Elementary Gates for Quantum Computation

Home , Norman Margolus

.ca. h.) z { e n )) y ber yy um it.edu. k y bits x; x x.ac.uk. tum com- [email protected] cs.m y of Calgary h as gener- estigate the eter Shor v )to( hard Clev P T&T Bell Labs einfurter ysics.o ersit A C5710) ein tum gates (U(2)) atson.ibm.com. x; y yu.edu. Ric Univ [email protected]. ysics.n alues ( Univ. of Innsbruc ysics.ucla.edu. (and IBM Researc Harald W t other gates, suc h 22, 1995 (A tum computation er b ounds on the exact n x y w esta.ph h [email protected] MIT yc , Alb erta, Canada T2N 1N4; clev h.att.com. k, Austria; harald.w y prop osed constructions of quan bridge MA 02139 USA; [email protected] Abstract UCLA Norman Margolus IBM Researc e upp er and lo X1 3PU, UK; [email protected] John Smolin Charles H. Bennett ysical Review A, Marc e deriv e-or gate (that maps Bo olean v ork, NY 10598, USA; b ennetc/divince@w tral role in man k , Oxford O e gates required to implemen ork, NY 10003 USA; t v y ysics, A-6020 Innsbruc h y o oli gates, that apply a sp eci c U(2) transformation to one orks. W y a cen ts, New Y w h-T ersit tary gates for quan o-bit exclusiv w that a set of gates that consists of all one-bit quan ork Univ. QUANT-PH-9503016 t of Computer Science, Calgary . DiVincenzo submitted to Ph w ho Sleator ersal in the sense that all unitary op erations on arbitrarily man wn Heigh )) can b e expressed as comp ositions of these gates. W y Hill, NJ 07974 USA; shor@researc e sho yc n b er of the ab o IBM Researc New Y W T vid P ysics Dept., Los Angeles, CA 90024; smolin@v ysics Dept., New Y orkto um Oxford Univ putational net is univ alized Deutsc input bit if andThese only gates pla if the logical AND of all remaining input bits is satis ed. and the t (U(2 n Inst. for Exptl. Ph Ph Murra Ph Lab oratory for Computer Science, Cam Departmen Y Clarendon Lab oratory Da x z y k { Adriano Barenco yy Elemen

of elementary gates required to build up a varietyoftwo- and three-bit quan-

tum gates, the asymptotic numb er required for n-bit Deutsch-To oli gates, and

make some observations ab out the numb er required for arbitrary n-bit unitary

op erations.

PACS numb ers: 03.65.Ca, 07.05.Bx, 02.70.Rw, 89.80.+h 2

1 Background

It has recently b een recognized, after ftyyears of using the paradigms of classical

physics (as emb o died in the Turing machine) to build a theory of computation, that

quantum physics provides another paradigm with clearly di erent and p ossibly much

more p owerful features than established computational theory. In quantum compu-

tation, the state of the computer is describ ed by a state vector , which is a complex

linear sup erp osition of all binary states of the bits x 2f0;1g:

X X

(t)= jx ;:::;x i; j j =1:

x 1 m x

x2f0;1g

The state's evolution in the course of time t is describ ed byaunitary op erator U on

this vector space, i.e., a linear transformation which is bijective and length-preserving.

This unitary evolution on a normalized state vector is known to b e the correct physical

description of an isolated system evolving in time according to the laws of quantum

mechanics[1].

Historically, the idea that the quantum mechanics of isolated systems should b e

studied as a new formal system for computation arose from the recognition twenty

years ago that computation could b e made reversible within the paradigm of clas-

sical physics. It is p ossible to p erform any computation in a way that is reversible

b oth logical ly |i.e., the computation is a sequence of bijective transformations|and

thermodynamical ly |the computation could in principle b e p erformed bya physi-

cal apparatus dissipating arbitrarily little energy [2]. A formalism for constructing

reversible Turing machines and reversible gate arrays (i.e., reversible combinational

logic) was develop ed. Fredkin and To oli[3 ] showed that there exists a 3-bit \univer- 3

sal gate" for reversible computation, that is, a gate which, when applied in succession

to di erent triplets of bits in a gate array, could b e used to simulate any arbitrary

reversible computation. (Two-bit gates like NAND which are universal for ordinary

computation are not reversible.) To oli's version[4] of the universal reversible gate

will gure prominently in the b o dy of this pap er.

Quantum physics is also reversible, b ecause the reverse-time evolution sp eci ed by

1 y

the unitary op erator U = U always exists; as a consequence, several workers rec-

ognized that reversible computation could b e executed within a quantum-mechanical

system. Quantum-mechanical Turing machines [5, 6], gate arrays [7], and cellular

automata [8] have b een discussed, and physical realizations of To oli's[9, 10 , 11]

and Fredkin's[12, 13 , 14 ] universal three-bit gates within various quantum-mechanical

physical systems have b een prop osed.

While reversible computation is contained within quantum mechanics, it is a small

subset: the time evolution of a classical reversible computer is describ ed by unitary

op erators whose matrix elements are only zero or one | arbitrary complex numb ers

are not allowed. Unitary time evolution can of course b e simulated by a classical

computer (e.g., an analog optical computer governed by Maxwell's equations)[15 ], but

the dimension of the unitary op erator thus attainable is b ounded by the number of

classical degrees of freedom|i.e., roughly prop ortional to the size of the apparatus.

By contrast a quantum computer with m physical bits (see de nition of the state

ab ove) can p erform unitary op erations in a space of 2 dimensions, exp onentially

larger than its physical size.

Deutsch[16] intro duced a quantum Turing machine intended to generate and op- 4

erate on arbitrary sup erp ositions of states, and prop osed that, aside from simulating

the evolution of quantum systems more economically than known classical meth-

o ds, it might also b e able to solve certain classical problems|i.e., problems with a

classical input and output|faster than on any classical Turing machine. In a se-

ries of arti cial settings, with appropriately chosen oracles, quantum computers were

shown to b e qualitatively stronger than classical ones [17, 18, 19, 20 ], culminating in

Shor's [21, 22 ] discovery of quantum p olynomial time algorithms for two imp ortant

natural problems, viz. factoring and discrete logarithm, for which no p olynomial-time

classical algorithm was known. The search for other such problems, and the physi-

cal question of the feasibility of building a quantum computer, are ma jor topics of

investigation to day[23 ].

The formalism we use for quantum computation, whichwe call a quantum \gate

array" was intro duced by Deutsch [24], who showed that a simple generalization of

the To oli gate (the three-bit gate ^ (R ), in the language intro duced later in this

2 x

pap er) suces as a universal gate for quantum computing. The quantum gate arrayis

the natural quantum generalization of acyclic combinational logic \circuits" studied

in conventional computational complexity theory. It consists of quantum \gates",

interconnected without fanout or feedbackby quantum \wires". The gates have the

same numb er of inputs as outputs, and a gate of n inputs carries a unitary op eration

n n

of the group U(2 ), i.e., a generalized rotation in a Hilb ert space of dimension 2 .

Each wire represents a quantum bit, or qubit [25, 26 ], i.e., a quantum system with a

2-dimensional Hilb ert space, capable of existing in a sup erp osition of Bo olean states

and of b eing entangled with the states of other qubits. Where there is no danger of 5

confusion, we will use the term \bit" in either the classical or quantum sense. Just as

classical bit strings can represent the discrete states of arbitrary nite dimensionality,

so a string of n qubits can b e used to represent quantum states in any Hilb ert space of

dimensionalityupto2 . The analysis of quantum Turing machines [20] is complicated

by the fact that not only the data but also the control variables, e.g., head p osition,

can exist in a sup erp osition of classical states. Fortunately,Yao has shown [27] that

acyclic quantum gate arrays can simulate quantum Turing machines. Gate arrays are

easier to think ab out, since the control variables, i.e., the wiring diagram itself and

the numb er of steps of computation executed so far, can b e thought of as classical,

with only the data in the wires b eing quantum.

Here we derive a series of results which provide new to ols for the building-up of

unitary transformations from simple gates. We build on other recent results which

simplify and extend Deutsch's original discovery[24] of a three-bit universal quantum

logic gate. As a consequence of the greater p ower of quantum computing as a formal

system, there are many more choices for the universal gate than in classical reversible

computing. In particular, DiVincenzo[28] showed that two-bit universal quantum

gates are also p ossible; Barenco[29] extended this to show than almost anytwo-bit

gate (within a certain restricted class) is universal, and Lloyd[30] and Deutsch et

al.[31 ] have shown that in fact almost anytwo-bit or n-bit (n 2) gate is also

universal. A closely related construction for the Fredkin gate has b een given[32]. In

the present pap er we take a somewhat di erent tack, showing that a non-universal,

classical two-bit gate, in conjunction with quantum one-bit gates, is also universal;

we b elieve that the presentwork along with the preceding ones cover the full range 6

of p ossible rep ertoires for quantum gate array construction.

With our universal-gate rep ertoire, we also exhibit a numb er of ecientschemes

for building up certain classes of n-bit op erations with these gates. Avarietyof

strategies for constructing gate arrays eciently will surely b e very imp ortant for

understanding the full p ower of quantum mechanics for computation; construction of

such ecientschemes have already proved very useful for understanding the scaling of

Shor's prime factorization[33]. In the presentwork we in part build up on the strategy

intro duced by Sleator and Weinfurter[9], who exhibited a scheme for obtaining the

To oli gate with a sequence of exactly vetwo-bit gates. We nd that their approach

can b e generalized and extended in a number of ways to obtain more general ecient

gate constructions. Some of the results presented here havenoobvious connection

with previous gate-assembly schemes.

We will not touch at all on the great diculties attendant on the actual phys-

ical realization of a quantum computer | the problems of error correction[34] and

quantum coherence[35, 36] are very serious ones. We refer the reader to [37 ] for a

comprehensive discussion of these diculties.

2 Intro duction

We b egin byintro ducing some basic ideas and notation. For any unitary

u u

00 01

U = ;

u u

10 11

(m+1)

and m 2f0;1;2;:::g, de ne the (m + 1)-bit (2 -dimensional) op erator ^ (U )as

u jx ;:::;x ;0i + u jx ;:::;x ;1i if x =1

y0 1 m y1 1 n k

k =1

^ (U)(jx ;:::;x ;yi)=

m 1 m

jx ;:::;x ;yi if x =0,

1 m k

k =1 7

for all x ;:::;x ;y 2f0;1g. (In more ordinary language, x denotes the AND of

1 m k

k =1

(m+1) (m+1)

the b o olean variables fx g.) Note that ^ (U ) is equated with U . The 2 2

k 0

matrix corresp onding to ^ (U )is

0 1

B C

u u

@ A

00 01

u u

10 11

(where the basis states are lexicographically ordered, i.e., j000i; j001i;::: ;j111i).

When

0 1

; U =

1 0

^ (U ) is the so-called To oli gate[4] with m + 1 input bits, which maps jx ;:::;x ;yi

m 1 m

to jx ;:::;x ;( x ) y i.For a general U , ^ (U ) can b e regarded as a general-

1 m k m

k=1

ization of the To oli gate, which, on input jx ;:::;x ;yi, applies U to y if and only

1 m

if x =1.

k =1

As shown by one of us [31, 29], \almost any" single ^ (U ) gate is universal in the

sense that: by successive application of this gate to pairs of bits in an n-bit network,

any unitary transformation may b e approximated with arbitrary accuracy. (It suces

for U to b e sp eci ed by Euler angles which are not a rational multiple of .)

We show that in some sense this result can b e made even simpler, in that any uni-

tary transformation in a network can always b e constructed out of only the \classical"

0 1

two-bit gate ^ along with a set of one-bit op erations (of the form ^ (U )).

1 0

This is a remarkable result from the p ersp ective of classical reversible computation

b ecause it is well known that the classical analogue of this assertion|which is that 8

0 1 0 1

and ^ all invertible b o olean functions can b e implemented with ^

0 1

1 0 1 0

gates[38 ] | is false. In fact, it is well known that only a tiny fraction of Bo olean func-

tions (those which are linear with resp ect to mo dulo 2 arithmetic) can b e generated

with these gates[39 ].

We will also exhibit a numb er of explicit constructions of ^ (U ) using ^ (U ),

m 1

which can all b e made p olynomial in m.Itiswell known[4] that the analogous family

0 1

from of constructions in classical reversible logic whichinvolve building ^

1 0

0 1

the three-bit To oli gate ^ , is also p olynomial in m.We will exhibit one

1 0

imp ortant di erence b etween the classical and the quantum constructions, however;

To oli showed[4] that the classical ^ 's could not b e built without the presence of

some \work bits" to store intermediate results of the calculation. By contrast, we show

that the quantum logic gates can always b e constructed with the use of no workspace

whatso ever. Similar computations in the classical setting (that use very few or no

work bits) app eared in the work of Cleve[40] and Ben-Or and Cleve[41]. Still, the

presence of a workspace plays an imp ortant role in the quantum gate constructions

|we nd that to implement a family of ^ gates exactly, the time required for our

implementation can b e reduced from (m )to(m) merely by the intro duction of

one bit for workspace.

3 Notation

We adopt a version of Feynman's[7] notation to denote ^ (U ) gates and To oli gates

in quantum networks as follows. 9

v v

v v v

l l

U U

In all the gate-array diagrams shown in this pap er, time pro ceeds from left to right The

rst network contains a ^ (U ) gate and the second one contains a 3-bit To oli gate[42 ].

The third and fourth networks contain a ^ (U ) and a 2-bit reversible exclusive-or

(simply called XOR henceforth) gate, resp ectively. The XOR gate is intro duced as

the \measurement gate" in [24 ], and will playa very prominent role in many of the

constructions we describ e b elow. Throughout this pap er, when we refer to a basic

op eration, we mean either a ^ (U ) gate or this 2-bit XOR gate.

In all the gate-array diagrams shown in this pap er, we use the usual convention

that time advances from left to right, so that the left-most gate op erates rst, etc.

4 Matrix Prop erties

Lemma 4.1: Every unitary 2 2 matrix can b e expressed as

! ! ! !

i i =2 i =2

e 0 e 0 cos =2 sin =2 e 0

;

i =2 i =2

0 e sin =2 cos =2

0 e 0 e

where , , , and are real-valued. Moreover, any sp ecial unitary 2 2 matrix (i.e.,

with unity determinant) can b e expressed as

! ! !

i =2 i =2

cos =2 sin =2 e 0 e 0

i =2 i =2

sin =2 cos =2

0 e 0 e 10

Pro of: Since a matrix is unitary if and only if its rowvectors and column vectors

are orthonormal, every 2 2 unitary matrix is of the form

i( + =2+ =2) i(+ =2 =2)

e cos =2 e sin =2

;

i( =2+ =2) i( =2 =2)

e sin =2 e cos =2

where , , , and are real-valued. The rst factorization ab ovenow follows imme-

diately. In the case of sp ecial unitary matrices, the determinant of the rst matrix

must b e 1, which implies e = 1, so the rst matrix in the pro duct can b e absorb ed

into the second one.2

De nition: In view of the ab ove lemma, we de ne the following.

cos =2 sin =2

R ( )= (a rotation by aroundy ^[43]).

sin =2 cos =2

i =2

e 0

R ( )= (a rotation by aroundz ^).

i =2

0 e

e 0

(a phase-shift with resp ect to ). Ph( )=

0 e

0 1

= (a \negation", or Pauli matrix).

1 0

I = (the identity matrix).

0 1

Lemma 4.2: The following prop erties hold:

1. R ( ) R ( )=R ( + )

y 1 y 2 y 1 2

2. R ( ) R ( )= R ( + )

z 1 z 2 z 1 2 11

3. Ph( ) Ph ( ) = Ph( + )

1 2 1 2

4. = I

x x

5. R ( ) =R ()

x y x y

6. R ( ) =R ( )

x z x z

Lemma 4.3: For any sp ecial unitary matrix W (W 2 SU (2)), there exist matrices

A, B , and C 2 SU (2) such that A B C = I and A B C = W .

x x

Pro of: By Lemma 4.1, there exist , , and such that W =R ( )R ()R ( ).

z y z

Set A =R ( )R ( ), B =R ( )R ( ), and C =R ( ). Then

z y y z z

2 2 2 2

A B C = R ( ) R ( ) R ( ) R ( ) R ( )

z y y z z

2 2 2 2

= R ( ) R ( )

z z

= I;

and

) R ( ) R ( ) R ( ) A B C = R ( ) R (

x y z x z x x z y

2 2 2 2

) R ( ) R ( ) R ( ) = R ( ) R (

x y x x z x z z y

2 2 2 2

= R ( ) R ( ) R ( ) R ( ) R ( )

z y y z z

2 2 2 2

= R ( ) R ( ) R ( )

z y z

= W:

2 12

5 Two-Bit Networks

5.1 Simulation of General ^ (U ) Gates

Lemma 5.1: For a unitary 2 2 matrix W ,a^ (W)gate can b e simulated bya

network of the form

v v v

l l

W A B C

where A, B , and C 2 SU (2), if and only if W 2 SU (2).

Pro of: For the \if " part, let A, B , and C b e as in Lemma 4.3. If the value of the

rst (top) bit is 0 then A B C = I is applied to the second bit. If the value of the

rst bit is 1 then A B C = W is applied to the second bit.

x x

For the \only if " part, note that A B C = I must hold if the simulation is

correct when the rst bit is 0. Also, if the network simulates a ^ (W ) gate then

A B C = W . Therefore, since det(A B C )= 1, W must also b e

x x x x

sp ecial unitary.2

Lemma 5.2: For any and S = Ph( ),a^ (S)gate can b e simulated by a network

of the form

S 13

where E is unitary.

Pro of: Let

1 0

E =R ()Ph ( )= :

0 e

Then the observation is that the 4 4 unitary matrix corresp onding to eachofthe

ab ove networks is

1 0

1 0 0 0

C B

0 1 0 0

C B

C : B

A @

0 0 e 0

0 0 0 e

Clearly, ^ (S ) comp osed with ^ (W ) yields ^ (S W ). Thus, by noting that any

1 1 1

unitary matrix U is of the form U = S W , where S = Ph( ) (for some ) and W is

2 SU (2), we obtain the following.

Corollary 5.3: For any unitary 2 2 matrix U ,a^ (U)gate can b e simulated by

at most six basic gates: four 1-bit gates (^ ), and twoXOR gates (^ ( )).

0 1 x

5.2 Sp ecial Cases

In Section 5.1, wehave established a general simulation of a ^ (U ) gate for an ar-

bitrary unitary U . For sp ecial cases of U that maybeofinterest, a more ecient

construction than that of Corollary 5.3 is p ossible. Clearly, Lemma 5.1 immediately

yields a more ecient simulation for all sp ecial unitary matrices. For example, the

\x-axis rotation matrix" (to use the language suggested by the mapping b etween

SU(2) and SO(3), the group of rigid-b o dy rotations[43 ])

cos =2 i sin =2

R ( )= =R ( )R ()R ( )

x z y z

2 2

i sin =2 cos =2 14

is sp ecial unitary.(R is of sp ecial interest b ecause ^ (iR ) is the \Deutsch gate"[24 ],

x 2 x

whichwas shown to b e universal for quantum logic.) For other sp eci c SU(2) matrices

an even more ecient simulation is p ossible.

Lemma 5.4: A ^ (W ) gate can b e simulated by a network of the form

v v v

l l

W A B

where A and B 2 SU (2) if and only if W is of the form

e cos =2 sin =2

; W =R ( )R ()R ( )=

z y z

sin =2 e cos =2

where and are real-valued.

Pro of: For the \if " part, consider the simulation of ^ (W ) that arises in Lemma 5.1

when W =R ( )R ()R ( ). In this case, A =R ( )R ( ), B =R ( )R ( )

z y z z y y z

2 2

and C = I .Thus, B = A and C can b e omitted.

For the \only if " part, note that B = A must hold for the simulation to b e valid

when when the rst bit is 0. Therefore, if the rst bit is 1 then A A is applied

x x

to the second bit. Now, the matrix A A has determinant 1 and is traceless (since

its trace is the same as that of ). By sp ecializing the characterization of unitary

matrices in Lemma 4.1 to traceless matrices with determinant 1, we conclude that

A A must b e of the form

sin =2 e cos =2

A A = :

e cos =2 sin =2 15

Therefore,

e cos =2 sin =2

A A = ;

x x

sin =2 e cos =2

as required.2

Examples of matrices of the form of Lemma 5.4 are R ( ) itself, as well as R ( )=

y z

R ( )R (0) R ( ). However, R ( ) is not of this form.

z y z x

2 2

Finally, for certain U ,we obtain an even greater simpli cation of the simulation

of ^ (U ) gates.

Lemma 5.5: A ^ (V ) gate can b e simulated by a construction of the form

v v

V A B

where A and B are unitary if and only if V is of the form

sin =2 e cos =2

V =R ( )R ()R ( ) = ;

z y z x

e cos =2 sin =2

where and are real-valued.

Pro of: If an additional ^ ( ) is app ended to the end of the network in Lemma 5.4

1 x

then, the network is equivalent to that ab ove (since ^ ( )isaninvolution), and also

1 x

simulates a ^ (W ) gate (since ^ (W ) comp osed with ^ ( )is^ (W )).2

1 x 1 1 x 1 x

Examples of matrices of the form of Lemma 5.5 are the Pauli matrices

0 i

= =R ( )R (2 ) R ( )

y z y z x

2 2

i 0 16

and

1 0

= =R (0) R ( ) R (0)

z z y z x

0 1

(as well as itself ).

Lemma 5.5 p ermits an immediate generalization of Corollary 5.3:

Corollary 5.6: For any unitary 2 2 matrix U ,a^ (U)gate can b e simulated by

at most six basic gates: four 1-bit gates (^ ), and two gates (^ (V )), where V is of

0 1

the form V =R ( )R ()R ( ) .

z y z x

A particular feature of the ^ ( ) gates is that they are symmetric with resp ect

1 z

to their input bits. In view of this, as well as for future reference, weintro duce the

following sp ecial notation for ^ ( ) gates.

1 z

= =

6 Three-Bit Networks

6.1 Simulation of General ^ (U ) Gates

Lemma 6.1: For any unitary 2 2 matrix U ,a^ (U)gate can b e simulated bya

network of the form 17

v v v v

v v l v l

U V V

where V is unitary.

Pro of: Let V b e such that V = U . If the rst bit or the second bit are 0 then the

transformation applied to the third bit is either I or V V = I . If the rst two bits

are b oth 1 then the transformation applied to the third is V V = U .2

Some of the intuition b ehind the construction in the ab ove Lemma is that, when the

rst two input bits are x and x , the sequence of op erations p erformed on the third

1 2

bit is: V i x =1, V i x = 1, and V i x x = 1. Since

1 2 1 2

x + x (x x )=2(x ^x )

1 2 1 2 1 2

(where \+", \", and \" are the ordinary arithmetic op erations), the ab ove sequence

of op erations is equivalent to p erforming V on the third bit i x ^ x = 1, whichis

1 2

2 2

the ^ (V ) gate. (This approach generalizes to pro duce a simulation of ^ (V ),

2 m

for m> 2, which is considered in Section 7.)

We can now combine Lemma 6.1 with Corollary 5.3 to obtain a simulation of

^ (U ) using only basic gates (^ ( ) and ^ ). The numb er of these gates is reduced

2 1 x 0

when it is recognized that a numb er of the one-bit gates can b e merged and eliminated.

In particular, the ^ (C ) from the end of the simulation of the rst ^ (V ) gate, and

0 1 18

y y

the ^ (C ) from the ^ (V ) gate combine to form the identity and are eliminated

0 1

entirely. This same sort of merging o ccurs to eliminate a ^ (A) gate and a ^ (A )

0 0

gate. We arrive at the following count:

Corollary 6.2: For any unitary 2 2 matrix U ,a^ (U)gate can b e simulated by

at most sixteen basic gates: eight 1-bit gates (^ ) and eightXOR gates (^ ( )).

0 1 x

A noteworthy case is when U = , where we obtain a simulation of the 3-bit To oli

gate ^ ( ), which is the primitive gate for classical reversible logic [4]. Later we will

2 x

use the fact that b ecause ^ ( ) is its own inverse, either the simulation of Lemma

2 x

6.1 or the time-reversed simulation (in which the order of the gates is reversed, and

each unitary op erator is replaced by its Hermitian conjugate) may b e used.

6.2 Three-bit gates congruentto^(U)

Wenow show that more ecient simulations of three-bit gates are p ossible if phase

shifts of the quantum states other than zero are p ermitted. If we de ne the matrix

W as

0 1

=Ph( W = ) ;

1 0

then the gates ^ (W ) and ^ ( ) can b e regarded as b eing \congruent mo dulo phase

2 2 x

shifts", b ecause the latter gate di ers only in that it maps j111i to j110i (instead

of j110i). This is p erfectly acceptable if the gate is part of an op eration which merely

mimics classical reversible computation, or if the gate is paired with another similar

one to cancel out the extra phase, as is sometimes the case in reversible gate ar-

rangements (see Corollary 7.4); however, this phase di erence is dangerous in general

if non-classical unitary op erations app ear in the computation. Gates congruentto 19

^ ( ) mo dulo phase shifts have b een previously investigated in [44 ].

2 x

The following is a more ecient simulation of a gate congruentto ^ ( ) mo dulo

2 x

phase shifts:

v v

v v v

l l l l

y y

A A

" indicates that the networks are not identical, where A =R ( ). In the ab ove, the \

but di er at most in the phases of their amplitudes, which are all 1 (the phase of

the j101i state is reversed in this case).

An alternative simulation of a gate congruentto^() mo dulo phase shifts

2 x

(whose phase shifts are identical to the previous one) is given by

y y

B B

). where B =R (

4 20

7 n-Bit Networks

The technique for simulating ^ (U ) gates in Lemma 6.1 generalizes to ^ (U ) gates

2 m

for m>2. For example, to simulate a ^ (U ) gate for any unitary U , set V so that

V = U and then construct a network as follows.

t t t t t t

t t t t t h h

t t t t t h h h h

y y y

U V V V V

V V V

The intuition b ehind this construction is similar to that b ehind the construction of

Lemma 6.1. If the rst three input bits are x , x , and x then the sequence of

1 2 3

op erations p erformed on the fourth bit is:

V i x =1 (100)

V i x x =1 (110)

1 2

V i x =1 (010)

V i x x =1 (011)

2 3

V i x x x =1 (111)

1 2 3

V i x x =1 (101)

1 3

V i x =1 (001).

The strings on the right enco de the condition for the op eration V or V at each step|

the \1"'s indicate which input bits are involved in the condition. For an ecient

implementation of ^ (U ), these strings form a grey co de sequence. Note also that the

parityofeach bit string determines whether to apply V or V . By comparing this

sequence of op erations with the terms in the equation

x + x + x (x x ) (x x ) (x x )+(x x x )= 4(x ^x ^x );

1 2 3 1 2 1 3 2 3 1 2 3 1 2 3 21

it can b e veri ed that the ab ove sequence of op erations is equivalent to p erforming

4 4

V on the fourth bit i x ^ x ^ x = 1, which is the ^ (V ) gate.

1 2 3 3

The foregoing can b e generalized to simulate ^ (U ) for larger values of m.

Lemma 7.1: For any n 3 and any unitary 2 2 matrix U ,a^ (U)gate can

n1 y

b e simulated byan n-bit network consisting of 2 1 ^ (V ) and ^ (V ) gates and

1 1

2 2 ^ ( ) gates, where V is unitary.

1 x

We omit the pro of of Lemma 7.1, but p oint out that it is a generalization of the

n = 4 case ab ove and based on setting V so that V = U and \implementing" the

identity

X X X

x (x x )+ (x x x ) +(1) (x x x )

k k k k k k 1 2 m

1 1 2 1 2 3

k k

1 1 2 1 2 3

=2 (x ^x ^ ^ x )

1 2 m

with a grey-co de sequence of op erations.

For some sp eci c small values of n (for n = 3, 4, 5, 6, 7, and 8 ), this is the most

ecient technique that we are aware of for simulating arbitrary ^ (U ) gates as

well as ^ ( ) gates; taking account of mergers (see Corollary 6.2), the simulation

n1 x

n1 n1 n

requires 3 2 4 ^ ( )'s and 2 2 ^ 's. However, since this numb er is (2 ), the

1 x 0

simulation is very inecient for large values of n.For the remainder of this section,

we fo cus on the asymptotic growth rate of the simulations with resp ect to n, and show

that this can b e quadratic in the general case and linear in many cases of interest. 22

7.1 Linear Simulation of ^ ( ) Gates on n-Bit Networks

n2 x

eg then a ^ ( ) gate can b e simulated Lemma 7.2: If n 5 and m 2f3;:::;d

m x

by a network consisting of 4(m 2) ^ ( ) gates that is of the form

2 x

1 t t t

2 t t t

3 t t t t t

4 t t t t t

5 t t t

6 t h t t h t

7 t h h t t h h t

8 t h h t h h

9 h h h

(illustrated for n =9 and m =5).

Pro of: Consider the group of the rst 7 gates in the ab ove network. The sixth bit

(from the top) is negated i the rst two bits are 1, the seventh bit is negated i the

rst three bits are 1, the eighth bit is negated i the rst four bits are 1, and the

ninth bit is negated i the rst ve bits are 1. Thus, the last bit is correctly set, but

the three preceding bits are altered. The last 5 gates in the network reset the values

of these three preceding bits.2

Note that in this construction and in the ones following, although many of the bits

not involved in the gate are op erated up on, the gate op eration is p erformed correctly 23

indep endent of the initial state of the bits (i.e., they do not have to b e \cleared" to

0 rst), and they are reset to their initial values after the op erations of the gate (as

in the computations which o ccur in [41] and [40 ]). This fact makes constructions like

the following p ossible.

Lemma 7.3: For any n 5, and m 2f2;:::;n3g a ^ ( ) gate can b e simulated

n2 x

by a network consisting of two ^ ( ) gates and two ^ ( ) gates whichisof

m x nm1 x

the form

t t t

t t h h

h h h

(illustrated for n =9 and m =5).

Pro of: By insp ection.2

Corollary 7.4: On an n-bit network (where n 7), a ^ ( ) gate can b e simulated

n2 x

by 8(n 5) ^ ( ) gates (3-bit To oli gates), as well as by 48n 204 basic op erations.

2 x

Pro of: First apply Lemma 7.2 with m = d e and m = n m 1tosimulate

1 2 1

2 24

^ ( ) and ^ ( ) gates. Then combine these by Lemma 7.3 to simulate the

m x m x

1 2

^ ( ) gate. Then, each ^ ( ) gate in the ab ove simulation may b e simulated by

n2 x 2 x

a set of basic op erations (as in Corollary 6.2). We nd that almost all of these To oli

gates need only to b e simulated mo dulo phase factors as in Sec. 6.2; in particular,

only 4 of the To oli gates, the ones whichinvolve the last bit in the diagram ab ove,

need to b e simulated exactly according to the construction of Corollary 6.2. Thus

these 4 gates are simulated by 16 basic op erations, while the other 8n 36 To oli

gates are simulated in just 6 basic op erations. A careful accounting of the mergers

of ^ gates which are then p ossible leads to the total count of basic op erations given

ab ove. 2

The ab ove constructions, though asymptotically ecient, requires at least one \extra"

bit, in that an n-bit network is required to simulate the (n 1)-bit gate ^ ( ). In

n2 x

the next subsection, we shall showhow to construct ^ (U ) for an arbitrary unitary

U using a quadratic numb er of basic op erations on an n-bit network, which includes

the n-bit To oli gate ^ ( ) as a sp ecial case.

n1 x

7.2 Quadratic Simulation of General ^ (U ) Gates on n-Bit

Networks

Lemma 7.5: For any unitary 2 2 matrix U ,a ^ (U)gate can b e simulated by

a network of the form 25

t t t t

t t h t h

U V V

(illustrated for n =9), where V is unitary.

Pro of: The pro of is very similar to that of Lemma 6.1, setting V so that V = U .2

Corollary 7.6: For any unitary U ,a^ (U)gate can b e simulated in terms of

(n ) basic op erations.

Pro of: This is a recursive application of Lemma 7.5. Let C denote the cost

of simulating a ^ (U ) (for an arbitrary U ). Consider the simulation in Lemma

7.5. The cost of simulating the ^ (V ) and ^ (V ) gates is (1) (by Corollary 5.3).

1 1

The cost of simulating the two ^ ( ) gates is (n)(by Corollary 7.4). The cost

n2 x

of simulating the ^ (V ) gate (by a recursive application of Lemma 7.5) is C .

n2 n2

Therefore, C satis es a recurrence of the form

C = C +(n);

n1 n2

which implies that C 2 (n ).2

n1 26

In fact, we nd that using the gate-counting mentioned in Corollary 7.4, the number

of basic op erations is 48n + O (n).

Although Corollary 7.6 is signi cant in that it p ermits any ^ (U )tobesimulated

with \p olynomial complexity", the question remains as to whether a sub quadratic

simulation is p ossible. The following is an (n)lower b ound on this complexity.

Lemma 7.7: Any simulation of a nonscalar ^ (U ) gate (i.e. where U 6= Ph ( ) I )

requires at least n 1 basic op erations.

Pro of: Consider any n-bit network with arbitrarily many 1-bit gates and fewer than

n 1 ^ ( ) gates. Call two bits adjacent if there there is a ^ ( ) gate b etween

1 x 1 x

them, and connected if there is a sequence of consecutively adjacent bits b etween

them. Since there are fewer than n 1 ^ ( ) gates, it must b e p ossible to partition

1 x

the bits into two nonempty sets A and B such that no bit in A is connected to any

bit in B . This implies that the unitary transformation asso ciated with the network is

jAj jB j

of the form A B , where A is 2 -dimensional and B is 2 -dimensional. Since the

transformation ^ (U ) is not of this form, the network cannot compute ^ (U ).2

n1 n1

It is conceivable that a linear size simulation of ^ (U ) gates is p ossible. Al-

though we cannot show this presently, in the remaining subsections, we show that

something \similar" (in a numb er of di erent senses) to a linear size simulation of

^ (U ) gates is p ossible.

n1 27

7.3 Linear Approximate Simulation of General ^ (U )

Gates on n-Bit Networks

De nition: Wesay that one network approximates another one within " if the

distance (induced by the Euclidean vector norm) b etween the unitary transformations

asso ciated with the two networks is at most ".

This notion of approximation in the context of reducing the complexity of quantum

computations was intro duced by Copp ersmith[33], and is useful for the following

reason. Supp ose that two networks that are approximately the same (in the ab ove

sense) are executed with identical inputs and their outputs are observed. Then the

probability distributions of the two outcomes will b e approximately the same in the

sense that, for anyevent, its probability will di er by at most 2" between the two

networks.

Lemma 7.8: For any unitary 2 2 matrix U and "> 0,a^ (U)gate can b e

approximated within " by (n log ( )) basic op erations.

Pro of: The idea is to apply Lemma 7.5 recursively as in Corollary 7.6, but to observe

that, with suitable choices for V , the recurrence can b e terminated after (log( ))

levels.

Since U is unitary, there exist unitary matrices P and D , such that U = P D P

and

e 0

D =

0 e

id id

1 2

where d and d are real. e and e are the eigenvalues of U .IfV is the matrix used

1 2 k

in the k recursive application of Lemma 7.3 (k 2f0;1;2;:::g) then it is sucient 28

2 y

that V = V for each k 2f0;1;2;:::g.Thus, it suces to set V = P D P ,

k k k

k +1

where

id =2

e 0

D = ;

id =2

0 e

for each k 2f0;1;2;:::g. Note that then

kV I k = kP D P I k

k 2 k 2

= kP (D I ) P k

k 2

kP k kD Ik kPk

2 k 2 2

= kD Ik

k 2

=2 :

Therefore, if the recursion is terminated after k = dlog ( )e steps then the discrepancy

between what the resulting network computes and ^ (U ) isan(nk)-bit trans-

formation of the form ^ (V ). Since k^ (V )^ (I)k = kV Ik

nk 1 k nk1 k nk1 2 k 2

)e dlog (

", the network approximates ^ (U ) within ".2 =2

7.4 Linear Simulation in Sp ecial Cases

Lemma 7.9: For any SU(2) matrix W ,a^ (W)gate can b e simulated bya

network of the form 29

t t t

t t t t

h h

W A B C

where A, B , and C 2 SU (2).

Pro of: The pro of is very similar to that of Lemma 5.1, referring to Lemma 4.3.2

Combining Lemma 7.9 with Corollary 7.4, we obtain the following.

Corollary 7.10: For any W 2 SU (2),a^ (W)gate can b e simulated by (n)

basic op erations.

As in Section 5, a noteworthy example is when

0 1

W = =Ph( ) :

1 0

In this case, we obtain a linear simulation of a transformation congruent mo dulo

phase shifts to the n-bit To oli gate ^ ( ).

n1 x 30

7.5 Linear Simulation of General ^ (U ) Gates on n-Bit

Networks With One Bit Fixed

Lemma 7.11: For any unitary U ,a^ (U)gate can b e simulated byan n-bit

network of the form

t t t

h t h

0 0

U U

(illustrated for n =9), where the initial value of one bit (the second to last) is xed

at 0 (and it incurs no net change).

Pro of: By insp ection.2

Combining Lemma 7.11 with Corollary 7.4, we obtain the following.

Corollary 7.12: For any unitary U ,a^ (U)gate can b e simulated by (n) basic

op erations in n-bit network, where the initial value of one bit is xed and incurs no

net change.

Note that the \extra" bit ab ovemay b e reused in the course of several simulations of 31

^ (U ) gates.

8 Ecient general gate constructions

In this nal discussion we will change the ground rules slightly by considering the

\basic op eration" to b e any two-bit op eration. This mayormay not b e a physically

reasonable choice in various particular implementations of quantum computing, but

for the moment this should b e considered as just a mathematical convenience which

will p ermit us to address somewhat more general questions than the ones considered

ab ove. When the arbitrary two-bit gate is taken as the basic op eration, then as

wehave seen, 5 op erations suce to pro duce the To oli gate (recall Lemma 6.1), 3

pro duce the To oli gate mo dulo phases (we p ermit a merging of the op erations in the

construction of Sec. 6.2), and 13 can b e used to pro duce the 4-bit To oli gate (see

Lemma 7.1). In no case do wehave a pro of that this is the most economical metho d

for pro ducing each of these functions; however, for most of these examples wehave

comp elling evidence from numerical study that these are in fact minimal[44].

In the course of doing these numerical investigations we discovered a number of

interesting additional facts ab out two-bit gate constructions. It is natural to ask, how

manytwo-bit gates are required to p erform any arbitrary three-bit unitary op eration,

if the two-bit gates were p ermitted to implementany memb er of U(4)? The answer

is six, as in the gate arrangement shown here. 32

28 46 64

16 37 55

We nd an interesting regularityinhow the U(8) op eration is built up by this se-

quence of gates, which is summarized by the \dimensionalities" shown in the diagram.

The rst U(4) op eration has 4 = 16 free angle parameters; this is the dimensionality

of the space accessible with a single gate, as indicated. With the second gate, this

dimensionality increases only by 12, to 28. It do es not double to 32, for two reasons.

First, there is a single global phase shared by the two gates. Second, there is a set

of op erations acting only on the bit shared by the two gates, which accounts for the

additional reduction of 3. Formally, this is summarized by noting that 12 is the di-

mension of the coset space SU(4)/SU(2). The action of the third gate increases the

dimensionalityby another 9 = 16 1 3 3. 9 is the dimension of the coset space

SU(4)/SU(2)SU(2). The further subtraction by 3 results from the duplication of

one-bit op erations on b oth bits of the added gate. At this p oint the dimensionality

increases by nine for each succeeding gate, until the dimensionality reaches exactly 64,

the dimension of U(8), at the sixth gate. In preliminary tests on four-bit op erations,

we found that the same rules for the increase of dimensionality applied. This p ermits

us to make a conjecture, just based on dimension counting, of a lower b ound on the

number of two-bit gates required to pro duce an arbitrary n-bit unitary transforma-

1 1 1

4 n . It is clear that \almost all" unitary transformations will b e tion: (n)=

9 3 9

computationally uninteresting, since they will require exp onentially many op erations 33

to implement.

Finally,we mention that by combining the quantum gate constructions intro duced

here with the decomp osition formulas for unitary matrices as used by Reck al.[15 ], an

explicit, exact simulation of any unitary op erator on n bits can b e constructed using

3 n

a nite numb er ((n 4 )) of two-bit gates, and using no work bits. In outline, the

pro cedure is as follows: Reck et al.[15 ] note that a formula exists for the decomp osition

of any unitary matrix into matrices only involving a U(2) op eration acting in the space

of pairs of states (not bits):

U =( T (x1;x2)) D:

x1;x22f0;1g ;x1>x2

T (x1;x2) p erforms a U(2) rotation involving the two basis states x1 and x2, and leaves

all other states unchanged; D is a diagonal matrix involving only phase factors, and

thus can also b e thought of as a pro duct of 2 matrices which p erform rotations in

two-dimensional subspaces. Using the metho ds intro duced ab ove, each T (x1;x2) can

b e simulated in p olynomial time, as follows: write a grey co de connecting x1 and x2;

for example, if n =8, x1 = 00111010 , and x2 = 00100111:

1 00111010 x1

2 00111011

3 00111111

4 00110111

5 00100111 x2

Op erations involving adjacent steps in this grey co de require a simple mo di cation of

the ^ gates intro duced earlier. The (n 1) control bits which remain unchanged

are not all 1 as in our earlier constructions, but they can b e made so temp orarily by

the appropriate use of \NOT" gates (^ ( )) b efore and after the application of the

0 x

^ op eration. Now, the desired T (x1;x2) op eration is constructed as follows: rst,

n1 34

p ermute states down through the grey co de, p erforming the p ermutations (1,2), (2,3),

(3,4), ... (m-2,m-1). These numb ers refer to the grey co de elements as in the table

ab ove, where m, the numb er of elements in the grey co de, is 5 in the example. Eachof

these p ermutations is accomplished by a mo di ed ^ ( ). Second, the desired U(2)

n1 x

rotation is p erformed by applying a mo di ed ^ (U )involving the states (m 1) and

(m). Third, the p ermutations are undone in reverse order: (m-2,m-1), (m-3,m-2), ...

(2,3), (1,2).

The numb er of basic op erations to p erform all these steps may b e easily estimated.

Each T (x1;x2) involves 2m 3 (mo di ed) ^ gates, each of which can b e done in

(n ) op erations. Since m, the numb er of elements in the grey co de sequence, cannot

exceed n + 1, the numb er of op erations to simulate T (x1;x2) is (n ). There are

O (4 ) T 's in the pro duct ab ove, so the total numb er of basic op erations to simulate

n 3 n

any U (2 ) matrix exactly is (n 4 ). (The numb er of steps to simulate the D matrix

is smaller and do es not a ect the count.) So, we see that this strict upp er b ound

di ers only by a p olynomial factor (which likely can b e made b etter than n ) from the

exp ected lower b ound quoted earlier, so this Reck pro cedure is relatively \ecient"

(if something which scales exp onentially may b e termed so). A serious problem with

this pro cedure is that it is extremely unlikely, so far as we can tell, to provide a

p olynomial-time simulation of those sp ecial U (2 ) which p ermit it, which of course

are exactly the ones which are of most interest in quantum computation. It still

remains to nd a truly ecient and useful design metho dology for quantum gate

construction. 35

Acknowledgments

We are very grateful to H.-F. Chau, D. Copp ersmith, D. Deutsch, A. Ekert and T.

To oli for helpful discussions. We are also most happy to thank the Institute for

Scienti c Interchange in Torino, and its director Professor Mario Rasetti, for making

p ossible the workshop on quantum computation in Octob er, 1994, at whichmuchof

this work was p erformed. A. B. thanks the nancial supp ort of the Berrow's fund in

Lincoln College (Oxford).

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