We consider the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant, and where even the unnormalized target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index Sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of Sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of MSE$^{-1}$, while single-level methods require MSE$^{-\xi}$ for $\xi>1$. This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in $1$ and $2$ spatial dimensions, where $\xi$ is $5/4$ and $3/2$, respectively. It is also illustrated on a more challenging log Gaussian Cox model from spatial statistics, where single level complexity is {approximately $\xi=9/4$} and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives {$\xi = 5/4 + \omega$, for $\omega > 0$}, whereas our method is again canonical.