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1 % { Eletroni Transations on Numerial Analysis. Volume 21, pp , Copyright 2005,. ISSN QR FACTORIZATIONS USING A RESTRICTED SET OF ROTATIONS DIANNE P. O LEARY AND STEPHEN S. BULLOCK Dediated to Alan George on the oasion o his 60th birthday Abstrat. Any matrix o dimension ( ) an be redued to upper triangular orm by multiplying by a sequene o appropriately hosen rotation matries. In this work, we address the question o whether suh a atorization exists when the set o allowed rotation planes is restrited. We introdue the rotation graph as a tool to devise elimination orderings in QR atorizations. Properties o this graph haraterize sets o rotation planes that are suiient (or suiient under permutation) and identiy rotation planes to add to omplete a deiient set. We also devise a onstrutive way to determine all easible rotation sequenes or perorming the QR atorization using a restrited set o rotation planes. We present appliations to quantum iruit design and parallel QR atorization. Key words. QR deomposition, Givens rotations, plane rotations, parallel QR deomposition, quantum iruit design, qudits AMS subjet lassiiations. 65F25, 81R05, 65F05, 65F20, 81P68 1. Introdution. The QR atorization o a matrix o dimension! (#"$ ) is a key tool in many matrix omputations, inluding inding a basis or the range or null spae o a matrix and solving linear least squares problems. The unitary or real orthogonal matrix is usually omputed in one o three ways: Givens rotations, Householder reletions, or Gram-Shmidt orthogonalization [8]. We ous in this paper on Givens rotations. The Givens-based deomposition is also an essential tool in quantum omputing. A quantum omputation applies a unitary matrix to a vetor [5], and quantum (logi) iruits are a notational shorthand or atorizations o suh unitary matries into more basi ones, orresponding to gates. These gates are either tensor produts o &'(& unitary matries with identity matries (one-qubit operators) or tensors o )*) unitary matries with identity matries (two-qubit operators.) It is also ommon to implement Givens rotations using iruit bloks that inlude up to +-,/. ontrols (102 '03&54 ), with the number o gates proportional to the number o ontrols. A (real) Givens rotation 687:9 is a matrix that is equal to the identity exept that our entries are modiied: ;<7=7>02;?9@90$ACBEDGF and ;<7:90H,I;?97J0KDLM F or some angle NPOQFRO3&5S. Forming 6T7:9U modiies only rows V and W o a matrix, and an appropriate hoie o F ores some entry in the W th row o the produt to be zero. The extension to the omplex ase is straightorward. The usual Givens QR algorithm redues to upper triangular orm by applying X!, ZY[R\3.^]_<& Givens rotations in the order 6*`bad 6*`bah adecgcgug ahi 6*`ej 6*`e ejklugcgcg ei*cgugcgm 6*`=no 6 `no ngp noqra?ugcgug nsp i where 6t`=uv 7w9 is used to zero out the entry in row W and olumn x o the produt. For example, or a )zy % matrix, we ould onstrut There are many dierent rotation sequenes that aomplish this same goal, however. by applying the rotations indiated Reeived May 21, Aepted or publiation July 11, Reommended by J. Gilbert. Mathematial and Computational Sienes Division, National Institute o Standards and Tehnology, Gaithersburg, MD ; and Dept. o Computer Siene and Institute or Advaned Computer Studies, University o Maryland, College Park, MD (oleary@s.umd.edu). The work o this author was supported in part by the National Siene Foundation under Grant CCR Mathematial and Computational Sienes Division, National Institute o Standards and Tehnology, Gaithersburg, MD (stephen.bullok@nist.gov). 20

2 above QR FACTORIZATIONS USING A RESTRICTED SET OF ROTATIONS 21 6 `bah 6 ` ad 6 `bah ahe ahk a@} or by applying this sequene o rotations: 6 `ej 6 `ej 6 `k ejk e} kh} 6*`bah 6*`bah 6*` ad 6*`e 6*`ej 6*`kj kh}~ ejk ahë kh}~ ejk k}lg Both sequenes produe an upper triangular using the minimal number o rotations. However, perormane an vary based on the hoie o ordering o Givens rotations. Moreover, some rotation planes might not be available. Commerial omputers have memory hierarhies (involving registers, ahe levels, main memory, disks, et.), so some pairs o rows are aster to aess than others. Thus, in order to obtain optimal perormane, the hoie o rotations must depend on the layout o the matrix in memory. For parallel or grid omputing with memory distributed among proessors, the hoie o rotations should minimize the amount o proessor ommuniation, and this again depends on how the matrix is distributed among proessors. In quantum iruit design, applying a rotation or whih the binary representations o V,. and WI,/. dier in a single bit an be aomplished by a single ully-ontrolled one-qubit rotation (a partiular Givens rotation) and hene osts a small number o gates. All other rotations require a permutation o data beore the rotation is applied and thus should be avoided. Sine many atoms and ions have more than two hyperine energy levels or the eletron into whih quantum data might be enoded, there is also interest in quantum multi-level logis (qudits.) However, seletion rules apply to these hyperine levels, meaning that some but not all hoies o Givens rotation planes an be aessed using laser pulses. Thereore, it is important to determine whether a given set o rotation planes ƒsy[v Ws] an be used to redue a matrix to upper triangular orm using a minimal number o rotations. We will all suh a set o rotation planes omplete. In this paper we provide a onstrutive answer to the ollowing question: Given a set o allowed rotation planes ƒsy[v Ws]v, is there an algorithm to redue any l matrix to upper triangular orm using l,8zy \.?]_o& rotations in those planes? In other words, is the set o allowed rotation planes omplete? In the next setion we answer this question, and then in Setion 3 we illustrate the appliation o our result. 2. Neessary and Suiient Conditions or Completeness o a Set o Rotations. The key to the solution o our problem is the rotation graph, an undireted graph o nodes with a onnetion between node V and node W i a rotation is allowed in plane YV Ws]. A sample rotation graph is given in Figure 2.1. There are 4 edges and thereore 4 allowed rotation planes. A easible rotation sequene or step x o a triangularization will be a sequene o ˆ, x allowed rotation planes that an be used to redue the elements in rows x \K. through o olumn x to zero. One other deinition is useul. Consider deleting the nodes o a tree one-by-one, with the root deleted last, in any order that leaves a onneted tree at eah stage. From suh an admissible node ordering, the list o edges Y[V Ws] in the order in whih node W was deleted deines an admissible edge ordering. For example, orm a tree rom the graph in Figure 2.1

3 Š g 22 D. O LEARY AND S. S. BULLOCK Œ FIG A sample rotation graph. by removing the edge Y& orderings are ye] and rooting the graph at node 1. Then the three admissible edge Yy Yy Yh. )s] )s] &<] Yh. Yh. Yy ye] &<] ) ] Notie that these are exatly the three easible rotation sequenes (x 0. ) or the rotation planes orresponding to the three edges that we kept. One interesting rotation graph is the star graph, with node onneted by an edge to eah other node. There are YH,'&E]v admissible edge orderings on the resulting tree rooted at, orresponding to easible rotation sequenes that put,z. zeros in olumn 1 o a matrix using rotations Y W ], W'0 & 1,3. in any order, ollowed by the rotation Yh.!]. It is easy to see that this set o 1,2. CgUgCg rotation planes is omplete, sine it an be used to redue any ( "( ) matrix to upper triangular orm. A seond interesting rotation graph is the hain. There are onstraints on the node numbering or the hain, though. I the rotation graph is the hain 1 2 3, then we an introdue zeros in the irst olumn using the rotations (2,3) and (1,2), and then in the seond olumn using the rotation (2,3), so these rotation planes are omplete. I the rotation graph is the hain 3 1 2, then we an introdue zeros in the irst olumn using the rotations (1,3) and (1,2) in either order. But then there are no rotations that an be used to zero the element in row 3, olumn 2 o the matrix, so this set o rotation planes is not omplete and is not suiient or triangularization. These examples lead us to understand that there is a orrespondene between easible rotation sequenes and admissible edge orderings o trees derived rom the rotation graph. We summarize this relation in the ollowing result. LEMMA 2.1. Suppose that an -node rotation graph is onneted ater removal o nodes. through x,q.. a. Then there exists an admissible node ordering on the remaining, x nodes. b. An ordering o the edges o a spanning tree o the Yˆ, x] -node graph, rooted at node x, is admissible i and only i the orresponding rotation sequene at step x o the redution o a matrix to upper triangular orm is easible. Proo. a. Create any spanning tree rooted at x. An admissible node ordering an be reated by a depth irst searh, deleting hildren beore parents. b. Suppose we have an admissible ordering o the edges o a spanning tree rooted at node x, and onsider the sequene o rotations orresponding to this ordering, where we use rotation Y[V Ws] to eliminate the element in row W o the matrix when node W o the graph is deleted. Consider the irst edge YV Ws] in the list. One o its nodes, say W, the one urther rom x, does not appear later in the list, sine it is being deleted. In the matrix, we an zero the element in row W, knowing that none o our later rotations will use this row and that either VZ0x or we will have a later opportunity to zero the element in row V, when its turn or node Yd. Yd. Yh. &<]m ye]m y ]

4 . % QR FACTORIZATIONS USING A RESTRICTED SET OF ROTATIONS 23 deletion omes. We repeat this argument on eah o the,tx edges in our list through the last edge, whih uses rotation Yšx Ws] to eliminate the last node other than x. Thus an admissible edge ordering indues a easible rotation sequene. Conversely, i we have a easible rotation sequene, then it deines an elimination order or the nodes o the rotation graph, with x eliminated last. These rotations orm a spanning tree rooted at x by direting eah edge toward the node that is not eliminated. Now we an state the riterion or determining ompleteness o a set o allowed rotation planes, related to the notion o reahable sets [7, p.97]. THEOREM 2.2. A set o allowed rotation planes ƒ YV Ws] with. O2V WXO2 is omplete i and only i or every node W there exists a path in the rotation graph rom node to node W that passes through no node numbered lower than W. Proo. Suppose we have a node W or whih suh a path does not exist. When nodes W,'. are eliminated, the remaining rotation graph will be disonneted. I we try to do elimination UgCgCgC in olumn W, we will be able at best to eliminate all but one o the nonzeros in the disonneted piee, but there is no rotation that will eliminate that last nonzero. Thereore, this set o allowed rotation planes is not omplete. ) o the elimination, the rotation graph is onneted and thus by Lemma 2.1, an admissible CgUgCgm edge ordering and a easible rotation sequene exists. Thereore the set o rotation planes is omplete. COROLLARY 2.3. A set o allowed rotation planes ƒ Y[V W ]v with.ro V W O is omplete i and only i the rotation graph as well as eah o the graphs ormed by deleting nodes. through x, or x0.,., are onneted. Proo. This onnetedness UgCgCgC property is neessary and suiient or the existene o a path in the rotation graph rom node to node W that passes through no node numbered lower than W. One obvious result is perhaps worth stating. COROLLARY 2.4. The minimal number o allowed rotation planes that an be used to Suppose these paths do exist. Then at eah stage W (W 0œ. redue a general X matrix to upper triangular orm is,q.. Proo. Sine a onneted graph on nodes must have at least 2,P. edges, the statement ollows rom Corollary 2.3. Thus the star graph and the hain graph with node numbers dereasing along eah path rom give minimal sets o allowable rotation planes. Finally, we generalize the onept o a omplete set o rotations by allowing reorderings o the rows and olumns o the matrix to be triangularized. We will all a set o allowed rotation planes permutation-omplete i there exists a permutation matrix ž suh that any Ÿ matrix an be atored as ž8 ~žÿ0 YžT ž8]d where is upper triangular, ž % is a permutation matrix, and is the produt o at most Y,ˆ.^]_<& rotations in the allowed rotation planes. (Equivalently, we ould require that an ' matrix ( " ) an be atored into a permutation o an upper triangular matrix using X,PZY[T\z.?]_o& suh rotations.) % Realizing that permutations o orrespond to renumberings o the nodes o the rotation graph, we haraterize permutation-ompleteness in the next theorem. THEOREM 2.5. A set o allowed rotation planes is permutation-omplete i and only i the rotation graph is onneted. Proo. Suppose that the rotation graph is onneted. As in the proo o Lemma 2.1, hoose a spanning tree o the graph and reate an admissible node ordering by depth-irst searh, deleting hildren beore parents. With this numbering o the nodes o the rotation graph, Theorem 2.2 tells us that the set o allowed rotation planes is omplete. Conversely, i the rotation graph is not onneted, there is no renumbering that reates a

5 } 9 u e a e e a 24 D. O LEARY AND S. S. BULLOCK path between nodes in the disjoint piees, so we onlude rom Theorem 2.2 that this set o allowed rotation planes is not permutation-omplete. 3. Three Examples. We apply our algorithms to two problems in onstruting the sequene o gates to implement a quantum iruit and to QR atorization o a matrix distributed among parallel proessors. For ease o notation, we number the rows and olumns o an ~ matrix rom N to,z., rather than rom. to, sine this makes the graph onnetivity more lear Designing Quantum Ciruits or 3 Qubits. In quantum omputing with 3 qubits, we transorm a state vetor o length & k 0 by multipliation by a unitary matrix. Due to the axioms o quantum mehanis, a data-state o a three quantum-bit quantum omputer is desribed by a vetor in the Kroneker produt spae, where is the one- a«a«qubit state spae spanned by vetors (or kets) NE and.^. The pratial impliation is that the most easily implemented unitary maps, whih orrespond to physial proesses loal to a single quantum-bit, are matries o the orm,, and e e e e, where is a &X/& m± unitary matrix and is the ²*!² identity matrix. In order to apply an arbitrary unitary matrix, we would like to design our quantum iruits by atoring the unitary matrix into a produt o matries drawn rom this olletion o tensor produts, but they are not suiient. Indeed, the set o all matries o the orm ³ is ten dimensional and annot ontain all T unitary matries. However, it is possible to obtain all unitary matries by interleaving two-qubit operators, tensors o the orm and e or varying over ) ) unitary matries [6]. Instead o adding two-qubit U> operators, we augment this olletion with ontrolled operations. The simplest o these is. All are equivalent to replaing ertain opies o in the tensor produt matries by identity matries. Physially, is applied to the target qubit i and only i the ontrol qubits arry a ixed logial state, usually hosen to have all qubits equal to one. Controlled gates may be implemented using a number o two-qubit gates that is linear in the number o ontrols [5, Fig.6]. Not all Givens rotations o an eight dimensional vetor spae are doubly-ontrolled gates, but every 6R` where the binary expansions o W and x dier in a single bit is. Early papers on quantum iruit design (e.g., [5]) used the traditional hoie o rotations. For example, to redue the irst olumn o an matrix to upper triangular orm, we apply the rotations in the order 6*` 6*` 6*` j 6*` 6*` 6*` 6*` j ma el jk } j¹ g Only the rotations 6, ma 6, and je 6 satisy the onstraint that the binary representations o j} the indies V and W dier in a single bit. There are 12 allowed rotation planes that satisy the onstraint, orresponding to the 12 edges o the rotation graph o Figure 3.1. Sine this is a onneted graph on the 8 nodes, and sine deleting nodes in numerial order preserves onnetivity, this set o rotation planes is omplete. We an derive many elimination strategies that use only the allowed rotations. For example, or x0., orming the spanning tree by breaking the edges (2,6), (6,7), (3,2), (4,5), and (3,7) and proessing highest numbered nodes irst gives the rotation sequene 6*` 6*` 6*` j 6*` 6*` 6*` 6*` j j¹ ad } ahk el va and this onstrution an be extended to hyperube rotation graphs o any size. One suh strategy has been desribed in [10], optimizing the use o the tensor produt struture. A onstrution or qudits, with a state spae o dimension ºE4 (º»(& ) instead o &?4, is given in [3, 2].

6 spae o suh a qudit is isomorphi to ÀrÁ with spanning kets NE :.? º¼,.^.) For physial reasons, only the 8 rotation planes orresponding to the edges o the UgCgCgC graph o Figure 3.2 are allowed [9, 1]. I we are onstrained to the ordering given in the graph, this set o rotation planes is not omplete; or example, there is no path rom 5 to 7 that does not pass through 0. Thereore, in order to do the redution, we must, or instane, add the ability to do swaps, permutations o two rows in order to position them or the allowed rotation planes. These permutations are themselves (trivial) rotations. Counting permutations, 54 rotations are required to redue a general matrix t matrix to upper triangular orm. In this appliation, however, ordering is not important, so the ritial property is permutation-ompleteness. The rule (Corollary 2.3) is to delete the nodes in an order that ensures that the remaining graph at eah stage is onneted. So, or example, 7 must be deleted beore 0, and 3 must be deleted beore 2. One a node is deleted rom the yle ( ), then the ordering on the remaining nodes must be suh that we delete rom one or both ends o the resulting hain o 5 nodes. The minimum number o rotations required is ~ÂlÃE_o& 0 &<. One ordering that requires only 28 allowed rotations (gates) is 7,0,6,5,3,2,4,1, and here is one minimal easible rotation sequene: QR FACTORIZATIONS USING A RESTRICTED SET OF ROTATIONS 25 Š¼Šs½ ŠG½¼½ ŠG½JŠ Š¾Š¼Š ½¼½¼½ ½JŠs½ ½¼½JŠ ½rŠ¼Š FIG The 3-qubit rotation graph (Setion 3.1). Also, the allowed rotation planes or a hyperube (Setion 3.2) 3.2. Parallel QR Fatorization. I a QR atorization is to be omputed on a hyperube multiproessor, where the matrix is distributed as one row per proessor, then the allowed rotation planes, the ones orresponding to ommuniation only among neighboring proessors, are exatly those orresponding to the edges o a hyperube, as illustrated in Figure 3.1 or!03 proessors. I a blok o rows is distributed on eah proessor, then the hyperube rotations are suiient to add to the rotations loal to eah proessor in order to orm a omplete set o rotation planes. One suh elimination strategy was developed by Chu and George [4]. In addition to using only the allowed rotations, it is important to minimize the height o the spanning tree and the degree o the nodes within it, sine this determines the time needed or the deomposition Designing Quantum Ciruits or Qudits. The alkali o ¹ Rb is being investigated or its use in quantum omputing. The 8 hyperine levels o its ground state an be used to enode quantum inormation as a qudit with º 0ˆy bits. (In the Dira notation, the state-

7 Å ½ Æ Ä g 26 D. O LEARY AND S. S. BULLOCK Š Œ FIG The qudit rotation graph. Step 1: Redue olumn 7 to a single nonzero by the rotation sequene 6 `=¹ 6 `¹j 6 `¹j 6 `¹ 6 `¹j 6 `¹ }Up a emp } ecp k mp e Up Cp 6 `¹j ¹Cp g Step 2: Redue olumn 0 to 2 nonzeros by the rotation sequene 6*` 6*` j 6*` j 6*` 6*` 6*` j }mai e}l ek jë Step 3: Redue olumn 6 to 3 nonzeros by the rotation sequene 6 ` 6 ` 6 ` 6 ` 6 ` ej ek }je ad} a g Step 4: Redue olumn 5 to 4 nonzeros by the rotation sequene 6*`= 6*` 6*` j }ma eh}~ ejk 6*` j jë g Step 5: Redue olumn 3 to 5 nonzeros by the rotation sequene 6 `=k 6 `kj }ma eh} 6 `kj kje g Step 6: Redue olumn 2 to 6 nonzeros by the rotation sequene 6 `e }vä 6 `ej e}lg Step 7: Redue olumn 4 to 7 nonzeros by the rotation sequene 6 ` } }va g 4. Conlusions. We have developed the rotation graph as a tool to devise elimination orderings in Givens QR atorizations. Properties o this graph haraterize sets o rotation planes that are omplete (or permutation-omplete) and identiy rotation planes to add to omplete an inomplete set. Through spanning trees, we also have a onstrutive way to determine all easible rotation sequenes or perorming the QR atorization. This onstrution ould be automated or quantum iruit design.

8 QR FACTORIZATIONS USING A RESTRICTED SET OF ROTATIONS 27 REFERENCES [1] G. K. BRENNEN, D. P. O LEARY, AND S. S. BULLOCK, Criteria or exat qudit universality, Phys. Rev. A, 71 (2005), [2] S. S. BULLOCK, D. P. O LEARY, AND G. K. BRENNEN, Asymptotially optimal quantum iruits or Ç -level systems, Phys. Rev. Lett., 94 (2005), [3], Unitary evolutions o Ç -level systems, teh. report, Otober 2004, [4] E. CHU AND A. GEORGE, QR atorization o a dense matrix on a hyperube multiproessor, SIAM J Si. Stat. Comput., 11 (1990), pp [5] G. CYBENKO, Reduing quantum omputations to elementary unitary operations, Computing in Siene and Engineering, 3 (2001), pp [6] D. P. DIVINCENZO, Two-bit gates are universal or quantum omputation, Phys. Rev. A, 51 (1995), pp [7] A. GEORGE AND J. W.-H. LIU, Computer Solution o Large Sparse Positive Deinite Systems, Prentie-Hall, Englewood Clis, NJ, [8] G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, Johns Hopkins Univ., Baltimore, Maryland, third ed., [9] C. LAW AND J. EBERLY, Synthesis o arbitrary superposition o Zeeman states in an atom, Optis Express, 2 (1998), pp [10] J. J. VARTIAINEN, M. MOTTONEN, AND M. M. SALOMAA, Eiient deomposition o quantum gates, Phys. Rev. Lett., 92 (2004),