Programmable quantum simulation of anharmonic dynamics

Abstract

Continuous-variable-discrete-variable (CV-DV) quantum simulators offer a natural route to simulating bosonic dynamics relevant to many branches of physics and chemistry. However, programmable simulation of arbitrary dynamics is an outstanding challenge. In particular, simulating anharmonic dynamics, which is ubiquitous across the physical sciences, is challenging due to the highly harmonic nature of oscillators used in CV-DV simulators. Here, we experimentally demonstrate programmable CV-DV quantum simulation of anharmonic dynamics in a range of double-well potentials, implemented in a trapped-ion system. We synthesise the time-evolution operators using a bosonic-quantum-signal-processing subroutine, which allows the potential to be tuned between experiments by controlling classical experimental parameters. We observe coherent dynamics in various double-well potentials, where a wavepacket tunnels through the potential barrier, and we suppress this effect by programmatically introducing asymmetry.

Figures (8)

• Figure 1: Quantum simulation of dynamics in an anharmonic potential.(a) A double-well potential is synthesised using a parabolic contribution and a cosine contribution from a trigonometric gate. (b) A trapped-ion quantum simulator, with the spin providing the qubit and the motion the oscillator. (c) Dynamics in a symmetric double well; a wavepacket initialised in one well tunnels through the barrier to the opposite well.
• Figure 2: Circuit diagram for simulating dynamics in double-well potentials.(a) The experiment consists of three steps. 1. Initialisation of the wavepacket at a chosen position. The potential is a symmetric double well which is not synthesised at this stage (dotted line). The initial position is the minimum of the potential at $-x_\mathrm{min}$. 2. Time evolution of the wavepacket in the double-well potential (solid line) is simulated using the trigonometric gate, $\widetilde{G}_\mathrm{c}$. 3. Measurement of the oscillator state. Dashed box: optional operation to extract the imaginary part of the characteristic function. (b) The trigonometric gate is constructed with two BQSP subroutines. (c) The BQSP subroutine consists of two SDDs and an SQR.
• Figure 3: Reconstruction of motional wavepackets and extraction of position expectation values. Given are representative measurements at $t=4ms$ for a symmetric double well. (a,b) Real and imaginary parts of the characteristic function $\chi(\beta)$. (c) Wigner function $W(x,p)$ obtained as the Fourier transform of $\chi(\beta)$. (d) Probability distribution $P(x)$, obtained as $\int dp\,W(x,p)$. (e) Slope of $\mathop{\mathrm{Im}}\nolimits[\chi(\beta)]$ at $\mathop{\mathrm{Im}}\nolimits[\beta]=0$ determines $\langle x\rangle$ (\ref{['eq:xexpect']}). (f) Finer measurement of $\partial\mathop{\mathrm{Im}}\nolimits[\chi(\beta)]/\partial\mathop{\mathrm{Im}}\nolimits[\beta]$ to determine $\langle x\rangle$. All theoretical models include dephasing and Trotter error.
• Figure 4: Quantum tunnelling in a synthesised symmetric double-well potential.(a) Expectation value $\langle x \rangle$ of the wavepacket's position through time, showing oscillations due to tunnelling through the barrier centred at $x=0$. Experimental points are determined as in \ref{['fig:tomography']}f, with error bars obtained from the uncertainty in the slope of the fit line. We also show the theoretical predictions for the exact dynamics under $H_\mathrm{sim}$ (solid line) and including dephasing and Trotter error (dashed line). (b--d) Reconstructed probability distributions $P(x)$ at $t=0ms$, $t=2ms$ and $4ms$, compared with theoretical predictions (including both dephasing and Trotter error). Also shown in (b) is the target potential and the associated energy levels. The combined probability of occupying the two lowest levels (both below the barrier) is 95%.
• Figure 5: Programmable anharmonic dynamics. Wavepacket evolution for various potentials and initial states. Position expectation values $\langle x \rangle$ are extracted, as functions of time, by 2PFD (\ref{['eq:xapprox']} with $h=0.4$). To engineer asymmetric potentials, we vary the trigonometric-gate angle $\varphi$ from (a)$\varphi=0$ to (b,c,d)$\varphi=-\pi/20$ and (e)$\varphi=-\pi/10$. Error bars derived from spin-measurement noise are smaller than the data markers. Theory includes dephasing, Trotter, and 2PFD errors.