Simulating hadronic physics on noisy intermediate-scale quantum devices using basis light-front quantization

The analogy between quantum chemistry and light-front quantum field theory, first noted by Wilson, serves as motivation to develop light-front quantum simulation of quantum field theory. We demonstrate how calculations of hadron structure can be performed on noisy intermediate-scale quantum devices within the basis light-front quantization (BLFQ) framework. Within BLFQ, relativistic quantum field theories take a form that permits direct application of methods for digital quantum simulation of quantum chemistry, which can be readily scaled into the quantum advantage regime. We calculate the light-front wave functions of pions using an effective light-front Hamiltonian in a basis representation on a current quantum processor. We use the variational quantum eigensolver to find the ground-state energy and the corresponding wave function, which is subsequently used to calculate pion mass radius, decay constant, elastic form factor, and charge radius.

Figure 1

Schematic of the variational quantum eigensolver (VQE). The parameter vector $\vec{\theta }$ completely specifies the Ansatz wave function.

Figure 2

Estimating the magnitude of the inner product $\langle v|\psi \left(\vec{\theta }\right)\rangle$ for fixed $|v\rangle$. Up to the first dashed line, the circuit prepares the VQE Ansatz state by applying the Ansatz circuit $U\left(\vec{\theta }\right)$ to the first set of registers, resulting in the state $|{\mathrm{\Phi }}_1\rangle =|\psi \left(\vec{\theta }\right)\rangle \otimes |0\rangle$. The next rotation $R_{|v\rangle \to {|1\rangle }^{\otimes n}}$ represents any unitary operator that maps the state $|v\rangle$ to the state ${|1\rangle }^{\otimes n}$. Thus, the state $|{\mathrm{\Phi }}_2\rangle$ at the second dashed line is given by $|{\mathrm{\Phi }}_2\rangle =\langle v|\psi \left(\vec{\theta }\right)\rangle |v\rangle \otimes |1\rangle +\left(\right|\psi \left(\vec{\theta }\right)\rangle -\langle v|\psi \left(\vec{\theta }\right)\rangle |v\rangle )\otimes |0\rangle$. The quantity $|\langle v|\psi \left(\vec{\theta }\right)\rangle |=\sqrt{{|{\langle v|\psi \left(\vec{\theta }\right)\rangle |}^2}^{\phantom{\int }}}$ is found as the square root of the probability for the ancilla qubit to collapse into the state $|1\rangle$.

Figure 3

Ansatz circuits for preparing an arbitrary superposition of single-particle Fock states with real coefficients. For the direct encoding (a), we use a generalization of a circuit from [55] for preparation of $W_N$ states. For the binary encoding (b), we use arbitrary state preparation, with all single-qubit rotations replaced by $R_y\left(\theta \right)$ gates, where $R_y\left(\theta \right)$ denotes a single-qubit rotation through an angle $\theta$ about the $y$ axis. After transpiling into native qiskit gates, the circuits look as in Fig. 4.

Figure 4

qiskit circuits for (a) preparing a superposition of single-occupancy states with real amplitudes, (b) preparing a two-qubit state with real amplitudes. Circuits before transpiling are shown in Fig. 3.

Figure 5

Precision vs number of samples for ground-state energy obtained via sampling from the exact distribution. Fitting gives $n\approx 382/{\epsilon }^{2.04}$ (direct encoding) and $n\approx 46/{\epsilon }^{2.1}$ in (compact encoding), confirming the theoretical $n\sim O(1/{\epsilon }^2)$ dependence. Compact encoding shows better convergence due to having shorter circuits on fewer qubits [compare Figs. 3 and 3].

Figure 6

The results of the VQE minimization algorithm in the compact and direct encodings. These were obtained from 8192 samples per term on IBM Vigo machine, with and without measurement error mitigation. The results for the case when the expectation values are evaluated using the state vector are not shown; they reach the exact result after $\sim 15$ steps.

Figure 7

Relative errors in estimates of various observables. These were obtained from 8192 samples per term on IBM Vigo machine, with and without measurement error mitigation. Physically significant observables have a significant contribution from the constant term in their multiqubit representation. Observables are shown with and without the contribution of the constant term. For the GS energy, the error was calculated relative to the second lowest eigenvalue $m_{\rho }^2$. For the compact encoding, measurement error mitigation consistently improves the results.

Figure 8

Pion elastic form factor, as defined in Eq. (27). Pion elastic form factor is used to calculate the charge radius, obtaining the values given in Table 5 [charge radius is defined in Eq. (28)]. Data points for the quantum simulation on the IBM Vigo processor used 8192 samples per term, with and without measurement error mitigation. The results measured on the quantum computer are in good agreement with the exact ones due to the strong contribution to the measurement operators from the identity term. The results for the compact mapping are not shown as they are visually indistinguishable from the exact lines.

Figure 9

Converting a four-qubit state from Jordan-Wigner to Bravyi-Kitaev encoding.