QPL 2005 Prelmnary Verson Paul measurements are unversal Vncent Danos 1 CNRS & Unversté Pars 7 Elham Kashef 2 IQC & Unversty of Waterloo Abstract We show that a varant of the one-way model where one only allows X and Y one qubt measurements s approxmately unversal. Key words: Quantum computng, measurements. 1 Introducton Whle the one-way model [,5,6] has been recognsed snce ts ncepton as an mportant theoretcal quantum computng model, t s only recently that some of ts measurement patterns have been realsed n the lab [7]. There are a number of dfferent questons to be addressed before such mplementatons on physcal substrates can be carred out on larger examples. One of these, whch s the one we are partcularly nterested n n ths note, s the effcent mplementaton of the feedforward mechansm by whch the measurements angles are allowed to depend on the outcomes of past measurements. We show that a varant of the orgnal one-way model, ncorporatng only Paul measurements s approxmately unversal. Thus, when executng a partcular pattern, the feedforward mechansm only has to deal wth a choce of two angles, 0 and π, and ths consttutes a potentally useful smplfcaton of the 2 underlyng hardware needed to realse the feedforward mechansm. We also show that ths varant stll admts a standardsaton procedure, meanng that all measurement patterns can be rewrtten n a way that entanglement s done frst, by adaptng the measurement calculus gven for the orgnal model [2]. Because our model s a varant, t s also nterestng from the theoretcal pont of vew, n that t helps n better chartng out the fundamental propertes 1 Emal: Vncent.Danos@pps.jusseu.fr 2 Emal: ekashef@qc.ca Ths s a prelmnary verson. The fnal verson wll be publshed n Electronc Notes n Theoretcal Computer Scence URL: www.elsever.nl/locate/entcs
Danos and Kashef one needs n a measurement-based model for t to be both unversal and standardsable. 2 Measurement patterns We use X, Z to denote the usual Paul matrces, H for the Hadamard transformaton, Z α for the Z-rotaton wth angle α, and the followng abbrevatons: P = Z π 2, Q = Z π. Note that Q 2 = P and P 2 = Z π = Z. The followng notatons wll also be useful: + := H 0, := H 1 = Z +, + α = Z α ( + ), α = Z α ( ) A measurement pattern, or smply a pattern, s defned by the choce of V a fnte set of qubts, two subsets (actually two maps to be precse) I and O determnng the pattern nputs and outputs (whch we don t suppose to have an empty ntersecton), and a fnte sequence of any of the followng nstructons (, j denote qubts n V where the nstructon apples): Measurements M α wth outcome wrtten s ; Correctons X s j, P s j ; Constant operatons E j and Q ; Preparatons +. Above, E j s just a notaton for Z j (controlled-z), also known as controlled phase, appled at qubt and j. Snce ths operator s symmetrc n, j, there s no need to say whch of, j s the control qubt, and whch s the target. Note that constant operatons, Q and E j (whch are called constant snce they never depend on any outcome) all commute together, so the specfc order n whch they appear n a measurement pattern s rrelevant to the result of a computaton. The nstructon M α stands for a one qubt measurement appled at qubt usng the orthonormal bass + α, α. One wrtes s for ths measurement outcome, wth the conventon that s = 0 when the measurement behaves as + α, and s = 1 when t behaves as α. A pattern s run by frst preparng non nputs n the + state, then by settng the nput qubts to a gven nput value, and fnally by executng each nstructon of the nstructon sequence. Wth that n mnd, t s natural to ask the followng: (D0) no command depends on an outcome not yet measured; (D1) no command acts on a qubt already measured; (D2) a qubt s measured f and only f s not an output. We see that the frst condton smply ensures that by the tme a correcton has to be done, one can actually compute ts exponent from the extant outcomes. The second condton ensures that qubts are not reused. Whle 86
Danos and Kashef reusng s legtmate to spare on the number of physcal qubts, t dsrupts the standardsaton property (because, obvously f qubts are reused, then they also have to be re-entangled, and therefore the whole entanglement can no longer be done frst), and s best left as an optonal optmsaton not taken nto account n our pattern language. Ths condton also ensures that there s at most one measurement done on any gven qubt, and therefore the notaton s, standng for the outcome of the measurement done at qubt s sensble. Fnally, the thrd condton ensures that by the tme all measurements have been done, the output qubts are no longer entangled wth the rest, so that they can effectvely be read out of the computaton space. 2.1 Actons on measurements In general, our set of correctons can be absorbed n measurements n the followng way: M α Xs = M ( 1)s α M α P s = M α s π 2 One readly sees that the subset of angles {0, π, π, 2 π } s closed under the 2 actons of the correctons. Therefore we may, and ths s what we do now, restrct to that partcular subset. Also, startng wth the same equatons, one can compute the effect of a sequence of correctons on the actual angle of measurement. For nstance: M α X s P t = M ( 1)s α t π 2 M α P t Xs = M ( 1)s α ( 1) s t π 2 Note that the obtaned angles are dfferent n case s = t = 1. In other words, P and X respectve actons on measurements don t commute. Ths makes the computaton of angle dependences a bt more complcated than n the orgnal model where one consdered only X and Z actons (whch do commute). Thereafter, a measurement preceded by a sequence of correctons on the same qubt wll be called a dependent measurement. Note that, by the absorpton equatons above, these ndeed can be seen as measurements, where angles depend on the outcomes of some other measurements made beforehand. Ths s the feedforward mechansm mentoned n the ntroducton. 3 Standardsaton Standardsaton s a procedure by whch nstructons n a pattern can be rearranged n a specfc order, where constant nstructons are done frst, then dependent measurements, then correctons on the outputs. Such patterns wll be called standard. 87
Danos and Kashef The frst needed ngredent for standardsaton s gven by the absorpton laws, where a correcton gets absorbed by a measurement as n the equatons gven above n the precedng secton. The second ngredent concerns the commutaton of a constant nstructon wth a correcton: E j X s = X s P s j P s j E j E j P s Q X s Q P s = P s E j = P s X s Q = P s Q Essentally standard rewrtng theory arguments show that by drectng the absorpton and commutaton equatons from left to rght, and usng free commutaton equatons (when nstructons apply on dfferent qubts) any sequence of nstructons n a pattern satsfyng (D1) can be rewrtten n a standard form. Moreover the other condtons (D0) and (D2) are stable under rewrtng, and the standard form s essentally unque. Standardsaton s partcularly useful when a bg pattern s gven by composng (ether sequentally or n parallel) smaller ones. It s also useful to reveal the ntrnsc depth complexty n a pattern, where ths depth s defned as that of the graph of dependences of ts standard form. Note that condton (D0) exactly amounts to sayng that ths dependency graph s acyclc, and thereby guarantees that the depth s fnte (and then, smaller than the total number of qubts). Unversalty Defne J α := HZ α. We know [3] that the famly of J α s together wth Z s unversal. We also know that the set consstng of J 0 (whch s H), J π, and Z s approxmately unversal. To prove the approxmate unversalty of our language of patterns, t s therefore enough to exhbt a pattern for each of these three untares. Let us begn wth J 0 and Z whch are computed by the followng 2-qubt patterns: J 0 := X s 1 2 M0 1 E 12 (1) Z:= E 12 (2) where the frst pattern has nputs {1}, and outputs {2}, whle the second one has {1, 2} both as nputs and outputs (note the overlappng). Now to mplement Jπ n the ordnary one-way model, one may use: J π := Xs 1 2 M π 1 E 12 (3) = X s 1 2 M 0 1E 12 Q 1 () wth nputs {1} and outputs {2} as n our pattern for J 0. We note that the frst form doesn t ft n the varant model, snce t uses a measurement wth 88
Danos and Kashef an angle π, but the second one does. It follows that any untary can be obtaned wth an arbtrary precson, by a pattern obtaned by tensorng and composng our three basc patterns. If we have a look back at our three generatng patterns, t seems that only X correctons are needed, but ths s not so. Indeed, from to the rules commutng constant nstructons and correctons, ones sees that X correctons generate Z π 2 correctons (frst and thrd equatons), so these are really needed, at least f one wants to work wth standard patterns. 5 Concluson We have proved that a varant of the one-way model where only Paul measurements are ever made s approxmately unversal. Ths s smlar n sprt to a result known from the crcut model, where one proves that magcal states preparatons, meanng Z π +, and Clfford operators are approxmately unversal [1]. However the proof gven here s much smpler. From the standardsaton procedure, we also see that restrctng to standard patterns, where entanglement s done frst, stll results n an approxmately unversal model. Therefore, durng a computaton, the physcal devce used to perform one qubt measurements needs only to be swtched to an X or a Y measurement. The prce to pay for ths smplfcaton s twofold. Frst, one needs to rotate some of the nputs usng a Z-rotaton wth angle π. Note however, that by standardsaton, ths phase may always be computed at the begnnng, concurrently wth the entanglement phase, and s done once and for all. It can also be dsposed of altogether by usng the teleportaton pattern, but ths wll ncur a cost of two extra qubts. Second, snce the X and Z π 2 actons on measurement angles don t commute, the arthmetc needed to compute at run-tme whether one should do an X or a Y measurement s more complcated than t s n the usual model. As ths arthmetc on the forwarded outcomes s purely classcal, ths doesn t seem to be a problem. As was suggested recently by Bregel, ths model could also prove useful to deal wth the mportant ssue of fault-tolerance n measurement-based quantum computng. Indeed, snce only X and Y measurements are ever used, the tradtonal error model, based on Z and X errors, wll affect the computaton n a partcularly smple way, only swappng the outcomes of a measurement. Furthermore, from the absorpton equatons, one sees that X errors are ether of no consequence (n the case the measurement angle s 0 or π), or equvalent to a Z error (n the case the measurement angle s π or 2 π ). Snce Z correctons are known to be easy both to detect and to correct, by usng repetton 2 codes, t seems ths model could mesh well wth error-correcton technques. To really see how fault-tolerant ths model could be made, further work s needed however. 89
Danos and Kashef References [1] Bravy, S. and A. Ktaev, Unversal quantum computaton wth deal Clfford gates and nosy ancllas (200), arxv:quant-ph/003025. [2] Danos, V., E. Kashef and P. Panangaden, The measurement calculus (200), arxv:quant-ph/012135. [3] Danos, V., E. Kashef and P. Panangaden, Robust and parsmonous realsatons of untares n the one-way model (200), arxv:quant-ph/011071. [] Raussendorf, R. and H.-J. Bregel, A one-way quantum computer, Physcal Revew Letters 86 (2001). [5] Raussendorf, R. and H.-J. Bregel, Computatonal model underlyng the one-way quantum computer, QIC 2 (2002). [6] Raussendorf, R., D. E. Browne and H.-J. Bregel, Measurement-based quantum computaton on cluster states, Phys. Rev. A 68 (2003). [7] Walther, P., K. J. Resch, T. Rudolph, E. Schenck, H. Wenfurter, V. Vedral, M. Aspelmeyer and A. Zelnger, Expermental one-way quantum computng, Nature 3 (2005), pp. 169 176. 90