Determining the proton content with a quantum computer

We present a first attempt to design a quantum circuit for the determination of the parton content of the proton through the estimation of parton distribution functions (PDFs), in the context of high energy physics (HEP). The growing interest in quantum computing and the recent developments of new algorithms and quantum hardware devices motivates the study of methodologies applied to HEP. In this work we identify architectures of variational quantum circuits suitable for PDFs representation (qPDFs). We show experiments about the deployment of qPDFs on real quantum devices, taking into consideration current experimental limitations. Finally, we perform a global qPDF determination from collider data using quantum computer simulation on classical hardware and we compare the obtained partons and related phenomenological predictions involving hadronic processes to modern PDFs.

Figure 1

Operational scheme of a variational quantum circuit. A unitary gate $\mathcal{U}$, depending on some parameters $\theta$, transforms the initial $|0⟩$ state into some output state. This state is measured and used to compute a loss function $\mathcal{L}(\theta )$. The classical optimizer performs an update on the parameters to minimize the value of $\mathcal{L}(\theta )$. New parameters are then sent to the quantum circuit and the loop starts again.

Figure 2

Schematic workflow for the implementation of qPDF.

Figure 3

On the left we show an example of one layer architecture. On the right we present the scheme of a full Ansatz circuit including 8 qubits and entangling gates. The $U^l({\theta }_l,{\gamma }_l,x)$ from the left figure enters the full ansatz as $U^l$. Note that the last layer does not have any entangling gate.

Figure 4

Multiflavor qPDF fits using the weighted Ansatz (orange curves) and the Fourier Ansatz (blue curves) with 5 layers and 8 qubits. The mean value and $1\sigma$ uncertainty of the target PDF data is shown by means of a solid black line and a shaded grey area.

Figure 5

Comparison between single-flavor fits (left) and multiflavor fits (right) for the gluon, up and strange quarks PDFs. For the single-flavor fits the weighted Ansatz (orange curves) and Fourier Ansatz (blue curves) are composed by 1 qubits and 6 layers. On the other hand for the multiflavor fits, the Ansätze are composed by 8 qubits and 5 layers. The mean value and $1\sigma$ uncertainty of the target PDF data is shown by means of a solid black line and a shaded grey area.

Figure 6

Single-flavor fit for all flavors, using the weighted Ansatz, for 5 layers and 8 qubits, that is one qubit per flavor. The red lines represent the prediction of the qPDF model with simulated noise from the IBM Athens processor [50]. Green points are the results of running the circuit on the Athens quantum processor. The mean value and $1\sigma$ uncertainty of the target PDF data is shown by means of a solid black line and a shaded grey area.

Figure 7

Multiflavor fit for all flavors, using the weighted Ansatz, for 5 layers and 8 qubits, that is one qubit per flavor. Blue lines are the mean and the blue shadowed area the $1\sigma$ uncertainty of the circuit measurement results for an ideal noise free quantum device. The red curve refers to simulated circuit measurements using the noise model for the IBM Melbourne processor [50]. Similarly, green and orange curves show simulation results with noise reduced by 50% and 90% respectively. The mean value and $1\sigma$ uncertainty of the target PDF data is shown by means of a solid black line and a shaded grey area.

Figure 8

The error as a function of the error interpolation parameter $t_{\text{error}}$. The $y$-axis is given as the ratio between the error, ${\chi }^2$, and the error on an ideal quantum computer, ${\chi }_{\text{ideal}}^2$.

Figure 9

Predictions for a toy $s\overline{s}$ initiated Drell-Yan process with qPDF and a simplified version of NNPDF3.1 where the positivity constraint has been removed.

Figure 10

${\chi }^2/N$ per experiment grouping. There is a deterioration of the goodness of the fit (measured by the ${\chi }^2$) for some of the experiments for the central value. The goodness of the fit is very similar between the reference and qPDF for most of the experiments being considered.

Figure 11

Distance [as defined by Eq. (14)] between qPDF and NNPDF3.1. When the distance is kept under $d(f_i,r_i)=10$ the two fits are $1\text{−}\sigma$ compatible. All partons except for $u$ and $s$ are below or around the $1\text{−}\sigma$ distance for the entire range considered. Note however, by comparing to Fig. 4 that the fits for both the $u$ and $s$ quarks are compatible in the most relevant regions for these particles.

Figure 12

Fit results for the gluon and the $u$ and $s$ quarks. As previously seen in Fig. 4, qPDF is able to reproduce the features of NNPDF3.1. We now see this is also true when the fit performed by comparing to data and not by comparing directly to the goal function. The differences seen at low-x can be attributed to the lack of data in that region.

Figure 13

Theoretical predictions computed with the method describe in [60] in order to compare the same prediction with three different PDF sets. We note that the predictions for the qPDF set is compatible with both the experimental measurements and the released PDF set. The parton-level calculation has been performed with the $\mathrm{NLOjet}++$ [63] and MCFM [64] tools. (a) Atlas jets data dierential in rapidity [65]. (b) CMS Z dierential in rapidity for xed value of pT [66]. (c) LHCb, Z cross section dierential in rapidity [67].

Figure 14

PDF correlation matrix for flavors in a grid of $x$ points for NNPDF3.1 NNLO (left) and the qPDF (right).