Fast thermal state preparation beyond native interactions

Abstract

While questions on quantum simulation of ground state physics are mostly focussed on the realization of effective interactions, most work on quantum simulation of thermal physics explores the realization of dynamics towards a thermal mixed state under native interactions. Many open questions that could be answered with quantum simulations, however, involve thermal states with respect to synthetic interactions. We present a framework based solely on unitary dynamics to design quantum simulations for thermal states with respect to Hamiltonians that include non-native interactions, suitable for both present-day digital and analogue devices. By classical means, our method finds the control sequence to reach a target thermal state for system sizes well out of reach of state-vector or density-matrix control methods, even though quantum hardware is required to explicitly simulate the thermal state dynamics. With the illustrative example of the cluster Ising model that includes non-native three-body interactions, we find that required experimental resources, such as the total evolution time, are independent of temperature and criticality.

Figures (10)

• Figure 1: The left panel (a) summarises the target thermal state $\varrho_\mathbf{T}^\beta$ preparation as a four-step process. (i) An optimization, with respect to a figure of merit $\mathcal{J}$, to determine the specific time-dependencies of the system Hamiltonian, $H(t)$, and the initial condition $K_0$ (Eq. \ref{['eq:optimization']}). (ii) The preparation of the thermal state of the initial condition $K_0$, $\varrho^\beta(0)$, either by sampling computation basis states as a chain of excited/unexcited spins, $\ket{z}$, from the corresponding Boltzmann distribution, $p(z)$, (suited to analogue devices), or by a gate sequence (suited to digital devices). (iii) The optimized control sequence, realized by dynamics induced by $H(t)$ found in (i), is applied to the spin-chain, or implemented via gates in the case of digital devices. (iv) An observable of interest is measured. Steps (ii),(iii),(iv) are repeated as many times as are needed to accurately capture the statistics. The right panel (b) shows a sketch of what can be achieved using the present method. The results in Sec. \ref{['sec:results']} demonstrate that one can prepare the unitarily evolved thermal state $\varrho^\beta(t_\text{f})\approx \varrho^\beta_\mathbf{T}$ for a broad range of temperatures, even in critical regions of a phase diagram. Strikingly, for a given number of spins, the required evolution time $t_\text{f}$ (to reach a state within some accuracy of the target) appears to be unaffected by criticality. As such, the results show that one can prepare high-fidelity thermal states (typically $\sim0.99$ in the worst case) even in critical regimes and low temperature.
• Figure 2: Schematic representation of the phase diagram of the cluster Ising model (Eqs. \ref{['eq:targetinvariant']}, \ref{['eq:scale']}), spanned by the vectors $\vec{\lambda}=(\bm{\lambda},0,0)$ (left bottom), $\vec{\lambda}=(0,\bm{\lambda},0)$ (right bottom) and $\vec{\lambda}=(0,0,\bm{\lambda})$ (center top) that define the limiting cases of paramagnetic, Ising-ordered and symmetry-protected topological (SPT) phase. The model exhibits non-critical behaviour with one dominant term in the Hamiltonian, i.e. domains close to the corners of the triangles depicted in green. If two or all three terms in the Hamiltonian are of comparable magnitude (as indicated in grey), the system shows critical behaviour. In the asymptotic limit $n\to\infty$, the critical and non-critical regions are separated by a sharp phase transition; for finite system size the crossover between critical and non-critical behaviour takes place over a finite interval in the parameter space, as indicated by the shaded domains. Seven selected points in the parameter space for which optimal control solutions are discussed in Sec. \ref{['sec:results']} are indicated by blue points $P_1$ to $P_7$, with $P_1$ to $P_4$ in critical and $P_5$ to $P_7$ in non-critical regimes.
• Figure 3: Thermal state infidelity (Eq. \ref{['eq:thermalinfidelity']}) as a function of $\boldsymbol{\lambda}\beta$, with the scale $\boldsymbol{\lambda}$ given in Eq. \ref{['eq:scale']}, for the two exemplary points $P_1$ and $P_7$ depicted in the phase diagram in Fig. \ref{['fig:labelledtriangle']} for a system of $n=11$ spins. The optimal control solution for the thermal state preparation is obtained by optimizing for the operator infidelity, and the final value obtained in the numerical optimization is depicted as a dashed line. The point $P_1$ (left inset) corresponds to critical behaviour with a closing gap; the state fidelity at low temperatures is thus larger than the operator fidelity in contrast to the case of non-critical behaviour depicted in the right inset. In both cases, it is evident that the actual figure of merit (the thermal state infidelity ${\cal J}(\varrho^\beta(t_{\mathrm{f}}),\varrho_\mathbf{T}^\beta)$) remains small even though the proxy (the operator infidelity $\mathcal{J}(K(t_{\mathrm{f}}),K_{\mathbf{T}})$) was optimized.
• Figure 4: Operator infidelity resultant from the optimization for the two exemplary points $P_1$ and $P_7$ depicted in Fig. \ref{['fig:labelledtriangle']} as a function of system size (green diamonds). State fidelities obtained with the corresponding optimal control solutions are depicted for system sizes up to $n=11$ spins both for the ground state ($\beta\to\infty$, red triangles) and for the finite temperature $\beta=2/\bm{\lambda}$ (purple crosses) with the scale $\bm{\lambda}$ (Eq. \ref{['eq:scale']}) of $K_\mathbf{T}$. The ground state infidelity upper bound $\mathcal{B}_\mathrm{gs}$ (Eq. \ref{['eq:gsBound']}) is depicted by empty square points. Similar to Fig. \ref{['fig:thermalstates']}, state infidelities tend to be larger than operator infidelities in critical regions (top panel) and smaller in non-critical regions (bottom panel). In both panels, the operator infidelity is a sufficiently good proxy for the state infidelity to be used as control target.
• Figure 5: Best operator infidelity obtained as result of the optimization as a function of the evolution time $g t_\text{f}$, where $g$ is the interaction constant in the system Hamiltonian $H(t)$ (Eq. \ref{['eq:explicitcontrolham']}). There is a well-defined drop in the infidelity, suggesting that there is a minimal duration of system dynamics required to realize the thermal state preparation. This minimal duration appears independent of the specific target within the cluster Ising model, and it seems to be growing only moderately with the system size $n$.