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quantum information science ( see e.g.  @xcite and references therein ) has received an increased attention in recent years due to the understanding that it enables to perform procedures not possible by purely classical resources . experimental techniques to manipulate increasingly complex quantum systems are also rapidly progressing . one of the central issues is on the one hand to control and manipulate delicate complex quantum states in an efficient manner , but on the other hand at the same time to prevent all uncontrollable influences from the environment . in order to tackle such problems , one has to understand the structure and properties of quantum states . this can be done either through studies of particular states in a particular setting , or through focusing on the properties of the most generic states . random quantum states , that is states distributed according to the unitarily invariant fubini - study measure , are good candidates for describing generic states . indeed , they are typical in the sense that statistical properties of states from a given hilbert space are well described by those of random quantum states . also , they describe eigenstates of sufficiently complex quantum systems  @xcite as well as time evolved states after sufficiently long evolution . not least , because random quantum states possess a large amount of entanglement they are useful in certain quantum information processes like quantum dense coding and remote state preparation  @xcite . they can be used to produce random unitaries needed in noise estimation  @xcite and twirling operations  @xcite . in addition , as random states are closely connected to the unitarily invariant haar measure of unitary matrices , the unitary invariance makes theoretical treatment of such states simpler . producing such states therefore enables to make available a useful quantum resource , and in addition to span the space of quantum states in a well - defined sense . therefore several works have recently explored different procedures to achieve this goal . it is known that generating random states distributed according to the exact invariant measure requires a number of gates exponential in the number of qubits . a more efficient but approximate way to generate random states uses pseudo - random quantum circuits in which gates are randomly drawn from a universal set of gates . as the number of applied gates increases the resulting measure gets increasingly close to the asymptotic invariant measure  @xcite . some bipartite properties of random states can be reproduced in a number of steps that is smaller than exponential in the number of qubits . polynomial convergence bounds have been derived analytically for bipartite entanglement  @xcite for a number of pseudo - random protocols . on the numerical side , different properties of circuits generating random states have been studied @xcite . in order to quantify how well a given pseudo - random scheme reproduces the unitarily invariant distribution , one can study averages of low - order polynomials in matrix elements  @xcite . in particular , one can define a state or a unitary @xmath0-design , for which moments up to order @xmath0 agree with the haar distribution  @xcite . although exact state @xmath0-designs can be built for all @xmath0 ( see references in  @xcite ) they are in general inefficient . in contrast , efficient approximate @xmath0-designs can be constructed for arbitrary @xmath0 ( for the specific case of 2-design see  @xcite ) . the pseudo - random circuit approach can yield only pseudo - random states , which do not reproduce exactly the unitarily invariant distribution . the method has been shown to be useful for large number of qubits , where exact methods are clearly inefficient . however , for systems with few qubits , the question of asymptotic complexity is not relevant . it is thus of interest to study specifically these systems and to find the most efficient way  in terms of number of gates  to generate random states distributed according to the unitarily invariant measure . this question is not just of academic interest since , as mentioned , few - qubit random unitaries are needed for e.g. noise estimation or twirling operations . optimal circuits for small number of qubits could also be used as a basic building block of pseudo - random circuits for larger number qubits , which might lead to faster convergence . in addition , systems of few qubits are becoming available experimentally , and it is important to propose algorithms that could be implemented on such small quantum processors , and which use as little quantum gates as possible . indeed , quantum gates , and especially two - qubit gates , are a scarce resource in real systems which should be carefully optimized .    in this paper we therefore follow a different strategy from the more generally adopted approach of using pseudo - random circuits to generate pseudo - random states , and try and construct exact algorithms generating random states for systems of three qubits . in the language of @xmath0-designs such algorithms are exact @xmath1-designs . we present a circuit composed of one - qubit and two - qubit gates which produces exact random states in an optimal way , in the sense of using the smallest possible number of cnot gates . the circuit uses in total @xmath2 cnot gates and @xmath3 one - qubit elementary rotations . our procedure uses results recently obtained  @xcite which described optimal procedures to transform a three - qubit state into another . our circuit needs @xmath4 random numbers which should be classically drawn and used as parameters for performing the one - qubit gates . the probability distribution of these parameters is derived , showing that it factorizes into a product of 10 independent distributions of one parameter and a joint distribution of the 4 remaining ones , each of these distributions being explicitly given . since we had to devise specific methods to compute these distributions , we explain the derivation in some details , as these methods can be useful in other contexts .    after presenting the main idea of the calculation in section [ metrictensor ] , we start by treating the simple case of two - qubit states in section [ 2qubits ] . we then turn to the three - qubit case and first show factorization of the probability distribution for a certain subset of the parameters ( section [ 7 - 14 ] ) , the remaining parameters being treated in section [ 1 - 6 ] . the full probability distribution for three qubits is summarized in section [ explicit_calc ] . formally , a quantum state @xmath5 can be considered as an element of the complex projective space @xmath6 , with @xmath7 the hilbert space dimension for @xmath8 qubits  @xcite . the natural riemannian metric on @xmath6 is the fubini - study metric , induced by the unitarily invariant haar measure on @xmath9 . it is the only metric invariant under unitary transformations . to parametrize @xmath6 one needs @xmath10 independent real parameters . such parametrizations are well - known , for instance using hurwitz parametrization of @xmath9 @xcite . however , they do not easily translate into one and two - qubit operations , as desired in quantum information . in ref . @xcite , optimal quantum circuits transforming the three - qubit state @xmath11 into an arbitrary quantum state were discussed . in the case of three qubits , a generic state can be parametrized up to a global phase by 14 parameters . the quantum circuit requiring the smallest amount of cnot gates has three cnots and 15 one - qubit gates depending on 14 independent rotation angles . from @xcite it is possible ( see appendix ) to extract the circuit depicted in fig . [ circuit ] , expressed as a series of cnot gates and single qubit rotations , where z - rotation is @xmath12 and y - rotation is @xmath13 with @xmath14 the pauli matrices . the circuit allows to go from @xmath11 to any quantum state ( up to an irrelevant global phase ) . it therefore provides a parametrization of a quantum state @xmath5 by angles @xmath15 .    in order to generate random vectors distributed according to the fubini - study measure , it would of course be possible to use e.  g.  hurwitz parametrization to generate classically a random state , and then use the procedure described in @xcite to find out the consecutive steps that allow to construct this particular vector from @xmath11 . however this procedure requires application of a specific algorithm for each realization of the random vector . instead , our aim here is to directly find the distribution of the @xmath16 such that the resulting @xmath5 is distributed according to the fubini - study measure . this is equivalent to calculating the invariant measure associated with the parametrization provided by fig . [ circuit ] in terms of the angles @xmath15 . geometrically , the fubini - study distance @xmath17 is the angle between two normalized states , @xmath18 . the metric induced by this distance is obtained by taking @xmath19 , getting @xmath20 where @xmath21 is the usual hermitian scalar product on @xmath22 @xcite . if a state @xmath5 is parametrized by some parameters @xmath23 then the riemannian metric tensor @xmath24 is such that @xmath25 and the volume form at each point of the coordinate patch , directly giving the invariant measure , is then given by @xmath26 . thus the joint distribution @xmath27 of the @xmath16 is simply obtained by calculating the determinant of the metric tensor given by with the parametrization @xmath28 , @xmath29 . unfortunately the calculation of such a @xmath30 determinant for @xmath31 qubits is intractable and one has to resort to other means . let us first consider the easier case of @xmath32 qubits , where by contrast the calculation can be performed directly . a normalized random 2-qubit state @xmath5 depends , up to a global phase , on 6 independent real parameters . a circuit producing @xmath5 from an initial state @xmath33 is depicted in fig . [ fig:2qcircuit ] . one can easily calculate the parametrization of the final state @xmath34 in terms of all six angles , thus directly obtaining the metric tensor @xmath24 from ( [ ds2 ] ) . square root of the determinant of @xmath35 then gives an unnormalized probability distribution of the angles as @xmath36 ( see also  @xcite ) . several observations can be made about this distribution . first , the three rotations applied on the first qubit ( top wire in fig .  [ fig:2qcircuit ] ) after the cnot gate represent a random su(2 ) rotation , for which the y - rotation angle is distributed as @xmath37 and the z - rotation angles are uniformly distributed @xcite . secondly , angle @xmath38 gives the eigenvalue of the reduced density matrix , @xmath39 , for which the distribution is well known , see , _ e.g. _ ,  @xcite . the third observation is that , remarkably , the joint distribution ( [ eq:2qp ] ) of all 6 angles factorizes into 6 independent one - angle distributions . let us now turn to our main issue , which is the distribution of angles in the three - qubit case . in order to have an indication whether the distribution of an angle @xmath16 factorizes , we numerically computed the determinant det@xmath42 of the metric tensor as a function of @xmath16 with the other angles fixed . we also numerically computed the marginal distribution of @xmath16 by using the procedure given in the appendix to find the angles corresponding to a sample of uniformly distributed random vectors . if the distribution for a given angle @xmath16 factorizes , these two numerically computed functions should match ( up to a constant factor ) . this is what we observed for all angles but four of them ( angles @xmath43 to @xmath44 ) .    in order to turn this numerical observation into a rigorous proof , we are going to show in this section that the distributions for angles @xmath45 to @xmath41 indeed factorize . in the next section we will complete the proof by dealing with the cases @xmath38 to @xmath44 . the explicit analytical expression of the probability distribution for individual angles will be given in section [ explicit_calc ] . let us denote by @xmath46 the circuit of fig . [ circuit ] and by @xmath47 the unitary operator corresponding to it , so that @xmath48 . because circuits @xmath46 span the whole space of 3-qubit states , any unitary 3-qubit transformation @xmath49 maps parameters @xmath50 to new parameters @xmath51 such that @xmath52 . we denote by @xmath53 the circuit parametrized by angles @xmath51 corresponding to performing @xmath54 followed by @xmath49 . it is associated with the unitary operator @xmath55 such that @xmath56 . unitary invariance of the measure implies for @xmath27 that @xmath57 with @xmath58 the jacobian of the transformation @xmath59 and @xmath60 denotes the determinant . note that eq . ( [ eq : invariance ] ) is not a simple change of variables , as the same function @xmath61 appears on both sides of the equation . the jacobian matrix @xmath58 for transformation @xmath49 from angles @xmath50 to @xmath51 , @xmath52 , tells how much do the angles @xmath51 of @xmath62 change if we vary angles @xmath50 in @xmath63 keeping transformation matrix @xmath49 fixed . choosing @xmath49 that sets some angles @xmath64 in circuit @xmath53 to a fixed value , say zero , and at the same time showing that @xmath65 depends only on these angles @xmath64 , would prove factorization of @xmath27 with respect to angles @xmath64 through eq . [ eq : invariance ] . the simplest case is that of gates at the end of the circuit @xmath54 of fig . [ circuit ] , _ e.g. _ , gate @xmath66 . for @xmath49 we take y - rotation by angle @xmath67 on the third qubit , @xmath68 . it defines a mapping @xmath59 such that @xmath69 for @xmath70 and @xmath71 . matrix elements of the jacobian , i.e. partial derivatives @xmath72 , are equal to the kronecker symbol @xmath73 . the jacobian is equal to an identity matrix and its determinant is one . equation   taken at @xmath74 then gives @xmath75 , from which one concludes that the distribution for @xmath41 factorizes and is in fact uniform ( unless noted otherwise @xmath61 s are not normalized ) . the same argument holds for the two other rotations by angles @xmath76 and @xmath77 applied at the end of each qubit wire . proceeding to angle @xmath78 one could use @xmath79 applied on the first qubit and show that the jacobian depends only on @xmath78 and @xmath80 , while at @xmath81 and @xmath82 one gets @xmath83 , from which factorization of @xmath78 would follow from eq . [ eq : invariance ] . there is however a simpler way . observe that the three single - qubit gates with angles @xmath84 and @xmath80 on the first qubit span the whole su(2 ) group . therefore , for any one - qubit unitary @xmath49 , gates @xmath85 can be rewritten as @xmath86 , without affecting other @xmath87 s . the distribution of these three angles must therefore be the same as the distribution of corresponding su(2 ) parameters . note that the same argument can be applied in the 2-qubit case of section [ 2qubits ] . as a consequence , the distribution of angles for gates @xmath88 at the end of the circuit should be the same in both cases , that is the distribution of @xmath40 is uniform while that of @xmath78 is proportional to @xmath89 . similarly , one can show that the distribution for the angles @xmath90 to @xmath76 is the same as for angles @xmath40 to @xmath80 . as opposed to gates 10 - 12 , for gate 13 we can not use the analogy with the 2-qubit circuit ( fig.[fig:2qcircuit ] ) because the two gates 13 and 14 on the third qubit do not span the whole su(2 ) group . therefore a different argument should be used . in what follows we show that the joint distribution for angles @xmath91 and @xmath41 can be factorized out of the full distribution . since it has been shown in subsection [ gate14 ] that angle @xmath41 factorizes , this will prove that the distribution for @xmath91 also factorizes . using @xmath92 on the third qubit we can set @xmath93 and @xmath94 to zero with the choice @xmath95 and @xmath96 . our goal is to show that @xmath65 depends only on @xmath91 and @xmath41 . we can formally consider each angle @xmath97 as being a function @xmath98 of the initial @xmath50 as well as of the parameters @xmath99 through @xmath100 . to calculate matrix elements of @xmath58 for our choice of @xmath49 evaluated at @xmath95 and @xmath96 , we must obtain the first - order expansion in @xmath101 of the quantities @xmath102 some angles @xmath103 are very simple . we immediately see that when varying angles @xmath104 , that is taking @xmath105 in eq.[limlim ] , angles @xmath106 do not change ( i.e @xmath107 ) . the corresponding @xmath108-dimensional subblock in @xmath58 is therefore equal to an identity matrix . similarly , taking @xmath109 in eq.[limlim ] we see that @xmath110 do not change . the corresponding column in @xmath58 is therefore zero apart from @xmath111 on the diagonal . the jacobian thus has a block structure of the form @xmath112 where @xmath113 is a @xmath114-dimensional identity matrix and @xmath115 is a @xmath116-dimensional block with partial derivative @xmath117 . the angle @xmath94 given by eq.([limlim ] ) is obtained by varying angle @xmath41 by @xmath101 . the condition that @xmath100 is @xmath118 , where @xmath119 is a state after the third cnot acts on @xmath120 ( counting from the left in fig .  [ circuit ] ) , explicitly given by @xmath121 with @xmath122 @xmath123 is therefore determined by @xmath124 and @xmath125 , where @xmath126 . similarly , @xmath127 is determined by @xmath128 and @xmath129 . projecting these conditions for @xmath130 and @xmath131 on the computational basis @xmath132 , eliminating unwanted variables , we get the following two equations : @xmath133 one with @xmath134 , one with @xmath135 . expanding these equations to first order in @xmath101 yields @xmath136 and @xmath137 . the derivative @xmath138 is therefore equal to @xmath139 , which completes the proof . incidentally , we also see that the distribution of @xmath91 is proportional to @xmath140 . in the preceding section , we have shown that the distribution for angles @xmath40 to @xmath41 factorizes . as was mentioned , numerical observations indicated us that the distribution for angles @xmath38 and @xmath141 should also factorize , but that it is not the case for the joint distribution of @xmath142 . as we were not able to directly prove by the same methods as above that the distributions for @xmath38 and @xmath141 factorize , we use a different strategy . namely , we first assume that this factorization is true , then we compute the distributions under this assumption , and the knowledge of the answer allows us to prove a posteriori that it is indeed the correct probability distribution .    if the factorization holds , the distribution for @xmath38 and @xmath141 is easily calculated from the matrix @xmath35 using symbolic manipulation software , by replacing angles @xmath64 , @xmath143 , in @xmath35 by suitably chosen simple values , so that the @xmath30 determinant giving the volume form can now be handled . this yields , up to a normalization constant , @xmath144    the joint distribution of @xmath142 can not be further factorized , and requires heavy calculations . indeed , even replacing all angles but @xmath145 by numerical values the determinant @xmath146 of the metric tensor given by still depends on 4 variables , which is too much for it to be evaluated by standard software . we thus proceed as follows . first one can show that @xmath146 can be put under the form @xmath147 with the sums running over all @xmath148 but only even values of @xmath149 and @xmath150 . because of the parity of @xmath151 , there are @xmath152 independent coefficients @xmath153 . evaluating numerically the determinant at @xmath154 random values of the angles one gets an @xmath155 linear system that can be solved numerically . if the values of the coefficients of the matrix @xmath24 are multiplied by a factor 4 , then one is ensured ( from inspection of @xmath146 ) that the @xmath153 are rationals of the form @xmath156 , @xmath157 . this allows to deduce their exact value from the numerical result . we are left with 6998 nonzero terms in @xmath146 , and terms with odd @xmath158 or @xmath159 do not exist . we then suppose that @xmath160 can be expanded as @xmath161 this assumption is validated a posteriori , since a solution of the form can indeed be found . there are 4851 coefficients @xmath162 , which can be obtained by identifying term by term coefficients in the expansion of @xmath163 and @xmath146 . we have to solve a system of quadratic equations @xmath164 the first equation is quadratic and fixes an overall sign . equation @xmath165 is linear once the values obtained from equations @xmath111 to @xmath0 are plugged into it . starting with the highest - degree term @xmath166 one can thus recursively solve all equations . there are only 1320 non - zero coefficients @xmath162 . gathering together terms @xmath167 one can simplify the sum to a sum of 96 terms of the form @xmath168 . expanding this expression in powers of @xmath169 and @xmath170 and simplifying separately each coefficient we finally get @xmath171 where @xmath172 and @xmath173 with @xmath174 , @xmath130 given by eq .  . recall that @xmath131 is the bit - flip transform of @xmath130 , @xmath126 . note that @xmath175 . angles @xmath176 can be obtained from @xmath177 where @xmath178 and @xmath179 . we do not have a general argument to explain this remarkable expression of the distribution in terms of the scalar products of @xmath174 , @xmath130 and @xmath131 .    to complete the proof for the joint distribution @xmath180 it remains to be checked that the determinant of the metric tensor @xmath35 with angles @xmath40 to @xmath41 replaced by constants is indeed proportional to @xmath181 . this a posteriori verification is easier to handle symbolically than the full a priori calculation of the @xmath30 determinant . indeed , the determinant can first be reduced to an @xmath182 determinant by gauss - jordan elimination . the remaining determinant can be expanded as a trigonometric polynomial . although symbolic manipulation software do not allow to simplify the coefficients of this polynomial , they are able to check that these coefficients match those of the expected distribution . we proved in that way that the difference between the determinant det@xmath42 and our expression is identically zero . this gives a computer - assisted but rigorous proof for the distribution of angles @xmath38 to @xmath44 . gathering together the results of the previous sections we obtain that the joint distribution @xmath27 can be factorized as @xmath183 the joint distribution @xmath184 has been derived in the previous section and is given by eq .  . the distribution for @xmath38 and @xmath141 is given by eqs .  - . given the factorization , it is easy to calculate the remaining @xmath185 for each @xmath186 as was done for @xmath38 and @xmath141 in the previous section : replacing angles @xmath64 , @xmath187 , in @xmath35 by suitably chosen simple values , the @xmath30 determinant giving the volume form can be easily evaluated by standard symbolic manipulation . this yields , up to a normalization constant , @xmath188    the knowledge of the angle distribution ( [ eq : p ] ) allows to easily generate random three - qubit vectors using the circuit of fig . [ circuit ] . angles @xmath189 and @xmath40 to @xmath41 can be drawn classically according to their individual probability distribution . angles @xmath190 can be obtained classically from the joint distribution by , for instance , monte - carlo rejection method ( that is , drawing angles @xmath43 to @xmath44 and a parameter @xmath191 $ ] at random , and keeping them if @xmath192 ) . bounding @xmath184 from above by @xmath193 yields a success rate of about @xmath194 . in this work , we constructed a quantum circuit for generating three - qubit states distributed according to the unitarily invariant measure . the construction is exact and optimal in the sense of having the smallest possible number of cnot gates . the procedure requires a set of @xmath4 random numbers classically drawn , which will be the angles of the one - qubit rotations , and whose distribution has been explicitly given . remarkably , we have shown that the distribution of angles factorizes , apart from that of four angles . the circuit can be used as a three - qubit random state generator , thus producing at will typical states on three qubits . it could be also used as a building block for pseudo - random circuits in order to produce pseudo - random quantum states on an arbitrary number of qubits . at last , it gives an example of a quantum algorithm producing interesting results which could be implemented on a few - qubit platform , using only @xmath195 quantum gates , of which @xmath3 are one - qubit elementary rotations much less demanding experimentally . we thank the french anr ( project infosysqq ) and the ist - fet program of the ec(project eurosqip ) for funding . m would like to acknowledge support by slovenian research agency , grant j1 - 7437 , and hospitality of laboratoire de physique th ' eorique , toulouse , where this work has been started . in this appendix , we explain how to obtain the angles @xmath16 of the circuit ( fig .  [ circuit ] ) for a given @xmath5 , based on the discussion in  @xcite . this justifies the use of these angles as a parametrization of the quantum states . we start from a state @xmath5 , and transform it by the inverse of the different gates of fig .  [ circuit ] to end up with @xmath11 , specifying how the angles @xmath16 are obtained in turn . more details can be found in  @xcite . a generic three - qubit state @xmath5 can be written in a canonical form as a sum of two ( not normalized ) product terms  @xcite , @xmath196 where @xmath197 are one - qubit states , @xmath198 is a one - qubit state orthogonal to @xmath199 and @xmath200 is a two - qubit state of the second and third qubits . the angle @xmath80 is chosen such that the z - rotation of angle @xmath201 eliminates a relative phase between the coefficients of the expansion of @xmath199 into @xmath202 and @xmath203 . ( note that because we are using the circuit in the reverse direction the angles of rotations have opposite signs ) . a subsequent y - rotation with angle @xmath204 results in the transformation @xmath205 ( up to a global phase ) . similarly , rotations of angles @xmath206 and @xmath207 rotate @xmath208 into @xmath202 . after applying rotations of angles @xmath204 , @xmath201 , @xmath207 and @xmath206 the state has become of the form @xmath209 ( up to normalization ) . two rotations on the third qubit of angles @xmath210 and @xmath211 are now chosen so as to rotate @xmath212 into some new state @xmath213 while @xmath214 is rotated , up to normalization , into @xmath215 . it was shown in @xcite that this can always be done by writing the normalized @xmath216 as @xmath217 , and then @xmath41 is a solution of @xmath218 while @xmath219 , where @xmath220 s are relative phases in @xmath221 . acting with a cnot@xmath222 gate on the resulting state one obtains a quantum state for the three qubits of the form @xmath223 , with @xmath224 . the z - rotation angle @xmath225 on the second qubit is now determined so as to eliminate a relative phase between the expansion coefficients of @xmath226 , making them real up to a global phase . on the third qubit we now apply three rotations of angles @xmath227 , @xmath228 , and @xmath229 to bring @xmath230 to @xmath231 and @xmath232 into @xmath233 , eliminating also a relative phase . then a cnot@xmath234 is applied . at this point ( after the second cnot in fig .  [ circuit ] , counting from right , but without the @xmath40 rotation ) , the state has become of the form @xmath235 , where the one - qubit states @xmath236 and @xmath237 are normalized and real . with @xmath40 we now eliminate the relative phase @xmath238 , and with an y - rotation of angle @xmath239 the third qubit is brought to the state @xmath202 . then the combination of two y - rotation of angles @xmath141 and @xmath240 with a cnot@xmath241 brings the second qubit to @xmath202 , and the last rotation of angle @xmath242 on the first qubit yields the final state @xmath11 . note that in the circuit of fig . [ circuit ] the two z - rotations of angles @xmath40 and @xmath90 commute with cnot gates if they act on the control qubit . this is the reason why the rotation of angle @xmath40 can be applied at any point between @xmath38 and @xmath78 and , similarly , @xmath90 can be applied at any point between @xmath141 and @xmath243 . weinstein and c.  s.  hellberg , phys . .  lett . * 95 * , 030501 ( 2005 ) ; m.  nidari , phys . a * 76 * , 012318 ( 2007 ) ; y.  most , y.  shimoni and o.  biham , phys . a * 76 * , 022328 ( 2007 ) ; d.  rossini and g.  benenti , phys . 100 * , 060501 ( 2008 ) ; y.  s.  weinstein , w.  g.  brown and l.  viola , phys . a * 78 * , 052332 ( 2008 ) ; l.  arnaud and d.  braun , phys . a * 78 * , 062329 ( 2008 ) .
we explicitly construct a quantum circuit which exactly generates random three - qubit states . 
 the optimal circuit consists of three cnot gates and fifteen single qubit elementary rotations , parametrized by fourteen independent angles . 
 the explicit distribution of these angles is derived , showing that the joint distribution is a product of independent distributions of individual angles apart from four angles .