In this work we present a method of decomposition of arbitrary unitary matrix
$U\in\mathbf U(2^k)$ into a product of single-qubit negator and
controlled-$\sqrt{\mbox{NOT}}$ gates. Since the product results with negator
matrix, which can be treated as complex analogue if bistochastic matrix, our
method can be seen as complex analogue of Sinkhorn-Knopp algorithm, where
diagonal matrices are replaced by adding and removing an one-qubit ancilla. The
decomposition can be found constructively and resulting circuit consists of
$O(4^k)$ entangling gates, which is proved to be optimal. An example of such
transformation is presented

Glos, Adam

Sadowski, Przemysław

English

arXiv

Adam Glos

Przemysław Sadowski

Crossref

Quantum Inf Process (2016) 15:5145–5154DOI 10.1007/s11128-016-1448-zConstructive quantum scaling of unitary matricesAdam Glos1,2 · Przemysław Sadowski1Received: 4 March 2016 / Accepted: 21 September 2016 / Published online: 12 October 2016© The Author(s) 2016. This article is published with open access at Springerlink.comAbstract In this work, we present a method of decomposition of arbitrary unitarymatrix U ∈ U(2k) into a product of single-qubit negator and controlled-√NOTgates. Since the product results with negator matrix, which can be treated as com-plex analogue of bistochastic matrix, our method can be seen as complex analogueof Sinkhorn–Knopp algorithm, where diagonal matrices are replaced by adding andremoving an one-qubit ancilla. The decomposition can be found constructively, andresulting circuit consists of O(4k) entangling gates, which is proved to be optimal. Anexample of such transformation is presented.Keywords Matrix decomposition · Negator matrix · Scaling matrix1 IntroductionScaling a real matrix O with non-negative entries means finding diagonal matricesD1, D2 such that B = D1OD2 is bistochastic. Sinkhorn theorem presents a necessaryand sufficient condition for existence of the decomposition of a matrix. Moreover,the iterative Sinkhorn–Knopp algorithm finds the bistochastic matrix B [1]. Suchdecomposition can be used for ranking web pages [2], preconditioning sparse matrices[3] and understanding traffic circulation [4].Since unitary matrices are complex analogue of orthogonal matrices, it is naturalto ask whether there exist a counterpart of Sinkhorn theorem for them. De Vos andB Adam Glosaglos@iitis.pl1 Institute of Theoretical and Applied Informatics, Polish Academy of Sciences,Bałtycka 5, 44-100 Gliwice, Poland2 Institute ofMathematics, SilesianUniversity ofTechnology,Kaszubska23, 44-100Gliwice, Poland1235146 A. Glos, P. SadowskiDe Baerdemacker considered whether it is possible, that for arbitrary unitary matrixU ∈ U(n), there exist two unitary diagonal matrices U1,U2 such, that matrix U1UU2has all lines sums equal to 1. Suchdecomposition exists for arbitrary unitarymatrix, andan algorithm for finding it approximately was presented [5]. Matrices called negatorswere treated as quantum counterpart of bistochastic matrices and form a group XU(n)under multiplication. Idel and Wolf propose an application of the quantum scaling inquantum optics [6].Algorithm converges for arbitrary unitary matrix U [7]. Similar decomposition ofunitary matricesU ∈ U(2m) called bZbXbZ decomposition was presented [8]. Theyshow that there always exist matrices A, B,C, D ∈ U(m) such thatU =[A 00 B]12[I + C I − CI − C I + C] [I 00 D], (1)where I is identity matrix. Matrix in the middle is a block-negator matrix (which isalso a negator matrix), while left and right matrices are block diagonal matrices. In[9], an algorithm of finding such decomposition was presented.Group XU(2n) is isomorphic to U(2n − 1) and can be generated by single-qubitnegator and controlled-√NOTgates [10]. However, the proof is non-constructive sincea decomposition designed for generating random matrices was used [11]. Althoughit is proved that it exists for any unitary matrix, obtaining such a decomposition is avery complex task. Therefore, another approach is needed for efficient decompositionprocedure.In this article, using similar method presented by de Vos and de Baerdemacker [10],we demonstrate an implementation of arbitrary k-qubit unitary operation using one-qubit ancilla with controlled-√NOT and single-qubit negator gates. Since product ofthese basic negator gates is still a negator matrix, our result can be seen as quantumanalogue of scaling matrix. More precisely, we prove that for arbitrary matrix U ∈U(2k), which is performed on system H, there exist a negator N ∈ XU(2k+1) suchthat for arbitrary state |ψ〉 ∈ H, we haveU |ψ〉 = (N(|ψ〉)). (2)Here,  denotes the operation of extending the system with an ancilla register in |−〉state and denotes partial trace over the ancilla system. Since after performing opera-tions and N the state is of the form |−〉⊗U |ψ〉, the partial trace is simply removingthe ancilla system giving a pure state U |ψ〉. We describe an efficient algorithm thatfor given U returns explicit and exact form of N with decomposition into a sequenceof single-qubit negator and controlled-√NOT gates only in contrast to results of deVos and de Baerdemacker [9,10].In Sect. 2, we recall basic facts. In Sect. 3, we show how to perform such transfor-mation efficiently and demonstrate the cost in terms of controlled-√NOT gates.To illustrate the transformation method, a transformation of Grover’s search algo-rithm is presented step by step in Sect. 4.123Constructive quantum scaling of unitary matrices 51472 Basic factsNegator gates of dimension 2 were introduced by de Vos and de Baerdemacker [10]as unitary matrices N ∈ U(2) which are also a convex combination of identity matrixand NOT gate. Simple calculation shows that they are of the formN (θ) = 12[1 + eiθ 1 − eiθ1 − eiθ 1 + eiθ],where θ ∈ [0, 2π). Negators form a subgroup of single-qubit unitary operations, i.e.,N (φ)N (ψ) = N (φ +ψ) for any values of φ and ψ . In the following, we will also usea 2-qubit negator operation controlled-√NOT gate (which is also controlled-N (π2 )gate)⎡⎢⎢⎣1 0 0 00 1 0 00 0 1+i21−i20 0 1−i21+i2⎤⎥⎥⎦ .As these gates are used as basic operators, we will use a simplified notation in circuit,respectivelyθ and •√ .These two kinds of unitary matrices will be called NCN gates (Negators-Controlled-Negator).In Sect. 3, decomposition of single-qubit unitary gateswill be needed. Every unitarymatrix U ∈ U(2) can be presented as a product of global phase, two z-rotators andone y-rotator [12]U = eiφ0[cos φ12 eiφ2 sin φ12 eiφ3− sin φ12 e−iφ3 cos φ12 e−iφ2]= eiφ0[eiφ2+φ32 00 e−iφ2+φ32][cos φ12 sinφ12− sin φ12 cos φ12][eiφ2−φ32 00 e−iφ2−φ32]= eiφ0 Rz(−φ2 − φ3)Ry(φ1)Rz(φ3 − φ2). (3)Since global phase is not measurable, we can simplify this representation without lossof informationU ∼= Rz(γ )Ry(β)Rz(α), (4)where ‘∼=’ means equality up to a global phase. The same applies in the case of globalphase change on one of the registers of a bigger systemU1 ⊗ eiφU2 ⊗ U3 = eiφ(U1 ⊗ U2 ⊗ U3) ∼= U1 ⊗ U2 ⊗ U3. (5)1235148 A. Glos, P. SadowskiUsing these two facts, we can say that in any situation, we can ignore global phasechange on any register.While it may lead to a conclusion that our transformation is mainly applied to groupSU(n), we decided to stay with the unitary matrices formalism, since negator gatesare not special unitary matrices. The result may be written using the special matrices;however, then the negators gates column and row sums will equal eiθ in general.3 Circuit transformation methodIn this section, we provide complete description of the transformation method. Werecall a sketch of a proof of universality theorem between quantum gates and negatorgates from the work of de Vos and de Baerdemacker [10]. Next, we present transfor-mation method of arbitrary single-qubit gate into NCN product. Then, we provide amethod of decomposition for arbitrary k-qubit circuit, based on the single-qubit case.Finally, we analyze the cost of presented transformation.3.1 Universality theoremDeVos and de Baerdemacker proved a universality theorem: Group XU(2k) generatedby negators and controlled-√NOT is isomorphic to U(2k −1) [10]. The proof consistsof several steps:1. Every matrix U ∈ U(2k − 1) can be decomposed into a product of m gatesU1U2 · · ·Um , where matrices Ui ∈ U(2k − 1) are of some special forms [11].2. Group U(2k − 1) is isomorphic to group1U(2k) ={[1 0T0 U]: U ∈ U(2k − 1)}, (6)because of the isomorphism h : U(2k − 1) → 1U(2k)h(U ) =[1 00 U]. (7)3. Function f : 1U(2k) → XU(2k) of the form f (U ) = (H ⊗ I2k )U (H ⊗ I2k ) is anisomorphism.4. Decomposition of every f (h(Ui )) into a product of NCN gates is possible, whereUi comes from point 1.The proof used the decomposition presented in the work of Poz´niak et al. [11], becauseit is proven that the decomposition exists for any unitary matrix. However, obtainingsuch decomposition is a very complex task. Therefore, we need to choose a differentdecomposition in order to find an efficient decomposition procedure.Obviously, group U(2k) is isomorphic to some subgroup of XU(2k+1). In otherwords, with ancilla (one additional qubit), every unitary matrix can be replaced with asequence of NCN gates. For our purpose, we choose function g : U(2k) → XU(2k+1)123Constructive quantum scaling of unitary matrices 5149g(U ) = 12H ⊗ I (|0〉〈0|⊗ I +|1〉〈1|⊗U )H ⊗ I = 12[I II −I] [I 00 U] [I II −I].(8)Using the function g, every gate U changes into controlled-U . Using circuit notation,we can present this fact asg	→H • H/k U /k U .Note that if we assume that the first qubit is set to |−〉, the control qubit does notinfluence the result (the condition is always ‘true’).3.2 Single-qubit gate transformationNow, we aim at decomposition of arbitrary single-qubit gate into NCN gates. WithEq. (4) for any (U ∈ U(2)), there exist real parameters α, β, γ such thatU ∼= Rz(γ )Ry(β)Rz(α). (9)Therefore, after applying function g, we haveH • H∼=H • H H • H H • HU Rz(α) Ry(β) Rz(γ ) .We change the rotators with neighboring Hadamard gates into NCN gates as shownin Fig. 1H • H ∼=α2 −α2 • • β2 • −β2 • • γ2 − γ2U • • • √† √† • • • .Let us note that the symbols of controlled-NOT, controlled-√NOT†and controlled-negator used in the decomposed circuit do not mean that these gates cannot betransformed. We use these symbols as a simplified notation for its decompositionwith use of controlled-√NOT gates as shown in Fig. 2.3.3 General transformation methodNow, we consider transformation of arbitrary k-qubit circuit. Let us assume that wehave a circuit which consists of unitary operation (U ∈ U(2k)), generalized mea-surement M = {Ma ∈ L(C2k ) : a ∈ }, where  is a set of classical outputs ofmeasurement, and starting state |φ0〉|φ0〉 /k U M .1235150 A. Glos, P. SadowskiH • H=• • θ2 • − θ2 • •Ry(θ) • √† √† •H • H=θ2 − θ2Rz(θ) • •H • H=−π2 • π2 • −π2 • π2 •√ −π4 • −π4 • π2• • −π2 • −π2√ √† • −π4 π2Fig. 1 Decomposition of controlled-y-rotator, controlled-z-rotator and Toffoli gate. Decompositions usethe simplified notation from Fig. 2•=• •√ √•=• • •√† √ √ √•=π4 • −π4 • θ2 • − θ2 • π4 • −π4θ θ2√† √†Fig. 2 Decomposition of controlled-NOT, controlled-√NOT†gates and controlled-negator [10]In order to construct a decomposition of unitary U into a sequence of negator gates,we begin with obtaining a decomposition of U into controlled-NOT and single-qubitgates|φ0〉 /k U M ∼= |φ0〉 /k V1 V2 · · · Vm M ,here denoted by a sequence of gates U = Vm · · · V1. Contrary to the decompositionpresented in the work of Poz´niak, Z˙yczkowski and Kus´, there exist efficient methodsfor constructing such circuit [13]. Next, we need to add an additional qubit, transformVi gates into controlled-Vi gates and add Hadamard gates as below (since H H = I )|1〉 H H • H H • H · · · H • H H |1〉|φ0〉 /k V1 V2 · · · Vm M .123Constructive quantum scaling of unitary matrices 5151Let us note that product H ·controlled-Vj · H is an image of homomorphism presentedin Eq. (8) on Vj . Next, we replace the product with the sequence of NCN gates (heredenoted by N j ) as in previous subsection (if Vj is controlled-NOT, then we chooseToffoli gate transformation from Fig. 1)|1〉 HN1 N2· · ·NmH |1〉|φ0〉 /k · · · M .For the sake of simplicity, we may change the starting state and resulting state on thefirst wire|−〉N1 N2· · ·Nm|−〉|φ0〉 /k · · · M .Now, we have an equivalent circuit which consists of negators and controlled-√NOT gates only.3.4 Transformation costNow,we consider upper bound of cost of decomposition into negator circuit. Twokindswill be discussed: memory complexity and number of single- and two-qubit gates. Inthe first case for arbitrary k-qubit circuit transformation requires one additional qubit.Let cCNOT(k) and cs(k) denote upper bound of the number of, respectively,controlled-NOT and single-qubit gates needed for the implementation of an arbitraryk-qubit circuit. Using the operation presented above, we need 17cCNOT(k) + 64cs(k)controlled-√NOT gates and 11cCNOT(k) + 34cs(k) negators to implement an equiv-alent circuit (up to global phase).Any circuit which consists of controlled-NOT and single-qubit gates can be sim-plified in such a way that cs(k) ≤ 2cCNOT(k) + k. This estimation is based on theworst case, when there are two single-qubit gates between every controlled-NOT gate.Taking this into account, we can express the previous result in terms of cCNOT only,because only 17cCNOT(k) + 64cs(k) ≤ 145cCNOT(k) + 64k controlled-√NOT gatesare needed. In fact, if cCNOT = O(4k), then so is the number of controlled-√NOTgates.4 Step-by-step transformation exampleTo illustrate the introduced decomposition, we will present Grover’s algorithm fork = 2 qubits as NCN circuit. The original circuit for this algorithm is presented inFig. 3, where ω denotes the searched state.As in the previous section, we will add one qubit, change every H and G gate intocontrolled-H and controlled-G, respectively, and add Hadamard gates on the ancillaregister. Former steps of the decomposition are explicitly presented in Fig. 4. Thefollowing facts were used1235152 A. Glos, P. SadowskiFig. 3 Original Grover’s searchalgorithm circuit in case k = 2.G is Grover diffusion operator,Uω is quantum black box andwe perform measurement M .Algorithm comes from [14]Repeat O(√2k) times|0H UωGM|0 M|1Repeat O(√2k) times|1 H H • H H • H H |1|0H UωGM|0 M|1|0|0|1H • H H • H H • HHHHMMπ2−π2 • •π4 • − π4 • •• • • √† √† •H • H H • H H • H H • H H • H H • H H • H H • HH• •HRz − π2 Rz π2− π2 • π2 • − π2 • π2 •√ − π4 • − π4 • π2• − π2 • − π2√ √† • − π4 π2− π4 π4• •H • H H • Hπ2−π2 • •π2 • − π2 • •• • • √† √† •a) c)d)b)c)d)d)c) d)Fig. 4 Grover’s search algorithm decomposition. Unnecessary Hadamard gates have already been removed(a) |1〉 changes into |−〉; (b) measurement M does not change, on first wire we end with |−〉 state; (c)subcircuit simplification; (d) subcircuit transforming into NCN gates. Any other transformation left can bedone similarly, except the Uω case123Constructive quantum scaling of unitary matrices 5153• the decomposition of Hadamard gate is H ∼= Rz(π)Ry(π2 )Rz(0) = Rz(π)Ry(π2 ),• the decomposition of NOT gate is NOT ∼= Rz(π)Ry(π)Rz(0) = Rz(π)Ry(π),• for any (U, V ∈ U(2)), we haveH • H H • H H • HU = UV V ,• Grover’s diffusion operator can be decomposed in the following way/kG ∼=/k H • • HH Rz(−π2 ) Rz (π2 ) H .Decomposition ofUω depends strictly on the value ofω; therefore, it is not presentedin the example. The full decomposition is presented in Fig. 4.5 Concluding remarksIn the presented work, we provide a constructive method of scaling arbitrary unitarymatrices U ∈ U(2k). More precisely, we proved that for arbitrary unitary matrixU ∈ U(2k), there exists unitary negator matrix N ∈ XU(2k+1) such that for arbitrarystate |ψ〉, we haveU |ψ〉 = (N(|ψ〉)). (10)Here,  denotes the operation of extending the system with an ancilla register in|−〉 state and  denotes partial trace over the ancilla system. We described efficientalgorithm of decomposing N into product of single-qubit negator and controlled-√NOT gates. Our decomposition consists of O(4k) entangling gates which is provedto be optimal and needs one-qubit ancilla.Our result can be seen as complex analogue of Sinkhorn–Knopp algorithm, whichis known to havewide applications. The result is in contrast to the previous results [10],which could be only used to prove the existence of such decomposition. Moreover, ourtransformation is exact and can be found constructively. In contrast to [9], our trans-formation consists only of negator gates. The main difference is that transformationneeds one-qubit ancilla.Acknowledgements The work was supported by the Polish National Science Center: A. Glos under theresearch Project Number DEC-2011/03/D/ST6/00413, P. Sadowski under the research Project Number2013/11/N/ST6/03030.Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changes were made.1235154 A. Glos, P. SadowskiReferences1. Sinkhorn, R., Knopp, P.: Concerning nonnegative matrices and doubly stochastic matrices. Pac. J.Math. 21(2), 343–348 (1967)2. Knight, P.A.: The Sinkhorn–Knopp algorithm: convergence and applications. SIAM J. Matrix Anal.Appl. 30(1), 261–275 (2008)3. 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Constructive quantum scaling of unitary matrices

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