever, ths s to e expected, one of the requrements of the refocusng processes was that the pulse lengths must e kept short [Leun99]. The tme that the pulses are eng appled s larger relatve to the total duraton of the refocusng, lkely due to the fact that not all quts are eng pulsed for the same amount of tme durng the refocusng sequence; we eleve 5

16 . Create Entangled Bell State. Intalze adamard J-couplng 3. Local Measurements Fgure 6 (top): Quantum Teleportaton Crcut Fgure 7 (ottom): Quantum Teleportaton Pulse Sequence that ths causes the effectveness of the J- couplng refocusng etween quts to reak down. The Zeeman refocusng should e unaffected. Ths suggests that smaller J- couplng strengths would mprove ths prolem, however, that would ncrease the total refocusng tme requred. In the end t s a alancng effort etween pulse length and refocusng tme. We dd not have a chance to examne as many quantum crcuts as we would lke to have. The man stumlng lock other than tme, was that most of the nterestng crcuts nvolve constructon of some set of U f s that act as oracles n the Deutsch-Jozsa prolem or the Grover Search prolem [Erma3]. It was not ovous to us how to construct these untary functons usng the gates we had avalale. owever, durng the early stages of development of QCNMR, we dd test out the untares provded for the three-qut Deutsch- Jozsa algorthm [Km]. These untary functons where not constructed usng the standard gate set, they we constructed usng NMR rotatons and nteractons. It s n the realm of QCNMR to perform such operatons as we descred n prevous sectons; that functonalty s prvately nternal to the kernel and for desgn choces was not exposed to the user. It would e smple to allow the constructon gates that allowed for artrary X, Y and Z rotatons, ut they were hdden n order to make the user s nterface cleaner. In future teratons of the QCNMR, ths functonalty wll lkely e exposed. Fnally, a note aout smulated and expermental accuracy; QCNMR s not ntended 6

17 to exactly smulate a nuclear magnetc resonance computer n complete detal. There are a numer of practcal assumptons that had to e made n order to make ths project tractale. The most mportant assumpton s that we consder the NMR system as a statstcal ensemle as opposed to a physcal n-ody system, whch s, tself, s a horrly ntractale prolem. Along wth that assumpton was the assumpton that the amltonan for our system was tme-ndependent, meanng the ulk relaxaton effects were not eng consdered, though they could e ncluded to zero-th order as a constrant on total computaton tme. Another assumpton that we consdered was that the only quts n the system were nucle that we nvolved n the computaton, ths s not strctly necessary, the user can add whatever extra quts they wsh to add, the only cost s that refocusng s performed unquely on each one of them. The fnal, and perhaps somewhat understated assumpton, s that we assumed that refocusng was necessary for atomc operaton, strctly speakng, refocusng s proaly not always necessary, n fact, t s proaly wasteful sometmes. The prolem s that t s dffcult to quantfy the effectveness of usng refocusng at some pont n the evoluton of the crcut. Whle always refocusng s good from a correctness pont of vew, t can often e duous from an effcency perspectve. Agan, t s a case where alance and optmzaton of parameters comes nto play. We decde to err on the sde of correctness as opposed to the sde of effcency, for t wll lkely e easy to approach an optmal soluton from a soluton that s already correct. V. Concluson A. Summary We presented our constructon of a nuclear magnetc resonance quantum computer smulator, QCNMR. We dscussed the necessary ackground for a layman to get a good physcal understandng of NMR and quantum computng at a asc level. Our algorthms for constructng refocusng sequences and for performng smultaneous pulses were also dscussed. We examned the structure and nterface to the open-source NMR smulaton lrary that we used as a ackend for QCNMR s nuclear magnetc resonance smulaton as well as ts techncal short-comngs and lmtatons. The asc structure and classes n our mplementaton were refly descred n ther form and functon. We have concluded wth a short example crcut that we mplemented on QCNMR. For the most part ths project constructed what t set out to do; to mplement an emulator that takes a classcal quantum crcut and performs t y usng NMR pulses. Fnally we commented on a numer of assumptons that were made n the formulaton and mplementaton of QCNMR. The next major step from QCNMR s to construct a tmecorrectness optmzaton solver for applyng pulses. B. Related Work There are relatvely few examples of smlar attempts to mplement a nuclear magnetc resonance quantum computer smulator. One such example s the Quantum Computer Emulator (QCE) y a group at the Unversty of Gronngen, Demark [Mch3]. Ther approach s at a much lower level, y allowng the user to choose all the pulses and the parameters of the pulses. They make ths slghtly more tractale to a novce y constructng nstructon sets of asc useful pulses wth pre-set parameters. Ther mplementaton, however, seems to focus on eng an deal quantum computer smulator as opposed to a NMR ased smulator. They to not seem to provde the automatc constructon of the necessary refocusng pulse for the user; t s up to the user to mplement them however they choose. Ths method s allows the user more control over how ther astract quantum crcut s mplemented n NMR hardware, though at the cost of consderaly more tme to 7

18 mplement wth the QCE system. QCNMR provdes a quck way for user to test and examne the asc propertes of crcuts that wll e on smulated NMR system at the loss of user control of the pulse sequencng. C. Future Work We would lke to have spent more tme testng some example crcuts to examne the effectveness of QCNMR, ut acklogs n the development schedule hndered the process. We dd not spend much tme tryng to optmze the pulse sequences eyond applyng multple pulses at the same tme. Future work could address optmzng pulse sequences oth on a per-operaton ass and on a gloal scale usng some lnear programmng technques. There was not enough tme to compare the results of the smulated computaton and the generated pulse sequences to the physcal mplementatons and actual pulse used sequences. There s one thng that remans to e mplemented, however, at the tme of pulshng ths paper, smultaneous homonuclear soft pulses are not yet mplemented due to complexty ssues and tme constrants. We also look to further teratons on the development of the GUI, n order to add make t a more powerful tool that allows t to exercse more features of the QCNMR lrary. The ssue of swappng quts (and optmzng the swappng of quts) for performng nformaton exchange on quts that have zero or very small J-couplng nteracton strength was not even dscussed. There are a numer of smple optmzatons that come to our mnds as ths paper was eng wrtten; mostly optmzatons n the maxmum tme duraton of a refocusng sequences and the organzaton of pulses wthn each refocusng pulse nterval. D. Acknowledgements Frstly, thanks to the research group at the Unversty of Florda and n Swtzerland that produced GAMMA and made t openly avalale to the pulc, for wthout whch ths project would have een mpossle. Secondly, thanks to Dr. Arvnd, for some useful ponters and dscussons aout usng NMR for quantum computaton. Thrdly, we would lke to thank Dr. Avrm Blum for sgnng as co-advsor to ths project and supportng t even though t s not hs area of expertse. Fourthly, we would lke to thank Mark Stehlk for organzng the CS Senor Thess program, as well as for some good advce concernng ths thess and other aspects of CMU CS undergraduate lfe. Fnally, and most mportantly, we would lke to Dr. Roert Grffths for eng the man advsor of ths research; there were defntely some rough spots durng the past two semesters, ut t seems we made t through no worse for wear havng completed a frustratng, though ultmately worthwhle, project. Thanks everyody. VI. Blography Cra95 Crac I J and Zoller P Quantum computatons wth cold trapped ons, Phys. Rev. Let., 74: , 995. Cory96 Cory, D. G., Fahmy, A. F. and avel, T. F., Nuclear magnetc resonance spectroscopy: an expermentally accessle paradgm for quantum computng. In Proceedngs of the 4th Workshop on Physcs and Computaton, Boston: New England Complex Systems Insttute, (996). DV95 DVncenzo, D. P., Quantum Computaton. Scence, pages 7:55,