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There needs to be a gap between the eigenvalue of the groundspace and the first excited state that is inverse-polynomial in the system size, and indeed such a promise is required for the QMA-complete …

The quantum computational model is formalized, equivalently, by the Quantum Turing Machine (QTM) model and by the quantum circuits model, which is predominantly used nowadays. A third, less frequently …

In general, I'd second Joe's advice. But for a quick intro, I'd put Lance Fortnow's and Stephen Fenner's texts on the reading list of computer scientists going quantum.

Circuits made out of general linear operators are $PP$-complete. See the PostBQP paper by Scott Aaronson or Schuch's paper on the computational power of PEPS and tensor network contraction.

Quantum Proofs for Classical Theorems by Andrew Drucker and Ronald de Wolf is a very nice survey on this topic. One of the first classical results obtained by thinking about the problem quantumly was …

On the second question: http://arxiv.org/abs/0905.4755v2 gives a classical QMA-complete eigenvalue problem related Markov chains.

Scott Aaronson has been addressing this topic: http://arxiv.org/abs/0910.4698 Related hardness results: Boson sampling: http://arxiv.org/abs/1011.3245 Commuting circuits: http://arxiv.org/abs/1005. …

This paper elaborates on the ideas sketeched above in detail.

The shortest proof for me is this: By definition of $PostBQP$, $PP=PostBQP$ implies that quantum circuits with deterministic projections onto a measurement outcome can be simulated in $PP$. Following …

The paper "BQP and the Polynomial Hierarchy" by Scott Aaronson directly addresses your question. If P=NP, then PH would collapse. If furthermore BQP were in PH, then no quantum speed-up would be possi …

In arXiv:quant-ph/0409035v2 Buhrman and Spalek present a quantum algorithm beating the Coppersmith-Winograd algorithm in cases where the output matrix has few nonzero entries. Update: There is also a …

There is A Simple Proof that Toffoli and Hadamard are Quantum Universal by Dorit Aharonov which first shows how complex amplitudes can be simulated by real amplitudes over a larger Hilbert space with …

Early distributed systems theory, especially papers by Leslie Lamport et al., has had some impact from Special Relativity to get the correct picture w.r.t. to (fault-tolerant) agreement on a global sy …

The minimum witness size over all witnesses of a span program for a given function equals the generalized adversary bound, as shown e.g. in Theorem 1.7 here. Further, the generalized adversary bound i …

The spectral norm $||H||$ determines the maximum energy involved in driving the evolution of the quantum system and thus the quantum computation. Any quantum evolution could be sped-up by a factor of …