The preservation of security policies goes beyond the classical notion of compiler correctness which only enforces the preservation of semantics. In contrast to the semantics which is defined by individual traces, many security properties are characterized by two or more traces. For example, indistinguishability-based properties like non-interference are characterized by two traces while robust non-interference as well as absence of speculative side-channel leaks are characterized by four traces. Several standard compiler optimisations are prone to break standard security policies. Existing approaches to secure compilation are very restrictive with respect to the compiler optimisations that they permit or to the security policies they support. We present hyperproperty simulations, a novel framework to secure compilation that models the preservation of arbitrary $k$-hyperproperties during compilation and overcomes several limitations of existing approaches. We show the expressiveness of hyperproperty simulations by designing and proving a leakage-based non-interference-preserving dead code elimination pass, a standard compiler optimisation that could be handled in a very limited way by existing approaches. Our results are verified in the Coq proof assistant.