Towards a Quantum Computing Algorithm for Helicity Amplitudes and Parton Showers
Khadeejah Bepari, Sarah Malik, Michael Spannowsky, Simon Williams
IIPPP/20/41
Towards a Quantum Computing Algorithm for HelicityAmplitudes and Parton Showers
Khadeejah Bepari, a Sarah Malik, b Michael Spannowsky a and Simon Williams b a Institute for Particle Physics Phenomenology, Department of Physics, Durham University, DurhamDH1 3LE, U.K. b High Energy Physics Group, Blackett Laboratory, Imperial College, Prince Consort Road, London,SW7 2AZ, United Kingdom
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
The interpretation of measurements of high-energy particle collisions reliesheavily on the performance of full event generators. By far the largest amount of time topredict the kinematics of multi-particle final states is dedicated to the calculation of thehard process and the subsequent parton shower step. With the continuous improvementof quantum devices, dedicated algorithms are needed to exploit the potential quantumcomputers can provide. We propose general and extendable algorithms for quantum gatecomputers to facilitate calculations of helicity amplitudes and the parton shower process.The helicity amplitude calculation exploits the equivalence between spinors and qubits andthe unique features of a quantum computer to compute the helicities of each particle in-volved simultaneously, thus fully utilising the quantum nature of the computation. Thisadvantage over classical computers is further exploited by the simultaneous computationof s and t-channel amplitudes for a 2 → a r X i v : . [ h e p - ph ] O c t ontents → → → n amplitude calculations 11 D.1 1 → E Detailed quantum circuit for collinear parton shower algorithm 25
E.1 Count gate 25E.2 Emission gate 26E.3 History gate 27E.4 Update gate 28
Modern collider experiments such as the Large Hadron Collider (LHC) at CERN dependheavily on the modelling of particle collisions and simulations of detector response to exam-ine physics processes within the experiments. This modelling is used to construct differentpossible outcomes from particle collisions, used both for the identification of certain physicalprocesses, and for the construction of event backgrounds. Consequently, such simulationsplay a crucial role in modern high energy physics, and are usually carried out by MonteCarlo event generators such as
Pythia [1],
Herwig [2] and
Sherpa [3].– 1 –he theoretical description of LHC events can be highly complex. In a typical event,hundreds of particles are produced as a result of the evolution of an event from the collisionof two protons to the formation of long-lived hadrons, leptons and photons. The collisionprocess can be separated into several stages. The protons consist of many partons, eachcarrying a fraction of the total proton energy. When protons collide, two of their partonscan interact with each other via a large momentum transfer, thereby giving rise to theso-called hard interaction. In this part of the collision, large interaction scales are probed,possibly accessing new physics. However, if color-charged particles are produced during thehard interaction process, they are likely to emit further partons. This results in a partonshower, providing a mechanism that evolves the process from the hard interaction scaledown to the hadronisation scale O (Λ QCD ), where non-perturbative processes rearrange thepartons into colour-neutral hadrons.The hard interaction and the parton shower are the two parts of the event evolutionthat can be described perturbatively and largely independently of non-perturbative pro-cesses, as a result of the factorisation theorem [4]. In addition, they are by far the mosttime-consuming parts of the event simulation and pose, therefore, the bottleneck in thegeneration of pseudo-data for ongoing measurements at the LHC.While a speed improvement in calculating the hard process and the parton shower iscrucial for the interpretation of high-energy collision experiments, the conceptual methodsused to calculate either of these two parts are distinctly different. For a mathematicaldescription of the hard interaction, scattering matrix elements are calculated, which nowa-days rely on helicity amplitude methods to cope with the ever-increasing complexity of thepartonic scattering process [5, 6]. Instead, the parton shower is technically implementedthrough a Markov chain algorithm ordered in some measure of showering time t , wheresplitting functions define the probability for a parton to branch into two partons and Su-dakov factors [7] determine the probability for the system not to change between two showertimes ∗ t in and t end . Recent developments in combining helicity amplitudes with the partonshower have shown to improve the theoretical description of scattering events includingmultiple jets [9–14], in hypothesis testing [15, 16] and in particular in the construction ofspin-dependent parton showers [17].With practical quantum computers becoming available, there has been growing inter-est in harnessing the power and advantages that these machines may provide. This interestextends to applying the abilities of quantum computers to describe processes in field the-ories, with the hope of exploiting the intrinsic ‘quantumness’ of these novel machines tocalculate quantum phenomena efficiently. Current quantum computers are divided intotwo classes: quantum annealers and universal gate quantum computers (GQC). The for-mer is based on the adiabatic theorem of quantum mechanics to find the ground state ofa complex system. Quantum annealers perform continuous-time quantum computationsand are therefore well-suited to study the dynamics of quantum systems, even quantumfield theories [18, 19], and in solving optimisation problems, e.g. applied to Higgs phe-nomenology [20]. However, they are not universal. Despite their severe limitation due ∗ For more details see [8] and references therein. – 2 –o the relatively small number of qubits of current machines, GQC are a popular choicefor the implementation of algorithms to calculate multi-particle processes [21–31], oftenwith field theories mapped onto a discrete quantum walk [32–35] or a combined hybridclassical/quantum approach [36–39].Here, we aim to provide a first step towards a generic implementation of quantumalgorithms, applicable to QGC devices, for the most time-consuming parts of the eventgeneration in high-energy collisions, i.e. the calculation of the hard process in terms ofhelicity amplitudes and the simulation of the parton shower † .As depicted in Fig. 1, QC calculations proceed in general in three stages: i) encoding ofthe initial state, i.e. an initial wavefunction, using a specific representation of the problem,ii) applying unitary operations on this state, which on a GQC is realised through circuits,and iii) measuring a specific property of interest, i.e. a projection onto the final statevector. . . . . ψ . . . . . .. . MM . . . U state U operation Figure 1 : Schematic setup of generic quantum computing calculations with the followingsteps: (i) encoding of the initial state, (ii) the application of (unitary) operations and (iii)the measurement of the transformed state.Following this structure, we will elucidate how the calculation of a hard process in termsof helicity amplitudes or the parton shower can be performed using a GQC. Specifically, weuse the IBM Q Experience [40], which provides access to a range of public access quantumcomputers and a 32-qubit Quantum Simulator [41]. We have designed the circuits with afocus on limiting the number of qubits needed to perform the calculations. While our codecan be run on a real quantum device, the current quantum machines cannot outperformclassical computers. The quantum circuits presented here, therefore, serve as a templateand nucleus for future developments.This paper is organised as follows: in Sec. 2, we motivate and detail the implementationof our QC algorithm for helicity amplitudes for a 1 → → † A first implementation of a parton shower algorithm was provided in [26], where interference effects inthe parton shower evolution were studied. – 3 –
Helicity amplitude algorithm
Scattering processes are calculated using conventional techniques by squaring the scat-tering amplitude and then performing a sum of all possible helicity processes using tracetechniques. For a process with N possible Feynman diagrams, this results in N termsin the squared amplitude. Therefore, for processes with a large number of Feynman di-agrams, such calculations become extremely complicated. In contrast, helicity amplitudecalculations provide a more efficient way of calculating such processes, as one calculates theamplitude for a specific helicity setup. The different helicity combinations do not interfere,and therefore the full amplitude can be obtained by summing the squares of all possiblehelicity amplitudes.Helicity amplitude calculations are based on the manipulation of helicity spinors. Asthe Lorentz group Lie algebra can be written as the direct sum of two SU (2) sub-algebras,i.e. so (3 ,
1) = su (2) ⊕ su (2), there are two specific complex representations each specifiedby two degrees of freedom which solve the massless Weyl equation: a right-handed Weylspinor, associated with the representation ( , , ). Consequently and for concreteness, the helicity spinor | p (cid:105) ˙ a for a massless state can be chosen to be expressed as | p (cid:105) ˙ a = √ E (cid:32) cos θ sin θ e iφ (cid:33) , (2.1)associated with momentum p µ and energy E , such that p µ p µ = − m using the η µν =diag(-1,+1, +1, +1) metric convention. This spinor is parametrised by the angles θ and φ , wherethe other spinors (cid:104) p | ˙ a , | p ] a and [ p | a are related by p a ˙ b = −| p ] a (cid:104) p | ˙ b and p ˙ ab = −| p (cid:105) ˙ a [ p | b . Thecorrespondence between the two-dimensional helicity spinors and four-component Diracspinors associated with Feynman rules is demonstrated in Appendix B.To facilitate and implement such calculations on a GQC, we use qubits , the quantumanalogue of the bit for classical computation. The state of the qubit is defined on a two-dimensional complex vector space with states | (cid:105) and | (cid:105) forming the orthonormal basisfor this space. A qubit can thus be formed by a linear superposition of these orthonormalbasis states. By considering a general qubit parametrized by two angles | ψ (cid:105) = cos θ | (cid:105) + e iϕ sin θ | (cid:105) = (cid:32) cos θ sin θ e iφ (cid:33) , (2.2)we can represent the qubit on a three-dimensional unit sphere called the Bloch sphere.Performing unitary operations on qubit states corresponds to rotating states in the Blochsphere.Remarkably, comparing Eqs. (2.1) and (2.2), helicity spinors can be represented througha qubit, modulo an overall normalisation factor √ E , and the calculation of helicity am-plitudes follows the identical structure shown in Fig. 1, i.e. quantum operators act on aninitial state to eventually perform the projection onto a final state. In contrast to classical– 4 –omputers, where all numerical quantities are converted into a binary system representa-tion, on which an algorithm is applied, and then transformed back into quantities thatcan be understood in terms of a numerical result, in a quantum computing algorithm, thehelicity spinor is a faithful representation of the object the circuit directly operates on.The spinors can be directly represented as vectors on the Bloch sphere, which provides themost efficient encoding of the state on which the algorithm operates. This indicates thatGQC provide an ideal framework for the calculation of helicity amplitudes.Consequently, we will exploit that the spinors used to calculate helicity amplitudesnaturally live in the same representation space as qubits. This motivates the manipulationof the direct correspondence of the θ and φ variables of the qubit states and helicityspinors to represent the spinors on a quantum circuit. We further encode operators actingon spinors as quantum circuits of unitary operations. These can be applied to qubits(rotating vectors on the Bloch sphere) to calculate helicity amplitudes. The helicity spinors | p (cid:105) ˙ a ,( (cid:104) p | ˙ a ) T , | p ] a and ([ p | a ) T are visualised for θ = π/ φ = π/ E = 1 /
2, as vectors onthe Bloch sphere in Fig. 2, in direct analogy to their respective qubit representation.This study aims to create the basic building blocks to encode spinor helicity calculationson a quantum circuit. These basic building blocks are then used to construct quantumalgorithms for two simple examples of helicity calculations: i) the contraction of an externalpolarisation vector corresponding to a g → q ¯ q vertex and ii) the construction of s and t-channel amplitudes for a q ¯ q → q ¯ q process with identical initial and final quark flavours.‘Helicity registers’ are crucially introduced into these circuits to control the helicity of eachparticle involved. In addition, we introduce a superposition state between the helicityqubits of | + (cid:105) = | (cid:105) and |−(cid:105) = | (cid:105) by applying Hadamard gates to the helicity registers.In doing so, we can calculate both helicities of each particle involved simultaneously, thusfully utilising the quantum nature of the computation. This advantage is further exploitedby the simultaneous computation of s and t-channel amplitudes for the q ¯ q → q ¯ q process.This section is organised as follows: a description of the quantum circuit for the 1 → g → q ¯ q is given in Sec. 2.2, together with a comparison of the results of thealgorithm as run on a real machine and a simulator, the quantum circuit and the results forthe 2 → q ¯ q → q ¯ q are given in Sec. 2.3, and a brief discussion of the generalisationof the algorithm to 2 → n processes follows in Sec. 2.4. (a) | p (cid:105) ˙ a (b) | p ] a (c) ( (cid:104) p | ˙ a ) T (d) ([ p | a ) T Figure 2 : A visualisation of the helicity spinors | p (cid:105) ˙ a , (cid:104) p | ˙ a , ( | p ] a ) T and ([ p | a ) T for θ = π/ φ = π/ E = 1 / .1 Constructing helicity spinors and scalar products on the Bloch sphere The helicity spinors have been implemented on the quantum circuit by constructing Blochsphere representations, like the ones shown in Fig. 2. The helicity spinor decompositionsare outlined in detail in Appendix C. They utilise the Qiskit U ( θ, φ, λ ) gate, which appliesa rotation to a single qubit. The rotation is defined by, U ( θ, φ, λ ) = (cid:32) cos (cid:0) θ (cid:1) − e iλ sin (cid:0) θ (cid:1) e iφ sin (cid:0) θ (cid:1) e i ( φ + λ ) cos (cid:0) θ (cid:1)(cid:33) . (2.3)A simple U gate acting on a | (cid:105) state has been used to create the | q (cid:105) ˙ a spinor, where θ and φ variables of the U gate corresponded to the θ and φ variables of the helicity spinor.The | q ] a spinor has been created by sequentially applying a U † rotation and a N OT gate,where here the θ and λ variables of the U gate corresponded to the θ and φ variables ofthe | q ] a spinor.To construct the scalar products (cid:104) pq (cid:105) or [ pq ] on a quantum computer, 2 × U (cid:104) p and U [ p were created such that, when they act on the | q (cid:105) ˙ a and | q ] a spinorsrespectively, the scalar product values correspond to the first component of the final qubitstate, i.e. the complex coefficient associated with the | (cid:105) state. It should be noted thatthe factors of √ E in the definition of the helicity spinors have not been accounted forsuch that the spinor-qubit states are normalized to one on the quantum register. As aconsequence, these factors must be added after the results have been obtained from thequantum computer. → A simple application of the helicity amplitude approach is the calculation of a 1 → q → gq by calculating the gqq vertex, M gqq = (cid:104) p f | ¯ σ µ | p f ] (cid:15) µ ± , (2.4)where p f and p f are the momenta associated with the fermon and anti-fermion respectively.The gluon polarisation vectors are defined as [42], (cid:15) µ + = − (cid:104) q | ¯ σ µ | p ] √ (cid:104) qp (cid:105) , (cid:15) µ − = − (cid:104) p | ¯ σ µ | q ] √ qp ] . (2.5)From this, it is possible to create a circuit where each four-vector present in the amplitude,i.e. the fermion anti-fermion vertex and polarisation vector, is calculated individually ona series of 4 qubits. This is done by using the corresponding Pauli gates for each four-vector component on each qubit. However, this will lead to a large circuit depth due tothe number of gates required to do such a calculation. Therefore it is useful to simplify theexpression for the amplitude using the Fierz identity, (cid:104) p | ¯ σ µ | q ] (cid:104) k | ¯ σ µ | l ] = 2 (cid:104) pk (cid:105) [ ql ] . (2.6)– 6 –ith this, the amplitude for the gqq vertex becomes M + = −√ (cid:104) p f q (cid:105) [ p f p ] (cid:104) qp (cid:105) , M − = −√ (cid:104) p f p (cid:105) [ p f q ][ qp ] . (2.7)As a consequence of this simplification, the number of qubits needed to calculate the am-plitude on the quantum computer can be reduced from 10 to 4. The circuit for calculatingthis amplitude is shown in Fig. 3. The three q i qubits calculate the three scalar productsfrom Eq. (2.7) using the gate decompositions outlined in Appendix C. These rotation gatesare controlled from the helicity register, h . If h is in the | (cid:105) state, then the helicity ispositive and the M + amplitude is calculated; if h is in the | (cid:105) state, then the helicity isnegative and the M − amplitude is calculated. The three calculation qubits, q i , are thenmeasured by the quantum machine. q h p f q i h p f p i q [ p f p ] [ p f q ] q h qp i [ qp ] h H Figure 3 : gqq vertex circuit. The amplitude for the process is calculated on the q i qubits,which are controlled from the helicity register. The q i qubits are then measured by thequantum computer.negative and the M amplitude is calculated. The three calculation qubits, q i , are thenmeasured by the quantum machine.Figure 4 shows the results of the algorithm for a random selection of small scatteringangles, with runs on the IBM Q 32-qubit Quantum Simulator [42] and the IBM Q 5-qubit Santiago Quantum Computer [44]; both of which have been compared to theoreticalpredictions of the probability distributions extrapolated directly from analytic calculationsof the helicity amplitude, calculated using the S@M software [45]. The simulator has beenrun without a noise profile for 10,000 shots, and has been shown to agree within 1 ofthe theoretically predicted values. From these distributions, one can determine the helicitysetup of the process and consequently reconstruct the helicity amplitudes of the process.The Santiago machine has been run on the maximum shot setting of 8192 for 100runs, leading to a total of 819,200 shots of the algorithm. From Fig. 4, it is clear that thequantum computer’s performance does not match that of a perfect machine. Although thehelicity of the process which has been calculated can be identified from the distinct prob-ability distributions, one cannot determine the explicit amplitude from the real machine.However, it should be noted that a comparison to a perfect machine may not be a faircomparison for modern quantum computers. Therefore a comparison between a simulatorrun with the Santiago device’s noise profile and the quantum computer results is shownin Appendix D. Section 2.4 explores the future of quantum computers for precise helicityamplitude calculations.The results from the quantum computer, shown in Fig. 4, have been achieved byisolating the individual helicity processes on the quantum circuit, and removing the su-perposition between the positive and negative processes. The full amplitude is achievedthrough the implementation of a Hadamard Gate on the helicity qubit, which puts thesystem into a superposition state of the positive and negative processes. The qubit setupchosen here has been used in order to best reduce the CNOT qubit errors and limits thenumber of
SWAP operations needed in the algorithm. The Santiago machine is a 5-qubitquantum computer, with all qubits connected inline to their adjacent qubit. The helicity– 7 –
Figure 3 : gqq vertex circuit. The amplitude for the process is calculated on the q i qubits,which are controlled from the helicity register. The q i qubits are then measured by thequantum computer.Figure 4 shows the results of the algorithm for a random selection of small scatteringangles, with runs on the IBM Q 32-qubit Quantum Simulator [41] and the IBM Q 5-qubit Santiago Quantum Computer [43]; both of which have been compared to theoreticalpredictions of the probability distributions extrapolated directly from analytic calculationsof the helicity amplitude, calculated using the S@M software [44]. The simulator has beenrun without a noise profile for 10,000 shots. The results agree well with theoreticallypredicted values, to within 1 σ . From these distributions, one can determine the helicitysetup of the process and consequently reconstruct the helicity amplitudes.The Santiago machine has been run on the maximum shot setting of 8192 for 100 runs,leading to a total of 819,200 shots of the algorithm. Figure 4 shows that the quantumcomputer’s performance does not match that of a perfect machine, as expected. Therefore,the simulator is rerun with the noise profile of the Santiago device and a comparisonbetween this and the quantum computer is shown and discussed in Appendix D.The results from the quantum computer, shown in Fig. 4, have been achieved by isolat-ing the individual helicity processes on the quantum circuit, and removing the superposition– 7 –etween the positive and negative processes. The full amplitude is achieved through theimplementation of a Hadamard gate on the helicity qubit, which puts the system into a su-perposition state of the positive and negative processes. The helicity of the process is thendetermined by measuring the helicity register. The qubit setup chosen here has been usedin order to best reduce the CNOT qubit errors and limits the number of
SWAP operationsneeded in the algorithm. The Santiago machine is a 5-qubit quantum computer, with allqubits connected inline to their adjacent qubit. The helicity qubit, h , from Fig. 3 has beenassigned to qubit 4 on the Santiago machine, with the q i qubits on the 2nd, 3rd and 5thqubits of the Santiago machine. The optimum qubit setup would have the h qubit fullyconnected to the q i qubits, thus fully minimising the SWAP operation errors. However,the available machines with such a qubit mapping on the public IBM Q experience havea lower quantum volume than the Santiago machine, which reports a quantum volume of32. Consequently, the trade of ideal qubit mapping for a better quantum volume has beenmade.
000 001 010 100 011 101 110 1110.00.20.40.60.81.0 P r o b a b ili t i e s gqq vertex, positive helicity Theory (S@M)SimSantiago000 001 010 100 011 101 110 1110.00.20.40.60.81.0 P r o b a b ili t i e s gqq vertex, negative helicity Theory (S@M)SimSantiago
Figure 4 : Results for the q → gq helicity amplitude calculation. Comparison between the-oretically calculated probability distribution, quantum simulator and real quantum com-puter.One of the key sources of error in the quantum computer is readout noise. Errormitigation methods have been used to optimise the output from the quantum computerand reduce readout noise effects. This has been done using the Qiskit Ignis software [40],which provides tools for noise characterisation and error correction based on noise models– 8 –f the quantum machines. The method involves testing simple qubit states on a series ofcalibration circuits, which are run using the quantum simulator with the noise profile ofthe Santiago machine. The response matrix created from this is shown in Fig. 5. Thisresponse matrix is calculated immediately before running the algorithm and then appliedto the machine results to obtain the error corrected results, as shown in Fig. 4.
000 001 010 011 100 101 110 111True000001010011100101110111 M e a s u r e d IBM Q Santiago Measurement Errors P e r c e n t a g e M e a s u r e d T r u e Figure 5 : IBM Q Santiago 5-qubit Quantum Computer Response Matrix for measurementerror correction on the 4 qubit helicity amplitude calculation algorithm. → Extending from the 1 → → ‡ . As anexample, we consider a qq → qq process. The initial state quark and antiquark are labelledas particles 1 and 2 respectively and the final state quark and antiquark as 3 and 4. Intotal, there are only 4 non-zero helicity configurations possible for each s and t-channelprocess. The relevant amplitudes are, M s (+ − + − ) = −(cid:104) | ¯ σ µ |
1] 1 s [3 | σ µ | (cid:105) , M s (+ −− +) = −(cid:104) | ¯ σ µ |
1] 1 s (cid:104) | ¯ σ µ |
4] (2.8)and M t (++ −− ) = −(cid:104) | ¯ σ µ |
1] 1 s [2 | σ µ | (cid:105) , M t (+ −− +) = −(cid:104) | ¯ σ µ |
1] 1 s (cid:104) | ¯ σ µ |
4] (2.9)where the +/- signs denote the helicity of the outgoing-particles 1, 2, 3 and 4 and s ij = − ( p i + p j ) = (cid:104) ij (cid:105) [ ji ] . (2.10)The other non-zero amplitudes are obtained by complex conjugation.The calculation is performed in the Centre-of-Mass (CM) frame and the momenta ofindividual particles is defined such that the only dependent input variable is the angle, ‡ Note, for the calculation of the 1 → – 9 – , through which the quark (and antiquark) is scattered. In the CM frame, the overallmagnitude of energy, E , associated with the momenta of each particle also drops out ofthe final helicity amplitude and is therefore not considered in this example.In the ‘all-outgoing’ convention of spinor-helicity formalism [42], the momenta of in-coming particles are flipped so that the incoming quark (1) (antiquark (2)) is mapped toan outgoing antiquark (quark) with opposite helicity. In the quantum algorithm, eachquark-antiquark vertex is calculated on a 4-qubit quantum register, q i . The outgoing an-tifermion spinor, q (cid:105) /q ], is implemented on the vertex quantum register, q ji , followed by thetwo dimensional representation of the gamma matrices, σ µ / ¯ σ µ , and then finally the vertexis closed with the opposite helicity outgoing fermion spinor, [ q/ (cid:104) q . A single qubit, s , is usedto calculate the denominator of the gluon propagator. The calculation is controlled bothfrom the helicity registers, h i , which determine what helicity configuration the particlesare in, and the amplitude qubit, p , which controls whether the s or t-channel process iscalculated. A schematic of the quantum circuit is shown in Fig. 6. Through this imple-mentation, each component of the helicity amplitude can be calculated and extracted fromthe machine. and M t (++ ) = h | ¯ µ |
1] 1 s [2 | µ | i , M t (+ +) = h | ¯ µ |
1] 1 s h | ¯ µ |
4] (2.9)where the +/- signs denote the helicity of the outgoing-particles 1, 2, 3 and 4 and s ij = ( p i + p j ) = h ij i [ ji ] . (2.10)The other non-zero amplitudes are obtained by complex conjugation.The calculation is performed in the Centre-of-Mass (CM) frame and the momenta ofindividual particles are defined such that the only dependent input variable is the angle, ✓ , through which the quark (and antiquark) is scattered. In the CM frame, the overallmagnitude of energy, E , associated with the momenta of each particle also drops out ofthe final helicity amplitude and is therefore not considered in this example.In the ‘all-outgoing’ convention of spinor-helicity formalism [ ? ], the momenta ofincoming particles is flipped so that the incoming quark (1) (antiquark (2)) is mappedto an outgoing antiquark (quark) with opposite helicity. In the quantum algorithm, eachquark-antiquark vertex is calculated on a 4-qubit quantum register, q i . The outgoingantifermion spinor, q i /q ], is implemented on the vertex quantum register, q ji , followedby the two dimensional representation of the gamma matrices, µ / ¯ µ , and then finallythe vertex is closed with the opposite helicity outgoing fermion spinor, [ q/ h q . A singlequbit, s , is used to calculate the denominator of the gluon propagator. The calculation iscontrolled both from the helicity registers, h i , which determine what helicity configurationthe particles are in, and the amplitude qubit, p , which controls whether the s or t-channelprocess is calculated. A schematic of the quantum circuit is shown in Fig. ?? . Through thisimplementation, each component of the helicity amplitude can be calculated and extractedfrom the machine. q ji i / µ / µ [2 / h h i Figure 6 : Circuit for the qq ! qq process helicity amplitude calculation. The q ji registersare used to calculate the qq vertices, and these are controlled from the helicity registers, h i , which dictate the helicity configuration of the process.This method is powerful as it allows for each component of the calculation to beextracted, however it leads to a complicated circuit, especially if one implements a method– 9 – Figure 6 : Circuit for the qq → qq process helicity amplitude calculation. The q ji registersare used to calculate the qq vertices, and these are controlled from the helicity registers, h i , which dictate the helicity configuration of the process.This method is powerful as it allows for each component of the calculation to beextracted, however it leads to a complicated circuit, especially if one implements a methodof dealing with incorrect helicity setups. As in Sec. 2.2, the circuit can be simplified bydirectly calculating the scalar products required for the final amplitudes. The amplitudesgiven in Eqs. (2.8) and (2.9) can be simplified using Eq. (2.6) (and that [ p | σ µ | q (cid:105) = (cid:104) q | ¯ σ µ | p ])to give the final forms, M s (+ − + − ) = 2 (cid:104) (cid:105) [31] (cid:104) (cid:105) [21] , M s (+ −− +) = 2 (cid:104) (cid:105) [41] (cid:104) (cid:105) [21] (2.11)– 10 –nd M t (++ −− ) = 2 (cid:104) (cid:105) [21] (cid:104) (cid:105) [31] , M t (+ −− +) = 2 (cid:104) (cid:105) [41] (cid:104) (cid:105) [31] . (2.12)Using these expressions, the number of qubits needed for the circuit is reduced from 17to 12 qubits. Another advantage is that the machine now only has to read out 3 qubits,where previously 8 qubits were read out per run. On these three qubits, each of the scalarproducts is calculated. The quark-antiquark vertex scalar products from the numeratorare calculated on the first two qubits, and the denominator of the gluon propagator iscalculated on the third qubit. Only one scalar product needs to be calculated for thedenominator since [42], (cid:104) ij (cid:105) = [ ji ] ∗ , (2.13)therefore the second scalar product can be determined from the same qubit.This simplified circuit is run on the IBM Q 32-qubit Quantum Simulator [41] for10,000 runs and compared to theoretically calculated probability distributions, extrapo-lated directly from analytic calculations of the helicity amplitude, calculated using theS@M software [44]. Using the equivalence between helicity spinors and orthogonal purestate qubits, these theoretical predictions have been obtained from the probabilities ofeach of the qubits to be in the | (cid:105) or | (cid:105) state, which correspond to the magnitude squaredof the upper and lower components of the helicity spinor respectively. The results fromthe quantum simulator show that the output of the quantum circuit lies within 1 σ of thetheoretically predicted probability distribution and are shown in Fig. 7 for both the s andt-channel in a specific helicity configuration.
000 001 010 100 011 101 110 1110.00.20.40.60.81.0 P r o b a b ili t i e s qq qq , s and t-channel (+,-,+,-) QC s-channelTheory s-channelQC t-channelTheory t-channel
Figure 7 : Comparison between theoretically predicted qubit final state probabilities and32-qubit quantum simulator output for the s and t-channel qq → qq process in the (+,-,+,-)helicity configuration. The quark (antiquark) scattering angle has been chosen as θ = π . → n amplitude calculations It can be shown, using the BCFW recursion formula [45, 46] and the relations in Eq. (2.5),that scattering amplitudes for massless partons can be reduced to a combination of scalar– 11 –roducts between helicity spinors § . Consequently, the algorithm presented in Secs. 2.2and 2.3 can be generalised to multi-particle amplitudes straightforwardly, as the tools arealready created, namely the circuit decompositions of the helicity spinors from Appendix C.The number of calculation qubits, q i , and the number of helicity qubits, h i , needed in thealgorithm both scale linearly with the number of final state particles, n . As the numberof helicity qubits, h i , scales linearly, then so does the number of work qubits needed inthe algorithm. Each scalar product calculation requires two spinor operations, and so thealgorithm can be easily extended without adding disproportionate complexity. The circuitdepth scales linearly with an increase in the number of scalar products, calculated on the q i qubits, and the number of helicity qubits, h i , added to the circuit.It is interesting and practical to consider the extension of the simple helicity amplitudealgorithms presented here to more complicated processes that are likely to be present inhigh energy collisions, such as those studied at the LHC. As we have seen in Sec. 2.2,modern public access quantum computers do not perform to a standard where one couldextrapolate accurate calculations of helicity amplitudes, even for a single vertex. However,the performance of public access computers is well below that of state of the art machines,such as the IBM 53-qubit machine and the Honeywell machine. The latter, in unpublishedwork, claims to have the world’s best Quantum Volume of 64 [47]. Such computers donot have the same restrictions as the smaller, less capable public access machines. Themore powerful machines offer more choice for qubit setup and mapping, and the abilityto perform more operations before decoherence in the machine starts to affect the circuitoutput. We can speculate that the algorithms presented here would be very accurate onthese machines, especially the vertex calculation, which comprises a maximum of only 33operations across 4 qubits.The main difficulty of extending such algorithms for helicity amplitude calculations onquantum computers comes not only from limitations due to the number of qubits, but alsothe machine’s fault tolerance. The more complicated the helicity amplitude calculation, themore operations are needed to calculate it. Therefore, a machine needs not only sufficientqubits but also the ability to implement many operations without excess noise. For thealgorithm proposed, the immediate challenge is not the number of qubits available, but thenumber of operations that can be reliably implemented on the circuit. With advancementsin the Quantum Volume of quantum computers [48], this limitation will likely be overcomeon current hardware. It is possible that near-future computers will have the ability toperform accurate and precise calculations and also have a large number of qubits. IBMrecently announced their roadmap for the future and the goal of having machines with thenumber of qubits exceeding 1,000 by 2023 [49]. Therefore, it is highly likely that these near-future devices will be able to perform precise helicity amplitude calculations for processes § A well-known example is the Parke-Taylor formula for a 2 → n gluon scattering process, where thegluons i and j have helicity (-) and all other gluons have helicity (+). Then the formula provides thefollowing expression for the amplitude A n , A n [1 + · · · i − · · · j − · · · n + ] = ( − g s ) n − (cid:104) ij (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) · · · (cid:104) n (cid:105) . (2.14) – 12 –ith a large number of particles. After the hard process is calculated, the next step in simulating a scattering event at ahigh-energy collider experiment is the parton shower stage. The parton shower evolves thescattering process from the hard interaction scale down to the hadronisation scale. Wepropose an algorithm for simulating a QCD parton shower using IBM Quantum Experi-ence [40] software and hardware. The quantum circuit has been implemented to simulate a2-step QCD parton shower with collinear splittings only. Section 3.1 provides the theoret-ical outline for the shower algorithm and discusses the splitting functions and probabilitycalculations implemented in the quantum circuit. A brief overview of the quantum circuitis given in Sec. 3.2, and a comparison between the results of the algorithm and theoreti-cally calculated probability distributions are discussed in Sec. 3.3. A glossary of quantumlogic gates is given in Appendix A and a detailed overview of the quantum circuit for thealgorithm in Appendix E.
We present a parton shower algorithm with the ability to simulate a general, discrete QCDparton shower, harnessing the quantum computer’s ability to remain in a quantum statethroughout the algorithm. In contrast to classical methods, the algorithm does not needto explicitly keep track of individual shower histories. Instead, our algorithm constructsand maintains a wavefunction that consists of a superposition of all possible shower his-tories, with the final measurement projecting out a specific quantity of the final state.Consequently, the algorithm presented inherently simulates the quantum interference be-tween all possible final states, without the need for extensive computational logic presentin current classical algorithms. In a classical algorithm, a physically meaningful quantitycan only be extracted from a parton shower calculation after summing over all possibleshower histories, requiring them to be stored on a physical memory device. Our quantumalgorithm avoids the need for such an intermediate step, as the measurement is performedon the superposition of all shower histories directly.The goal is to create the foundation for constructing a general quantum algorithm thatcan simulate a full QCD parton shower. To comply with the current capabilities of publicaccess quantum computers and simulators provided by IBM Quantum Experience [40], thealgorithm presented here uses a simplified model consisting of one flavour of quark and agluon. This reduces the number of qubits needed, and the algorithm can be run on the IBMQ 32-qubit Quantum Simulator [41]. To further reduce the number of required qubits, onlycollinear splittings are considered within the model. By neglecting the soft-limit, there isno need to keep track of the detailed kinematics of the particles in the shower history.Collinear emission occurs when a parton splits into two massless particles which haveparallel 4-momenta, such that the total momentum, P , is distributed between the particlesas p i = xP, p j = (1 − x ) P, (3.1)– 13 –hus, ( p i + p j ) = P = 0 [50].The emission probabilities in the algorithm are calculated using the collinear splittingfunctions outlined in [51–54]. A consequence of the collinear limit being a semi-classicalinterpretation with 1-to-2 splittings leads to the presence of a diagonal colour charge in thesplitting functions, C ii . The splitting for a quark to a gluon and a quark, with momentumfractions z and 1 − z respectively, is described at Leading Order (LO) by P q → qg ( z ) = C F − z ) z , (3.2)with C F = 4 /
3. The gluon splitting can be divided into two parts, with the first describingthe splitting of a gluon to a quark-antiquark pair and the second describing the splittingof a gluon to two gluons, P g → qq ( z ) = n f T R ( z + (1 − z ) ) , P g → gg ( z ) = C A (cid:104) − zz + z (1 − z ) (cid:105) , (3.3)where C A = 3 and T R = 1 /
2. Here, n f is the number of massless quark flavours, and T R isthe colour factor. It should be noted that both splitting functions have a soft singularityat z = 0; the hard-collinear limit only takes into account finite z .Further to calculating the splitting functions, the Sudakov factors have been used todetermine whether an emission occurred in the step. The Sudakov factors for a QCDprocess are given by [7] ∆ i,k ( z , z ) = exp (cid:104) − α s (cid:90) z z P k ( z (cid:48) ) dz (cid:48) (cid:105) , (3.4)and are used to calculate the non-emission probability. The running of the strong coupling, α s , is not simulated in this algorithm and for ease has been set to 1. For any given step N , there are N possible particles present, and so the probability that none of the particlessplit is given by ∆ tot ( z , z ) = ∆ n g g ( z , z )∆ n q q ( z , z )∆ n q q ( z , z ) . (3.5)Finally, the probability of a certain splitting is therefore obtained fromProb k → ij = (cid:0) − ∆ k (cid:1) × P k → ij ( z ) . (3.6)To implement the algorithm efficiently, preference has been given to gluons splitting toa quark-antiquark pair. This splitting preference implementation is explained in depthin Appendix E, but, for definiteness, the probability of a gluon splitting to two gluons iscalculated as Prob g → gg = (cid:0) − ∆ g (cid:1) × (cid:0) − P g → qq ( z ) (cid:1) × P g → gg ( z ) . (3.7)For the energy scale considered here, this should have a small affect on the results as P g → qq ( z ) (cid:28) P g → gg ( z ). – 14 – .2 Implementation on quantum circuit A quantum circuit has been constructed to simulate a parton shower with collinear split-tings. The circuit comprises of particle registers, emission registers and history registersand uses a total of 31 qubits. The algorithm is discretised into individual steps. An emis-sion can occur in each step, and the probabilities are calculated from the splitting functionsand Sudakov factors. To meet the 32 qubit limit of the IBM Q Quantum Simulator [41],the algorithm has been limited to two steps, but it is generally extendable. Figure 8 showsthe circuit diagram for a single step. in depth in Appendix ?? , but, for definiteness, the probability of a gluon splitting to twogluons is calculated asProb g ! gg = g ⇥ P g ! qq ( z ) ⇥ P g ! gg ( z ) . (3.7)For the energy scale considered here, this should have a small a↵ect on the results as P g ! qq ( z ) ⌧ P g ! gg ( z ). A quantum circuit has been constructed to simulate a parton shower with collinear split-tings. The circuit comprises of particle registers, emission registers and history registersand uses a total of 31 qubits. The algorithm is discretised into individual steps. An emis-sion can occur in each step, and the probabilities are calculated from the splitting functionsand Sudakov factors. In order to meet the 32 qubit limit of the IBM Q Quantum Simulator[ ? ], the algorithm has been limited to two steps, but it is generally extendable. Figure ?? shows the circuit diagram for a single step. p i Update p j ... n Count | i Resetfornext step e Emission | i h History...
Figure 8 : Circuit diagram for one step of the algorithm. The circuit comprises particleregisters, emission registers and history registers.The algorithm follows a similar method to that described by Bauer et al. in [ ? ],first counting the particles present in the simulation, determining whether an emission hasoccurred and if so assessing which splitting did occur, then finally updating the particlecontent of the simulation. In contrast to the method shown by [ ? ], the algorithm presentedhere has the ability to simulate a QCD process, with splittings for both gluons and quarksimplemented using the splitting functions outlined in Eqs. ( ?? ) and ( ?? ). The addition ofsuch splitting functions leads to significant changes to the algorithm presented in Baueret al., specifically in the History and Update Gates of the algorithm, shown in Fig. ?? . Theimplementation of these gates is outlined in detail in Appendix ?? . Unlike the algorithmpresented by Bauer et al., we have chosen not to introduce flavour mixing at the start of the– 14 – Figure 8 : Circuit diagram for one step of the algorithm. The circuit comprises particleregisters, emission registers and history registers.The algorithm follows a similar method to that described in [26], first counting theparticles present in the simulation, determining whether an emission has occurred andif so, assessing which splitting did occur, then finally updating the particle content ofthe simulation. In contrast to the method shown by [26], the algorithm presented herehas the ability to simulate a QCD process with splittings for both gluons and quarksimplemented using the splitting functions outlined in Eqs. (3.2) and (3.3). The additionof such splitting functions leads to significant changes to the algorithm compared to thatpresented in [26], specifically in the History and Update gates of the algorithm, shownin Fig. 8. The implementation of these gates is outlined in detail in Appendix E. Unlikethe algorithm presented in [26], we have chosen not to introduce flavour mixing at thestart of the algorithm. Instead, the superposition and interference between the possibleoutput states are introduced in the tailored History and Update gates. With the ability tosimulate gluon and quark splittings, the algorithm is thus well suited to hadronic partonshower simulation and provides the foundations for a general parton shower algorithm foruse on a GQC.The parton shower algorithm is designed to operate on the public access IBM Q 32-qubit Quantum Simulator [41], which allows for a total of two steps to be simulated onthe machine. As the machine is a simulator, it does not suffer from noise or a limit on the– 15 –umber of operations due to qubit decoherence effects, therefore giving a simulation of aperfect machine. As a consequence, error checking is easily done with direct comparisonto theoretically predicted probability distributions, and this is discussed in Sec. 3.3.One of the main benefits of using a quantum computing (QC) algorithm for the sim-ulation of QCD parton showers over classical methods is the computational simplicity ofthe algorithm. When dealing with interference of different splittings in the shower process,the algorithm presented here offers a much less computationally complex approach thanthat provided by modern Monte Carlo event generators. This is achieved by utilising theunique ability to maintain the quantum computer in a fully quantum state throughoutthe algorithm, and only collapse to a classical circuit by measurement at the end of theprocess. This allows for the system to account for all possible parton shower historiessimultaneously. In contrast, modern Monte Carlo methods must manually keep track ofthe particle splitting histories to consider all possible contributions to a specific final state.For a two-step, discrete parton shower, this is a relatively easy task for a modern MonteCarlo generator. However, the quantum computing field is still in its infancy; the truepotential of quantum computing for simulating QCD parton showers will become apparentwith the advancement of quantum technologies. With more available qubits and machineswith improved hardware, the algorithm presented here will have the ability to simulatequantum effects, without the extensive and complex computational logic that a classicalcomputer would need. Therefore, quantum computers offer an avenue to explore processesthat contain a large number of shower particles, thus requiring complicated parton historiesand computing power, not currently achievable with modern classical techniques. BeyondQCD parton showers, this feature of a quantum computing algorithm can be of particu-lar interest for cosmic-ray air showers, where millions of long-lived particles are simulated[55, 56].
A comparison of the output from the parton shower algorithm and theoretical predictionsof the splitting probabilities is made, and the results are shown in Fig. 9. The algorithm wasrun for 10,000 shots using the IBM Q 32-qubit Quantum Simulator [41], with a momentuminterval of z lower = 0.3 to z upper = 0.5, and no noise simulation. Here the theoretical predic-tions have been calculated using the collinear splitting functions from Eqs. (3.2) and (3.3),using the method outlined in Sec. 3.1. The z value used for the particle splitting probabil-ities from Eq. (3.7) is the mid-point of the momentum interval used in the algorithm. Theresults are in agreement with the theoretically calculated probabilities to within 1 σ .A consequence of running the algorithm on a quantum simulator is that there will beno noise in the results, unlike a real quantum computer. Therefore, problems with thealgorithm can be identified through direct comparison with the theoretical calculations. Inthe future, if the algorithm can be run on a real quantum computer with enough qubits,then IBM Q offers a range of noise reducing schemes for its devices through the Qiskitsoftware [40]. Another advantage of using the quantum simulator is that it automaticallychooses an optimum qubit setup. In a real quantum computer, the user has to select a– 16 –ubit mapping in order to optimise the operation of the computer. For future use of thealgorithm, this can be done using the calibration data provided by IBM Q. g g g qq g gg g ggg g gqq P r o b a b ili t i e s Two step parton shower: z up = 0.5, z low = 0.3 (10000 shots) QC simTheory (a) Initial particle a gluon. q q q qg q qgg q qqq P r o b a b ili t i e s Two step parton shower: z up = 0.5, z low = 0.3 (10000 shots) QC simTheory (b) Initial particle a quark. q q q qg q qgg q qqq P r o b a b ili t i e s Two step parton shower: z up = 0.5, z low = 0.3 (10000 shots) QC simTheory (c) Initial particle an antiquark.
Figure 9 : Results from the quantum circuit compared to theoretical predictions for twosteps of the parton shower with momentum interval of z lower = 0.3 to z upper = 0.5 and theinitial state particle of (a) gluon, (b) quark and (c) antiquark.– 17 – Summary and conclusions
The accurate modelling of the complexity of collisions at experiments, such as the LargeHadron Collider, relies on theoretical calculations of multi-particle processes. Such calcu-lations can be factorised into: the hard interaction, which models the large momentumtransfer, and the parton shower, which models the evolution from the hard interactionscale down to the hadronisation scale.We present general and extendable quantum computing algorithms that provide cal-culations of the hard interaction process and the parton shower, as a first step towards aquantum computing algorithm to describe the full collision event at the LHC.The hard interaction calculation uses helicity amplitudes by exploiting the equivalenceof spinors and qubits, and encoding operators as a series of unitary operations in thequantum circuit, thus demonstrating an excellent use case of quantum computers to modelthe intrinsic quantum behaviour of the system. A quantum algorithm is constructed fortwo simple examples of helicity calculations; a gq ¯ q vertex and the qq → qq process. Byapplying Hadamard gates to helicity registers, we introduce a superposition state betweenthe helicity qubits and can therefore calculate the positive and negative helicities of eachparticle involved simultaneously. This is further exploited in the simultaneous computationof s and t-channel amplitudes for the q ¯ q → q ¯ q process, thus fully utilising the quantumnature of the computation. A comparison between the theoretical predictions and theoutput of the quantum algorithm shows very good agreement. Furthermore, the successfulimplementation of the gq ¯ q vertex algorithm on a real machine is also demonstrated bycomparing results from the machine with a simulator.We also present a quantum algorithm for simulating collinear emission in a two-step,discrete parton shower with a maximum of three final state particles, utilising the quantumcomputer’s ability to remain in a quantum state throughout the simulation. In contrastto classical implementations of parton showers, where individual shower histories have tobe stored on a physical memory device, our quantum computing algorithm constructs awavefunction for the whole parton shower process, which contains a superposition of allshower histories. As a result, we do not need to keep track of individual shower historiesexplicitly, and a physical quantity of the shower process can be obtained through a mea-surement of the wavefunction. The results from the algorithm, as performed on the IBM Q32-qubit Quantum Simulator [41], show good agreement with theoretical predictions. Thealgorithm builds on previous work [26] by including a vector boson and boson splittings,which leads to significant changes in its implementation. The ability to simulate gluon andquark splittings makes the algorithm presented here well suited to hadronic parton showersimulation and provides the foundations for developing a general parton shower algorithm.With advancements in quantum technologies, this algorithm can be extended to includeall flavours of quarks without adding disproportionate computational complexity.With IBM recently setting their goal of exceeding 1,000 qubits by 2023 [49] and ad-vancements in the development of devices with better Quantum Volume [48], we are onthe brink of a quantum revolution. These developments would allow the algorithms pre-sented in this paper to be extended to reflect the processes seen in experiments such as the– 18 –HC. The consequence of such advancements would be algorithms that can fully modelthe dynamics of quantum field theories to provide accurate and precise helicity amplitudecalculations and simulations of parton showers. Acknowledgements:
We would like to acknowledge the use of the IBM Q for this work.We are grateful to the authors of [26] for answering questions on the circuit presented intheir work and for sharing their preliminary codes. M.S would like to thank Steve Abeland Daniel Maitre for helpful discussions. K.B and M.S are supported by the STFC undergrant ST/P001246/1. S.M and S.W are supported by a grant from the Royal Society. – 19 –
Quantum logic gate definitions • NOT gate– a NOT gate is a single qubit operation which flips the state of the qubit.NOT | (cid:105) = | (cid:105) , NOT | (cid:105) = | (cid:105) . The circuit representation of a NOT gate is: presented in this paper to be extended to reflect processes seen in experiments such as theLHC. The consequence of such advancements would be algorithms which can fully modelthe dynamics of quantum field theories and provide accurate and precise helicity amplitudecalculations and simulations of parton showers.
Acknowledgements:
We would like to acknowledge the use of the IBM Q for this work.We are grateful to the authors of [ ? ] for answering questions on the circuit presentedin their work and for sharing their preliminary codes. M.S would like to thank Steve Abeland Daniel Maitre for helpful discussions. K.B and M.S are supported by the STFC undergrant ST/P001246/1. S.M and S.W are supported by a grant from the Royal Society. A Quantum Logic Gate Definitions • NOT Gate– a NOT Gate is a single qubit operation which flips the state of the qubit.NOT | i = | i , NOT | i = | i . The circuit representation of a NOT Gate is: • CNOT Gate– a controlled -NOT (CNOT) Gate is a two qubit operation which flips the stateof a target qubit dependent on the state of a control qubit.CNOT | i = | i , CNOT | i = | i , CNOT | i = | i , CNOT | i = | i . Here, the first qubit is the control. The circuit representation of a CNOT Gateis: • To↵oli Gate (CCNOT)–
A To↵oli Gate is a three qubit operation, which is just a further extension ofthe NOT gate with two control qubits.– 18 – • CNOT gate– a controlled -NOT (CNOT) gate is a two qubit operation which flips the stateof a target qubit dependent on the state of a control qubit.CNOT | (cid:105) = | (cid:105) , CNOT | (cid:105) = | (cid:105) , CNOT | (cid:105) = | (cid:105) , CNOT | (cid:105) = | (cid:105) . Here, the first qubit is the control. The circuit representation of a CNOT gateis: presented in this paper to be extended to reflect processes seen in experiments such as theLHC. The consequence of such advancements would be algorithms which can fully modelthe dynamics of quantum field theories and provide accurate and precise helicity amplitudecalculations and simulations of parton showers.
Acknowledgements:
We would like to acknowledge the use of the IBM Q for this work.We are grateful to the authors of [ ? ] for answering questions on the circuit presentedin their work and for sharing their preliminary codes. M.S would like to thank Steve Abeland Daniel Maitre for helpful discussions. K.B and M.S are supported by the STFC undergrant ST/P001246/1. S.M and S.W are supported by a grant from the Royal Society. A Quantum Logic Gate Definitions • NOT Gate– a NOT Gate is a single qubit operation which flips the state of the qubit.NOT | i = | i , NOT | i = | i . The circuit representation of a NOT Gate is: • CNOT Gate– a controlled -NOT (CNOT) Gate is a two qubit operation which flips the stateof a target qubit dependent on the state of a control qubit.CNOT | i = | i , CNOT | i = | i , CNOT | i = | i , CNOT | i = | i . Here, the first qubit is the control. The circuit representation of a CNOT Gateis: • To↵oli Gate (CCNOT)–
A To↵oli Gate is a three qubit operation, which is just a further extension ofthe NOT gate with two control qubits.– 18 – • Toffoli gate (CCNOT)–
A Toffoli gate is a three qubit operation, which is just a further extension of theNOT gate with two control qubits.CCNOT | (cid:105) = | (cid:105) , CCNOT | (cid:105) = | (cid:105) , CCNOT | (cid:105) = | (cid:105) , CCNOT | (cid:105) = | (cid:105) , CCNOT | (cid:105) = | (cid:105) , CCNOT | (cid:105) = | (cid:105) . The circuit representation of a Toffoli gate is:
CCNOT | i = | i , CCNOT | i = | i , CCNOT | i = | i , CCNOT | i = | i , CCNOT | i = | i , CCNOT | i = | i . The circuit representation of a To↵oli Gate is:– 19 – – 20 –
Hadamard gate– a Hadamard gate is a purely quantum logic gate and does not have a classicallogic gate equivalent. A Hadamard gate is a single qubit operation which putsa qubit into a superposition.H | (cid:105) = 1 √ (cid:0) | (cid:105) + | (cid:105) (cid:1) , H | (cid:105) = 1 √ (cid:0) | (cid:105) − | (cid:105) (cid:1) . The Hadamard gate can be controlled, and so is only applied depending on thestate of the control qubit. The circuit representation of a Hadamard gate is: • Hadamard Gate– a Hadamard gate is a purely quantum logic gate and does not have a classicallogic gate equivalent. A Hadamard gate is a single qubit operation which putsa qubit into a superposition.H | i = 1 p | i + | i , H | i = 1 p | i | i . The Hadamard gate can be controlled, and so is only applied depending on thestate of the control qubit. The circuit representation of a Hadamard Gate is: H B Dirac and Helicity Spinor correspondence
The following demonstration of the correspondence between Dirac spinors and Helicityspinors can be seen in Chapter 2 of [ ? ].Fermion and anti-fermion spinors satisfy the Dirac equations such that,( /p + m ) u ( p ) = 0 , ( /p + m ) ⌫ ( p ) = 0 . (B.1)where both equations have independent solutions which can be labelled by subscripts s = ± .One can move to a basis where the ± denotes spin up/down along the z-axis, by ensuringthat spinors u ± and ⌫ ± are eigenstates of the z-component of the spin-matrix in the restframe. For massless fermions, ± denotes the helicity; the projection of the spin alongthe momentum of the particle. These spinors are also associated with the conventionalFeynman rules for external fermions, e.g. ⌫ ± ( p ) for an outgoing anti-fermion and ¯ u ± ( p ) foran outgoing fermion.For the massless case, the Dirac equations reduce to /p⌫ ± ( p ) =0 ¯ u ± ( p ) /p = 0 , (B.2)where ⌫ ± ( p ) and u ± ( p ) are the wave functions associated with outgoing anti-fermions andfermions respectively. For this case the wavefunctions are related as u ± = ⌫ ⌥ and ¯ ⌫ ± = ¯ u ⌥ .The two independent solutions of the Dirac equations can be written as ⌫ + ( p ) = | p ] a ! , ⌫ ( p ) = | p i ˙ a ! (B.3)and ¯ u ( p ) = ⇣ h p | ˙ a ⌘ , ¯ u + ( p ) = ⇣ [ p | a ⌘ (B.4)where the angle and square spinors are 2-component spinors that satisfy the massless Weylequation. – 20 – B Dirac and helicity spinor correspondence
The following demonstration of the correspondence between Dirac spinors and Helicityspinors can be seen in Chapter 2 of [42].Fermion and anti-fermion spinors satisfy the Dirac equations such that,( /p + m ) u ( p ) = 0 , ( − /p + m ) ν ( p ) = 0 . (B.1)where both equations have independent solutions which can be labelled by subscripts s = ± .One can move to a basis where the ± denotes spin up/down along the z-axis, by ensuringthat spinors u ± and ν ± are eigenstates of the z-component of the spin-matrix in the restframe. For massless fermions, ± denotes the helicity, the projection of the spin alongthe momentum of the particle. These spinors are also associated with the conventionalFeynman rules for external fermions, e.g. ν ± ( p ) for an outgoing anti-fermion and ¯ u ± ( p ) foran outgoing fermion.For the massless case, the Dirac equations reduce to /pν ± ( p ) =0 ¯ u ± ( p ) /p = 0 , (B.2)where ν ± ( p ) and u ± ( p ) are the wavefunctions associated with outgoing anti-fermions andfermions respectively. For this case the wavefunctions are related as u ± = ν ∓ and ¯ ν ± = ¯ u ∓ .The two independent solutions of the Dirac equations can be written as ν + ( p ) = (cid:32) | p ] a (cid:33) , ν − ( p ) = (cid:32) | p (cid:105) ˙ a (cid:33) (B.3)and ¯ u − ( p ) = (cid:16) (cid:104) p | ˙ a (cid:17) , ¯ u + ( p ) = (cid:16) [ p | a (cid:17) (B.4)where the angle and square spinors are 2-component spinors that satisfy the massless Weylequation. – 21 – Helcity amplitude gate decompositions • U a (cid:105) gate– The U a (cid:105) takes the form of a conventional U rotation gate, U a (cid:105) = U ( θ, φ, λ ) = (cid:32) cos (cid:0) θ (cid:1) − e iλ sin (cid:0) θ (cid:1) e iφ sin (cid:0) θ (cid:1) e i ( φ + λ ) cos (cid:0) θ (cid:1)(cid:33) . (C.1)Therefore, the circuit representation is just a qiskit U rotation, C Helcity Amplitude Gate Decompositions • U a i Gate–
The U a i takes the form of a conventional U rotation gate, U a i = U ( ✓, , ) = cos ✓ e i sin ✓ e i sin ✓ e i ( + ) cos ✓ ! . (C.1)Therefore, the circuit representation is just a qiskit U rotation,= U a i U ✓, , ) • U a ] Gate–
The U a ] has the matrix form, U a ] ( ✓, , ) = e i sin ✓ e i ( + ) cos ✓ cos ✓ e i sin ✓ ! (C.2)Therefore, this gate has the circuit representation,= U a ] U ✓, ⇡ ,⇡ ) • U h b Gate–
The U h b has the matrix form, U h b ( ✓, ) = e i sin ✓ cos ✓ cos ✓ e i sin ✓ ! (C.3)Therefore, this gate has the circuit representation,= U h b U ⇡ ✓, ⇡ , ) U , , + ⇡ ) U , , + ⇡ ) – 21 – • U a ] gate– The U a ] has the matrix form, U a ] ( θ, φ, λ ) = (cid:32) − e − iλ sin (cid:0) θ (cid:1) e − i ( φ + λ ) cos (cid:0) θ (cid:1) cos (cid:0) θ (cid:1) e − iφ sin (cid:0) θ (cid:1) (cid:33) (C.2)Therefore, this gate has the circuit representation, C Helcity Amplitude Gate Decompositions • U a i Gate–
The U a i takes the form of a conventional U rotation gate, U a i = U ( ✓, , ) = cos ✓ e i sin ✓ e i sin ✓ e i ( + ) cos ✓ ! . (C.1)Therefore, the circuit representation is just a qiskit U rotation,= U a i U ✓, , ) • U a ] Gate–
The U a ] has the matrix form, U a ] ( ✓, , ) = e i sin ✓ e i ( + ) cos ✓ cos ✓ e i sin ✓ ! (C.2)Therefore, this gate has the circuit representation,= U a ] U ✓, ⇡ ,⇡ ) • U h b Gate–
The U h b has the matrix form, U h b ( ✓, ) = e i sin ✓ cos ✓ cos ✓ e i sin ✓ ! (C.3)Therefore, this gate has the circuit representation,= U h b U ⇡ ✓, ⇡ , ) U , , + ⇡ ) U , , + ⇡ ) – 21 – • U (cid:104) b gate– The U (cid:104) b has the matrix form, U (cid:104) b ( θ, φ ) = (cid:32) − e iφ sin (cid:0) θ (cid:1) cos (cid:0) θ (cid:1) cos (cid:0) θ (cid:1) e − iφ sin (cid:0) θ (cid:1)(cid:33) (C.3)Therefore, this gate has the circuit representation, C Helcity Amplitude Gate Decompositions • U a i Gate–
The U a i takes the form of a conventional U rotation gate, U a i = U ( ✓, , ) = cos ✓ e i sin ✓ e i sin ✓ e i ( + ) cos ✓ ! . (C.1)Therefore, the circuit representation is just a qiskit U rotation,= U a i U ✓, , ) • U a ] Gate–
The U a ] has the matrix form, U a ] ( ✓, , ) = e i sin ✓ e i ( + ) cos ✓ cos ✓ e i sin ✓ ! (C.2)Therefore, this gate has the circuit representation,= U a ] U ✓, ⇡ ,⇡ ) • U h b Gate–
The U h b has the matrix form, U h b ( ✓, ) = e i sin ✓ cos ✓ cos ✓ e i sin ✓ ! (C.3)Therefore, this gate has the circuit representation,= U h b U ⇡ ✓, ⇡ , ) U , , + ⇡ ) U , , + ⇡ ) – 21 – – 22 – U [ b gate– The U [ b has the matrix form, U [ b ( θ, φ ) = (cid:32) cos (cid:0) θ (cid:1) e − iφ sin (cid:0) θ (cid:1) − e iφ sin (cid:0) θ (cid:1) cos (cid:0) θ (cid:1) (cid:33) (C.4)Therefore, this gate has the circuit representation, • U [ b Gate–
The U [ b has the matrix form, U [ b ( ✓, ) = cos ✓ e i sin ✓ e i sin ✓ cos ✓ ! (C.4)Therefore, this gate has the circuit representation,= U [ b U ✓, ⇡ + ,⇡ ) D Helicity Amplitude Calculation Circuit Diagrams and Further Results
D.1 ! Amplitude Calculation
Here we present the detailed circuit diagram for the q ! gq process, shown in Fig. ?? , whichis implemented using the helicity amplitude gate decompositions outlined in Appendix ?? .This demonstrates the simplification achieved by using fully contracted helicity amplitudesin the calculation, with a scalar product calculated on each qubit. The first slice in thecircuit diagram calculates the positive helicity, controlling from the h register in the | i state. The second slice controls from the h register in the | i state and calculates thenegative helicity process. A superposition of both the positive and negative processes,and thus the full amplitude, is achieved by implementing a Hadamard gate on the helicityqubit, h . q U q i U h p f U p i U h p f q U p ] U [ p f U q ] U [ p f q U p i U h q U p ] U [ q h Figure 10 : Explicit circuit for q ! gq helicity amplitude calculation, using gate decom-positions outlined in Sec. ?? . The amplitude for the process is calculated on the q i qubits,which are controlled from the helicity register. The q i qubits are then measured by thequantum computer.In Fig. ?? , a comparison between the output of the IBM Q Santiago 5-qubit QuantumComputer [ ? ] and the IBM Q 32-qubit Quantum Simulator [ ? ] run with the Santiagodevice’s noise profile is presented. The quantum computer has been run for 100 runs of8192 shots, giving a total of 819,200 shots on the circuit and the simulator has been run– 22 – D Helicity amplitude calculation circuit diagrams and further results
D.1 → amplitude calculation Here we present the detailed circuit diagram for the q → gq process, shown in Fig. 10, whichis implemented using the helicity amplitude gate decompositions outlined in Appendix C.This demonstrates the simplification achieved by using fully contracted helicity amplitudesin the calculation, with a scalar product calculated on each qubit. The first slice in thecircuit diagram (to the left of the vertical dashed line) calculates the positive helicity,controlling from the h register in the | (cid:105) state and the second slice (to the left of the verticaldashed line) controls from the h register in the | (cid:105) state and calculates the negative helicityprocess. A superposition of both the positive and negative processes, and thus the fullamplitude, is achieved by implementing a Hadamard gate on the helicity qubit, h . • U [ b Gate–
The U [ b has the matrix form, U [ b ( ✓, ) = cos ✓ e i sin ✓ e i sin ✓ cos ✓ ! (C.4)Therefore, this gate has the circuit representation,= U [ b U ✓, ⇡ + ,⇡ ) D Helicity Amplitude Calculation Circuit Diagrams and Further Results
D.1 ! Amplitude Calculation
Here we present the detailed circuit diagram for the q ! gq process, shown in Fig. 10, whichis implemented using the helicity amplitude gate decompositions outlined in Appendix C.This demonstrates the simplification achieved by using fully contracted helicity amplitudesin the calculation, with a scalar product calculated on each qubit. The first slice in thecircuit diagram calculates the positive helicity, controlling from the h register in the | i state. The second slice controls from the h register in the | i state and calculates thenegative helicity process. A superposition of both the positive and negative processes,and thus the full amplitude, is achieved by implementing a Hadamard gate on the helicityqubit, h . q U q i U h p f U p i U h p f q U p ] U [ p f U q ] U [ p f q U p i U h q U p ] U [ q h H Figure 10 : Explicit circuit for q ! gq helicity amplitude calculation, using gate decom-positions outlined in Sec. C. The amplitude for the process is calculated on the q i qubits,which are controlled from the helicity register. The q i qubits are then measured by thequantum computer.In Fig. 11, a comparison between the output of the IBM Q Santiago 5-qubit QuantumComputer [44] and the IBM Q 32-qubit Quantum Simulator [42] run with the Santiagodevice’s noise profile is presented. The quantum computer has been run for 100 runs of8192 shots, giving a total of 819,200 shots on the circuit and the simulator has been runfor 10,000 shots. Here we see more reasonable agreement between the noisy simulator– 22 – Figure 10 : Detailed circuit diagram for the q → gq helicity amplitude calculation, usinggate decompositions outlined in Sec. C. The amplitude for the process is calculated on the q i qubits, which are controlled from the helicity register. The q i qubits are then measuredby the quantum computer.In Fig. 11, a comparison between the output of the IBM Q Santiago 5-qubit QuantumComputer [43] and the IBM Q 32-qubit Quantum Simulator [41] run with the Santiagodevice’s noise profile is presented. The quantum computer has been run for 100 runs of8192 shots, giving a total of 819,200 shots on the circuit and the simulator has been run– 23 –or 10,000 shots. Here we see a more reasonable agreement between the noisy simulatorand the output from the quantum computer than the comparison to the perfect machinesimulator from Sec. 2.2. However, it should be noted that the noise profile used in the noisysimulation is only an approximation of the real quantum computer errors. Noise profiles arebuilt from a limited number of parameters and are based on average measurements of qubiterrors [40]. As a result, some discrepancies are present between the quantum simulator withthe Santiago device’s noise profile and the real device. These can be attributed to noisenot accounted for in the quantum computer.
000 001 010 100 011 101 110 1110.00.20.40.60.81.0 P r o b a b ili t i e s gqq vertex, positive helicity Noisy SimMitigated QCQC000 001 010 100 011 101 110 1110.00.20.40.60.81.0 P r o b a b ili t i e s gqq vertex, negative helicity Noisy SimMitigated QCQC
Figure 11 : Results for the q → gq helicity amplitude calculation. Comparison betweenthe results from a quantum simulator with the relevant machine noise profile, the resultsfrom the Santiago quantum computer and the error mitigated results from the quantumcomputer.While obtaining the results, we noticed a discrepancy between separate runs on thenegative helicity case. By changing the qubit mapping in the measurement process, thiswas identified as a tuning error on the entangling gate between qubits 2 and 3 of theSantiago machine. This error was later fixed, and the results shown were obtained fromruns with a fully functioning machine. To further validate the results, a series of runs wereperformed on the IBM Q Valencia 5-qubit machine [57], which has a Quantum Volume of16. The results confirmed that the Santiago machine was working correctly.– 24 – Detailed quantum circuit for collinear parton shower algorithm
The algorithm presented here follows a similar method to that outlined in [26]. In contrast,the algorithm does not introduce flavour mixing, but does simulate a vector boson withthe possibility of boson splittings. As a result, the algorithm presented here includestailored History and Update gates to deal with the increased splitting channels. Shownin Fig. 8, the circuit comprises of four tailored gate operations: Count, Emission, History,and Update gate. The particle identity is encoded using a 3-qubit base, and the followingqubit combinations have been chosen for each type of particle:gluon quark antiquarkp p p p
100 001 011 (E.1)Using a 3-qubit base, it is possible to simulate 7 different types of particle and 1 null state.Therefore, the algorithm could be easily extended to accommodate more quark flavours ifmore qubits were available.
E.1 Count gate
The count gate comprises of three individual counting mechanisms for each type of particle,and is applied to each particle register individually. The algorithm utilises a series of
NOT , controlled-NOT ( CNOT ) and Toffoli (
CCNOT ) gates to update the count registers, n i ,depending on the type of particle represented in the particle register. Fig. 12 shows thecounting mechanism for a gluon, controlling only from the gluon state outlined in E.1. E Detailed Quantum Circuit for Collinear Parton Shower Algorithm
The algorithm presented here follows a similar method to that outlined by Bauer et al. [ ? ]. In contrast, the algorithm does not introduce flavour mixing, but does simulate a vectorboson with the possibility of boson splittings. As a result, the algorithm presented hereincludes tailored History and Update gates to deal with the increased splitting channels.Shown in Fig. ?? , the circuit comprises of four tailored gate operations: Count, Emission,History and Update gate. The particle identity is encoded using a three qubit base, andthe following qubit combinations have been chosen for each type of particle:gluon quark antiquarkp p p p
100 001 011 (E.1)Using a 3-qubit base, it is possible to simulate 7 di↵erent types of particle and 1 null state.Therefore, the algorithm could be easily extended to accommodate more quark flavours ifmore qubits were available.
E.1 Count Gate
The count gate is made up from three individual counting mechanisms for each type ofparticle, and is applied to each particle register individually. The algorithm utilises aseries of
NOT , controlled-NOT ( CNOT ) and To↵oli (
CCNOT ) gates to update the countregisters, n i , depending on the type of particle represented in the particle register. Fig. ?? shows the counting mechanism for a gluon, controlling only from the gluon state outlinedin ?? . p k p p p work w w n g n q n q Figure 12 : Count Gate circuit decomposition for counting a gluon in the particle register.To complete the count gate, this is repeated for all other possible particle types by applyingdi↵erent combinations of
NOT gates. – 24 –
Figure 12 : Count gate circuit decomposition for counting a gluon in the particle register.To complete the count gate, this is repeated for all other possible particle types by applyingdifferent combinations of
NOT gates. – 25 –he total number of count registers, n i , used in the algorithm is 4. As the particlecount registers are updated at the beginning of a step, the maximum number of gluonsthat can be present is 2, and the maximum number of quarks/antiquarks is 1. Therefore,for this algorithm, only 2 gluon count registers and 1 quark/antiquark count register arerequired. Ideally, one would have the same number of count registers for each particle type,which would be equal to the step number. However, due to the limited number of availablequbits, this has not been possible here. E.2 Emission gate
The emission gate implements the Sudakov factors from Eq. (3.5) by defining a U rotationthat can be applied to the emission register, e . The structure of this rotation takes thesame form as that presented in [26], U e = (cid:32) (cid:112) ∆ tot ( z , z ) − (cid:112) − ∆ tot ( z , z ) (cid:112) − ∆ tot ( z , z ) (cid:112) ∆ tot ( z , z ) (cid:33) . (E.2)This rotation changes the state of the emission gate, e , to | (cid:105) if there is an emission, andkeeps it in state | (cid:105) if there is no emission. Non-emission probabilities (Sudakov factors)are used due to the Qiskit [40] definition of a qubit state, | (cid:105) = (cid:32) (cid:33) , | (cid:105) = (cid:32) (cid:33) . (E.3) The total number of count registers, n i , used in the algorithm is 4. As the particlecount registers are updated at the beginning of a step, the maximum number of gluons thatcan be present is 2 and the maximum number of quarks/antiquarks is 1. Therefore, for thisalgorithm only 2 gluon count registers and 1 quark/antiquark count register are required.Ideally, one would have the same number of count registers for each of the particle types,and this would be equal to the step number. However, due to the limitation on the numberof available qubits, this has not been possible here. E.2 Emission Gate
The emission gate implements the Sudakov factors from Eq. ( ?? ) by defining a U rotationthat can be applied to the emission register, e . The structure of this rotation takes thesame form as that presented by Bauer et al. in Reference [ ? ], U e = p tot ( z , z ) p tot ( z , z ) p tot ( z , z ) p tot ( z , z ) ! . (E.2)This rotation changes the state of the emission gate, e , to | i if there is an emission, andkeeps it in state | i if there is no emission. Non-emission probabilities (Sudakov factors)are used due to the Qiskit [ ? ] definition of a qubit state, | i = ! , | i = ! . (E.3) n g n g n g n q n q work w w w emission e U e Figure 13 : Emission Gate for a single gluon in the first particle register. Here the U e isa U rotation is used to implement the Sudakov Factors.Similarly to the Count Gate, the Emission Gate is constructed from a series of NOT gates which determine the target state, and a series of
CCNOT gates which implement theoperation if the target state is present. Here, the emission is determined by controlling from– 25 –
Figure 13 : Emission gate for a single gluon in the first particle register. Here the U e is a U rotation is used to implement the Sudakov factors.Similarly to the Count gate, the Emission gate is constructed from a series of NOT gates which determine the target state, and a series of
CCNOT gates which implement the– 26 –peration if the target state is present. Here, the emission is determined by controlling fromthe particle count gates. If the desired particles are present, then the emission rotationfrom Eq. (E.2) is applied to the emission register. As only one emission can occur in asingle step, then only one emission qubit is needed per step.
E.3 History gate
The history gate is the most complicated implementation in the algorithm. This is largelydue to the fact that a gluon can split to either a gluon pair, or a quark-antiquark pair. As aconsequence this requires two calculations of splitting probabilities for a gluon, as outlinedin Eq. (3.7). These probabilities are implemented by controlling from present particles andapplying a rotation to the relevant history register; again taking a form similar to the onepresented in [26], U h = (cid:113) − P k → ij ( z ) P tot ( z ) − (cid:113) P k → ij ( z ) P tot ( z ) (cid:113) P k → ij ( z ) P tot ( z ) (cid:113) − P k → ij ( z ) P tot ( z ) , (E.4)where P tot is defined as, P tot ( z ) = n g ( P g → qq + P g → gg ) + n q P q → qg + n q P q → qg . (E.5)Here the non-splitting probabilities are used in the diagonal elements due to the definitionof the qubit states outlined in Eq. (E.3). the particle count gates. If the desired particles are present, then the emission rotationfrom Eq. ( ?? ) is applied to the emission register. As only one emission can occur in a singlestep, then only one emission qubit is needed per step. E.3 History Gate
The history gate is the most complicated implementation in the algorithm. This is largelydue to the fact that a gluon can split to either a gluon pair, or a quark-antiquark pair. As aconsequence this requires two calculations of splitting probabilities for a gluon, as outlinedin Eq. ( ?? ). These probabilities are implemented by controlling from present particles andapplying a rotation to the relevant history register; again taking a form similar to the onepresented by Bauer et al. [ ? ], U h = P k ! ij ( z ) P tot ( z ) q P k ! ij ( z ) P tot ( z ) q P k ! ij ( z ) P tot ( z ) q P k ! ij ( z ) P tot ( z ) , (E.4)where P tot is defined as, P tot ( z ) = n g ( P g ! qq + P g ! gg ) + n q P q ! qg + n q P q ! qg . (E.5)Here the non-splitting probabilities are used in the diagonal elements due to the definitionof the qubit states outlined in Eq. ( ?? ). p k p p p emission ework w w w history h h U g h U g Figure 14 : History Gate for a single gluon in the first step. Here the U h gate is a U rotation used to implement the splitting probabilities.The history gate used in this algorithm di↵ers from [ ? ], such that it controls fromthe particle registers and not the count registers. This is to reduce the number of count– 26 – Figure 14 : History gate for a single gluon in the first step. Here the U h gate is a U rotation used to implement the splitting probabilities.The history gate used in this algorithm differs from [26], such that it controls fromthe particle registers and not the count registers. This is to reduce the number of count– 27 –egisters needed in the algorithm. For this algorithm, the history rotation needs to knowwhich particle is being considered and which particle register it is in so that the correctrotation can be applied to the correct history qubit. This could be done by increasing thecount registers by one qubit per particle type every step, to have a count register for eachpossible particle in a specific particle register. However, this need for more counting qubitscan be reduced by simply controlling from the particle registers themselves; this is shownfor a gluon in Fig. 14. This can be done without impacting the rest of the algorithm, aslong as there are enough count qubits to count the number of present particles correctly.This is because the emission gate does not need to know what specific particle is presentin which particle register, just how many particles are present.Once the particle content of the simulation has been assessed, the history rotations, U h , from Eq. (E.4) are applied to the relevant history registers. The first, labelled g , isfor the g → qq splitting, and the second, labelled g , is for the g → gg . Note that bothof these rotations could result in a splitting, and thus rotate the history qubit to the | (cid:105) state. Therefore, they are applied to different history registers. In general, one could havea condition on the second rotation, such that it is not applied if the first rotation yieldsa | (cid:105) , but in this algorithm, this condition was carried forward to the update gate, seeSec. E.4. As a result of these different splittings, the required number of qubits needed forthe history register in each step is 3 N , where N is the step number. Figure 14 shows thehistory gate for the first step, thus 3 qubits are needed for the history register: two for thegluon splittings, and one for the quark/antiquark splittings. E.4 Update gate
The final gate in the algorithm is the update gate, which, if an emission has occurred,changes the particle content of the particle registers accordingly. Figure 15 shows theupdate gate from the first step, which is the simplest update gate in the algorithm. Thecircuit shown is sliced into individual updates. The first slice on the left shows the additionof a new gluon to the particle register. This is controlled from the quark/antiquark historygate, and so corresponds to the ( − ) q → ( − ) q g process.The middle slice in Fig. 15 shows the update of a gluon splitting to a quark-antiquarkpair controlled from the g → qq history register. The first three CNOT gates of thisslice put the particle registers into a state of two quarks. The update gate then utilises a controlled-Hadamard gate, putting the p j qubit in a superposition of | (cid:105) and | (cid:105) states.The final gate of the slice controls from the history register, but also controls on a | (cid:105) stateon the p j . At the point of measurement, if p j is measured as a | (cid:105) state, then the p k register represents an antiquark and p j represents a quark. If p j is measured in the | (cid:105) state, then the p k register represents a quark and the p j register represents an antiquark.In a sense, this controlled-Hadamard gate and subsequent CCNOT gate put the particleregisters p j and p k into a superposition of quark-antiquark and antiquark-quark states.The final slice on the right of the circuit diagram in Fig. 15 shows the update gatecorresponding to the g → gg process. This simple update changes the p j qubit to a | (cid:105) state controlled from the history register, like the quark/antiquark update gate. However,this is where the algorithm adds a preference to g → qq process over the g → gg process.– 28 – i p Particlespassed on tonext step p p p j p p Hp history h h h Figure 15 : Update Gate for the first step of the algorithm. Each slice is a di↵erent updatemechanism: far left slice updates q ! qg splittings, centre slice updates g ! qq and thefar right slice updates g ! gg .The CCNOT gate for the final slice in Fig. ?? also controls from a | i state on the g ! qq history qubit. Therefore a gluon can only split to a gluon pair if the history gate for a gluonsplitting to a quark-antiquark pair is in the | i state. This is an acceptable approximationbecause the splitting probabilities for g ! qq are a lot less than for g ! gg . Consequently,there is only a small probability that they are both in the | i state at any one time.However, it is possible that this may be a limitation in comparison to current classicalparton shower algorithms provided by packages such as Pythia [ ? ], Herwig [ ? ] and Sherpa [ ? ], as these give more complex weightings to the di↵erent splitting channels. References [1] T. Sj¨ostrand et. al. , An introduction to PYTHIA 8.2 , Comput. Phys. Commun. (2015)159–177, [ arXiv:1410.3012 ].[2] M. Bahr et. al. , Herwig++ Physics and Manual , Eur. Phys. J. C (2008) 639–707,[ arXiv:0803.0883 ].[3] T. Gleisberg, S. Hoeche, F. Krauss, A. Schalicke, S. Schumann, and J.-C. Winter, SHERPA1. alpha: A Proof of concept version , JHEP (2004) 056, [ hep-ph/0311263 ].[4] J. C. Collins, D. E. Soper, and G. F. Sterman, Factorization of Hard Processes in QCD ,vol. 5, pp. 1–91. 1989. hep-ph/0409313 .[5] S. J. Parke and T. Taylor,
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Figure 15 : Update gate for the first step of the algorithm. Each slice is a different updatemechanism: far left slice updates q → qg splittings, centre slice updates g → qq and thefar right slice updates g → gg .The CCNOT gate for the final slice in Fig. 15 also controls from a | (cid:105) state on the g → qq history qubit. Therefore a gluon can only split to a gluon pair if the history gate for a gluonsplitting to a quark-antiquark pair is in the | (cid:105) state. This is an acceptable approximationbecause the splitting probabilities for g → qq are a lot less than for g → gg . Consequently,there is only a small probability that they are both in the | (cid:105) state at any one time. However,it is possible that this may be a limitation in comparison to current classical parton showeralgorithms provided by packages such as Pythia [1],
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NuQS
Collaboration, S. Harmalkar, H. Lamm, and S. Lawrence,
Quantum Simulation ofField Theories Without State Preparation , [ arXiv:2001.11490 ].[38] A. Y. Wei, P. Naik, A. W. Harrow, and J. Thaler,
Quantum Algorithms for Jet Clustering , Phys. Rev. D (2020), no. 9 094015, [ arXiv:1908.08949 ].[39] K. T. Matchev, P. Shyamsundar, and J. Smolinsky,
A quantum algorithm for modelindependent searches for new physics , [ arXiv:2003.02181 ].[40] IBM Research,
Qiskit, an open source computing framework , .[41] IBM Q Team,
IBM Q 32 Simulator v0.1.547 , .[42] H. Elvang and Y.-t. Huang,
Scattering Amplitudes in Gauge Theory and Gravity . CambridgeUniversity Press, 2015.[43] IBM Q Team,
IBM Q Santiago 5 Qubit Quantum Computer v1.0.3 , .[44] D. Maˆıtre and P. Mastrolia,
S@M, a Mathematica implementation of the spinor-helicityformalism , Comput. Phys. Commun. (2008), no. 7 501–534.[45] F. Cachazo, P. Svrcek, and E. Witten,
MHV vertices and tree amplitudes in gauge theory , JHEP (2004) 006, [ hep-th/0403047 ]. – 31 –
46] R. Britto, F. Cachazo, B. Feng, and E. Witten,
Direct proof of tree-level recursion relation inYang-Mills theory , Phys. Rev. Lett. (2005) 181602, [ hep-th/0501052 ].[47] B. Yirka, Honeywell claims to have built the highest-performing quantum computer available ,Jun, 2020.[48] P. Jurcevic, A. Javadi-Abhari, L. Bishop, I. Lauer, D. F. Bogorin, M. Brink, L. Capelluto,O. Gunluk, T. Itoko, N. Kanazawa, A. Kandala, G. Keefe, K. D. Krsulich, W. Landers, E. P.Lewandowski, D. McClure, G. Nannicini, A. Narasgond, H. Nayfeh, E. Pritchett, M. B.Rothwell, S. Srinivasan, N. Sundaresan, C. Wang, K. X. Wei, C. Wood, J.-B. Yau, E. Zhang,O. Dial, J. Chow, and J. Gambetta,
Demonstration of quantum volume 64 on asuperconducting quantum computing system , 2020.[49]
Ibm’s roadmap for scaling quantum technology , Sep, 2020.[50] T. R. Taylor,
A course in amplitudes , Physics Reports (May, 2017) 1–37.[51] Y. L. Dokshitzer,
Calculation of the Structure Functions for Deep Inelastic Scattering ande+ e- Annihilation by Perturbation Theory in Quantum Chromodynamics. , Sov. Phys. JETP (1977) 641–653.[52] V. Gribov and L. Lipatov, Deep inelastic e p scattering in perturbation theory , Sov. J. Nucl.Phys. (1972) 438–450.[53] G. Altarelli and G. Parisi, Asymptotic freedom in parton language , Nuclear Physics B (1977), no. 2 298 – 318.[54] S. Marzani, G. Soyez, and M. Spannowsky,
Looking inside jets , Lecture Notes in Physics (2019).[55] J. Wentz, I. M. Brancus, A. Bercuci, D. Heck, J. Oehlschlager, H. Rebel, and B. Vulpescu,
Simulation of atmospheric muon and neutrino fluxes with CORSIKA , Phys. Rev. D (2003) 073020, [ hep-ph/0301199 ].[56] P. Schichtel, M. Spannowsky, and P. Waite, Constraining strongly coupled new physics fromcosmic rays with machine learning techniques , EPL (2019), no. 6 61002,[ arXiv:1906.09064 ].[57] IBM Q Team,
IBM Q Valencia 5 Qubit Quantum Computer v1.3.1 , ., .