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We present a quasi-polynomial time classical algorithm that estimates the partition function of quantum many-body systems at temperatures above the thermal phase transition point. It is known that in the worst case, the same problem is NP-hard below this point. Together with our work, this shows that the transition in the phase of a quantum system is also accompanied by a transition in the hardness of approximation. We also show that in a system of n particles above the phase transition point, the correlation between two observables whose distance is at least Ω(log n) decays exponentially. We can improve the factor of log n to a constant when the Hamiltonian has commuting terms or is on a 1D chain. The key to our results is a characterization of the phase transition and the critical behavior of the system in terms of the complex zeros of the partition function. Our work extends a seminal work of Dobrushin and Shlosman on the equivalence between the decay of correlations and the analyticity of the free energy in classical spin models. On the algorithmic side, our result extends the scope of a recent approach due to Barvinok for solving classical counting problems to quantum many-body systems.

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We propose a new no-go theorem by proving the impossibility of constructing a deterministic quantum circuit that iterates a unitary oracle by calling it only once. Different schemes are provided to bypass this result and to approximately realize the iteration. The optimal scheme is also studied. An interesting observation is that for large number of iterations, a trivial strategy like using the identity channel has the optimal performance, and preprocessing, postprocessing, or using resources like entanglement does not help at all. Intriguingly, the number of iterations, when being large enough, does not affect the performance of the proposed schemes.

In this paper, a simple memory limited transmitter for molecular communication is proposed, in which information is encoded in the diffusion rate of the molecules. Taking advantage of memory, the proposed transmitter reduces the ISI problem by properly adjusting its diffusion rate. The error probability of the proposed scheme is derived and the result is compared with the lower bound on error probability of the optimum transmitter. It is shown that the performance of introduced transmitter is near optimal (under certain simplifications). Simplicity is the key feature of the presented communication system: the transmitter follows a simple rule, the receiver is a simple threshold decoder and only one type of molecule is used to convey the information. I. INTRODUCTIONNew applications such as smart drug delivery and health monitoring give rise to the importance of molecular communication, a new paradigm for communication between nanomachines over a short (nanoscale or microscale) range. In molecular communication, information is carried by molecules, rather than electrons or electromagnetic waves [1], [2]. Several types of molecular communication have been considered, among them, diffusion based communication, which corresponds to traditional wireless communication [3], is of great interest, since it does not require any prior communication link infrastructure. In diffusion based communication, the transmitter nanomachine releases information molecules in the environment. These released molecules diffuse randomly until they hit the receiver nanomachine.[4], [5]. Due to the random nature of molecular propagation, diffusion based communication suffers from inter symbol interference (ISI). Several solutions have been proposed to mitigate ISI (e.g. see [6]-[9]). In [10], a new modulation technique, named Molecular Concentration Shift Keying (MCSK), is suggested. Exploiting two types of molecules, while MCSK eliminates the interference from the last transmitted symbol and reduces the error probability, it suffers from interference due to earlier transmissions. A solution based on adding intelligence to receiver is suggested in [11]. where the receiver stores the last decoded bits in memory to make an estimation of current interference level, and uses this estimation to adjust the threshold for decoding the current bit. In [12], a linear and time invariant model is presented and the optimal receiver is derived, under this model. However, this receiver is too complex to be implemented in practice. In [13], the authors considered a deterministic noiseless diffusion channel with memory, and proposed using different symbol durations to deal with ISI by taking into account the channel binary concentration state. They then computed the channel capacity by adapting the Shannon telegraph channel method. In this paper, we propose a simple transmitter which significantly reduces ISI by adaptively adjusting transmission rates to stabilize the rate of molecules at the receiver, enabling the use of a simple fixed threshold receiver. To...