Elementary Gates for Quantum Computation
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PHYSICAL REVIEW A VOLUME 52, NUMBER 5 NOVEMBER 1995 Elementary gates for quantum computation Adriano Barenco, ' Charles H. Bennett, '~ Richard Cleve, '~ David P. DiVincenzo, '~ Norman Margolus, '~ Peter Shor, Tycho Sleator, ~ John A. Smolin, and Harald Weinfurter 'Clarendon Laboratory, Oxford University, Oxford OXI 3PU, United Kingdom IBM Research, Yorktown Heights, ¹wYork 10598 Department of Computer Science, University of Calgary, Calgary, Alberta, Canada T2N IN4 Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ATckT Bell Laboratories, Murray Hill, New Jersey 07974 Physics Department, New York University, New York, New York 10003 Physics Department, University of California at Los Angeles, Los Angeles, California 90024 Institute for Experimental Physics, University of Innsbruck, A-6020 Innsbruck, Austria (Received 22 March 1995) We show that a set of gates that consists of all one-bit quantum gates [U(2)] and the two-bit exclusive-oR gate [that maps Boolean values (x,y) to (x,x&y)] is universal in the sense that all unitary operations on arbitrarily many bits n [U(2")] can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations. PACS number(s): 03.65.Ca, 07.05.Bx, 02.70.Rw, 89.80.+h I. BACKGROUND Historically, the idea that the quantum mechanics of iso- lated systems should be studied as a new formal system for It has recently been recognized, after 50 years of using the computation arose from the recognition 20 years ago that paradigms of classical physics (as embodied in the Turing computation could be made reversible within the paradigm machine) to build a theory of computation, that quantum of classical physics. It is possible to perform any computa- another with different physics provides paradigm clearly and tion in a way that is reversible both logically, i.e., the com- possibly much more powerful features than established com- putation is a sequence of bijective transformations, and ther- In the state the putation theory. quantum computation, of mody nami cally, i.e., the computation could in principle be computer is described a state vector 0', which is a com- by performed by a physical apparatus dissipating arbitrarily plex linear superposition of all binary states of the bits x little A formalism for constructing reversible Tur- c energy [2]. (0, 1}: ing machines and reversible gate arrays (i.e., reversible com- binational logic) was developed. Fredkin and Toffoli [3] X ~ ~ ~ showed that there exists a three-bit "universal gate" for re- +(t)= X Ag j y yXI x e (0, 1) versible computation, that is, a gate that, when applied in succession to different triplets of bits in a gate array, could be The state's evolution in the course of time t is described by a used to simulate any arbitrary reversible computation. (Two- unitary operator U on this vector space, i.e., a linear trans- bit such as NAND that are universal for ordinary com- formation that is bijective and length preserving. This unitary gates Toffoli's uni- evolution on a normalized state vector is known to be the putation are not reversible. ) version [4] of the correct physical description of an isolated system evolving in versal reversible gate will figure prominently in the body of time according to the laws of quantum mechanics [1]. this paper. Quantum physics is also reversible because the reverse- time evolution specified by the unitary operator U '= U~ Electronic address: a.barenco Imildred. physics. ox.ac.uk always exists; as a consequence, several workers recognized ~Electronic address: bennetc/divinceowatson. ibm. corn that reversible computation could be executed within a ~ Electronic address: eleve Icpsc.ucalgary. ca quantum-mechanical system. Quantum-mechanical Turing ~ Electronic address: nhmim. lcs.mit. edu machines [5,6], gate arrays [7], and cellular automata [8] Electronic address: shorresearch. att. corn have been discussed and physical realizations of Toffoli's ~ Electronic address: tycho sleator. physics. nyu. edu [9—11] and Fredkin*s [12—14] universal three-bit gates Electronic address: smolin vesta. physics. ucla. edu within various quantum-mechanical physical systems have Electronic address: harald. weinfurter Iuibk. ac.at been proposed. 1050-2947/95/52 (5)/3457(11)/$06. 00 52 3457 1995 The American Physical Society 3458 ADRIANO BARENCO et a(. While reversible computation is contained within quan- Here we derive a series of results that provide tools for tum mechanics, it is a small subset: the time evolution of a the building up of unitary transformations from simple gates. classical reversible computer is described by unitary opera- We build on other recent results that simplify and extend tors whose matrix elements are only zero or one, arbitrary Deutsch's original discovery [24] of a three-bit universal complex numbers are not allowed. Unitary time evolution quantum logic gate. As a consequence of the greater power can of course be simulated by a classical computer (e.g., an of quantum computing as a formal system, there are many analog optical computer governed by Maxwell's equations more choices for the universal gate than in classical revers- [15]), but the dimension of the unitary operator thus attain- ible computing. In particular, DiVincenzo [28] showed that two-bit universal quantum gates are also possible; Barenco able is bounded by the number of classical degrees of free- [29] extended this to show than almost any two-bit gate dom, i.e., roughly proportional to the size of the apparatus. In (within a certain restricted is universal and contrast, a quantum computer with m physical bits (see the class) Lloyd [30] in two-bit definition of the state above) can perform unitary operations and Deutsch [31]have shown that fact almost any or n-bit is universal. A related con- in a space of 2 dimensions, exponentially larger than its (n~ 2) gate also closely struction for the Fredkin has been In the physical size. gate given [32]. Deutsch [16] introduced a quantum Turing machine in- present paper we take a somewhat different tack, showing tended to generate and operate on arbitrary superpositions of that a nonuniversal, classical two-bit gate, in conjunction one-bit states and proposed that, aside from simulating the evolution with quantum gates, is also universal; we believe that the with the of quantum systems more economically than known classical present work, along preceding ones, covers the full con- methods, it might also be able to solve certain classical range of possible repertoires for quantum gate array struction. problems, i.e., problems with a classical input and output, With our universal-gate we also exhibit a num- faster than on any classical Turing machine. In a series of repertoire, ber efficient schemes artificial settings, with appropriately chosen oracles, quan- of for building up certain classes of -bit tum computers were shown to be qualitatively stronger than n operations with these gates. A variety of strategies for classical ones [17—20], culminating in Shor's [21,22] discov- constructing gate arrays efficiently will surely be very impor- tant for understanding the full mechanics ery of quantum polynomial-time algorithms for two impor- power of quantum for computation; the construction of such efficient schemes tant natural problems, viz. , factoring and discrete logarithm, for which no polynomial-time classical algorithm was has already proved very useful for understanding the scaling Shor's factorization In the known. The search for other such problems and the physical of prime [33]. present work we in Wein- question of the feasibility of building a quantum computer part build upon the strategy introduced by Sleator and are major topics of investigation today [23]. furter [9], who exhibited a scheme for obtaining the Toffoli The formalism we use for quantum computation, which gate with a sequence of exactly five two-bit gates. We find that their can be generalized and extended in a we call a quantum "gate array, " was introduced by Deutsch approach con- [24], who showed that a simple generalization of the Toffoli number of ways to obtain more general, efficient gate structions. Some of the results presented here have no obvi- gate [the three-bit gate A2(R, ), in the language introduced ous connection with schemes. later in this paper] suffices as a universal gate for quantum previous gate-assembly We will not touch at all on the attendant computing. The quantum gate array is the natural quantum great difficulties generalization of acyclic combinational logic "circuits" on the actual physical realization of a quantum computer; the studied in conventional computational complexity theory. It problems of error correction [34] and quantum coherence are serious We consists of quantum "gates, " interconnected without fanout [35,36] very ones. refer the reader to [37] for a comprehensive discussion of these difficulties. or feedback by quantum "wires. " The gates have the same number of inputs as outputs and a gate of n inputs carries a unitary operation of the group U(2"), i.e., a generalized ro- II. INTRODUCTION tation in a Hilbert space of dimension 2". Each wire repre- sents a quantum bit, or qubit [25,26], i.e., a quantum system We begin by introducing some basic ideas and notation.