A Study of Optimal 4-bit Reversible Toffoli Circuits and Their Synthesis

Optimal synthesis of reversible functions is a non-trivial problem. One of the major limiting factors in computing such circuits is the sheer number of reversible functions. Even restricting synthesis to 4-bit reversible functions results in a huge search space (16! {\approx} 2^{44} functions). The output of such a search alone, counting only the space required to list Toffoli gates for every function, would require over 100 terabytes of storage. In this paper, we present two algorithms: one, that synthesizes an optimal circuit for any 4-bit reversible specification, and another that synthesizes all optimal implementations. We employ several techniques to make the problem tractable. We report results from several experiments, including synthesis of all optimal 4-bit permutations, synthesis of random 4-bit permutations, optimal synthesis of all 4-bit linear reversible circuits, synthesis of existing benchmark functions; we compose a list of the hardest permutations to synthesize, and show distribution of optimal circuits. We further illustrate that our proposed approach may be extended to accommodate physical constraints via reporting LNN-optimal reversible circuits. Our results have important implications in the design and optimization of reversible and quantum circuits, testing circuit synthesis heuristics, and performing experiments in the area of quantum information processing.Comment: arXiv admin note: substantial text overlap with arXiv:1003.191

Programming Quantum Computers Using Design Automation

Recent developments in quantum hardware indicate that systems featuring more than 50 physical qubits are within reach. At this scale, classical simulation will no longer be feasible and there is a possibility that such quantum devices may outperform even classical supercomputers at certain tasks. With the rapid growth of qubit numbers and coherence times comes the increasingly difficult challenge of quantum program compilation. This entails the translation of a high-level description of a quantum algorithm to hardware-specific low-level operations which can be carried out by the quantum device. Some parts of the calculation may still be performed manually due to the lack of efficient methods. This, in turn, may lead to a design gap, which will prevent the programming of a quantum computer. In this paper, we discuss the challenges in fully-automatic quantum compilation. We motivate directions for future research to tackle these challenges. Yet, with the algorithms and approaches that exist today, we demonstrate how to automatically perform the quantum programming flow from algorithm to a physical quantum computer for a simple algorithmic benchmark, namely the hidden shift problem. We present and use two tool flows which invoke RevKit. One which is based on ProjectQ and which targets the IBM Quantum Experience or a local simulator, and one which is based on Microsoft's quantum programming language Q$\#$.Comment: 10 pages, 10 figures. To appear in: Proceedings of Design, Automation and Test in Europe (DATE 2018

Remarks on Quantum Modular Exponentiation and Some Experimental Demonstrations of Shor's Algorithm

An efficient quantum modular exponentiation method is indispensible for Shor's factoring algorithm. But we find that all descriptions presented by Shor, Nielsen and Chuang, Markov and Saeedi, et al., are flawed. We also remark that some experimental demonstrations of Shor's algorithm are misleading, because they violate the necessary condition that the selected number $q=2^s$, where $s$ is the number of qubits used in the first register, must satisfy $n^2 \leq q < 2n^2$, where $n$ is the large number to be factored.Comment: 12 pages,5 figures. The original version has 6 pages. It did not point out the reason that some researchers took for granted that quantum modlar exponentiation is in polynomial time. In the new version, we indicate the reason and analyze some experimental demonstrations of Shor's algorithm. Besides, the author Zhenfu Cao is added to the version for his contribution. arXiv admin note: text overlap with arXiv:1409.735