The main focus of my research revolves around the quantum PCP conjecture. The current status is that we believe we have a candidate proof for the unnecessary robustness requirement regarding the quantum Tanner code construction. An initial version of the proof can be found in my master thesis at the good qLDPC codes chapter - My master thesis. The next goal I head to is to show that one can embed computation in these codes. Namely, the existence of a computational problem family (hopefully also NP-hard) that can be reduced to $k$-Local Hamiltonians having the property that their ground space lies inside the good qLDPC space.

We have used a known lower bound on the number of rounds required to compute the disjointness in a two-party (quantum) computation setting to show the impossibility of projecting over a given general state (literally given, without the classical description). This result, though seemingly trivial, was difficult to prove using standard methods (e.g., demonstrating non-linearity as in the no cloning theorem). However, the contradiction of previous communication results is obtained almost immediately arxiv, demonstrating the impossibility of generalized diffusion.

We are attempting to develop an efficient simulation circuit for a given Hamiltonian in the computational basis. Our goal is to create a technique that can be used for practical computations in the future. The draft is still a work in progress. One direction for further research is to ask what properties are expected to characterize the most Hamiltonians in nature and how one can take advantage of having these properties.

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