On two subgroups of U(n), useful for quantum computing

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS On two subgroups of Un, useful for quantum computing To cite this article: Alexis De Vos and Stijn De Baerdemacker 05 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - THE GROUP OF UNITS OF GOLDIE PRIME RINGS I Z Golubchik - ON FINITE SIMPLE GROUPS CONTAINING STRONGLY ISOLATED SUBGROUPS N D Podufalov - Duality and the geometry of quantum mechanics José M Isidro Recent citations - Alexis De Vos and Stijn De Baerdemacker - Block- ZXZ synthesis of an arbitrary quantum circuit A. De Vos and S. De Baerdemacker This content was downloaded from IP address on /03/09 at 07:54

2 On two subgroups of Un, useful for quantum computing Alexis De Vos and Stijn De Baerdemacker Cmst, Vakgroep elektronika en informatiesystemen, Imec v.z.w., Universiteit Gent, Sint Pietersnieuwstraat 4, B Gent, Belgium Center for Molecular Modeling, Vakgroep fysica en sterrenkunde, Universiteit Gent, Technologiepark 903, B Gent, Belgium Ghent Quantum Chemistry Group, Vakgroep anorganische en fysische chemie, Universiteit Gent, Krijgslaan 8 - S3, B Gent, Belgium alex@elis.ugent.be, Stijn.DeBaerdemacker@UGent.be Abstract. As two basic building blocks for any quantum circuit, we consider the -qubit PHASOR circuit Φθ and the -qubit NEGATOR circuit Nθ. Both are roots of the IDENTITY circuit. Indeed: both Φ0 and N0 equal the unit matrix. Additionally, the NEGATOR is a root of the classical NOT gate. Quantum circuits acting on w qubits consisting of controlled PHASORs are represented by matrices from ZU w ; quantum circuits consisting of controlled NEGATORs are represented by matrices from XU w. Here, ZUn and XUn are subgroups of the unitary group Un: the group XUn consists of all n n unitary matrices with all n line sums i.e. all n row sums and all n column sums equal to and the group ZUn consists of all n n unitary diagonal matrices with first entry equal to. Any Un matrix can be decomposed into four parts: U = expiαz XZ, where both Z and Z are ZUn matrices and X is an XUn matrix. We give an algorithm to find the decomposition. For n = w it leads to a four-block synthesis of an arbitrary quantum computer.. Introduction The unitary group Un is important for quantum computing, because all quantum circuits acting on w qubits can be represented by a member of the unitary group U w. For example, all quantum circuits, acting on a single qubit, are represented by a matrix from U. The simplest U matrix is the unit matrix. It represents the IDENTITY gate or I gate: = I. 0 This gate is trivial, as it performs no action on the qubit: the output qubit equals the input qubit. Within U, the I matrix has a lot of square roots: four diagonal matrices,,, and, 0 as well as an infinity of anti-diagonal matrices: 0 e iχ e iχ 0 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the authors and the title of the work, journal citation and DOI. Published under licence by Ltd,

3 where χ is an arbitrary real number. From this set, we choose two elements: 0 and. 0 The latter matrix results from the choices = i and χ = π/ and represents the NOT gate or X gate []. The former matrix represents the Z gate. Whereas the X gate is a classical computer gate, inverting the incoming bit, the Z gate is a true quantum gate. We now introduce a generalization of both the X and the Z matrix: Nt = ti+tx Φt = ti+tz, where t is an interpolation parameter []. We can easily prove that these matrices are unitary, iff t is of the form e iθ /, resulting in Nθ = +eiθ I+ eiθ X Φθ = +eiθ I+ eiθ Z. We thus have constructed two -dimensional subgroups of the 4-dimensional group U: Nθ = +e iθ e iθ e iθ +e iθ Φθ = 0 e iθ. The gate represented by the matrix Nθ, we call the NEGATOR gate. It thus constitutes a generalization of the NOT gate. The gate represented by the matrix Φθ, we call the PHASOR gate. We use the following symbols for these quantum gates: Nθ and Φθ, respectively. In the literature [3] [4] [5] [6], some of these gates have a particular notation: N0 = I Nπ/4 = W Nπ/ = V Nπ = X Nπ = I Φ0 = I Φπ/4 = T Φπ/ = S Φπ = Z Φπ = I. In particular, the V gate is known as the square root of NOT [7] [8] [9] [0]. The subgroup of all Nθ matrices, we denote by XU; the subgroup of all Φθ matrices, we denote by ZU. These two subgroups of U have three interesting properties.

4 i Their intersection is minimal, as it equals U s trivial subgroup consisting of merely one matrix, i.e. the identity matrix I. ii Their closure is maximal, as it equals U itself. Indeed, an arbitrary member of U, i.e. U = e iα cosϕe iψ sinϕe iχ sinϕe iχ cosϕe iψ can be synthesized by cascading merely NEGATORs and PHASORs. The reader may verify the following two matrix decompositions: U = e iα+iϕ+iψ 0 ie iψ iχ U = e iα iϕ+iψ 0 ie iψ iχ +e iϕ e iϕ e iϕ +e iϕ 0 ie iψ+iχ +e iϕ e iϕ e iϕ +e iϕ 0 ie iψ+iχ. Because, moreover, any phase factor can be decomposed as e iβ 0 0 e iβ = 0 0 e iβ 0 0 e iβ, we conclude that U equals the cascade of three NEGATORs and three PHASORs: U = NπΦα+ϕ+ψNπΦα+ϕ χ+π/n ϕφ ψ +χ π/ U = NπΦα ϕ+ψnπφα ϕ χ π/nϕφ ψ +χ+π/. iii We note that the two subgroups are each other s Hadamard conjugate: ZU = H XU H and XU = H ZU H, where H denotes the Hadamard matrix: H =.. Decomposition of an arbitrary unitary matrix Two-qubit circuits are represented by matrices from U4. We may apply either the NEGATOR gate or the PHASOR gate from the previous section to either the first qubit or the second qubit. Here are two examples: Φθ and Nθ, i.e. a PHASOR acting on the first qubit and a NEGATOR acting on the second qubit, respectively. These circuits are represented by the 4 4 unitary matrices respectively e iθ e iθ and +e iθ e iθ 0 0 e iθ +e iθ e iθ e iθ 0 0 e iθ +e iθ, 3

5 However, we also introduce more sophisticated gates: the so-called controlled PHASORs and controlled NEGATORs. Two examples are Nθ and Φθ, i.e. a positive-polarity controlled NEGATOR acting on the first qubit, controlled by the second qubit, and a negative-polarity controlled PHASOR acting on the second qubit, controlled by the first qubit, respectively. The former symbol is read as follows: if the second qubit equals, then the NEGATOR acts on the first qubit; if, however, the second qubit equals 0, then the NEGATOR is inactive, i.e. the first qubit undergoes no change. The latter symbol is read as follows: if the first qubit equals 0, then the PHASOR acts on the second qubit; if, however, the first qubit equals, then the PHASOR is inactive, i.e. the second qubit undergoes no change. The matrices representing these circuit examples are: eiθ 0 eiθ eiθ 0 +eiθ respectively. We now give two examples of a 3-qubit circuit: Nθ and and Φθ, e iθ i.e. a positive-polarity controlled NEGATOR acting on the first qubit and a mixed-polarity controlled PHASOR acting on the third qubit. The 8 8 matrices representing these circuit examples are: eiθ eiθ eiθ eiθ respectively. We note the following properties: and, e iθ the former matrix has all eight row sums and all eight column sums equal to ; the latter matrix is diagonal and has upper-left entry equal to., Because the multiplication of two square matrices with all line sums equal to automatically yields a third square matrix with all line sums equal to, we can easily demonstrate that an arbitrary quantum circuit like, 4

6 consisting merely of uncontrolled NEGATORs and controlled NEGATORs is represented by a w w unitary matrix with all line sums equal to. The n n unitary matrices with all line sums equal to form a group XUn, subgroup of Un. We thus can say that an arbitrary NEGATOR circuit is represented by an XU w matrix. A laborious proof [] demonstrates that the converse theorem is also valid: any member of XU w can be synthesized by an appropriate string of uncontrolled NEGATORs. Because the multiplication of two diagonal square matrices yields a third diagonal square matrix and because the multiplication of two unitary matrices with first entry equal to yields a third unitary matrix with first entry equal to, we can easily demonstrate that an arbitrary quantum circuit like, consisting merely of uncontrolled PHASORs and controlled PHASORs is represented by a w w unitary diagonal matrix with first entry equal to. The n n unitary diagonal matrices with upper-left entry equal to form a group ZUn, subgroup of Un. We thus can say that an arbitrary PHASOR circuit is represented by a ZU w matrix. The converse theorem is also valid: any member of ZU w can be synthesized by an appropriate string of uncontrolled PHASORs. We summarize: the study of NEGATOR and PHASOR circuits leads to the introduction of two subgroups of the unitary group Un: the subgroup XUn, consisting of all n n unitary matrices with all of their n line sums are equal to ; the subgroup ZUn, consisting of all n n diagonal unitary matrices with upper-left entry equal to. These subgroups have properties similar to the three properties of XU and ZU, discussed in the previous section. i The intersection of XUn and ZUn is minimal, as it is the trivial subgroup consisting of a single matrix, i.e. the n n unit matrix. ii The closure of XUn and ZUn is Un. Indeed, any Un matrix U can be decomposed as U = e iα Z X Z X Z 3...Z p X p Z p, 3 with p n and where all Z j are ZUn matrices and all X j are XUn matrices []. Because we have the identity diaga,a,a,a,a,...,a,a = P 0 diag,a,,a,,...,,a P 0 diag,a,,a,,...,,a, where a is a short-hand notation for e iα and P 0 is the circulant permutation matrix we can conclude that U equals the product of n+ or less XUn members and n+ or less ZUn members., 5

7 iii We have that ZUn H n XUn H n and XUn H n ZUn H n, where H n is an n n complex Hadamard matrix [3]. In particular, the group H n ZUn H n is the subgroup of XUn consisting of all circulant XUn matrices. For arbitrary n, one may choose for H n the n n discrete Fourier transform F n : F n =... ω ω ω 3... ω n ω ω 4 ω 6... ω n. ω n ω n ω 3n... ω n n, where ω is the primitive n th root of unity. For n = w, also the real Hadamard matrix H w Kronecker product of Hadamard matrices H is a possible choice:..., with w factors in the tensor product. Because of the identity diag,a,a 3,...,a n,a n = diag,a,,...,,diag,,a 3,...,,... diag,,,...,a n, diag,,,...,,a n, it is clear that the group ZUn is isomorphic to the direct product U n and thus is an n -dimensional Lie group. The group XUn is isomorphic to its conjugate F n XUnF n. One can verify that, if X is an XUn matrix, then the matrix F n XFn is an n n unitary matrix with first row,0,0,...,0 and thus also first column,0,0,...,0. Conversely, it is possible to prove that a matrix of the form X = F n n F 0 n Y n, 4 where 0 a b denotes the a b zero matrix and Y is an arbitrary member of Un, is an XUn matrix []. Thus XUn is isomorphic to Un and therefore is an n -dimensional subgroup of the n -dimensional group Un. We summarize by noting the beautiful symmetry ZUn = U n and XUn = Un. In the past, properties of the subgroup XUn of Un have not been studied. Here, we note in particular the group chain Pn XUn Un, where Pn denotes the finite group of n n permutation matrices. As all classical reversible circuits acting on w bits are represented by a matrix from P w, we may say that the group XU w represents computers, situated between classical computers and full-fledged quantum computers. 6

8 Decomposition 3 should be compared with other factorizations of Un, described in the literature. Diţă [4] has proposed a product D n O n D n O n...d O D, where all D j are diagonal unitary matrices and all O j are orthogonal matrices. Reck et al. [5] have derived a decomposition of the form T T...T p D, where all T j are -parameter unitary matrices describing a beam splitter plus phase shifter, D is a diagonal matrix, and p nn /. Rowe et al. [6] have applied two mutually commuting subgroups of Un. Finally, Patera and Zassenhaus [7] have introduced the generalized Pauli group P n. We note that, for odd n, this finite group is generated by the permutation matrix P 0 together with a particular member of ZUn, i.e. diag,ω,ω,...,ω n, where ω is the primitive n th root of unity. 3. Short decomposition of an arbitrary unitary matrix In Reference [8], it is conjectured that a shorter decomposition 3 exists: with p. Thus an arbitrary member U of Un may be decomposed as U = expiαz XZ, 5 with X a member of XUn and both Z and Z member of ZUn. For n =, the conjecture is proved by eqns. For an arbitrary unitary matrix U with n >, a recent non-constructive proof is provided by Idel and Wolf [9], based on symplective topology. For a given unitary matrix U, De Vos and De Baerdemacker [8] provide a numerical procedure reminiscent of Sinkhorn s construction [0] of a doubly stochastic matrix in order to find the scalar expiα, as well as the three matrices Z, X, and Z. For thousands of examples from 3 n 3, the algorithm converges to a solution. We conjecture that the algorithm always converges. Thus any quantum circuit looks like Z X Z e iα, i.e. the cascade of an overall phase factor, an input section consisting merely of uncontrolled PHASORs, a core section consisting merely of uncontrolled NEGATORs, and an output section consisting merely of uncontrolled PHASORs. In turn, the phase factor e iα may be decomposed into two NEGATOR circuits i.e. two classical cyclic-shift circuits [] [] and two uncontrolled PHASORs. We note that this circuit decomposition is not unique, because the matrix decomposition 5 is not unique. See Appendix. Instead of synthesizing a quantum circuit with X and Z building-blocks, one can also opt for X and Hadamard blocks. Indeed, introducing the circulant XU w matrices we have X = H w Z H w X = H w Z H w, H H H H Z X Z = H H X H H X X H H H H. Thus any quantum computer up to an overall phase consists of three or less NEGATOR circuits and 4w or less HADAMARD gates. 7

9 By applying alternatingly 4 and 5, again and again, i.e. for n = w, n = w, n = w,..., and finally for n =, we find yet another decomposition of an arbitrary quantum computer [9] [3]: R 4 T 4 R 3 T 3 R T L T L T 3 L 3 T 4 L 4 e iα 4. Here the circuits T, T 3,..., and T w implement the constant block-diagonal matrices T j = w j, F j where a is the a a unit matrix and F j is the j j discrete Fourier transform. The w circuits R j and the w circuits L j are either ZU w circuits or circulant XU w circuits. 4. Conclusion Like the qubit being a quantum counterpart of the classical bit, the NEGATOR gate is the quantum counterpart of the classical NOT gate and the controlled NEGATOR is the quantum counterpart of the classical controlled NOT a.k.a. the TOFFOLI gate. Any quantum circuit can be built from uncontrolled NEGATORs, supplemented with either uncontrolled PHASOR gates or HADAMARD gates. Appendix The matrix decomposition 5 is not unique. For n =, eqns show two different solutions. E.g. the discrete Fourier transform a.k.a. the Hadamard transform F = has two and only two decompositions: i 0 i +i i i +i 0 i and +i 0 i i +i +i i 0 i However, if, in, either cosϕ or sinϕ is zero, decompositions constitute an infinity of decompositions. This is confirmed by the identities e ix 0 0 e iy = e ix 0 e ix+iz 0 e iy iz and 0 e ix e iy = e iz e iy iz 0 e ix iz, where, in both cases, z may take any value. If n >, then an infinity of decompositions of the matrix U arises whenever the matrix contains a row with n zeroes and thus automatically also a column with n zeroes. 8

10 provided that row is not the first row and that column is not the first column. If no such row and column is present, there may be either a finite or an infinite number of decompositions. The former is illustrated by the example of the 3 3 discrete Fourier transform: F 3 = ω ω, 3 ω ω where ω is the primitive cubic root of unity, i.e. +i 3. Indeed, for this matrix six and only six decompositions are possible: and iω iω i i iω iω ω 0 0 ω ω ω 0 0 ω ω 0 0 ω ω i 3 i i 3 i i 3 i 0 0, i 3 i 3 i 0 0 ω 3 i i 3 i i 3 i 3 i 0 0 ω 0, 3 i 3 i i 0 0 i 3 i 3 i 3 i 3 i i 0 0 ω 0, 3 i i 3 i 0 0 ω 3+i 3+i i i 3+i 3+i 0 0, 3+i i 3+i 0 0 ω 3+i i 3+i 3+i 3+i i 0 0 ω 0, i 3+i 3+i 0 0 i 3+i 3+i 3+i i 3+i 0 0 ω 0. 3+i 3+i i 0 0 ω The latter is illustrated by the decomposition of the 4 4 Fourier matrix: F 4 = i i i i = = b a a b b ia ia /a /a ia ia /a /a b b i/b i/b b b i/b i/b i/a i/a b b, 9

11 where both a and b are allowed to have any value on the unit circle of the complex plane. The Fourier matrices F n not only are important in quantum computing [], but also in quantum optics. In the latter application, they constitute the canonical form of the n n unitary matrices with all entries having the same modulus i.e. having modulus / n. Constantmodulus unitary matrices are important for multiports particle beam splitters [4]. References [] Nielsen M and Chuang I 000 Quantum Computation and Quantum Information Cambridge: Cambridge University Press [] De Vos A and De Baerdemacker S 04 Matrix calculus for classical and quantum circuits A.C.M. Journal on Emerging Technologies in Computing Systems 9 [3] Wille R and Drechsler R 00 Towards a Design Flow for Reversible Logic Dordrecht: Springer [4] Sasanian Z and Miller D 0 Transforming MCT circuits to NCVW circuits Proc. 3 rd Int. Workshop on Reversible Computation Gent pp [5] Selinger P 05 Efficient Clifford+T approximations of single-qubit operators Quantum Information & Computation 5 59 [6] Amy M, Maslov D and Mosca M 03 Polynomial-time T-depth optimization of Clifford+T circuits via matroid partitioning IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems [7] Deutsch D 99 Quantum computation Physics World 5 57 [8] Galindo A and Martín-Delgado M 00 Information and computation: classical and quantum aspects Review of Modern Physics [9] De Vos A, De Beule J and Storme L 009 Computing with the square root of NOT Serdica Journal of Computing [0] Vandenbrande S, Van Laer R and De Vos A 0 The computational power of the square root of NOT Proc. 0 th Int. Workshop on Boolean Problems Freiberg pp 57 6 [] De Vos A and De Baerdemacker S 03 The NEGATOR as a basic building block for quantum circuits Open Systems & Information Dynamics [] De Vos A and De Baerdemacker S 04 The decomposition of Un into XUn and ZUn Proc. 44 th Int. Symposium on Multiple-Valued Logic Bremen pp [3] Tadej W and Życzkowski K 006 A concise guide to complex Hadamard matrices Open Systems & Information Dynamics 3 33 [4] Diţă P 003 Factorization of unitary matrices Journal of Physics A: Mathematical and General [5] Reck M, Zeilinger A, Bernstein H and Bertani P 994 Experimental realization of any discrete unitary operator Physical Review Letters [6] Rowe D, Sanders B and de Guise H 999 Representations of the Weyl group and Wigner functions for SU3 Journal of Mathematical Physics [7] Patera J and Zassenhaus H 988 The Pauli matrices in n dimensions and finest gradings of simple Lie algebras of type A n Journal of Mathematical Physics [8] De Vos A and De Baerdemacker S 04 Scaling a unitary matrix Open Systems & Information Dynamics [9] Idel M and Wolf M 04 Sinkhorn normal form for unitary matrices arxiv:math-ph [0] Sinkhorn R 964 A relationship between arbitrary positive matrices and doubly stochastic matrices The Annals of Mathematical Statistics [] Beth T and Rötteler M 00 Quantum algorithms: applicable algebra and quantum physics In: Alber G, Beth T, Horodecki M, Horodecki P, Horodecki R, Rötteler M, Weinfurter H, Werner R and Zeilinger A 00 Quantum Information Berlin: Springer Verlag pp [] De Vos A 00 Reversible Computing Weinheim: Wiley VCH Verlag p 49 [3] De Vos A and De Baerdemacker S 04 The synthesis of a quantum circuit Proc. th Int. Workshop on Boolean Problems Freiberg pp 9 36 [4] Mattle K, Michler M, Weinfurter H, Zeilinger A and Zukowski M 995 Non-classical statistics at multiport beam splitters Applied Physics B 60 S 0