Quantum computation of mass gap in an asymptotically free theory

TL;DR

This work introduces a quantum-computation strategy to extract the mass gap $m=E_1-E_0$ in a lattice field theory by measuring a dipole observable that directly couples the ground and first excited states, thereby avoiding precision loss from subtracting large energies. The authors implement this method on a finite-dimensional fuzzy $O(3)$ sigma-model (the fuzzy sphere) and validate it through strong-coupling hardware experiments and weak-coupling ideal simulations, using Trotterized time evolution and Pauli-string decompositions. Strong-coupling results on current quantum devices show damped oscillations from which a dominant frequency is extracted, albeit with a small discrepancy relative to the exact value due to noise and imperfect state preparation; weak-coupling simulations confirm agreement with exact results when extrapolating to zero time step. Collectively, the work demonstrates a viable, though hardware-limited, approach to probing nonperturbative spectra in lattice field theories and highlights the need for error-corrected quantum computers to scale to larger systems and closer to the continuum limit.

Abstract

In relativistic field theories, the mass spectrum is given by the difference between the energy of the vacuum and the excited states. Near the continuum limit, the cancellation between these two values leads to loss of precision. We propose a method to extract the mass gap directly using quantum computers and apply it to a particular version of the nonlinear $σ$-model with the correct continuum limit and perform calculations in quantum hardware (at strong coupling) and simulation in classical computers (at weak coupling).

Figures (6)

• Figure 1: Diagram depicting the "Heisenberg comb" form of the fuzzy $\sigma$-model Bhattacharya:2020gpm. The "head" qubits are denote by $h_n$ (red squares) and the "fuzz" qubits are denoted by $f_n$ (blue circles) at site $l=0$, $1,\dots$, $L-1$.
• Figure 2: Circuit for the singlet Bell state. Wires represent the head ($2l$) and fuzz ($2l+1$) qubits for the site $l$. $H$ is the Hadamard gate.
• Figure 3: Circuit for $\exp[-i\alpha(X_lX_k+Y_lY_k+Z_lZ_k)]$.
• Figure 4: Fits to QPU data. Only the largest 3 lattice box sizes shown. The complete set of fits is in Table \ref{['table:QPU']}. Data up to $t=5.2$ were used in the fits.
• Figure 5: Top left panel: Large volume $L=4$ ideal simulation data (green data points) for $\langle d\rangle$ vs time $t$ at weak coupling $g=0.6$ with $\lambda=0.5$ in Eq. (\ref{['eq:extrapolate']}), $t_\text{prep}=0.1$, and Trotter steps $\Delta t =0.1$. The solid (red) curve is a fit of the form in Eq. (\ref{['eq:fit_damped']}) whose fit parameters are provided in the text. Top right panel: Extrapolation of the gap $\omega$ to $\Delta t\rightarrow 0$ for $g=0.6$, $L=4$. The (black) data points are the gaps extracted from ideal simulations at $\Delta t$ values as indicated. The solid (red) curve is a polynomial fit whose coefficients are indicated in the boxed insert. The extracted value $\omega=0.05461\pm0.0001$ is to be compared with the exact numerical result $\omega_\text{exact}=0.0541$. We have indicated only the errors from the $\chi^2$ fit. As discussed in the text, a cubic $\Delta t^3$ term in the fit brings the extracted $\omega$ value in better agreement with the exact result. Bottom left panel: Large volume $L=10$ ideal simulation data (green data points) for $\langle d\rangle$ at weak coupling $g=0.6$ with $\lambda=0.1$ in Eq. (\ref{['eq:extrapolate']}), $t_\text{prep}=0.01$, and Trotter steps $\Delta t =0.4$. Rest of the notation is the same as the top left panel. The solid (red) curve fit parameters are provided in the text. Bottom right panel: Extrapolation of the gap $\omega$ to $\Delta t\rightarrow 0$ for $g=0.6$, $L=10$. Rest of the notation is the same as the top right panel. The solid (red) curve is a polynomial fit with coefficients as indicated in the boxed insert. The extracted value $\omega=0.043\pm0.0003$ is in agreement with the exact numerical result $\omega_\text{exact}=0.0428$. We have indicated only the errors from the $\chi^2$ fit.