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Large-scale quantum information processing requires the use of quantum error correcting codes to mitigate the effects of noise in quantum devices. Topological error-correcting codes, such as surface codes, are promising candidates as they can be implemented using only local interactions in a two-dimensional array of physical qubits. Procedures such as defect braiding and lattice surgery can then be used to realize a fault-tolerant universal set of gates on the logical space of such topological codes. However, error correction also introduces a significant overhead in computation time, the number of physical qubits, and the number of physical gates. While optimizing fault-tolerant circuits to minimize this overhead is critical, the computational complexity of such optimization problems remains unknown. This ambiguity leaves room for doubt surrounding the most effective methods for compiling fault-tolerant circuits for a large-scale quantum computer. In this paper, we show that the optimization of a special subset of braided quantum circuits is NP-hard by a polynomial-time reduction of the optimization problem into a specific problem called Planar Rectilinear 3SAT., Comment: 9 pages, 9 figures

Quantum communication technologies will play an important role in quantum information processing in the near future as we network devices together. However, their implementation is still a challenging task due to both loss and gate errors. Quantum error correction codes are one important technique to address this issue. In particular, the Quantum Reed-Solomon codes are known to be quite efficient for quantum communication tasks. The high degree of physical resources required, however, makes such a code difficult to use in practice. A recent technique called quantum multiplexing has been shown to reduce resources by using multiple degrees of freedom of a photon. In this work, we propose a method to decompose multi-controlled gates using fewer $\rm{CX}$ gates via this quantum multiplexing technique. We show that our method can significantly reduce the required number of $\rm{CX}$ gates needed in the encoding circuits for the quantum Reed-Solomon code. Our approach is also applicable to many other quantum error correction codes and quantum algorithms, including Grovers and quantum walks., Comment: 9 pages, 11 figures

Distributed quantum information processing is based on the transmission of quantum data over lossy channels between quantum processing nodes. These nodes may be separated by a few microns or on planetary scale distances, but transmission losses due to absorption/scattering in the channel are the major source of error for most distributed quantum information tasks. Of course quantum error detection (QED) /correction (QEC) techniques can be used to mitigate such effects but error detection approaches have severe performance limitations due to the signaling constraints between nodes and so error correction approaches are preferable -assuming one has sufficient high quality local operations. Typically, performance comparisons between loss-mitigating codes assume one encoded qubit per photon. However single photons can carry more than one qubit of information and so our focus in this work is to explore whether loss-based QEC codes utilizing quantum multiplexed photons are viable and advantageous, especially as photon loss results in more than one qubit of information being lost. We show that quantum multiplexing enables significant resource reduction: in terms of the number of single photon sources while at the same time maintaining (or even lowering) the number of two-qubit gates required. Further, our multiplexing approach requires only conventional optical gates already necessary for the implementation of these codes., Comment: 5 pages, 2 figures

We propose a scheme for performing an entanglement-swapping operation within a quantum communications hub (a Bell like measurement) using an anNV-centre’s ∣1ñ « ∣A2ñoptical transition. This is based on the heralded absorption of a photon resonant with that transition. The quantum efficiency of a single photon absorption is low but can be improved by placing the NV centre inside a micro cavity to boost the interaction time and further by recycling the leaked photon back into the cavity after flipping its phase and/or polarization. Throughout this process, the NV is repeatedly monitored via a QND measurement that heralds whether or not the photon absorption has succeeded. Upon success we know a destructive Bell measurement has occurred between that photon and NV centre. Given low losses and high per-pass absorption probability, this scheme should allow the total success probability to approach unity. With long electron spin coherence times possible at low temperatures, this component could be useful within a memory-based quantum repeater or relay.

Quantum networks will support long-distance quantum key distribution (QKD) and distributed quantum computation, and are an active area of both experimental and theoretical research. Here, we present an analysis of topologically complex networks of quantum repeaters composed of heterogeneous links. Quantum networks have fundamental behavioral differences from classical networks; the delicacy of quantum states makes a practical path selection algorithm imperative, but classical notions of resource utilization are not directly applicable, rendering known path selection mechanisms inadequate. To adapt Dijkstra's algorithm for quantum repeater networks that generate entangled Bell pairs, we quantify the key differences and define a link cost metric, seconds per Bell pair of a particular fidelity, where a single Bell pair is the resource consumed to perform one quantum teleportation. Simulations that include both the physical interactions and the extensive classical messaging confirm that Dijkstra's algorithm works well in a quantum context. Simulating about three hundred heterogeneous paths, comparing our path cost and the total work along the path gives a coefficient of determination of 0.88 or better., Comment: 12 pages, 8 figures

We propose an approach to implement quantum repeaters for long-distance quantum communication. Our protocol generates a backbone of encoded Bell pairs and uses the procedure of classical error correction during simultaneous entanglement connection. We illustrate that the repeater protocol with simple Calderbank-Shor-Steane encoding can significantly extend the communication distance, while still maintaining a fast key generation rate.

In game theory, players have continuous expected payoff functions and can use fixed point theorems to locate equilibria. This optimization method requires that players adopt a particular type of probability measure space. Here, we introduce alternate probability measure spaces altering the dimensionality, continuity, and differentiability properties of what are now the game's expected payoff functionals. Optimizing such functionals requires generalized variational and functional optimization methods to locate novel equilibria. These variational methods can reconcile game theoretic prediction and observed human behaviours, as we illustrate by resolving the chain store paradox. Our generalized optimization analysis has significant implications for economics, artificial intelligence, complex system theory, neurobiology, and biological evolution and development., Comment: 11 pages, 5 figures. Replaced for minor notational correction