1000 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 25, NO. 6, 2006 Synthesis of Quantum-Logic Circuits Vivek V. Shende, Stephen S. Bullock, and Igor L. Markov, Senior Member, IEEE Abstract—The pressure of fundamental limits on classical com- I. INTRODUCTION putation and the promise of exponential speedups from quantum effects have recently brought quantum circuits (Proc. R. Soc. S THE ever-shrinking transistor approaches atomic pro- Lond. A, Math. Phys. Sci., vol. 425, p. 73, 1989) to the atten- A portions, Moore’s law must confront the small-scale tion of the electronic design automation community (Proc. 40th granularity of the world: We cannot build wires thinner than ACM/IEEE Design Automation Conf., 2003), (Phys.Rev.A,At.Mol. atoms. Worse still, at atomic dimensions, we must contend with Opt. Phy., vol. 68, p. 012318, 2003), (Proc. 41st Design Automation Conf., 2004), (Proc. 39th Design Automation Conf., 2002), (Proc. the laws of quantum mechanics. For example, suppose 1 bit Design, Automation, and Test Eur., 2004), (Phys. Rev. A, At. Mol. is encoded as the presence or the absence of an electron in a Opt. Phy., vol. 69, p. 062321, 2004), (IEEE Trans. Comput.-Aided small region.1 Since we know very precisely where the electron Des. Integr. Circuits Syst., vol. 22, p. 710, 2003). Efficient quantum– is located, the Heisenberg uncertainty principle dictates that logic circuits that perform two tasks are discussed: 1) implement- we cannot know its momentum with high accuracy. Without ing generic quantum computations, and 2) initializing quantum registers. In contrast to conventional computing, the latter task a reasonable upper bound on the electron’s momentum, there is nontrivial because the state space of an n-qubit register is not is no alternative but to use a large potential to keep it in place, finite and contains exponential superpositions of classical bit- and expend significant energy during logic switching. A quan- strings. The proposed circuits are asymptotically optimal for re- titative analysis of these phenomena leads experts from North spective tasks and improve earlier published results by at least a Carolina State University (NCSU), Semiconductor Research factor of 2. The circuits for generic quantum computation constructed by Corporation (SRC), and Intel [38] to derive fundamental lim- the algorithms are the most efficient known today in terms of itations on the scalability of any computing device that moves the number of most expensive gates [quantum controlled-NOTs electrons. (CNOTs)]. They are based on an analog of the Shannon decom- However, these same quantum effects also facilitate a radi- position of Boolean functions and a new circuit block, called cally different form of computation [14]. Theoretically, quan- quantum multiplexor (QMUX), which generalizes several known constructions. A theoretical lower bound implies that the circuits tum computers could outperform their classical counterparts cannot be improved by more than a factor of 2. It is additionally when solving certain discrete problems [17]. For example, a shown how to accommodate the severe architectural limitation successful large-scale implementation of Shor’s integer factor- of using only nearest neighbor gates, which is representative of ization [31] would compromise the Rivest Shamir Adelman current implementation technologies. This increases the number (RSA) cryptosystem used in electronic commerce. On the other of gates by almost an order of magnitude, but preserves the asymptotic optimality of gate counts. hand, quantum effects may also be exploited for public-key cryptography [4]. Indeed, such cryptography systems, based on Index Terms—Application specific integrated circuits, circuit single-photon communication, are commercially available from analysis, circuit optimization, circuit synthesis, circuit topology, design automation, logic design, matrix decompositions, quantum MagiQ Technologies in the U.S. and IdQuantique in Europe. effect semiconductor devices, quantum theory. Physically, a quantum bit might be stored in one of a variety of quantum-mechanical systems. A broad survey of these implementation technologies, with feasibility estimates and forecasts, is available in the form of the quantum computing roadmap [1]. Sample carriers of quantum information include top electrons in hyperfine energy levels of either trapped atoms or trapped ions, tunneling currents in cold superconductors, Manuscript received October 31, 2004; revised March 7, 2005. This work was supported by the Defense Advanced Research Projects Agency (DARPA) nuclear spin polarizations in nuclear magnetic resonance, and Quantum Information Science and Technology (QuIST) program and by a polarization states of single photons. A collection of n such National Science Foundation (NSF) grant. The work of S. S. Bullock was systems would comprise an n-qubit register, and quantum-logic supported by a National Research Council (NRC) postdoctoral fellowship. This paper was recommended by Associate Editor L. Stok. gates (controlled quantum processes) would then be applied to V. V. Shende is with the University of California, Berkeley, CA 94720 the register to perform a computation. In practice, such gates USA, on leave from the Department of Electrical Engineering and Computer might result from rotating the electron between hyperfine levels Science, The University of Michigan, Ann Arbor, MI 48109-2212 USA (e-mail: [email protected]; [email protected]). by shining a laser beam on the trapped atom/ion, tuning the S. S. Bullock was with the Mathematical and Computational Sciences tunneling potential by changing voltages and/or current in a Division, National Institute of Standards and Technology, Gaithersburg, MD superconducting circuit, or perhaps passing multiple photons 20899-8910 USA. He is now with Institute for Defense Analyses Center for Computing Sciences, 17100 Science Drive, Bowie, MD 20715-4300 USA through very efficient nonlinear optical media. (e-mail: [email protected]; [email protected]). I. L. Markov is with the Department of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48109-2212 USA. (e-mail:[email protected]). 1Most current computing technologies use electron charges to store informa- Digital Object Identifier 10.1109/TCAD.2005.855930 tion; exceptions include spintronics-based techniques, e.g., magnetic RAM. 0278-0070/$20.00 © 2006 IEEE SHENDE et al.: SYNTHESIS OF QUANTUM-LOGIC CIRCUITS 1001 The logical properties of qubits also differ significantly from Existing algorithms for n-qubit circuit synthesis remain a those of classical bits. Bits and their manipulation can be factor of 4 away from lower bounds and fare poorly for small n. described using two constants (0 and 1) and the tools of Boolean These algorithms require at least eight CNOT gates for n =2, algebra. Qubits, on the other hand, must be discussed in terms while three CNOT gates are necessary and sufficient in the of vectors, matrices, and other linear algebraic constructions. worst case [28], [29], [35], [36]. Further, a simple procedure We will fully specify the formalism in Section II, but give a exists to produce two-qubit circuits with minimal possible rough idea of the similarities and differences between classical number of CNOT gates [26]. In contrast, in three qubits, the and quantum information below. lower bound is 14, while the generic n-qubit decomposition in [22] achieves 48 CNOTs, and a specialty three-qubit circuit in 1) A readout (observation, measurement) of a quantum reg- [34] achieves 40. ister results in a classical bitstring. In this paper, we focus on identifying useful quantum- 2) However, identically prepared quantum states may yield circuit blocks. To this end, we analyze quantum conditionals different classical bitstrings upon observation. Quantum and define quantum multiplexors (QMUXs) that generalize physics only predicts the probability of each possible CNOT, Toffoli, and Fredkin gates. Such QMUXs implement readout, and the readout probabilities of different bits in if–then–else conditionals when the controlling predicate evalu- the register need not be independent. ates to a nonclassical state, e.g., coherent superposition of |0 3) After readout, the state “collapses” onto the classical and |1. We find that QMUXs prove amenable to recursive bitstring observed. All other quantum data are lost. decomposition and vastly simplify the discussion of many These differences notwithstanding, quantum-logic circuits, results in quantum-logic synthesis (cf., [9], [22], and [33]). from a high level perspective, exhibit many similarities with Ultimately, our analysis leads to a quantum analog of the their classical counterparts. They consist of quantum gates, Shannon decomposition, which we apply to the problem of connected (though without fan-out or feedback) by quantum quantum-logic synthesis. wires that carry quantum bits. Moreover, logic synthesis for We contribute the following key results. quantum circuits is as important as for the classical case. In 1) An arbitrary n-qubit quantum state can be prepared by a current implementation technologies, gates that act on three or circuit containing no more than 2n+1 − 2n CNOT gates. more qubits are prohibitively difficult to implement directly. This lies a factor of 4 away from the theoretical lower Thus, implementing a quantum computation as a sequence of bound. two-qubit gates is of crucial importance. Two-qubit gates may, 2) An arbitrary n-qubit