Quantum computation of magnon spectra

We demonstrate quantum computation of two-point correlation functions for a Heisenberg spin chain. Using the IBM Q 20-qubit quantum machines, we find that, for two sites, the correlation functions produce the exact results reliably. For four sites, results from the IBM Q 20-qubit Tokyo quantum computer are noisy due to read out errors and decoherence. Nevertheless, the correlation functions retain the correct spectral information. This is illustrated in the frequency domain by accurately extracting the magnon energies from peaks in the spectral function.

Figure 1

(a) Quantum circuit for the two-point correlator and (b) layout used on the IBM Q 20-qubit Tokyo machine. $H$ is the Hadamard rotation.

Figure 2

Quantum circuit for the evaluation of $exp(-i{\mathcal{H}}_{12}t).X$ denotes a Pauli $X$, whereas $R_{\alpha }\left(\theta \right)$ denotes a rotation by $\theta$ about the axis $\alpha$.

Figure 3

Real and imaginary parts of the (a) $zz$ and (b) $xx$ correlation functions for the two-site (a) antiferromagnetic and (b) ferromagnetic Heisenberg models. The solid lines are the exact solution, and the circles and squares are the results obtained from the IBM Q hardware. Note that the period of the quantum computation is well reproduced, even though the amplitude of the oscillations is reduced. The computations were performed on IBM Q Tokyo and Almaden for (a) and (b), respectively.

Figure 4

Quantum circuit for the evaluation of $exp(-i\mathcal{H}t)$ for a four-site ferromagnetic Heisenberg chain. For this particular system, the full time evolution can be factorized into four pairwise applications of $exp(-i{\mathcal{H}}_{12}t)$. Due to the topology of the layout, swap gates may be eliminated.

Figure 5

$xx$ spin-spin correlation function for the four-site ferromagnetic Heisenberg model. The measured raw data (the circles, from the IBM Q 20-qubit Tokyo) are compared with the analytic (solid lines) results.

Figure 6

$xx$ spin-spin correlation function. (a) Comparison of the error-mitigated data after just a readout correction (green) with both a read out and phase-and-scale [labeled (PaS)] correction (blue) and the corresponding analytical result (red). (b) Comparison of the phase-and-scale and readout corrected data (blue), the Fourier fit as discussed in the main text (orange), and the analytic result (red). The top and bottom rows show the real and imaginary parts of the function, respectively. Clearly, full error mitigation is required to extract meaningful results.

Figure 7

The dynamic spin susceptibility ${\left|S(q,\omega )\right|}^2$. It is obtained from the $xx$ spin-spin correlation functions for the four-site ferromagnetic Heisenberg model. Readout corrected, and readout $+$ phase-and-scale corrected data are compared with the analytic result.

Figure 8

${\left|S(q,\omega )\right|}^2$ for all the four cases along with the magnon dispersion curve. The fit plot has peaks at the frequencies obtained by fitting the data. The raw-data and phase-and-scale error mitigated plots have a finite width because of the finite number of data points in the time domain. The raw-data plot is scaled up. In all cases, peaks at the expected magnon frequencies are clearly visible; for the raw-data and phase-and-scale corrected data, there is leakage to other channels.