In this paper, the problem of constructing an efficient quantum circuit for the imple-mentation of an arbitrary quantum computation is addressed. To this end, a basic blockbased on the cosine-sine decomposition method is suggested which contains l qubits.In addition, a previously proposed quantum-logic synthesis method based on quantumShannon decomposition is recursively applied to reach unitary gates over l qubits. Then,the basic block is used and some optimizations are applied to remove redundant gates.It is shown that the exact value of l affects the number of one-qubit and CNOT gates inthe proposed method. In comparison to the previous synthesis methods, the value of l isexamined consequently to improve either the number of CNOT gates or the total numberof gates. The proposed approach is further analyzed by considering the nearest neighborlimitation. According to our evaluation, the number of CNOT gates is increased by atmost a factor of 5/3 if the nearest neighbor interaction is applied.
In this paper, the problem of constructing an efficient quantum circuit for the imple-mentation of an arbitrary quantum computation is addressed. To this end, a basic blockbased on the cosine-sine decomposition method is suggested which contains l qubits.In addition, a previously proposed quantum-logic synthesis method based on quantumShannon decomposition is recursively applied to reach unitary gates over l qubits. Then,the basic block is used and some optimizations are applied to remove redundant gates.It is shown that the exact value of l affects the number of one-qubit and CNOT gates inthe proposed method. In comparison to the previous synthesis methods, the value of l isexamined consequently to improve either the number of CNOT gates or the total numberof gates. The proposed approach is further analyzed by considering the nearest neighborlimitation. According to our evaluation, the number of CNOT gates is increased by atmost a factor of 5/3 if the nearest neighbor interaction is applied.
In this paper, the problem of constructing an efficient quantum circuit for the implementation of an arbitrary quantum computation is addressed. To this end, a basic block based on the cosine-sine decomposition method is suggested which contains l qubits. In addition, a previously proposed quantum-logic synthesis method based on quantum Shannon decomposition is recursively applied to reach unitary gates over l qubits. Then, the basic block is used and some optimizations are applied to remove redundant gates. It is shown that the exact value of l affects the number of one-qubit and CNOT gates in the proposed method. In comparison to the previous synthesis methods, the value of l is examined consequently to improve either the number of CNOT gates or the total number of gates. The proposed approach is further analyzed by considering the nearest neighbor limitation. According to our evaluation, the number of CNOT gates is increased by at most a factor of 35 if the nearest neighbor interaction is applied.
In this paper, the problem of constructing an efficient quantum circuit for the implementation of an arbitrary quantum computation is addressed. To this end, a basic block based on the cosine-sine decomposition method is suggested which contains l qubits. In addition, a previously proposed quantum-logic synthesis method based on quantum Shannon decomposition is recursively applied to reach unitary gates over l qubits. Then, the basic block is used and some optimizations are applied to remove redundant gates. It is shown that the exact value of l affects the number of one-qubit and CNOT gates in the proposed method. In comparison to the previous synthesis methods, the value of l is examined consequently to improve either the number of CNOT gates or the total number of gates. The proposed approach is further analyzed by considering the nearest neighbor limitation. According to our evaluation, the number of CNOT gates is increased by at most a factor of 35 if the nearest neighbor interaction is applied.
The pressure of fundamental limits on classical computation and the promise of exponential speedups from quantum effects have recently brought quantum circuits (Proc. R. Soc. Lond. A, Math. Phys. Sci., vol. 425, p. 73, 1989) to the attention of the electronic design automation community (Proc. 40th ACM/IEEE Design Automation Conf., 2003), (Phys. Rev. A, At. Mol. Opt. Phy., vol. 68, p. 012318, 2003), (Proc. 41st Design Automation Conf., 2004), (Proc. 39th Design Automation Conf., 2002), (Proc. Design, Automation, and Test Eur., 2004), (Phys. Rev. A, At. Mol. Opt. Phy., vol. 69, p. 062321, 2004), (IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., vol. 22, p. 710, 2003). Efficient quantum-logic circuits that perform two tasks are discussed: 1) implementing generic quantum computations, and 2) initializing quantum registers. In contrast to conventional computing, the latter task is nontrivial because the state space of an n-qubit register is not finite and contains exponential superpositions of classical bitstrings. The proposed circuits are asymptotically optimal for respective tasks and improve earlier published results by at least a factor of 2. The circuits for generic quantum computation constructed by the algorithms are the most efficient known today in terms of the number of most expensive gates [quantum controlled-NOTs (CNOTs)]. They are based on an analog of the Shannon decomposition of Boolean functions and a new circuit block, called quantum multiplexor (QMUX), which generalizes several known constructions. A theoretical lower bound implies that the circuits cannot be improved by more than a factor of 2. It is additionally shown how to accommodate the severe architectural limitation of using only nearest neighbor gates, which is representative of current implementation technologies. This increases the number of gates by almost an order of magnitude, but preserves the asymptotic optimality of gate counts.
We show how to implement an arbitrary two-qubit unitary operation using any of several quantum gate libraries with small a priori upper bounds on gate counts. In analogy to library-less logic synthesis, we consider circuits and gates in terms of the underlying model of quantum computation, and do not assume any particular technology. As increasing the number of qubits can be prohibitively expensive, we assume throughout that no extra qubits are available for temporary storage. Using quantum circuit identities, we improve an earlier lower bound of 17 elementary gates by Bullock and Markov to 18, and their upper bound of 23 elementary gates to 18. We also improve upon the generic circuit with six CNOT gates by Zhang et al. (our circuit uses three), and that by Vidal and Dawson with 11 basic gates (we use 10). We study the performance of our synthesis procedures on two-qubit operators that are useful in quantum algorithms and communication protocols. With additional work, we find small circuits and improve upon previously known circuits in some cases.
Quantum circuits currently constitute a dominant model for quantum computation according to M. Nielsen and I. Chuang (2000). Our work addresses the problem of constructing quantum circuits to implement an arbitrary given quantum computation, in the special case of two qubits. We pursue circuits without ancilla qubits and a small number of elementary quantum gates by A. Barenco et al. (1995) and G. Song and A. Klappenecker (2003) as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 17 gates: 15 one-qubit rotations and 2 CNOTs. To this end, we constructively prove a worst-case upper bound to 23 elementary gates, of which at most 4 (CNOTs) entail multi-qubit interactions. Our analysis shows that previous known synthesis algorithms, although more general, entail much larger quantum circuits than ours in the special case of two qubits. One such algorithm according to G. Cybenko (2001) has a worst case of 61 gates of which 18 may be CNOTs. Our technique rely on the KAK decomposition from Lie theory as well as the polar and spectral (symmetric Shur) matrix decompositions from numerical analysis. They are related to the canonical decomposition of two-qubit gate with respect to the "magic basis" of phase-shifted Bell states by N. Khaneja et al. (2001) and M. Lewenstein et al. (2001). We extend this decomposition in terms of elementary gates for quantum computation.
This paper proposes an optimized quantum block-ZXZ decomposition method that results in more optimal quantum circuits than the quantum Shannon decomposition, which was presented in 2005 by M. M\"ott\"onen, and J. J. Vartiainen [in Trends in quantum computing research, edited by S. Shannon (Nova Science Publishers, 2006) Chap. 7, p. 149, arXiv:quant-ph/0504100]. The decomposition is applied recursively to generic quantum gates, and can take advantage of existing and future small-circuit optimizations. Because our method uses only single-qubit gates and uniformly controlled rotation-Z gates, it can easily be adapted to use other types of multi-qubit gates. With the proposed decomposition, a general three-qubit gate can be decomposed using 19 cnot gates (rather than 20). For general n-qubit gates, the proposed decomposition generates circuits that have $\frac{22}{48}{4}^{n}\ensuremath{-}\frac{3}{2}{2}^{n}+\frac{5}{3}$ cnot gates, which is less than the best-known exact decomposition algorithm by $({4}^{n\ensuremath{-}2}\ensuremath{-}1)/3$ cnot gates.