The Problem of Constructing Efficient Universal Sets of Quantum Gates

Transcription

1 The Problem of Constructing Efficient Universal Sets of Quantum Gates Qingzhong Liang and Jessica Thompson Abstract The purpose of this report is threefold. First, we study the paper [Letter] in detail providing several new proofs and explanations there. Secondly we provide a new result on constructing efficient universal sets of quantum gates. Thirdly, we study two papers which give algorithms for the Solovay- Kitaev theorem. The paper [D-L] provides full details of parts and of this report. Acknowledgement We would like thank Dr Steven Damelin for his help and encouragement. This research was supported via RTG grant and the Department of Mathematics, University of Michigan. REU webpage: damelin/reu.html Introduction to Quantum Computing. Classical Bits and Quantum Bits A classical bit is a basic unit of information in classical computing, with two possible states, 0 or. The basic state in quantum computation is a single quantum bit or qubit for short, which is a unit vector in C. Different from classical bits, a quantum bit or qubit for short can be in a linear combination of states, often called superpositions, denoted by ψ : ψ = α 0 + β, where α, β C, and α + β =. For example, ψ = , when measuring the above qubit, 0 is given about 33% of the time and about 66% of the time.. Quantum Gates A quantum gates are the quantum computing s equivalent to a classical computing s logic gate. It is an operation that acts linearly on a qubit. Since qubits can be represented as unit vectors in C, quantum gates can be thought of as linear maps from C to C. Because it must be linear, a quantum gate can be represented by a matrix with complex coefficients. For a single qubits, it is a matrix. Because qubits must be unit vectors, the results of applying a quantum gate to a qubit, must also be a unit vector. This implies that the matrix representing a quantum gate must be unitary. Thus, G = P SU or SU represents all possible quantum gates that act on a single qubit. An example of quantum gates is the Hadamard gate, [ ], which sends ψ = α 0 + β to ψ = α+β 0 + α β.

2 Special Unitary Group A n n matrix P is called special unitary if it is a unitary matrix with det P =. All special unitary matrices form a group called the special unitary group denoted by SUn. SUn = {P Un det P = } = {P GL n C P P = I, det P = } For example, to find the general form of a n n matrix P SU, let P = det P =, by Cramer s Rule, we have [ P = d det P c ] [ b d = a c ] b. a [ a c ] b SU. Since d Since P SU, we have P = P. Thus, [ d c ] [ b ā = a b d = ā c = b ] c d Thus, for P SU, P is in the form [ a b ] b ā where a, b C and aā + b b =. If we set a = x + x w i and b = x 3 + x 4 i, then the matrix P becomes, [ x + x i x 3 + x 4 i ] x 3 + x 4 i x x i Since a, b C and aā+b b =, x +x +x 3 +x 4 =. Then, x, x, x 3, x 4 is a point on the unit 3-sphere in R 4 and each P SU corresponds to a point on S 3. This is a bijective and continuous map; hence, SU is homeomorphic to S 3. 3 Approximation Approach 3. Universal Subset Let G be a group, S = {s, s,, s n } be a subset of G. Recall that S is universal if Γ, the group generated by s i s is topologically dense in G. 3. Some Useful Functions For G = SU, let S = {s, s,, s n } be a finite universal subset of G, and Γ be the group generated by the s is. Then, Γ is topologically dense in G. Weight Function w The weight function is a non-negative function on S, w : S R 0,

3 where the non-negative value ws i is the cost of s i. Height h Since Γ is generated by S, for any γ Γ, there is at least one way to write γ in the form γ = s i s i s in, where s i, s i,, s in S. Then, the cost of γ in this form is n ws ik. If γ = s i s i s in = s j s j s jm,, we would like to compare the cost of each form. If k= k= n m ws ik ws jk, we will prefer to use γ = s i s i s in to construct γ. In other words, for each γ γ, we want to construct it in a way with minimal cost. Hence the height of each γ Γ, denoted by hγ, is defined as: k= { n } hγ = min ws in : γ = s i s i s in, k= which is the minimal cost to construct γ. For example, let S = {s, s, s, s, s 3, s 3 }, where s = + i 0 0 i, s = i i, s 3 = Recall that from [B-G-S], S is a universal subset of SU. Let Γ be the group generated by S. If + i 4 + i the weights ws i are all taken to be, then for γ Γ and γ =, we have 4 + i i For γ = s s, the cost is γ = s s = s s s s. and for γ = s s s s, the cost is ws + ws = + =, ws + ws + ws + ws = = 4. Since the cost of the first way is less, we prefer to construct γ by γ = s s, instead of γ = s s s s. According to the definition of height, hγ ws +ws =. Also, since s, s, s 3, s, s, s 3 γ, we have hγ >. Hence, hγ =. The Set with Bounded Height V t With the definition of height, the set V t is defined as 3

4 V t = {γ Γ : hγ t}, which is the subset, in which every element can be constructed with cost less or equal to t. It is a set with restriction to height. We will use subsets in this form to approximate any element x G. 4 Invariant Metric in SU Definition. Let G be a group and d G be a metric on G. d G is called left invariant if d G gx, gy = d G x, y for any x, y, g G, right invariant if d G xg, yg = d G x, y for any x, y, g G, and invariant if d G is both left invariant and right invariant. For G = SU, define the metric on G by d G : G G R 0, d G x, y = tracex y, for x, y SU. To prove d G is invariant, notice that for any h SU, we have h h = I. Hence, hx hy = x h hy = x h hy = xy. Also, since h = h, we have xh yh = h x yh = h x yh. Also, since traceh x yh = tracex y, we have Then, tracexh yh = traceh x yh = tracex y. Hence, d G is a invariant metric on SU. d Gx, y = d Ghx, hy = d Gxh, yh Approach Now, our goal is using elements of a subset V of G to approximate each point x G. To guarantee the accuracy, the error should not be greater than a given value ε > 0. As we have mentioned, we want to use an appropriate subset V t to approximate G. That is, for any x G, there exists γ V t that d G x, γ < ε, and we use γ to approximate x. In other words, for any x G, there exists γ V t that x is within the ball centered at γ with radius ε, i.e. x B G γ, ε. Actually, for a given ε > 0, there is a least t = t ε > 0 that V t ε can be used to approximate G = SU. That is, G is covered by the balls: 6 Haar Measure 6. Definition G γ V t ε B G γ, ε. Let G be a compact group, a Haar measure on G is a measure µ : Σ R 0, with Σ a σ-algebra containing all Borel subsets of G, that i µg = 4

5 ii For all g G and S Σ, µgs = µs, where gs = {gs s S}. 6. Haar Measure on G = SU Since G = SU is a compact group, there is a normalized Haar measure µ on G = SU. Since for given ε > 0, we have Then, G B G γ, ε. γ V t ε µ B G γ, ε = µg =. γ V t ε Then consider any two elements γ, γ V t, and suppose B G γ, ε and B G γ, ε are the two balls centered at them. Let g = γ γ G, then gγ = γ and γ = g γ. Then, for any h B G γ, ε, we have d G γ, h < ε. Since d G x, y is an invariant metric on G = SU, we have Hence, gh B G γ, ε. Similarly, if h B G γ, ε, we have d G γ, gh = d G gγ, gh = d G γ, h < ε. Then, d G γ, h < ε. Hence, g h B G γ, ε. Hence, d G γ, g h = d G g γ, g h = d G γ, h < ε. Since the Haar measure on µ on G is left invariant B G γ, ε = gb G γ, ε. µb G γ, ε = µgb G γ, ε = µb G γ, ε. Hence, the Haar measure of any two ball in SU satisfies: µb G γ, ε = µb G γ, ε = µb G ε, for any γ, γ SU. Here, µb G ε is any ball with radius ε in G. Also, since there are totally V t ε balls, and any two of the balls may have non-empty intersection. Hence then, V t ε µb G ε = µb G γ, ε µ B G γ, ε = µg =, γ V t ε γ V t ε

6 V t ε µb G ε. Moreover, the Haar measure µ can be thought of as the volume of a ball with radius ε on the surface of S 3. Hence, when ε is small. µb G ε ε 3, 7 Efficiency of Approximations The efficiency of a universal gate set S is measured by its ability to approximate any x G by γ s with small height. Also, for a given ε > 0, we have proved that V t µb G ε. Hence, V t µb G ε for any ε and universal subset S. When the values of V t and µb G ε become closer, the measure of the intersections among the balls is decreased, which means the approximation is more efficient. Mathematically, we express the efficiency of an approximation with universal subset S by Since V t ε µb G ε, we have log V t ε KS := lim ε 0 log µb G ε. KS. And, if KS =, the generating gates S are optimally asymptotically efficient. Hence, the goal now is to find how small KS we can make, by choosing a universal subset S suitably. 8 Approximation Example for q mod 4 8. The Universal Subset For convenience, we will sometimes use q = as an example, but the results are true for q mod 4. Consider the Pauli matrices I, X, Y, Z, where I = 0 0, X = 0 0, Y = 0 0, Z = 0 i i 0. Then, let S = {s, s, s, s, s 3, s 3 }, where 6

7 s = I + ix = s = I + iy = s 3 = I + iz = + i 0 0 i i i, s = I ix =, s = I iy =, s 3 = I iz = i i i i. Since s, s, s 3 SU, S is a subset of SU. Also, from [B-G-S], Γ, the subgroup generated by S, is dense in SU. Hence, S = {s, s, s, s, s 3, s 3 } is universal. If the weights ws i are all taken to be, then for γ Γ, the height hγ is the reduced word length of γ. Define Ut = {γ Γ hγ = t}. There are 6 elements with reduced word length : s, s, s, s, s 3, s 3, hence, V = U = 6. Also, if γ Γ has reduced word length t with the last symbol a S, then we can add any symbols in S except a to obtain a reduced word with length t +. Hence, Ut + = Ut, for t. Hence, Ut = {, t = 0 6 t, t. Hence, for t, Hence, In general, V t = t Uk = t = 3 t. k= V t = Ut = { { t = 0 3 t t., t = 0 q + q t, t, and, V t = { t = 0 q+ q qt q t 7

8 8. Representation From [B-G-S], we can represent the elements of the group as projections of integer solutions of spherical functions on S 3. Theorem. In the setting of this example q =, each point of V t actually corresponds to an integer solution of x + x + x 3 + x 4 = h, h t. The theorem is based on the following lemma and the following isomorphism. Lemma. U = { p = a + bi + cj + dk Np = l, l 0 }. Let { } p U = p U. Np Then U is generated by {i, j, k, + i, + j, + k}. By the mapping property, there exists a homomorphism ϕ : U W with ±i ±ix, ± j ±iy, ± k ±iz, ± i ± ix = s, s ± j ± iy = s, s ± k ± iz = s 3, s 3. Also, since W is generated by ix, iy, iz, + ix, + iy, + iz, ϕ is surjective. And since ker ϕ = {}, ϕ is injective, and hence is an isomorphism. Then, since Γ is a subgroup of W, Γ is isomorphic to the subgroup of U, ± i, ± j, ± k. Hence, for any γ V t Γ, we have hγ t, and ϕ γ = q q q hγ = hγ where q i { ± i, ± j, ± k}. Since, Nϕ γ =, we have q q q hγ, and ϕ γ = a hγ + b i + hγ c hγ j + d hγ k, a b c d hγ hγ hγ Hence, γ V t corresponds to an integer solution of =, hγ t. hγ a + b + c + d = l, where l = hγ t. Example. Let γ = s s = + i 4 + i 4 + i i V t, for t. 8

9 We can check that = = = = hγ. Hence, γ corresponds to the integer solution,, 4, of x + x + x 3 + x 4 =. 8.3 The Range of KS 8.3. A Useful Fact From [Letter], we have the following useful fact for the lower bound of KS. It is based on the fact that SU is homeomorphic to S 3. See [D-L]. Fact. If y Z 4, y 0, then the ball B S 3 y y, contains a ball in S 3 with radius r =, 0 y t 0t y t which has no points corresponds to elements in V t Lower Bound of KS Then, we can calculate the lower bound of KS. By definition, for a given ε, t ε is the least t for which G B G γ, ε. γ V t ε Since there is y Z 4 and a ball in S 3 of radius which has no points of V t, we have 0t ε y tε ε >, 0t ε y tε Hence, ε < 0t ε y tε = 0tε y tε 4. log V t ε KS = lim ε 0 log µb G ε = lim ε 0 log 3 tε log ε 3 lim ε 0 log 3 tε 3 log 0t ε y tε 4 log = lim tε + log 3 tε tε ε 0 3 log tε log 0t ε y = lim t ε = 4 3. t ε + log 3 tε tε 3t ε log 0t ε y An Equivalent Expression of KS Now, we calculate the upper bound of KS. Recall that in order to approximate any x G = SU, for given accuracy ε, we choose t ε to be the least t for which 9

10 G B G γ, ε. γ V t Note that SU is a double cover of SO3, which is the rotation group of R 3. That is SU/{I, I} = SO3. Since I / Γ, if P V t,then P / V t. Hence, if we use V 0 t to denote the corresponding elements of V t in SO3, then V 0 t = V t = 3 t. Then, approximating SU can be done by approximating SO3. Again, let t ε be the smallest t that V 0 t can be used to approximate every element of SO3. Then, where log V 0 t ε KS = lim ε 0 log µb S ε, µb S ε cε Hecke Operator Recall that U 0 t = Ut = {, t = 0 q + q t, t, and V 0 t = V t = t Ut i=0 = q + q qt q. Hence, Denote q t q = V t + q + q +. U 0 t = { A, A,, A N, A, A,, } A N SO3, where N = q + q t. Then the Hecke operator is T : L S L S, and T t fx = N fa i x + fa i x = i= γ U 0t fγx, T V0tfx = t fγx = T i fx, γ V 0t i=0 where L S = { f : S C S fx dx < }. For example, when q =, 0

11 T fx = s U 0 fsx = fs x + fs x + fs x + fs x + fs 3x + fs 3 x. Also, from [L-P-S ], L S decomposes under the Laplace operator as L S = H n, where H n is the space of spherical harmonics of degree n. Each H n is an eigenspace for with eigenvalue nn + and the dimension of H n is n +. Since the Hecke operator is self-adjoint, S R 3, and T H n = H n. By the Spectral theorem, there exists a sequence of eigenvalues n=0 and an orthonormal basis of L S λ 0 t, λ t, λ t, R, φ 0, φ, φ, consisting of eigenfunctions of the Hecke operator such that T t φ j = λ j φ j, and each φ j is an orthonormal basis of some H n. That is, {φ 0 } is the orthonormal basis of H 0, {φ, φ, φ 3 } is the orthonormal basis of H, {φ 4, φ, φ 6, φ 7, φ 8 } is the orthonormal basis of H, 8.3. Eigenvalues of Hecke Operator Let x 0 S be a fixed point, then the Hecke orbit of x 0 in S is O x0 = {sx 0 s V 0 t}. By the definition of eigenfunction, we have T t φ j x 0 = s U 0t φ j sx 0 = λ j tφ j x 0. In particular, H 0 is -dimensional with eigenvalue 0, and the orthonormal basis of H 0 is the constant spherical function Hence, we have φ 0 x = 4π. Then, T t φ 0 x 0 = φ 0 sx 0 = U 0 t = λ 0 tφ 0 x 0 = λ 0 t. 4π 4π s U 0t λ 0 t = U 0 t = q + q t. We have already seen that λ 0 T t = Ut with constant eigenfunction. We want to bound the other eigenvalues. From [L-P-S ], or from the Ramanujan Conjectures, we have the following theorem Theorem. For a prime q mod 4, the eigenvalues of T t satisfy that

12 λ j T q, for j. Now, let us find more relations between the eigenvalues. From the properties of the Hecke Operator, it can be verified that we have the following Theorem 8.. For a prime q mod 4, the Hecke operator T t satisfies that T t+ fx = T T t fx qt t fx, for t, T fx = T T fx q + fx, T 0 fx = fx. 3 Also, the eigenfunction corresponding to λ j T t+ is the nonconstant spherical harmonic function φ j. Since L S = n=0 H n and the orthonormal bases of H k do not depend on t, φ j is also the eigenfunction of T n, for n t with eigenvalue λ j T n. Proposition. For t N, the Hecke operators T t have the same orthonormal eigenfunctions. Moreover, an eigenfunction is not a zero function. Combining with the proof of Proposition, we have the following: Theorem 8.. Let φ j be a eigenfunction, then the corresponding eigenvalues satisfy λ j T t+ = λ j T λ j T t qλ j T t, for t, 4 λ j T = λ j T q +, λ j T 0 =. 6 Then, we consider T V0t, the Hecke operator of elements with height less or equal to t. definition, T V0tfx = γ V fγx = t 0t i=0 T ifx, hence From the On the other hand, from 4 we have t λ j T V0t = λ j T i. i=0 t t λ j T i+ = λ j T λ j T i qλ j T i i= i= = λ j T t t λ j T i q λ j T i. i= i= Combining with and first few terms, we have the following recurrence relation Proposition. The eigenvalues of T V0t satisfy the recurrence relation and λ j T V00 =. λ j T V0t+ = λ j T λ j T V0t qλ j T V0t, for t 7

13 Since λ j T is a constant, and from Theorem, λ j T q for j. function of the sequence {λ j T V0t Hence the characteristic x λ j T x + q = 0 has two equal real solutionswhen λ j T = q or two conjugative solutions when λ j T < q. Then, we can check that If λ j T < q, we have If λ j T = q, we have Hence, we have the following: λ j T V0t = q t sint + θ j + sin tθ j sin θ j q sin θ j for j. 8 λ j T V0t = q t [ + + t], for j. 9 q Theorem. For a prime q mod 4, the eigenvalues of T t satisfy that λ j T V0t q t t, for j. From this theorem and q t = q q+ V t + q+, we have q λ j T V0t t V t +, for j. q + q Point-Pair Invariant Let k ε x, y be a point-pair invariant on S such that and with the properties, k ε σx, σy = k ε x, y, for σ O f, k ε x, y 0, k ε x, ydy =, S k ε x, y = 0 if and only if d S x, y ε. From the last two properties, we have k ε x, ydy = S B S x,ε k ε x, ydy =. And since dy = µb S x, ε, and B S x,ε 3

14 µb S x, ε c ε, when ε 0, we can choose For example, we can choose k ε x, x c, for a fixed constant c. ε k ε = { cε = µb S x,ε, d Sx, y < ε 0, d S x, y ε A Useful Fact From [Letter], [L-P-S ], and the above discussion, we have the following fact now. Fact. V t ε 4πct ε ε Upper Bound of KS From [D-L], we can prove the following: and ε V 0t ε 4πct ε 4 KS. 8.4 Discussion Now, we have 4 3 Since V t ε = q+ q qtε q, we have log V t ε KS = lim ε 0. log µb G ε log q V t ε = log q q + q qtε q = t ε + log q q + q. tε q q Hence, for ε > 0 and x SU, there exists γ Γ such that d G x, γ < ε and hγ t ε t ε + log q q + q q q tε 6 log q ε. Also, there exists some y SU such that if γ Γ and d G y, γ < ε, then From [Letter], we have the following log q V hγ log µb G ε 4 3, hγ 4 log q ε log µb G ε log q q + q. tε q q 4

15 Theorem. Most points y with y SU have optimally good approximation. Or equivalently, most of ρ with ρ SO3 have good approximation. That is, for ε > 0 there exists an explicit t ε with log V 0 t ε log ε as ε 0, and for most points y SU with respect to Haar measure, there is γ Γ such that d G γ, y < ε and hγ t ε. Remark. In the proof of the theorem, we construct t ε explicitly as t ε = log q = + ε ε log ε q +εε ε, The proof is in [D-L], and the following diagram gives the measure of B, which is the set of points that can not be approximated well in this case. 6 Plot x xx x x Log x, x, 0, Figure : Measure of B 9 An Arbitrary Set of Generators In the last example, the universal subset we use is S = {s, s, s, s, s 3, s 3 }, with efficiency 4 3 KS. In order to improve this estimation, we use a larger universal subset: T = {s, s, s, s, s 3, s 3, ix, iy, iz}. Together with T and certain assumptions of covering radius on S 3 we have: Theorem 9.. KT = 4/3 and so evidently the strict KT <. See [D-L].

16 0 Solovay-Kitaev Alogrithm One of the biggest challenges to creating a functional quantum computer is designing a working faulttolerance system. A quantum computer must be able to correct environmental and hardware errors. An efficient method to control these errors is by using an algorithm to correct them. However, the computer is limited to only a finite set of gates, a library gate set. A simple example might be the Hadamard gate, the π/8-gate, and its inverse. Thus, the gates in the algorithm would need to be decomposed into these library gates. The Solvay-Kitaev S-K theorem and algorithm provide an effective way to decompose the gates from the correcting algorithm into those from the library set. 0. Solvay-Kitaev Theorem The S-K theorem states that for a given ɛ > 0 and subset G of SU, such that the group generated by G is dense in SU, there is a constant c such that for any U SU there exists a finite sequence S of length Olog c /ɛ of gates from G with du, S < ɛ. Remark: The value of c depends on the proof of the theorem. The smallest possible value is c = ; however, the proof is typically non-constructive. Thus, currently there is no algorithm that gives c =. The algorithm below from [D-N] gives a value of c = ln / ln3/ The calculations for this c are shown in section.3. The S-K theorem does not often provide much information on K but it is useful to construct polynomial time algorithms. 0. S-K Alogrithm for qubits Here is the SK algorithm in pseudocode, See [K-M-M]: function Solvay-KitaevGate U, depth n if n == 0 else The first line Return Basic Approximation to U Set U n = Solvay-KitaevU, n Set V, W = GC-DecomposeUU n Set V n = Solvay-KitaevV, n Set W n = Solvay-KitaevW, n Return U n = V n W n V n W n U n function Solvay-KitaevGate U, depth n shows that the algorithm depends on two inputs: the single-qubit quantum gate U the algorithm is approximating and an non-negative integer n which relates to the accuracy of the approximation. The function returns a sequence of instructions which approximates U to an accuracy of ɛ n, where ɛ n is a decreasing function of n, i.e. ɛ n 0 as n. The function is recursive and terminates when n = 0, if n == 0 Return Basic Approximation to U 6

17 Calculations must be done before using the algorithm, so that one can find a basic ɛ 0 -approximation to any U SU. Since ɛ 0 is a constant, the calculation can be done by enumerating and storing a large number of sequences of gates in G of length l 0, and creating a search algorithm to find the closet approximation to U. If n is not equal to 0, the first step is to find an ɛ n -approximation to U. else Set U n = Solvay-KitaevU, n In the next line, Set V, W = GC-DecomposeUU n one must find unitary gates V and W such that UU n = V W V W and di, V, DI, W < c gc ɛn, where c gc /. This decomposition is called a balanced group commutator. Let M := UU n. M is a rotation, by an angle θ about some axis ˆn on the Bloch sphere. Consider φ satisfying, sinθ/ = sin φ/ sin 4 φ/ [ ] φ = sin 4 cosθ/ 4. Let Ṽ be a rotation by φ about the ˆx axis of the Bloch sphere, and W be a rotation by φ about the ŷ axis. Then, Ṽ W Ṽ W is conjugate to M, i.e. M = SṼ W Ṽ W S for some unitary S. Let N := Ṽ W Ṽ W. Because M and N are unitary, they are both diagonalizable. Furthermore, because they are conjugates, they have the same eigenvalues. Thus, there are a diagonal matrix D, and two unitary matrices S M and S N ; such that, M = S M DS M N = S N DS N. This implies, M = S M S N NS N S M Letting S = S M S N, V = SṼ S and W = S W S, V and W satisfy UU n = V W V W. In the next line of the algorithm, a ɛ n -approximation is found for V and W. Set V n = Solvay-KitaevV, n Set W n = Solvay-KitaevW, n In the final line, Return U n = V n W n V n W n U n Because V n W n V n W n is an ɛ n-approximation of U n U n, V n W n V n W n U n = U n U n U n = U n is an ɛ n -approximation of U n. 7

18 0.3 Analysis Let ɛ n be as above, l n be the length of the sequence returned by the algorithm, and t n be the corresponding runtime. From the pseudocode, there are the recurrences, ɛ n = c approx ɛ 3/ n l n = l n t n 3t n + const, The first equation comes from the fact that V n W n V n W n is an c approxɛ 3/ n -approximation of U n U n. Because one wants this to be an ɛ n-approximation, one sets ɛ n = c approx ɛ 3/ n. For the second equation, the returned sequence is made up of sequences of accuracy n ; thus, l n = l n. Finally, because the function calls itself 3 times with an accuracy n, t n 3t n + const The constant in t n comes from the need to compute the balanced g group commutators, etc. It does not depend on n. The above recursions imply: ɛ n = c approx ɛ 3/ n = c approx c approx ɛ 3/ n 3/ = c approx c approx... c approx ɛ 3/ 0 3/... 3/ = c n j=0 3/n approx ɛ 3/n Since n j=0 3/n = 3/ 3/ = 3/ n, ɛ n = c approx 3/ ɛ 0 3 n = c approx ɛ 0 c approx 3 n l n = l n = l n = n l 0 = O n t n = 3t n + const = 3 t n + const = 3 n t 0 + const = O3 n 8

19 To obtain a given accuracy ɛ, ɛ = c approx ɛ 0 c approx 3 n c approxɛ = ɛ 0 c approx 3 n ln c approxɛ = 3 n ln ɛ 0 c approx 3 n = ln c approxɛ ln ɛ 0 c approx 3 n = ln /c approxɛ ln /ɛ 0 c approx n ln 3 ln /c = ln approx ɛ ln /ɛ 0 c approx ln/c ln ɛ approx ln/ɛ 0c approx n = ln 3 Thus, n must satisfy, ln n = ceil [ ln/ɛc ] approx ln/ɛ 0c approx ln3/. Note: ceil means to round up the decimal number to the nearest integer. Substituting this value into the equation of l n gives, l n = O n e lnn = O [ ] ln/ln3/ ln/ɛc = O approx ln/ɛ 0 c approx = O ln ln / ln3/ /ɛ Similarly, t n = O ln ln 3/ ln3/ /ɛ where ln / ln3/ References [B-G-S] A.Bocharov, Y.Gurevich, and K.Svore. Efficient decomposition of single-qubit gates into V basis circuits. Physical Review A : 033. [D-N] C.M. Dawson, M.A. Nielsen, The Solovay-Kitaev Algorithm, Quantum Information and Information, Vol 6, No 006, pp [K-M-M] V.Kliuchnikov, M.Dmitri, and M.Michele. Practical approximation of single-qubit unitaries by single-qubit quantum Clifford and T circuits. arxiv preprint arxiv: [D-L] Q. Liang and S.B. Damelin. On a Problem of Sarnak Related to Constructing Efficient Universal Sets of Quantum Gates, arxiv

20 [L-P-S ] A.Lubotzky, R.Phillips, and P.Sarnak, Hecke operators and distributing points on the sphere I. Communications on Pure and Applied Mathematics 39.S 986: S49-S86. [L-P-S ] A.Lubotzky, R.Phillips, and P.Sarnak, Hecke operators and distributing points on S II. Communications on Pure and Applied Mathematics : [Letter] P.Sarnak, Letter to Scott Aaronson and Andy Parlington on the Solovay-Kitaes Theorem and Golden Gates. [T] S.Tornier. Haar Measures. 04. [W] D.B.Westra. The Haar Measure on SU