discrete orthogonal transforms and discrete unitary transforms have found various applications in signal , image , and video processing , in pattern recognition , in biocomputing , and in numerous other areas [ ] .well - known examples of such transforms include the discrete fourier transform , the walsh - hadamard transform , the trigonometric transforms such as the sine and cosine transform , the hartley transform , and the slant transform .all these different transforms find applications in signal and image processing , because the great variety of signal classes occuring in practice can not be handeled by a single transform .  on a classical computer , the straightforward way to compute a discrete orthogonal transform of a signal vector of length in general operations .an important aspect in many applications is to achieve the best possible computational efficiency .the examples mentioned above allow an evaluation with as few as operations or in the case of the wavelet transforms even with as little as operations .in view of the trivial lower bound of operations for matrix - vector - products , we notice that these algorithms are optimal or nearly optimal .  the rules of the game change dramatically when the ultimate limit of computational integration is approached , that is , when information is stored in single atoms , photons , or other quantum mechanical systems .the operations manipulating the state of such a computer have to follow the dictum of quantum mechanics . however , this is not necessarily a limitation .a striking example of the potential speed - up of quantum computation over classical computation has been given by shor in 1994 .he showed that integers can be factored in polynomial time on a quantum computer .in contrast , there are no polynomial time algorithms known for this problem on a classical computer.  the quantum computing model does not provide a uniform speed - up for all computational tasks .in fact , there are a number of problems which do not allow any speed - up at all . for instance, it can be shown that a quantum computer searching a sorted database will not have any advantage over a classical computer .on the other hand , if we use our classical algorithms on a quantum computer , then it will simply perform the calculation in a similar manner to a classical computer . in order for a quantum computer to showits superiority one needs to design new algorithms which take advantage of quantum parallelism .a quantum algorithm may be thought of as a discrete unitary transform which is followed by some i / o operations .this observation partially explains why signal transforms play a dominant role in numerous quantum algorithms .another reason is that it is often possible to find extremely efficient quantum algorithms for the discrete orthogonal transforms mentioned above .for instance , the discrete fourier transform of length can be implemented with operations on a quantum computer .the basic unit of information in classical computation is a bit , a system with two distinguishable states representing logical values 0 or 1 .we mentioned in the introduction that a quantum computer will store such information in the states of a quantum mechanical system .suppose that the system has two distinguishable states .we will denote these states by and , where the notation reminds us that these states represent the logical values 0 and 1 .  a potential candidate for the storage of a single bit is given by a spin- particle , such as an electron , proton , or neutron .we can choose the state with the rotation vector pointing upward ( spin - up ) and the state with the rotation vector pointing downward ( spin - down ) to represent 0 and 1 , respectively .however , we know from quantum mechanics that quantum system can be in a superposition of states .in the case of a spin- particle , a superposition yields a state which rotates about a different axis .the coefficients in this superposition are complex numbers , which determine this spin axis .  the consequent abstraction of the preceding example leads to the notion of a quantum bit , or shortly qubit , the basic unit of information in quantum computation .a quantum bit is given by a superposition of the states and such as the value of a quantum bit remains uncertain until it is measured .a measurement will collapse to either the state or to the state .the coefficients and determine the probability of outcome of this measurement , namely { \left|\psi\right\rangle } \mbox { collapses to } { \left|0\right\rangle } & |a|^2/(|a|^2+|b|^2)\\[1ex ] { \left|\psi\right\rangle } \mbox { collapses to } { \left|1\right\rangle } & |b|^2/(|a|^2+|b|^2 )   \end{array}\ ] ] in either case , we will learn the outcome of the measurement .since proportional states lead to the same measurement results , it is conventially assumed that the state is normalized to length 1 , i.e. , it is assumed that holds .the measurement allows to implement a fair coin flip on a quantum computer . indeed , preparing a quantum bit in the state , and measuring the result yields either 0 or 1 .according to the above rule , either event will occur exactly with probability 1/2 .this example might suggest that computations on a quantum computer are indeterministic and maybe even somewhat fuzzy .however , this is not the case .we will see in a moment that all operations apart from measurements are completely deterministic .the only operations that might introduce some randomized behaviour are the measurements , which as penrose puts it ` magnify an event from the quantum level to the classical level ' [ , pp . 7 - 8 ] .we discuss now the deterministic operations on a quantum computer .we begin with the simplest case , the operations which manipulate the state of a single quantum bit .first of all , it should be noted that the states and can be understood as an orthonormal basis of the complex inner product space .it is customary to associate the base states and with the standard basis vectors and , respectively .therefore , a quantum bit in the state is represented by the state vector a deterministic operation has to realize a unitary evolution of the quantum state , following the rules of quantum mechanics . in other words ,a single quantum bit operation is given by a unitary operator acting on the state of the quantum bit .there is a graphical notation for quantum operations .the schematic for such a single qubit operation is shown in figure [ fig : single ] .( 50,50 ) setunit 1.5 mm ; qubits(1 ) ;  labels.lft(0,btex etex ) ; wires(0.5 cm ) ; gate(gpos 0 , btex etex ) ; wires(0.5 cm ) ; labels.rt(0,btex etex ) ;  examples of single qubit operations are given by the operation realizes a not operation , and .the operation implements a phase shift operation , and .an extremely useful operation is given by the hadamard gate  , which is for instance used to create superpositions , the hadamard gate should be familiar to readers with a background in signal processing or coding theory . in the following , we will keep the notations for these gates without further notice .the operations get more interesting in the case of multiple quantum bits .quantum mechanics tells us that the state space of a combined quantum system is given by the tensor product of the state spaces of its parts .a remarkable consequence of this rule is that the state space of a system with quantum bits is given by the vector space ( -fold tensor product ) .this simply means that the dimension of the state space _ doubles _ with the addition of a single quantum bit .the state of a system with two quantum bits can thus be described by a vector or , isomorphically , by the vector the latter notation is often abbreviated to .the label in the dirac ket notation specifies a location in the quantum memory .a dramatic consequence of the tensor product structure of the quantum memory can be illustrated with a single qubit operation .suppose that we apply a single qubit operation , say the hadamard gate , on the least significant bit of ( [ eq : twostate ] ) .the resulting state is in more traditional mathematical notation , we can formulate this as the action of the matrix therefore , the resulting operation is .although we act only on one quantum bit , we see every single position of the state vector is manipulated .this is a striking example of quantum parallelism .we observe that a butterfly structure , well - known from many signal processing algorithms , can be implemented with a single operation on a quantum computer .the direct generalization to arbitrary single qubit operations is shown in figure [ fig : twoqubit ] .in general , a single qubit operation is specified by a unitary matrix , and the position of the target qubit to which is applied .suppose that the target qubit position is , then each state is unconditionally transformed to , where .( 50,50 ) setunit 1.5 mm ; qubits(2 ) ; wires(0.5cm ) ; gate(gpos 0 , btex etex ) ; wires(0.5 cm ) ; qcxcoord : = qcxcoord+2qcstepsize ; wires(0.5 cm ) ; gate(gpos 1 , btex etex ) ; wires(0.5 cm ) ;  we can specify more elaborate gates , which allow to create an interaction between quantum bits .let and be two disjoint sets of quantum bit positions , neither of which contains the target bit position .a conditional -operation maps the state to the state , in case for all and for all .the state remains unchanged in all other cases .the set describes the set of zero - conditions and the set of one - conditions . in the schematics , we will use the symbol to denote a zero - condition and the symbol to denote a one - condition .figure [ fig : cnot ] shows the simplest , but most important , conditional quantum gate the controlled not operation .( 50,50 ) setunit 1.5 mm ; qubits(2 ) ; cnot(icnd 1 , gpos 0 ) ; label(btex etex , ( qcxcoord+1/2qcstepsize,(qcycoord[0]+qcycoord[1])/2 ) ) ; qcxcoord : = qcxcoord + qcstepsize ; gate(icnd 1 , gpos 0 , btex etex ) ;  we can use controlled not gates to get an interaction between different quantum bits . for example , consider the circuit in figure [ fig : swap ] .this circuit swaps the states of the two quantum bits .( 50,50 ) setunit 1.5 mm ; qubits(2 ) ; labels.lft(0,btex etex,1,btex etex ) ; cnot(icnd 0 , gpos 1 ) ; cnot(icnd 1 , gpos 0 ) ; cnot(icnd 0 , gpos 1 ) ; labels.rt(0,btex etex,1,btex etex ) ;  engineering controlled operations is in general a difficult task. we will refer to controlled not gates with a single control bit and to single qubit operations as _elementary gates_. elementary quantum gates are available in all mature quantum computing technologies .it can be shown that it is possible to implement a general controlled operation with elementary gates. we will always refer to elementary gates in gate counts , but we will use multiply controlled gates for the sake of brevity in circuit descriptions . there exist standard algorithms which transform these more general gates into a sequence of elementary gates. have seen that a number of powerful operations are available on a quantum computer .suppose that we want to implement a unitary or orthogonal transform on a quantum computer. the goal will be to find an implementation of in terms of elementary quantum gates .usually , our aim will be to find first a factorization of in terms of sparse structured unitary matrices , where , of course , should be small .the philosophy being that it is often very easy to derive quantum circuits for structured sparse matrices .for example , if we can find an implementation with few multiply controlled unitary gates for each factor , then the overall circuit will be extremely efficient .the success of this method depends of course very much on the availability suitable factorization of . however , in the case orthogonal transfroms used in signal processing , there are typically numerous classical algorithms available , which provide the suitable factorizations .it should be noted that , in principle , an exponential number of elementary gates might be needed to implement even a diagonal unitary matrix .fortunately , we will see that most structured matrices occuring in practice have very efficient implementations .in fact , we will see that all the transforms of size discussed in the following can be implemented with merely elementary quantum gates .we present a simple but novel approach to derive such efficient implementations .this approach is based on a divide - and - conquer technique .assume that we want to implement a family of unitary transforms , where denotes the length of the signal .suppose further the family can be recursively generated by a recursive circuit construction , for instance , such as the one shown in figure [ fig : rec1 ] .we will give a generic construction for the family of precomputation circuits pre_ _ and the family of postcomputation circuits post_. this way , we obtain a fairly economic description of the algorithms .( 50,50 ) setunit 1.5 mm ; qubits(5 ) ; dropwire(1,2 ) ; wires(2 mm ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; wires(2 mm ) ; circuit(2qcheight)(gpos 0,1,2 , btex etex ) ; wires(2 mm ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; wires(2 mm ) ; label(btex etex , ( qcxcoord+1/2qcstepsize , qcycoord[0]+3/2qcheight ) ) ; qcxcoord : = qcxcoord + qcstepsize ; wires(2 mm ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; wires(2 mm ) ; circuit(2.5qcheight)(gpos 0,1,2,btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; circuit(2qcheight)(gpos 0,1 , btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; circuit(2.5qcheight)(gpos 0,1,2,btex etex ) ; wires(2 mm ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; wires(2 mm ) ;  assume that a total of elementary operations are necessary to implement the precomputation circuit pre and the postcomputation circuit post .then the overall number of elementary operations can be estimated from the recurrence equation the number of operations for the recursive implementation can be estimated as follows :  lemma . if , then .we will illustrate the general approach by way of some examples .our first example is the discrete fourier transform .a quantum algorithm implementing this transform found a most famous application in shor s integer factorizationalgorithm. recall that the discrete fourier transform of length can be described by the matrix where denotes a primitive -th root of unity , .and denotes a square root of .  the main observation behind the fast quantum algorithm dates at least back to work by danielson and lanczos in 1942 ( and is implicitly contained in numerous earlier works ) .they noticed that the matrix might be written as where denotes the permutation of rows given by with an -bit integer , and a single bit , and denotes the matrix of twiddle factors .this observation allows to represent by the following product of matrices : this factorization yields an outline of an implementation on a quantum computer .the overall structure is shown in figure [ fig : fourierrec ] .( 50,50 ) setunit 1.5 mm ; qubits(5 ) ; dropwire(1,2 ) ; wires(2 mm ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; circuit(3qcheight)(gpos 0,1,2,btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; wires(2 mm ) ; qubits(5 ) ; label(btex etex,(qcxcoord+qcstepsize/2,qcycoord[2 ] ) ) ; dropwire(1,2 ) ; qcxcoord : = qcxcoord + qcstepsize ; wires(2 mm ) ; label(btex etex,(qcxcoord+1/2qcstepsize , qcycoord[0]+10 mm ) ) ; gate(gpos 2,btex etex ) ; circuit(2qcheight)(icnd 2 , gpos 0,1,btex etex ) ;label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; qcfgcolor : = 0.7white ; circuit(2qcheight)(gpos 0,1 , btex etex ) ; qcfgcolor : = black ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; qubits(5 ) ; dropwire(2 ) ; draw ( qcxcoord , qcycoord[3])(qcxcoord+qcstepsize , qcycoord[0 ] ) ; draw ( qcxcoord , qcycoord[0])(qcxcoord+qcstepsize , qcycoord[1 ] ) ; draw ( qcxcoord , qcycoord[2])(qcxcoord+qcstepsize , qcycoord[3 ] ) ; dropwire(2 ) ; qcxcoord : = qcxcoord + qcstepsize ; wires(2 mm ) ; label(btex etex , ( qcxcoord-2mm , qcycoord[1]+10 mm ) ) ;  it remains to detail the different steps in this implementation .the first step is a single qubit operation , implementing a butterfly structure .the next step is slightly more complicated .we observe that is a tensor product of diagonal matrices .indeed , thus , can be realized by controlled phase shift operations , see figure [ fig : twiddle ] for an example .we then recurse to implement the fourier transform of smaller size .the final permutation implements the cyclic rotation of the quantum wires .( 50,50 ) setunit 1.9 mm ; qubits(4 ) ; circuit(2qcheight)(icnd 3 , gpos 0,1,2 , btex etex ) ; label(btex etex , ( qcxcoord + 1/2qcstepsize , qcycoord[1]+1/4qcheight ) ) ; qcxcoord : = qcxcoord + qcstepsize ; gate(icnd 3 , gpos 2 , btex etex ) ;  gate(icnd 3 , gpos 1 , btex etex ) ; gate(icnd 3 , gpos 0 , btex etex ) ;  the complexity of the quantum fourier transform can be estimated as follows .if we denote by the number of gates necessary to implement the dft of length on a quantum computer , then figure [ fig : fourierrec ] implies the recurrence relation which leads to the estimate .it should be noted that all permutations at the end can be combined into a single permutation of quantum wires .the resulting permutation is the bit reversal , see figure [ bitreversal ] .( 50,50 ) setunit 1.5 mm ; qubits(4 ) ; perm(cyc 0,3 , cyc 1,2 ) ; label(btex etex , ( qcxcoord+1/2qcstepsize , qcycoord[1]+1/2qcheight ) ) ; qcxcoord : = qcxcoord + qcstepsize ; perm(cyc 0,1 ) ; wires(1.5 mm ) ; draw ( qcxcoord , qcycoord[0]-1/2qcheight)(qcxcoord , qcycoord[3]+1/2qcheight ) dashed evenly ; wires(1.5 mm ) ; perm(cyc 2,0,1 ) ; wires(1.5 mm ) ; draw ( qcxcoord , qcycoord[0]-1/2qcheight)(qcxcoord , qcycoord[3]+1/2qcheight ) dashed evenly ; wires(1.5 mm ) ; perm(cyc 3,0,1,2 ) ;  remark .another explanation of the discrete fourier transform algorithm is contained in [ ] .note that the row permutations are mistaken in that article .an approximate version of the discrete fourier transform has been proposed by coppersmith , which saves some operations .the walsh - hadamard transform is maybe the simplest instance of the recursive approach .this transform is defined by the hadamard gates in the case of signals of length 2 . for signals of larger length ,the transform is defined by this yields the recursive implementation shown in figure [ fig : walsh ] .( 50,50 ) setunit 1.5 mm ; qubits(5 ) ; dropwire(1,2 ) ; wires(2 mm ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; circuit(2qcheight)(gpos 0,1,2,btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; wires(2 mm ) ;  label(btex etex , ( qcxcoord + 1/2qcstepsize , qcycoord[0]+2qcheight ) ) ; qcxcoord : = qcxcoord + qcstepsize ; wires(2 mm ) ; label(btex etex,(qcxcoord+1/2qcstepsize , qcycoord[0]+10 mm ) ) ; gate(gpos 2 , btex etex ) ; circuit(2qcheight)(gpos 0,1 , btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; wires(2 mm ) ;  since , the lemma in section 3 shows that the number of operations .it is of course trivial to see that in this case exactly operations are needed .the slant transform is used in image processing for the representation of images with many constant or uniformly changing gray levels .the transform has good energy compaction properties .it is used in intel s ` indeo ' video compression and in numerous still image compression algorithms .the slant transform is defined for signals of length by the hadamard matrix and for signals of length , , by where denotes the all - zero matrix , and is given by the matrix product the matrices in ( [ eq : slant ] ) are defined as follows ( see also [ ] ) : is the identity matrix , is the hadamard matrix , and realizes the transposition , that is , the matrix is defined by for all except in the case , where it yields the phase change .finally where and are recursively defined by and it is easy to check that is a unitary matrix .the definition of the slant transform suggests the following implementation .equation ( [ eq : slantrec ] ) tells us that the input signal of a slant transform of length is first processed by two slant transforms of size , followed by a circuit implementing .we can write equation ( [ eq : slantrec ] ) in the form the tensor product structure is compatible with our decomposition into quantum bits .this means that a single copy of the circuit acting on the lower significant bits will realize this part .it remains to give an implementation for .equation ( [ eq : slant ] ) describes as a product of four sparse matrices , which are easy to implement .indeed , the matrix is realized by conditionally excerting the phase gate . the matrix is implemented by a hadamard gate acting on the most significant bit .a conditional application of implements the matrix .a conditional swap of the least and the most significant qubit realizes , that is , three multiply controlled not gates implement . the quantum circuit realizing this implementation is depicted in figure [ fig : slant ] .( 50,50 ) setunit 3 mm ; qubits(5 ) ; dropwire(2 ) ;  wires(unit ) ; label(btex etex , ( qcxcoord , qcycoord[1]+14 mm ) ) ; circuit(3*qcheight)(gpos 0,1,2,3,btex etex ) ; label(btex etex , ( qcxcoord , qcycoord[1]+14 mm ) ) ; wires(unit ) ; qubits(5 ) ;label(btex etex,(qcxcoord+1/2qcstepsize , qcycoord[2 ] ) ) ; qcxcoord : = qcxcoord + qcstepsize ; dropwire(2 ) ; wires(unit ) ; label(btex etex , ( qcxcoord , qcycoord[1]+14 mm ) ) ; circuit(2.5*qcheight)(gpos 0,1,2,btex etex ) ; label(btex etex , ( qcxcoord , qcycoord[1]+14 mm ) ) ; wires(unit ) ; xsave : = qcxcoord ; gate(icnd 3 , ocnd 1,2 , gpos 0,btex etex ) ; gate(gpos 3 , btex etex ) ; gate(icnd 3 , ocnd 1,2 , gpos 0 , btex etex ) ; gate(icnd 0 , ocnd 1,2 , gpos 3 , `` not '' ) ; gate(icnd 3 , ocnd 1,2 , gpos 0 , `` not '' ) ; gate(icnd 0 , ocnd 1,2 , gpos 3 , `` not '' ) ; wires(unit ) ;  qubits(5 ) ; path rec ; rec : = ( xsave , qcycoord[1]+2unit)(xsave+6qcstepsize , qcycoord[1]+2unit)(xsave+6qcstepsize , qcycoord[3]-2unit)(xsave , qcycoord[3]-2unit)cycle ; fill rec withcolor white ; draw rec withcolor white ; label(btex etex,(xsave+1/2qcstepsize , qcycoord[1]+14 mm ) ) ; label(btex etex,(xsave+5/2qcstepsize , qcycoord[1]+14 mm ) ) ; label(btex etex,(xsave+7/2qcstepsize , qcycoord[1]+14 mm ) ) ; label(btex etex,(xsave+9/2qcstepsize , qcycoord[1]+14 mm ) ) ; label(btex etex,(xsave+11/2qcstepsize , qcycoord[1]+14 mm ) ) ;  the slant transform of length can be realized on a quantum computer with at most elementary operations ( that is , controlled not gates and single qubit gates ) , assuming that additional workbits are available .recall that a multiply controlled gate can be expressed with at most elementary operations as long as additional workbits are available. it follows from the lemma in section 3 that at most elementary operations are needed to implement the slant transform .the discrete hartley transform is defined for signals of length by the matrix the discrete hartley transform is very popular in classical signal processing , since it requires only real arithmetic but has similar properties . in particular , there are classical algorithms available , which outperform the fastest fourier transform algorithms .we derive a fast quantum algorithm for this transform , again based on a recursive divide - and - conquer algorithm . a fast algorithm forthe discrete hartley transform based on a completely different approach has been discussed by klappenecker and rtteler .the hartley transform can be recursively represented as where is the permutation , with a single bit , separating the even indexed samples and the odd indexed samples ; for instance , .the matrix is given by the equation ( [ eq : hartley ] ) leads to the implementation sketched in figure [ fig : hartley ] .( 50,50 )  setunit 1.5 mm ; qubits(5 ) ; dropwire(1,2 ) ; wires(2 mm ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; circuit(2qcheight)(gpos 0,1,2 , btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; wires(2 mm ) ; label(btex etex , ( qcxcoord+1/2qcstepsize , qcycoord[0]+2qcheight ) ) ; qcxcoord : = qcxcoord + qcstepsize ;  qubits(5 ) ; dropwire(2,3 ) ; wires(2 mm ) ; label(btex etex,(qcxcoord , qcycoord[1]+10 mm ) ) ; draw ( qcxcoord , qcycoord[0])(qcxcoord+qcstepsize , qcycoord[2 ] ) ; draw ( qcxcoord , qcycoord[0]+qcheight)(qcxcoord+qcstepsize , qcycoord[0 ] ) ; draw ( qcxcoord , qcycoord[2])(qcxcoord+qcstepsize , qcycoord[2]-qcheight ) ; qcxcoord : = qcxcoord + qcstepsize ; qubits(5 ) ; dropwire(1,2 ) ; wires(2 mm ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; circuit(2qcheight)(gpos 0,1 , btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; circuit(2qcheight)(icnd 2 , gpos 0,1 , btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; gate(gpos 2,btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+10 mm ) ) ; wires(2 mm ) ;  it remains to describe the implementation of .it will be instructive to detail the action of the matrix on a state vector of qubits .we will need a few notations first .denote by a state vector of qubits , where denotes a single bit and an bit integer .we denote by the two s complement of .we mean by the number and by the number , that is , has all bits set and has no bit set .then the action of on is given by where and .( 50,50 ) setunit 1.8 mm ; qubits(5 ) ; dropwire(1,2 ) ; wires(2 mm ) ; label(btex etex,(qcxcoord , qcycoord[0]+12 mm ) ) ; circuit(2qcheight)(gpos 0,1,2 , btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+12 mm ) ) ; wires(2 mm ) ; label(btex etex , ( qcxcoord+1/2qcstepsize , qcycoord[0]+1.5qcheight ) ) ; qcxcoord : = qcxcoord + qcstepsize ;  wires(2 mm ) ; label(btex etex,(qcxcoord , qcycoord[0]+12 mm ) ) ; circuit(2qcheight)(icnd 2 , gpos 0,1 , btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+12 mm ) ) ; gate(gpos 2,btex etex ) ; gate(ocnd 0,1 , gpos 2 , btex etex ) ;  path rec ; rec : = ( qcxcoord-2/3qcstepsize , qcycoord[0]+2 mm ) (qcxcoord-1/3qcstepsize , qcycoord[0]+2 mm ) (qcxcoord-1/3qcstepsize , qcycoord[1]-2 mm ) (qcxcoord-2/3qcstepsize , qcycoord[1]-2mm)cycle ; fill rec withcolor white ; draw rec withcolor white ; draw ( qcxcoord-1/2qcstepsize , qcycoord[0]+2mm)(qcxcoord-1/2qcstepsize , qcycoord[1]-2 mm ) dashed withdots scaled 1/2 ; gate(icnd 1 , gpos 2 , btex etex ) ; draw ( qcxcoord , qcycoord[0 ] ) (qcxcoord+1/2qcstepsize , qcycoord[0 ] ) dashed withdots scaled 1/2 ; draw ( qcxcoord , qcycoord[1 ] ) (qcxcoord+1/2qcstepsize , qcycoord[1 ] ) dashed withdots scaled 1/2 ; draw ( qcxcoord , qcycoord[2 ] ) (qcxcoord+1/2qcstepsize , qcycoord[2 ] ) dashed withdots scaled 1/2 ; qcxcoord : = qcxcoord + 1/2qcstepsize ; gate(icnd 0 , gpos 2 , btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+12 mm ) ) ; circuit(2qcheight)(icnd 2 , gpos 0,1 , btex etex ) ; label(btex etex,(qcxcoord , qcycoord[0]+12 mm ) ) ; wires(2 mm ) ;  we are now in the position to describe the implementation of shown in figure [ fig : bc ] . in the first step ,the least qubits are conditionally mapped to their two s complement .more precisely , the input signal is mapped to if , and does not change otherwise .thus , the circuit implements the involutary permutation corresponding to the two s complement operation .this can be done with elementary gates , provided that sufficient workspace is available. in the next step , a sign change is done if , that is , , unless the input was equal to zero , .the next step is a conditioned cascade of rotations .the least significant bits determine the angle of the rotation on the ( ) most significant qubit .the qubits exerts a rotation , on the most significant qubit .finally , another two s complement circuit is conditionally applied to the state .one readily checks that the implementation indeed maps to and to .the input is mapped to , as desired .assume that the input is with .then the state is changed to by the circuit , and after that its sign is changed , which yields .the rotations map this state to .the final conditional two s complement operation yields the state , which is exactly what we want .the inital permutation , the circuit and the hadamard gate in figure [ fig : hartley ] can be implemented with elementary gates .it is crucial that additional workbits are available , otherwise the complexity will increase to the lemma in section 3 then completes the proof of the following theorem :  there exists a recursive implementation of the discrete hadamard transform on a quantum computer with elementary gates ( that is , controlled not gates and single qubit gates ) , assuming that additional workbits are available .we have presented a new approach to the design of quantum algorithms .the method takes advantage of an divide - and - conquer approach .we have illustrated the method in the design of quantum algorithms for the fourier , walsh , slant , and hartley transforms .the same method can be applied to derive fast algorithms for various discrete cosine transforms .it might seem surprising that divide - and - conquer methods have not been previously suggested in quantum computing ( to the best of our knowledge ) .one reason might be that the quantum circuit model implements only straight - line programs .we defined recursions on top of that model , similar to macro expansions in many classical programming languages .the benefit is that many circuits can be specified in a very lucid way .it should be emphasized that our divide - and - conquer approach is completely general .it can be applied to a much larger class of circuits , and is of course not restricted to signal processing applications . moreover, it should be emphasized that many variations of this method are possible .we would like to encourage the reader to work out a few examples quite often this is a simple exercise .d. r. simon , `` on the power of quantum computation , '' in _ proceedings of the 35th annual symposium on foundations of computer science _ , pp . 116123 , ieee computer society press , ( los alamitos , ca ) , 1994 .a. klappenecker and m. rtteler , `` on the irresistible efficieny of signal processing methods in quantum computing , '' in _ proceedings of first international workshop on spectral techniques and logic design for future digital systems , tampere , finland , june 2 - 3 , 2000 _ , r. creutzburg and k. egiazarian , eds . , _ ticsp _ * 10 * , pp .483497 , ttkk , ( monistamo ) , 2000 .

Abstract: a quantum computer directly manipulates information stored in the state of quantum mechanical systems . the available operations have many attractive features but also underly severe restrictions , which complicate the design of quantum algorithms . we present a divide - and - conquer approach to the design of various quantum algorithms . the class of algorithm includes many transforms which are well - known in classical signal processing applications . we show how fast quantum algorithms can be derived for the discrete fourier transform , the walsh - hadamard transform , the slant transform , and the hartley transform . all these algorithms use at most operations to transform a state vector of a quantum computer of length  .


the making of metrical verse lines is a pattern - matching exercise , and in this respect contrasts sharply with the generation of ordinary language output .the composer must first generate source text , and then search through it for elements which form , or may be accumulated to form , the desired pattern .this pattern is usually specified in relation to a selection from the available surface features of the language , those in english verse , for example , being based principally on the patterning of stressed and unstressed syllables to form arrangements of beats and offbeats ( , ) . in earlier work constable ( ) has observed that these rules lead to the implication of a simpler and more readily studied rule , of which a composer is usually unaware , namely that a line should consist of complete words totaling syllables , where can be a range .this may be put as a rule thus :  _ ` in every consecutive section of syllables there must be only complete words'_.  the composition of word strings to fit a line definition rule of this type involves finding sequences of complete words totaling syllables , or constructing them from sequences of less than syllables . in either case the activity is a pattern - matching search for a target , and the frequency of the target in the source language is of crucial interest to the composer , since a smaller number of found elements will more severely restrict the communicative options open .that is to say , the more targets are present in the source text , the more probable it is that the composer will find pieces that function adequately with regard to his or her purpose .therefore , in order to facilitate line composition , authors are expected to take whatever action they can to increase the frequency of target elements .an effective way of doing this is to reduce the mean number of syllables per word in the source language being searched .recent work in empirical metrics ( : 182 ) has demonstrated the relation between mean word length and target frequency empirically from small samples of english prose , and has claimed that the average frequency of a target object ( a sequence of words totaling syllables ) is given by the total number of words divided by the mean number of syllables per word ( and this leads to the prediction that when composing in verse authors will tend to choose shorter words than they would otherwise have selected , a point also explored in ( ) . the mathematical reformulation required to substantiate and explain the factors underlying this effect involves detailed observations regarding the frequency and sequential word length distributions typical of english , and we will now turn to this task .in approaching word - length data , it is of crucial importance to form an analytic strategy suitable for the purpose in hand . given a distribution , either of frequencies or probabilities , it is possible to fit it with any degree of accuracy to a function , as long as one has a large number of functions , each with a large enough number of parameters .previous research on word length frequency undertaken by the gttingen word length project ( see for an overview of the project and a bibliography ; prominent papers in the project include : , , , , , , , , ) has collected data relating to word length frequencies , usually from rather small samples , and then used software , the altmann fitter , to compute a best fit description , which in most cases proves to be an adjusted poisson distribution ( ) .such a fitting procedure is guaranteed to work : the number of words with more than seven syllables is small , and they are infrequent in output , so there are , normally , only approximately seven data points , and , therefore , it will not be difficult to find an accurate fit if we have several hundred functions , each with several parameters , from which to choose . mathematically , if there are seven data items , a general function with 7 parameters would suffice for a _ perfect _ fit .we stress that this is not relevant : although such a fit may serve certain technical purposes , it does not bring any insight into or understanding of the nature of the language under consideration .it is only when we can find a fit with a much smaller number of parameters than that of the data , that we can move towards abstraction and understanding .any deviation from this simple ansatz should be regarded as a subtle variation from the basic finding .methodology of this type is quite common in exact science .for example , in high - energy physics , where the purpose of experiments , say proton - electron collisions , is not to fit the experimental data with one of several thousand functions , but to grasp the nature of protons and their constituent quarks .once we know the fundamental properties , further exploration of any deviation in the experimental data opens the way to advance our understanding .  our research is directed with this principle in mind , and we have therefore chosen not to fit the distributions using a large set of functions and parameters . rather , we will first extract a simple property that describes the overall , global , structure of the data , from which we aim to obtain a deeper knowledge of the english language , if only in its syllabic organization . with this in handwe can then proceed to the study of deviations from this global structure , that is to the fine structure of the data .we have analyzed the word - length structure of almost two million words of prose by various authors as listed in table [ tbl : authors ] .the texts were chosen with no other view than convenience , constable having marked them in the course of other research .we acknowledge that a more principled choice would be desirable , and indeed hope to undertake such work ourselves , but we believe that this corpus is sufficient for present purposes. the syllabic data were obtained by using a simple marking program , constructed by constable , which reads text and uses a custom - built lexicon to determine the syllabic count of each word form .when new word forms are encountered , the program requests human intervention , and the form is added to the lexicon .consistency of syllable counting was ensured by the fact that only one user ( constable ) has been responsible for building the lexicon , which at the time of writing contains 71,666 items .abbreviations were expanded , and numbers were counted as if they were pronounced as hyphenated words ( 1,920 = one - thousand - nine - hundred - and - twenty ) , with the exception of years and dates which were counted in their normal pronounced form ( 1920 = nineteen twenty ) .none of these categories are , as it happens , frequent in our corpus .   in order to present the method of analysis in definite terms ,let us introduce several mathematical notations and facilities .we denote the syllabic data obtained as described above by a series of integers ( ) , where is the number of syllables in the -th word and the total number of words in the data .the number of sequences in a series which satisfy a line definition rule of the type introduced in the previous section is represented as follows ; the upper limit of this sum implies that the line definition rule is applied with a ` periodic boundary condition ' ; namely , the data is treated as a circle by connecting the end of data with the beginning .this is a technical definition justifiable by its utility in the following mathematical treatment .alternatively , one could use a dirichlet boundary condition , in which one simply terminates the data sequence at .these boundary conditions , however , do not significantly affect the results as long as the data size is large , which is true for all the data we have analyzed .it is straightforward to count the number which match eq.([eqn : ld ] ) for words : the numbers are obtained simply by counting the numbers equal to among the series .next , the numbers are obtained by counting similarly for , from , and so forth . by definition , the following identity is satisfied : since there are no zero - syllable words in english , the quantity we are interested in is the number of sequences matching the line definition rule for _ any _ number of words , which is given by the following : this counting algorithm has been coded in _mathematica _ by aoyama and has been found to work much faster than the original algorithm used by constable ( :181 ) . in table[ tbl : match ] we give the partial list of and obtained for all the data listed in table [ tbl : authors ] .  for theoretical reasons it is best to deal with quantities independent of the data size .therefore , we introduce the following normalized quantities : due to the identity ( [ eqn : lnorm ] ) , the following is satisfied : in this sense , a set of of a given defines a probability distribution . on the other hand, does not have this property .we call a ( normalized ) ` frequency ' . corresponding to eq.([eqn : lndef ] ) , we have the following relation : in fig.[fig : flat ] we give the plot of and for all the data listed in table [ tbl : authors ] . as is seen in fig.[fig : flat ], the most remarkable global feature of the frequency distribution is its flatness , that is , its independence from . in order to analyze this structure, we may define an idealized constant distribution , where is the average value of the actual distribution for . in the following section we will examine what lies behind this constant distribution .   as a working hypothesis we might assume that the word - length series is randomly ordered , or more accurately , that  _ ` the number of syllables in a word is independent of the number of syllables in preceding words . ' _  in other words, this hypothesis suggests that there is no correlation between the syllable - count values in the data series .this random - ordering hypothesis allows us to express in terms of , the probability of a word having syllables ( hereafter we denote this quantity by ) .for example , two consecutive words that satisfy the two - syllable line definition rule can be obtained by having two one - syllable words in a row .the number of one - syllable words is , and according to the random - ordering hypothesis the probability of having a one - syllable word after a one - syllable word is not affected by the first word having one - syllable , and therefore is .thus the number of 2-syllable lines of this form is given by .dividing this by the total number of words , , we obtain the normalized frequency : for larger and , combinatoric considerations must be addressed .for example , a three - syllable line can be created by having a two - syllable word and a one - syllable word in sequence , or vice versa .counting all possibilities , we obtain , some of the other relations are listed in table [ tbl : random ] .the general expression for can be obtained in a straightforward manner , but is complicated in written from , and can be handled most simply by the use of generating functions .we define a generating function to represent for by the following : knowledge of all is equivalent to knowing the behaviour of near the origin , as can be expressed as the -th order derivative of at : the normalization condition ( [ eqn : pnorm ] ) of is expressed as . the various moments of ( expectation values of powers of ) can be expressed in terms of derivatives of at .for example , the average and the standard deviation of are given by the following : we also define a generating function as follows : in terms of these generating functions , the relation ( [ eqn : qndef ] ) is written as a relation similar to eq.([eqn : qneq ] ) holds also for .the general expression of in terms of induced by the random ordering condition can be summarized very simply . in terms of the generating functionsit is expressed as follows : we note that the normalization is trivial in the above equation : . the relation ( [ eqn : simple ] ) leads to the following expression of the generating function : thus the reason for introducing the random - ordering hypothesis becomes evident : if that hypothesis is valid , the features of the frequency distribution can be explained by the features of the one - word probability distribution by using the relation ( [ eqn : qnfx ] ) .we will now turn to the verification of the random - ordering hypothesis . in table[ tbl : corr ] we list the number of -syllable words following immediately after -syllable words . the corresponding probability distribution is plotted in fig . [ fig : true ] . from this figure, we readily observe that this distribution is almost independent from the value of .therefore we confirm that the random - ordering hypothesis is valid to a reasonable degree of accuracy.   now that the random - ordering hypothesis is confirmed , we can obtain the probability that induces the constant frequency distribution . the generating function for is as follows : by solving eq .( [ eqn : qnfx ] ) in terms of , we obtain , therefore , which is the geometric probability distribution . in fig . [fig : geo ] we compare the actual probability distribution ( dots ) and the geometric distribution in eq .( [ eqn : geo ] ) ( dash - dotted line ) . as is seen in this plot ,the agreement is close .the geometric distribution ( [ eqn : geo ] ) yields the average number of the syllables per word ( mean word length ) as follows : this relation was observed earlier by constable ( :182 ) .we stress that our new finding of the relation between the constant distribution and the geometric distribution , which we reached through the application of the random - ordering hypothesis gives a sound theoretical basis to this observation .the two global properties we have found above , the random - ordering and the geometric distribution ( [ eqn : geo ] ) , allow a definite characterization of the word - length data , since these are the properties typical of a system with a given probability of termination at any point : namely , if one assumes that sequences of syllables are constructed such that after any syllable , the end of a word happens with probability , the above geometric distribution ( [ eqn : geo ] ) is obtained .putting this in a slightly different manner , if one has a large number of syllables and word boundaries ( spaces ) with to ratio and randomly places them in sequence , the same distribution is obtained .we call this _ random segmentation_.  the fact that this geometric distribution is not found in the lexicon itself , which is plotted in fig .[ fig : dic ] , has been noted by other researchers ( ) , and provoked explanation in terms of attractors and control cycles in the composition process . strictly speaking it is beyond the scope of our paper to engage deeply with this question .however , since we believe that these researchers have been misled by the presumption of order in the output distribution , there is some point in observing that hypotheses based on randomness could , in principle , account for the relations between these very different distributions , and explain the stability of the geometric distribution in output .for example , we might hypothesize that the concept of ` word ' or ` word boundary ' is a relatively late ( though perhaps prehistoric ) analytic category , and has been arrived at by segmenting the verbal output stream in such a way that word boundaries are placed with a fixed probability in relation to syllable boundaries .the resulting word - forms are used to compile a lexicon .since the sound system of a language is not infinitely extendible , there will be more unique and acceptable disyllabic forms than monosyllables , more trisyllables than monosyllables , and so on .thus , although in its early stages the lexicon would , obviously , follow the geometric distribution of the output it was drawn from ( fig.[fig : geo ] ) , eventually it would , temporarily , adopt the sort of curve seen in fig.[fig : dic ] .this is not to suggest that the segmentation of english is fundamentally random , or that ` words ' have a low linguistic reality , interesting though both speculations are . in line with the data examined so far , our hypothesis merely notes that whatever principle of regular order may be operating elsewhere , perhaps in relation to stress or phonemes , word boundaries and syllable boundaries are related with a fixed probability .readers will have noticed the differences between the global structure and the actual distributions .the most notable is the small dip of below the average value seen in fig.[fig : flat ] , and the small differences between of different in fig.[fig : true ] .when we discuss these differences , we need first to guard against statistical errors . in other words, we first need to see whether these differences are meaningful quantities or can be attributed to statistical fluctuations .only when the former is more likely , do we need to study the fine structures that explain these differences .we stress that this discussion of statistical errors is of the first importance . as we see below small data sets , such as those employed by the gttingen group ,do in fact suffer from large statistical errors .detailed study of such data is either irrelevant or misleading . in the following ,we first discuss the handling of statistical errors and then proceed to the discussion of features of the fine structure .   the standard estimate for statistical errorsmay be applied to the individual probabilities that we deal with this paper .the 3- error range for the probability would be , in other words , the true value lies in this range with about 99.7% probability .the estimate of the error range for the frequencies require more extensive discussion .such an estimate is made possible by relation ( [ eqn : qndef ] ) : it is trivial for , as it is actually a probability , . therefore its standard deviation is given by .the next is .the probability is given by dividing the observed number of 2-syllable words by the total number of words . from fig.[fig : geo ] we see that , therefore the standard deviation of can be approximated as .similarly , we approximate that .thus the total standard deviation for the observed number of strings is given as follows : where we used the statistical independence of and .that is , the standard deviation of is given simply by , just as if it is a probability by itself .the same is true for the rest of .the estimate of the statistical errors for the average value of is simplified because of the flatness of the distribution .given the fact that the value of is independent from , the statistical errors follow from statistical errors of any _ one _ value of , which is almost independent from , as explained above .therefore , we simply estimate the standard deviation of to be of the same order as a typical value of that of ; namely , we estimate the -range of to be between .   in order to study the small deviation of the frequency from the flat distribution ,we can plot the difference between the actual frequency and the flat distribution itself , , with vertical bars showing the 3 error ranges , as in fig.[fig : sig ] .  from this figure, we find that the 2 syllable depression plot is statistically significant , as are other deviations at .these fine deviations can be explained from underlying deviations ; namely , ( 1 ) deviation from the geometric distribution , ( 2 ) deviation from random - ordering . in order to examine these fig.[fig : nongeom ] plots ( a ) ( solid line ) and ( b ) ( dotted line ) . the line ( a ) is a measure of the deviation from the geometric distribution , while the difference between ( a ) and ( b ) is a measure of the deviation from random - ordering . in these figureswe find that ( 1 ) monosyllabic words have a slightly higher probability than that predicted by the geometric distribution , while disyllabic words have a slightly lower probability , and that ( 2 ) in the sequential distribution there is a slightly enhanced probability of a polysyllable after a monosyllable ( relative to random - ordering ) .these are the important , indeed the only , exceptions to the overall randomness in the distributions .these deviations , being small compared to unity , can be mathematically treated as perturbations in a randomly - ordered geometric distribution . in that manner , it is straightforward to show that these small deviations for smaller do not affect for large , thus explaining the fact that deviation of from the flat distributin is localized to small .we have not examined the underlying linguistic causes of these deviations , and are not in a position to do more than speculate .the corpus is predominantly of high status literary writing , mostly nineteenth - century , and of that a substantial portion comes from one author , henry james .it might therefore be suggested that these deviations are characteristic of an output type , or a period , or even of an author .however , some of the deviations observed overall are consistently found across authors and works , though in the case of kipling we found a significant depression at instead of an enhancement .we predict , therefore , that some of these deviations , the depression at for example , are universal characteristics , while some others will prove to be particular to an author , a work , a genre type , or a period . with regard to the deviation from random sequencing , we suggest that the relation between commonly occurring function terms , which are predominantly monosyllabic , and content terms , which are somewhat more likely to be polysyllabic , is the likeliest explanation .  whatever the best causal account , it should be recognized that deviations such as these are subtle variations from a strong fundamental trend , and it is not safe to conclude that they are evidence which ` confirm[s ] the assumption of a non - accidental distribution of word lengths ' ( :73 ) .the global structures we noted in the preceding section , the flatness of the frequency and the randomness of sequencing are also true for individual authors and works .in other words , it is not a result of averaging over large fluctuations among authors . in fig.[fig: flat - ge ] and fig.[fig : true - ge ] we give the plot of the frequency and the probability for george eliot s novel _middlemarch_. the global features are readily apparent from these figures .the average frequency is for the eliot data and is for all the data in table [ tbl : authors ] , and readers may wonder whether this difference is meaningful .as explained in a previous subsection , the standard deviation of the average frequency is estimated to be .this means that for the corpus listed in table [ tbl : authors ] .thus at the -confidence level , the true value of lies between and .similarly , the -confidence range of for the george eliot data is .since these ranges do not overlap , we conclude that indeed the mean number of syllables per word in eliot is significantly larger than that of the corpus . in fig.[fig : authorq ] we give a plot of the values of and the ranges of all the authors and the whole corpus .apart from the deviation observable in the prose texts in our corpus , we are aware of one large class of texts which routinely exhibit significant deviations in the distribution , namely isometrically lineated verse texts .this is hardly surprising .texts composed in regular lines are by definition ordered with respect to lineation , and this order will be detected by such a procedure as ours . if a poem is composed in lines of ten syllables , for example , then , and all multiples of ten , will be substantially above the flat distribution , and if it is composed in two core line lengths , as limericks and spenserian stanzas are , then it will exhibit two series of peaks .since we intend to discuss this matter at greater length elsewhere we will present only one example , the final version of william wordsworth s _ prelude _ , in fig.[fig : ww ] and fig.[fig : wwd ] .this poem , which was completed in 1839 but not published until 1850 , is composed in blank verse , that is unrhymed five - beat lines in duple rhythm , with a range of between 9 and 12 syllables per line ( in duple rhythm the offbeat position is usually a single syllable , but is sometimes filled with two syllables , or even left unfilled ) .the poem contains 7,849 lines and 57,570 words , with a mean number of syllables per word of 1.4 .approximately 77.5 % of the text is composed in ten syllable lines , with 19.4 % being of eleven syllables and 2.3 % of twelve syllables , together with a scattering of other lengths .this degree of concentration into the core line length is by no means abnormal , and if anything it is somewhat more distributed than other texts examined .the distribution reveals , apart from the expected depression at , significant peaks at ten and all multiples of ten , though the subsequent peaks are of course of lesser size , since long , uninterrupted , runs of ten - syllable lines are rare .it should be noted that this test is of theoretical rather than practical value .it is unlikely that we will often wish to test in order to detect lineation , since the fact is usually visually evident , or clear from other features such as rhythm .however , as a contribution to the theoretical definition of verse lines and texts , particularly as distinguished from prose , the procedure is of considerable interest . although it has been long obvious that lineation is not merely ` a visual or typographical fact ' but a ` fact of the language ' ( :591 ) , to use one well - known formula , there has been , to our knowledge , no conclusive empirical demonstration of the presence of this fact , or any explanation of its character .the deviation from the flat distribution performs both these functions .previous research on word length distribution ( , , , , , , , , , ) has attempted to infer significance from the non - geometric curve found , and held that it supports the belief that ` language is [ ... ] a self - regulating system , which is controlled by the needs of the language community ' ( ) , or is an organism of interrelated control cycles ( ) . furthermore , these researchers have incautiously borrowed terms from chaos theory , and been , in our view , misled by them . for example , in :98 it was claimed on the basis of a handful of data , that ` the sequence of words is clearly chaotic ' , and that the distribution of word length in a text could be explained by reference to ` attractors ' .however , in the current context , there is in fact no chaos , in its mathematical sense , and what we observe in our study is randomness : when the sample size is small , any distribution , height or weight in a human population , or , to mention something fundamentally random , quantum theoretical events , will exhibit large fluctuations , and as the data size grows , the distributions become smoother .we do not rule out the discovery of the sort of order sought by the synergetic linguists , but observe that our findings give little support to its existence in relation to word length .thus , in approaching frequency data of this type we find ourselves generally in sympathy with those such as mandelbrot ( ) and li ( ) , who are of course studying different linguistic features , in their advocation of interpretations grounded in randomness , and we are less drawn to positions such as those proposed by zipf ( :48 ) , where statistical regularities are seen to arise from some deep principle of order .  in conclusion, however , we should like to to emphasize that the theory and data outlined here are of more than negative value , or purely self - sufficing interest .our investigation was derived from empirical observations and hypotheses offered in an earlier paper ( ) with regard to the construction of verse lines , and those remarks are confirmed by our results .the relation between the mean number of syllables per word and the number of sequences of words totalling a given number of syllables ( :182 ) is dependent on the geometric frequency of word length totals , and the random distribution of the words in the text sequence , which we have shown here to be solid findings .thus , the apparently arcane facts of word length distribution in english output can be seen to deepen our understanding of one , and a historically very important , area of language output , isometrical verse .   1 .we note that these features , flatness of the frequency and random - ordering are also true for individual authors and works , and are not a result of averaging over them .these issues will be addressed in subsequent sections .one might think that having a number of data points for would reduce the standard deviation for by a factor equal to the square root of the number of .however , since the flat distribution has for , this argument would yield a zero standard deviation , which is clearly wrong .99 attridge , d. 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Abstract: recent observations in the theory of verse and empirical metrics have suggested that constructing a verse line involves a pattern - matching search through a source text , and that the number of found elements ( complete words totaling a specified number of syllables ) is given by dividing the total number of words by the mean number of syllables per word in the source text . this paper makes this latter point explicit mathematically , and in the course of this demonstration shows that the word length frequency totals in english output are distributed geometrically ( previous researchers reported an adjusted poisson distribution ) , and that the sequential distribution is random at the global level , with significant non - randomness in the fine structure . data from a corpus of just under two million words , and a syllable - count lexicon of 71,000 word - forms is reported . the pattern - matching theory is shown to be internally coherent , and it is observed that some of the analytic techniques described here form a satisfactory test for regular ( isometric ) lineation in a text . +  _ keywords : word length , metrics _  cite#1#2#1 , # 2 biblabel#1  = 1.2