Relaxation versus adiabatic quantum steady-state preparation

Adiabatic preparation of the ground states of many-body Hamiltonians in the closed-system limit is at the heart of adiabatic quantum computation, but in reality systems are always open. This motivates a natural comparison between, on the one hand, adiabatic preparation of steady states of Lindbladian generators and, on the other hand, relaxation towards the same steady states subject to the final Lindbladian of the adiabatic process. In this work we thus adopt the perspective that the goal is the most efficient possible preparation of such steady states, rather than ground states. Using known rigorous bounds for the open-system adiabatic theorem and for mixing times, we are then led to a disturbing conclusion that at first appears to doom efforts to build physical quantum annealers: relaxation seems to always converge faster than adiabatic preparation. However, by carefully estimating the adiabatic preparation time for Lindbladians describing thermalization in the low-temperature limit, we show that there is, after all, room for an adiabatic speedup over relaxation. To test the analytically derived bounds for the adiabatic preparation time and the relaxation time, we numerically study three models: a dissipative quasifree fermionic chain, a single qubit coupled to a thermal bath, and the “spike” problem of $n$ qubits coupled to a thermal bath. Via these models we find that the answer to the “which wins” question depends for each model on the temperature and the system-bath coupling strength. In the case of the “spike” problem we find that relaxation during the adiabatic evolution plays an important role in ensuring a speedup over the final-time relaxation procedure. Thus, relaxation-assisted adiabatic preparation can be more efficient than both pure adiabatic evolution and pure relaxation.

Figure 1

Adiabatic vs relaxation-based quantum-state preparation in a quasifree dissipative system. Gaps are in units of $J$ while times are in units of $J^{-1}$. (a) TTSS as a function of system size. The fit gives ${\tau }_{\mathrm{relax}}\sim n^{3.016}$ for the relaxation in accordance with $\tau \sim 1/{\mathrm{\Delta }}_{\mathrm{relax}}$. For the adiabatic case the exponent ranges between 4.4 using all 16 sizes (from 10 to 60), to 6.5 using the largest 5 sizes. (b) Location of the minimum gap as a function of system size. Lower panels: Adiabatic gap ${\mathrm{\Delta }}_{\mathrm{adia}}$ (c) and relaxation gap ${\mathrm{\Delta }}_{\mathrm{relax}}$ (d) as a function of system size. Both fits give $\mathrm{\Delta }\sim n^{-3}$ in accordance with [44].

Figure 2

(a) The ${log}_{10}$ of the ratio ${\tau }_{\mathrm{adia}}/{\tau }_{\mathrm{relax}}$ in the $(g,T)$ plane for $\epsilon ={10}^{-2}$. Blue indicates that adiabatic preparation is faster, red that relaxation is faster. (b) The ${log}_{10}$ of the ratio ${\mathrm{\Delta }}_{\mathrm{relax}}/{\mathrm{\Delta }}_{\mathrm{adia}}$, which roughly correlates as expected with ${\tau }_{\mathrm{adia}}/{\tau }_{\mathrm{relax}}$. The Hamiltonian minimum gap ${\delta }_{min}=1$. Both $T$ and $g$ are measured in units of ${\delta }_{min}$.

Figure 3

(a) $ln\left({\tau }_{\mathrm{adia}}\right)$ as a function $ln\left({\mathrm{\Delta }}_{\mathrm{adia}}\right)$ for $\epsilon ={10}^{-2}$. The slope to the (left) right of the kink is $-0.85$ ($-0.95$) from a linear fit. (b) $ln\left({\tau }_{\mathrm{adia}}\right)$ as a function $ln\left({\mathrm{\Delta }}_{\mathrm{relevant}}\right)$ for $\epsilon ={10}^{-2}$. The slope for the $T=0.025$ case is $-0.975$ from a linear fit. Here ${\tau }_{\mathrm{adia}}$ is measured in units of ${\delta }_{\min }^{-1}$ and ${\mathrm{\Delta }}_{\mathrm{relevant}}$ is measured in units of ${\delta }_{\min }$, where ${\delta }_{\min }$ is the minimum Hamiltonian gap.

Figure 4

(a) ${\mathrm{\Delta }}_{\mathrm{adia}}$ as a function of the system-bath coupling strength $g$, for two different temperatures. (b) Modulus of the eigenvalues ${\lambda }_{1,0}$ and ${\lambda }_{1,1}$ at their respective minimum points ($s=1/2$), as a function of coupling $g$, for $T=0.025$. Their crossing explains the kink seen in (a). The quantities ${\mathrm{\Delta }}_{\mathrm{adia}}$, $\lambda$, and $g$ are measured in units of the minimum Hamiltonian gap ${\delta }_{\min }$.

Figure 5

(a) The ${log}_{10}$ of the ratio ${\mathrm{\Delta }}_{\mathrm{adia}}/{\mathrm{\Delta }}_{\mathrm{relevant}}$. (b) The ${log}_{10}$ of the ratio ${\mathrm{\Delta }}_{\mathrm{relax}}/{\mathrm{\Delta }}_{\mathrm{relevant}}$. The Hamiltonian minimum gap ${\delta }_{min}=1$. Both $T$ and $g$ are measured in units of ${\delta }_{min}$.

Figure 6

Closed-system simulation results for the spike problem. (a) The time required to reach a TND of $\epsilon =0.01$ from the ground state (the zero-temperature Gibbs state) via evolution generated by the Hamiltonian (31). The solid line is the best fit of $ln\tau =1.485lnn+2.051$. (b) The TND from the instantaneous zero-temperature Gibbs state as a function of the dimensionless parameter $s$ for $n=20$ and $\tau =759.5$. The system remains very close to the instantaneous ground state ($>0.99$ overlap squared) during the evolution. Here $\tau$ is measured in terms of the inverse of the minimum gap of $f\left(z\right)$.

Figure 7

(a) Time to reach a TND of $\epsilon ={10}^{-2}$ for adiabatic preparation and relaxation, for spectral density [Eq. (27)] parameters $g=1$ and $\beta =1,10$. The relaxation time grows exponentially over the range of sizes shown, while the adiabatic preparation time decreases with growing system size. (b) The product of the relaxation gap and the relaxation time for $\beta =1,g=1$ exhibits polynomial scaling. The quantities $\tau$ and $\beta$ are measured in terms of the inverse of the minimum gap of $f\left(z\right)$, while $g$ is measured in terms of the minimum gap of $f\left(z\right)$.

Figure 8

(a)–(c) Time to reach a given TND of $\epsilon$ for adiabatic preparation of the “spike” thermal state. (a) $\beta =1$ and $\epsilon ={10}^{-2}$, (b) $\beta =10$ and $\epsilon ={10}^{-2}$, (c) $\beta =1$ and $g=1$. (d) TND between the evolved state and the final thermal state for $n=20$ and $g=\beta =1$. The solid line is a best fit to $a+b/\tau$. Same units as in Fig. 7.

Figure 9

TND of the evolved state from the instantaneous thermal state for $\beta =10$ (left) and $\beta =1$ (right) and system-bath coupling $g^2=1$ (top) and $g^2={10}^{-2}$ (bottom). For all $n$ values shown, $\tau$ is chosen so that $\epsilon ={10}^{-2}$. Same units as in Fig. 7.

Figure 10

Real part of the Lindbladian gap as a function of the dimensionless time $s$ for $\beta =10$ (left) and $\beta =1$ (right). The system-bath coupling is $g^2=1$ (top) and $g^2={10}^{-2}$ (bottom). The vertical dashed line is the location of the minimum Hamiltonian gap. The slope discontinuities are due to eigenvalue crossings. Same units as in Fig. 7.

Figure 11

(a) Scaling of the estimates given by Eqs. (21a) (denoted by $B_{\mathrm{int}}$) and (21b) (denoted by $B_{max}$), in the low-temperature setting ($\beta =10$) and $g^2=1$. Also shown is ${\tau }_{\epsilon }$, the minimum time such that the TND from the instantaneous Gibbs state is $\le \epsilon ={10}^{-2}\forall s\in [0,1]$. This quantity and $B_{\mathrm{int}}$ exhibit very similar scaling. (b) Time to reach a TND of $\epsilon ={10}^{-2}$ for adiabatic preparation and relaxation, with $g^2={10}^{-2}$ and $\beta =0.1$. At this relatively high temperature relaxation scales better than adiabatic preparation. Same units as in Fig. 7.