Quantum computation of hadron scattering in a lattice gauge theory

TL;DR

This work establishes the potential of quantum computers in simulating hadron-scattering processes in strongly interacting gauge theories in 1+1 dimensions and demonstrates the critical role of high-fidelity initial states for precision measurements of state-sensitive observables, such as S-matrix elements.

Abstract

We present a digital quantum computation of two-hadron scattering in a $Z_2$ lattice gauge theory in 1+1 dimensions. We prepare well-separated single-particle wave packets with desired momentum-space wavefunctions, and simulate their collision through digitized time evolution. Multiple hadronic wave packets can be produced using the efficient, systematically improvable algorithm of this work, achieving high fidelity with the target initial state. Specifically, employing a trapped-ion quantum computer (IonQ Forte), we prepare up to three meson wave packets using 11 and 27 system qubits, and simulate collision dynamics of two meson wave packets for the smaller system. Results for local observables are consistent with numerical simulations at early times, but decoherence effects limit evolution into long times. We demonstrate the critical role of high-fidelity initial states for precision measurements of state-sensitive observables, such as $S$-matrix elements. Our work establishes the potential of quantum computers in simulating hadron-scattering processes in strongly interacting gauge theories.

Figures (16)

• Figure 1: The $j^{\rm th}$-order ansatz infidelity $1-F$ (in logarithmic scale) as a function of Hamiltonian parameters $\epsilon$ and $m_f$ for $N=10$. Each column corresponds to a non-negative momentum $k$ in the Brillouin zone, and each row corresponds to the order of the ansatz $j$. Higher fidelity is observed for $\ket{k^{(j)}_{\rm op}}$ when larger values of $m_f, |\epsilon|$, and a higher ansatz order $j$ are used. For example, $F=0.99$ is achieved at $j=3$ in the entire parameter space for all $k$. The region to the left of the cyan contour corresponds to the existence of a low-energy non-mesonic excitation that is not captured by our mesonic ansatz.
• Figure 2: Shown is the circuit used in this work for simulating scattering in a (1+1)D $Z_2$ LGT in the MGF. The circuit acts on $N+1$ system qubits representing the lattice associated with $N$ staggered sites, and one or more ancilla qubits denoted by $a_{1,2,t}$. The system qubits are initialized in the SCV state $\ket{\Omega}_\text{SCV}$, as shown by the first vertical dotted line, while each ancilla starts in the $\ket{0}$ state. The circuit consists of three subcircuit modules. The vertical dotted lines indicate the state of the system qubits after the application of each module. The first module prepares the ground state, $\ket{\Omega}$, of the Hamiltonian in Eq. \ref{['eq: Z2 Ham MGF JW in Hh Hm He']} using the circuit block $\mathcal{Q}_{\rm GS}$ with parameters $\theta^{h*}$ and $\theta^{m*}$. The second module prepares the initial scattering state composed of two well-separated input wave packets, resulting in the state $\ket{\Psi_1,\Psi_2}$. It requires at least one ancilla qubit to prepare the initial state, shown here with a black solid line. Alternatively, it can also be applied using an extra ancilla qubits, shown with a dotted black line, to improve the accuracy of preparing the target state, as explained in the text. Finally, the last module, $\mathcal{Q}_{\rm Trott}$, performs the unitary time evolution $U(t)=e^{-itH}$ under the Hamiltonian $H$ in Eq. \ref{['eq: Z2 Ham MGF JW in Hh Hm He']}. The circuit components in red are to perform a Hadamard test to compute the return probability of the initial state as a function of time; they can be omitted when measuring only the expectation values of diagonal operators. The Hadamard test requires an additional ancilla and a controlled application of $\mathcal{Q}_{\rm Trott}$ with control on the ancilla. Here, $\mathsf{H}$ denotes the Hadamard gate and $R^{\textbf{x}}(\theta) \coloneq e^{-\frac{i}{2}\theta \sigma^{\textbf{x}}}$. To compute the return probability, as shown in Appendix \ref{['app: Hadamard test']}, one needs to separately implement either the $R^{\textbf{x}}(\frac{\pi}{4})$ or the $\mathsf{H}$ gate as the last operation on the $a_t$ ancilla, which is denoted in the circuit by $R^{\textbf{x}}(\frac{\pi}{4})/\mathsf{H}$. Details of each module and their constituent circuits are discussed in Sec. \ref{['sec: Quantum Algorithm and Circuit Design']}.
• Figure 3: Shown is the circuit $\mathcal{Q}_{\rm GS}$ that prepares the interacting ground state $\ket{\Omega}$ of the Hamiltonian in Eq. \ref{['eq: Z2 Ham MGF JW in Hh Hm He']}. This circuit is parameterized by two parameters, $\theta^h$ and $\theta^m$. The circuit $\mathcal{Q}_{\rm GS}$ acts on the strong-coupling vacuum $\ket{\Omega}_\text{SCV}$, which is given by alternating $\ket{0}$ and $\ket{1}$ states on qubits that represent fermion lattice sites, labeled here with a subscript $f_i$ for the $i^{\rm th}$ site, and $\ket{0}$ state for the qubit that represents the bosonic link in the MGF, denoted here by the subscript $b_{N-1}$. The two-qubit gate are defined as $R^{\textbf{xx}}(\theta) \coloneq e^{-\frac{i}{2}\theta\,\sigma^{\textbf{x}}_{f_{i_1}} \sigma^{\textbf{x}}_{f_{i_2}}}$ and $R^{\textbf{yy}}(\theta) \coloneq e^{-\frac{i}{2}\theta \,\sigma^{\textbf{y}}_{f_{i_1}} \sigma^{\textbf{y}}_{f_{i_2}}}$, where $i_1$ and $i_2$ denote fermion lattice sites each gate involves. The three-qubit gates $R^{\textbf{x}\tilde{\textbf{x}}\textbf{x}}(\theta)$ and $R^{\textbf{y}\tilde{\textbf{x}}\textbf{y}}(\theta)$ are defined in the similar manner with the overhead tilde indicating the qubit operation on the bosonic link qubit $b_{N-1}$, and $\alpha_N$ is defined below Eq. \ref{['eq: Hh Hm Hepsilon defs set']}. Finally, the parameters $\theta^{h*}$ and $\theta^{m*}$ are obtained using a VQE method that minimize the energy of the state prepared by the circuit with respect to the Hamiltonian in Eq. \ref{['eq: Z2 Ham MGF JW in Hh Hm He']}.
• Figure 4: Two different ways of creating the two wave-packet initial scattering state are shown here in (a) and (b). In the former case, the $\mathcal{Q}_{\rm WP}$ circuit that prepares a single-particle wave packet is acted twice using the same ancilla $a_1$ with different wave-packet profiles, $\Psi_1$ and $\Psi_2$, as input. In the latter case, the two applications of $\mathcal{Q}_{\rm WP}$ circuit use two different ancilla $a_1$ and $a_2$. The non-participating ancilla in a $\mathcal{Q}_{\rm WP}$ is indicated with a line crossing over the corresponding circuit block in (b). The method in (a) yields a larger systematic error compared to (b), leading to a lower-fidelity wave-packet state, as discussed in the text. Detailed decomposition of each $\mathcal{Q}_{\rm WP}(\Psi)$ is presented in Appendix \ref{['app: QWP MGF circuit']}.
• Figure 5: (a) Shown is the Trotterized time-evolution module $\mathcal{Q}_{\rm Trott}$ from Fig. \ref{['fig: circuit flow']}, in terms of its constituents according to Eq. \ref{['eq: Trotter time evolution operator U def']}. The circuit blocks denote one Trotter step and the dots inside the circuit indicate repeated application of this structure for $n_t$ Trotter step. Circuits for $e^{-\frac{i}{2}H^h \delta t}$ and $e^{-\frac{i}{2}H^m \delta t}$ can be obtained from Fig. \ref{['fig: GS circuit']} as discussed in the text, and the circuit for $e^{-i \epsilon H^{\epsilon}\delta t}$ is given in (b). The qubit labels for both (a) and (b), and the single-qubit gate $R^{\textbf{z}}$ have been defined in the caption of Fig. \ref{['fig: GS circuit']}. The dotted rows in (b) indicate similar circuit structure for qubits from $f_2$ to $f_{N-3}$. The filled (unfilled) CNOT control indicates that the target $\tilde{\sigma}^{\textbf{x}}$ gate on the boson-link qubit is applied if the control even (odd) matter-site qubit is in $\ket{1}$ ($\ket{0}$) implying a presence of fermion (anti-fermion); the $R^{\textbf{z}}$ rotation that follows this operation adds the appropriate phase to the state. A detailed explanation of its structure is given in the text.