Quantum Processors and Controllers

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by CERN Document Server

Alexander Yu. Vlasov ∗† Jan, May 2003

Abstract [2] or any other introduction in quantum information science).

In this paper is presented an abstract theory of quan- In quantum networks bits are changed to qubits tum processors and controllers, special kind of quan- (quantum bits) and yet another quantum gate with tum computational network defined on a composite two qubits is controlled-U gate [2], when quantum quantum system with two parts: the controlling and gate U is applied to second qubit if first one is 1 controlled subsystems. Such approach formally dif- ( a is notion for state of qubit in Dirac notation),| i fers from consideration of quantum control as some but| i if first qubit is 0 then second one is unchanged. external influence on a system using some set of | i Hamiltonians or quantum gates. The model of pro- It is possible to write controlled-U as 4 4 matrix grammed quantum controllers discussed in present × paper is based on theory of universal determinis- 1000  0100 tic quantum processors (programmable gate arrays). , (2)  00U11 U12  Such quantum devices may simulate arbitrary evolu-   tion of quantum system and so demonstrate an ex-  00U21 U22  ample of universal quantum control. where Uij are components of quantum gate for second qubit, i.e., 2 2 matrix U. Keywords: quantum, computer, control, processor, × universal To describe development of the idea, programmable quantum gate arrays are used in Sec. 2. Such quantum devices also are called quantum processors, but 1 Introduction may be used as quantum controllers as well, it is discussed in Sec. 3. More formal mathematical description of programmable quantum controllers provided Let us consider simple example of control using a in Sec. 4. Universality of quantum computations and Hilbert space of composite quantum system with two control are briefly recollected in Sec. 5. Some discus- parts: sion on universal control with continuous quantum = . (1) H Hc ⊗Hd variables is presented in Sec. 6. Here d is a Hilbert space of quantum system con- sideredH as data, (controlled variables, subject of control) and is Hilbert space of control (“manager”, 2 Programmable quantum gate ar- Hc program of changes). The approach is close analogue rays of conditional quantum dynamics [1].

It is possible to start with classical example with re- It is possible to use decomposition Eq. (1) with more versible Controlled-NOT gate defined on set of two bi- general quantum networks and describe process of nary variables as (a, b) (a, a XOR b), i.e., if first bi- control as nary variable is a = 0, then→ second one is unchanged, but if a = 1,thenb NOT b. Reversible logical Ctrl: C Ψ C0 (uC Ψ ), (3) | i⊗| i 7→ | i⊗ | i gates are usual tools in→ quantum computations; here XOR is eXclusive OR, a+b (mod 2). Construction of i.e., Ctrl is some fixed network and different control strategies ensured by different states C of control quantum Controlled-NOT gate is straightforward (see | i registers: for each such state different operator uC ∗Federal Radiological Center (IRH), 197101 Mira Street 8, acts on second system. The expression Eq. (3) co- St.–Petersburg, Russia incides with definition of special kind of quantum †A. Friedmann Lab. Theor. Physics, 191023, Griboedov Can. 30/32, St.–Petersburg, Russia network, programmable quantum gate array [3, 4].

1 m Let us use notation C n for dim c = m and 3 Quantum processors as controllers dim = n. & H Hd

It should be mentioned, that Ctrl as any other quan- It is clear, that such approach has some difference tum computational network [5] with pure states must with other methods of consideration of quantum con- be described as linear unitary operator acting on the trol [6, 7, 8], there controlled system is also described Hilbert space Eq. (1) (similarly with example Eq. (2) as some state, but control is introduced as set of “ex- above for simplest case with two qubits). Here the ternal” controlling operators; gates or Hamiltonians, quantum laws have serious implications denoted al- i.e., control and data are described from different ready in [3]: any two states of “program” (first, con- points of view (semiclassical coherent control). trol register) must be orthogonal, i.e., maximal number of different operators u availiable for applica- C Note Another attempt of joint quantum description of tion to second, controlled system is equal to dimen- control and controlled system, using same term sion of Hilbert space c, i.e., for universal control “quantum controller,” was included in [9], as some dim = , becauseH number of different quantum Hc ∞ perspective for above-mentioned semiclassical co- gates is infinite. herent control. It was not suggested a general model of such joint quantum description, but few illus- To explain this result, let us consider two different trative examples were presented. But here is dis- “control strategies” A and B cussed an alternative approach, it is enough to recall | i | i some distinctive principles of consistent framework for quantum control with pure states considered in Ctrl A Ψ = A0 (uA Ψ ), | i⊗| i | i⊗ | i present paper: Ctrl B Ψ = B0 (uB Ψ ), | i⊗| i | i⊗ | i 1. Control and controlled system must not be entangled. It follows directly from definition but because Ctrl is unitary operator, it may not change scalar product of two vectors, i.e., Eq. (3). 2. The consequence of such definition is impossi- bility to use superposition of states in control A B = A0 B0 Ψ u† u Ψ (4) h | i h | ih | A B| i register. 3. So, there is noticeable asymmetry between control and controlled system in such approach. In Eq. (4) A B and A0 B0 are fixed numbers, but h | i h | i 4. Final development of the principles is origi- for uA = uB term Ψ uA† uB Ψ depends on Ψ .But Eq. (4)6 must be satisfiedh | for| anyi Ψ and so| i nal three-level design of programmable quan- | i tum controller discussed below in Sec. 4, Fig. 1.

A B = A0 B0 =0, (5) h | i h | i The construction of programmable quantum gate ar- i.e., states corresponding to different programs are rays, or quantum processors [3, 4, 10, 11, 12, 13, 14] orthogonal. provides more unified description of control and controlled system. It is in agreement with principles just For example even for one controlled qubit, set of all mentioned above. Term “quantum controller” may possible gates may be described by continuous three- be also justified for such a system, because for uni- dimensional family, i.e., even for this simple case with versal quantum processor on controlled system d dim = 2, for universal control it is necessary to H Hd formally may be simulated practically any physical have dim c = with control register described process, if to use tradition of consideration of such H ∞ 3 by three continuous quantum variables (C ∞ ). systems suggested by R. Feynman and D. Deutsch &2 [15, 16]. For classical processors difference in sizes of It is interesting, that such enormous difference be- program and data is not such a radical and this new tween size of control and controlled system is rather property of quantum processor (controller) is related subtle property of quantum dynamics, for example, with infinite amount of different quantum programs it may be found linear, but non-unitary operator like (algorithms of control). Eq. (3) for universal control and with size of control register only in two times bigger than for controlled Despite of discussed above result about infinite- n2 quantum system C n, it is simply operator of mul- dimensional controlling register, universal quantum tiplication of a matrix& on a vector written as for- controllers with finite dimension of control space Hc mal linear map ( ∗) ( ∗) ; are also quite usefull. It should be mentioned first A v 1 Av{) H⊗H, but it would⊗H → contradictH⊗H to⊗H laws so-called stochastic quantum processors [3, 4, 12, 13]. of⊗ quantum7→ ⊗ mechanics.} Such quantum processor does not produce correct

2 answer each time, but provides special “check bit” 1 , 2 , ..., i.e., “no operation,” “operation #1,” displaying if answer is correct or not. If answer is “operation| i | i #2,” . . . not correct, it is suggested to perform calculations again and again. Probability of correct answer is re- If dim c = m and dim d = n in Eq. (1), then H H duced with size of data register and increases with dim = mn and in suggested new basis Ctrl may be H number of tries. written as block-diagonal mn mn matrix ×

u0 Seems idea of stochastic quantum processor quite in-   teresting, but has lot of problems, for example it is u1 0 Ctrl =  .  , (6) not even clear if it is possible to use composition  ..  of such networks for few-steps process due to un-    um 1  speciﬁed time of each step and it is certainly some 0 − problem for application of such system as quantum where u are n n matrices and it is convenient to k × controllers. It is also not quite clear, if it is always choose u0 = 1 (“no operation”). It is example of possible to “discard” incorrect result of action for conditional quantum dynamics described in [1] and general controller and start all again. using Dirac notation it may be rewritten as [1]

In addition, the “ideal limit” of such design resembles C = 0 0 u + 1 1 u + (7) trl | ih |⊗ 0 | ih |⊗ 1 ··· non-possible linear (but non-unitary) operator discussed earlier and it is similar with some other known Such approach may be appropriate for simple quan- models of quantum systems (“relaxation” gate, “in- tum controller, but for more difficult operations it is stantaneous” reduction, etc.), then balance between reasonable to consider an advanced design [10, 11] “arduous” and “impossible” is too fine and linked of quantum processor that can be used as apro- with deep problems of quantum mechanics. grammable quantum controller. Instead of two sys- Anyway, the idea of stochastic gates seems useful, for tems Eq. (1) here is used design with three “buses” example in [14] was shown, that continuous limit of = . (8) some special stochastic network discussed in [4, 13] H Hp ⊗Hc ⊗Hd coincides with continuous limit of some “determinis- Here p, c, d are Hilbert spaces of program, con- tic” quantum gate, despite of very different behavior trollerH andH dataH ,orpseudo-classical, intermediate in finite, discrete case. It should be mentioned also, and quantum buses respectively (see Fig 1). that main efforts of many authors last time were applied rather to the stochastic design, but determin- The idea is to use composition of two operators. First istic one seems more appropriate for present consid- one was described earlier, it is quantum controller Ctrl eration of quantum controllers. acting on intermediate and quantum buses Hc ⊗Hd and second one acts on p c andoneachstep Another construction with finite control register uses H ⊗H provides intermediate bus c with new state k used “universality in approximate sense”. It is quite rea- H | i as index k of operator uk by quantum controller. sonable approach and based on idea, that in realistic tasks always possible instead of continuous infinite Let us consider simplest example with “cyclic mem- space of parameters to use only finite set of points ory (ROM)”. Let dim c = m and it is necessary to H p 1 for approximation. The more dense set, the more perform program with p steps. Then dim p = m − accurate such a method. Some basic papers about and element K of mp-dimensional HilbertH space | i universality in quantum computation uses such ap- p c may be described as proach [16, 17, 18]. H ⊗H K k ,...,k ; k (9) | i≡| p 2 1i and “program” is simply operator of cyclic shift 4 More rigor mathematical S : K k ,k ,...,k ; k . (10) description hft | i7→| 1 p 3 2i Finally, for p steps of the programmable quantum con- p troller with cyclic ROM (C m ), it is possible to write Let us consider quantum processors and controllers &n with more details [10, 11]. It was already mentioned, p (ShftCtrl) : K Ψ K (ukp uk1 Ψ ) (11) that all different states of controlling register must | i| i 7→ | i ··· | i be orthogonal. Let us use for simplicity finite con- and because set of operators uk contains identity trolling register and choose such orthogonal states as (unit), it is possible to implement any sequence with new basis. It is possible to denote it simply as 0 , up to p operators using different programs K . | i | i

3 Pseudo-classical bus ( ) Hp ··· ···

Reversible program (Shft)

Intermediate bus ( ) Hc ··· ···

Quantum controller (Ctrl)

Quantum bus ( ) Hd ··· ···

Figure 1: Design of programmable quantum controller with three buses (cf [11]).

One problem here is huge size of program register. A 5 Universality of quantum control method to reduce the size is to use instead of shifted array more complicated algorithm for generation of indexes. For example, instead of each sequence of n When method of generation of arbitrary sequence of equivalent indexes k it could use pair (n, k). It should operators like Eq. (11) is given, ideas of implemen- be mentioned yet, that only reversible algorithms are tation of universal control follows to standard proce- appropriate here due to common principles of quan- dures[2,5,6,10,11,16,17,18,20,21]. tum computations — otherwise dynamics would not be unitary. Really there are some methods of auto- Let us consider case with finite size of control regis- matic conversion of any algorithm to reversible one, ter. For good approximation it is possible to choose but in such a case each step generates a “garbage” uk near identity operator, i.e. and size of program register may be even bigger, than u (ε)=expiH ε 1+iH ε (ε 0). (12) for ROM. So the area is related with classical theory k k ≈ k → of optimal reversible computations. Here Hk are Hermitian operator and corresponds to On the other hand, it was already mentioned earlier, Hamiltonians in some other approach to quantum that all states of program register are orthogonal. control [6]. Then small parameter ε is analogue of It is not necessary to use superposition of different minimal time of action of some external influence by states. It was a reason to call the register “pseudo- the control Hamiltonian. classical.” Such systems may be more simple for implementation [19] and so problem with size may be Due to general theory it is enough to have posi- not such essential, as for data register. bility to generate full Lie algebra of Hermitian operators as linear span of Hk and all possible con- Yet another advantage of such pseudo-classical pro- sequent commutators, but this part coinsides with gram register is possibility to use “halt bit” and al- general theory of universal quantum computations gorithms with variable length. It is mentioned here, and control and does not discussed here in details because such an opportunity is not very common for [2,5,6,10,11,16,17,18,20,21,22,23]. general quantum algorithms due to quantum paral- lelism and interference of different branches. Algorithms of generation of indexes for application of different gates uk often may be described using few nested cycles with repeating series of states-indexes k [10, 22, 23] due to general algebraic approach with Lie| i algebras and commutators mentioned above.

4 Despite of such analogue in mathematical expres- Hybrid controller bus ( ) s Hc sions, discussed approach has some advantages due ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ s to closed description of controlling and controlled ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ systems. Really the quantum controller uses only s iHC δt ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ one fixed Hamiltonian HC, Ctrl = e and Hk r are rather formal operators. s ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ s On the other hand, such quantum description has ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ some difficulties, because despite of pseudo-classical s ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ character of program register, it is anyway some spe- r ... cial kind of quantum system. It is not look reasonable, that for control of such program register may f? f? f? r be used some standard silicon chip. It was already θ1 θ2 θ3 mentioned, that for presented model only reversible programs are compatible with laws of quantum me- ? f? f? f? r chanics. Really it could be simply shown, that ir- θ1 θ2 θ3 reversible operator for some state of quantum con- ? troller has absurd property: it “shrinks” to zero wave Quantum bus ( ) ... vector of the system (and all environment, due to lin- Hd earity). One method of more correct modeling is to use mixed states and density matrices. But it is not iσkϕ Figure 2: Hybrid quantum control; θk = e , only problem of given approach, most other models where ϕ is continuous parameter of control and σk of interface between classical devices and quantum is Pauli matrices. Wavy ( ) and straight (—) lines system always provides some challenge and may be are continuous and finite∼∼ quantum variables respec- much more nontrivial [24]. tively (cf [14]).

6 Continuous quantum variables One simple method of description of quantum computation with continuous variables is to consider Quantum computations with continuous variables is some functional spaces and space of linear differen- also active area of research [25]. The ideas presented tial operators. Well known example are operators of here also possible to use in case of control described coordinate and momentum p, q. Draft of universal by continuous quantum variables. For such a case quantum controller, based on design of hybrid quan- direct sum in Eq. (7) should be changed to direct tum processor [14] depicted on Fig. 2. integral [14]. For such a system quantum control variables are continuous, but controlled system is de- Here controlled system is anyway finite-dimensional scribed by finite-dimensional Hilbert space. It is par- and only some subset of controlling variables de- ticular case of hybrid quantum computing [26]. scribed by infinite-dimensional Hilbert space. Inter- esting question is problem of universal control of con- Here pseudo-classical character of program bus pro- tinuous variables. Such models were described yet vides some simplification. It may be described using only in semiclassical approach to quantum compu- classical terms and it is in agreement with relative tation and control. It was shown, that Hamiltonian success of usual semiclassical description of quantum with (bi)linear combinations of coordinate and mo- control. But intermediate and quantum buses may mentum are not enough [25] for universal quantum not be considered using only classical ideas. Here computation (control) and so some nonlinear (third- intermediate bus could provide some challenge as order) [25] or exponential [27] expressions may be an “interface” between classical and quantum world. used instead. In presented approach it is not so critical, because pseudo-classical bus is also described as a quantum It is clear from previous consideration, that it is sim- system and was called so due to “recommendation” pler to use some analogue of universality in approx- to use here only orthogonal set of states. It is princi- imate sense for control of quantum continuous vari- pally possible to apply any superposition of states to ables — it was discussed earlier, dimension of Hilbert such “pseudo-classical” bus, but in such a case states space for universal control must coincide with cardi- of control and controlled system became entangled af- nality (“number of points”) in Hilbert space of con- ter application of Ctrl Eq. (3) and it is not considered trolled system and so for controlled system with con- ∞ as prescribed functioning of considered device. tinuous variables (C ∞ ) such idea would produce &∞

5 very different problems related with mathematical [8] P. Zanardi and S. Lloyd, “Universal con- theory of infinite cardinal numbers. trol of quantum subspaces and subsystems,” quant-ph/0305013 (2003). So, (precisely) universal control of N-dimensional system is possible using continuous quantum vari- [9] S. Lloyd, “Quantum controllers for quantum ables (C ∞ ), but quite likely, that control of contin- systems,” quant-ph/9703042 (1997); “Coher- &N uous quantum system (C ∞ ) may be universal only ent quantum feedback,” Phys. Rev. A 62, in approximate sense. On&∞ the other hand, distinc- 022108 (2000). tion between approximate and rigor universality in last case has rather theoretical significance, because [10] A. Yu. Vlasov, “Classical programmability is it is not clear, how to find a difference between such enough for quantum circuits universality in approximate sense,” , (2001). C ∞ -controllers during any finite amount of time. quant-ph/0103119 Anyway,&∞ both tasks discussed below are difficult and [11] A. Yu. Vlasov, “Universal quantum processors out of scope of presented paper. with arbitrary radix n > 2,” (Proc. ICQI 2001); In addition, more accurate consideration of models quant-ph/0103127, (2001). of quantum computations and control with continu- [12] M. Hillery, V. Buzek, and M. Ziman, “Prob- ous variables is not complete without necessary at- abilistic implementation of universal quantum tention to principles of quantum field theory. This processors,” Phys. Rev. A 65, 022301 (2002); difficult area is still in state of development, espe- quant-ph/0106088, (2001). cially because correct description of quantum fields is possible only using relativistic theory [28, 29, 30]. [13] M. Hillery, M. Ziman, and V. Buzek, “Imple- mentation of quantum maps by programmable quantum processors,” quant-ph/0205050 References (2002). [14] A. Yu. Vlasov, “ALEPH-QP: Universal hy- [1] A. Barenco, D. Deutsch, A. K. Ekert, and brid quantum processors,” (Proc. QPC 2002); R. Jozsa, “Conditional quantum dynamics and quant-ph/0205074, (2002). logic gates,” Phys. Rev. Lett. 74, 4083–4086 (1995). [15] R. P. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys. 21, 467–488 [2] R. Cleve, “An introduction to quantum com- (1982). plexity theory,” quant-ph/9906111, (1999). [16] D. Deutsch, “Quantum theory, the Church- [3] M. A. Nielsen and I. L Chuang, “Programmable Turing principle and the universal quantum quantum gate arrays,” Phys. Rev. Lett. 79 321– computer,” Proc. R. Soc. London Ser. A 400, 324, (1997). 97–117 (1985).

[4] G. Vidal, L. Masanes, and J. I. Cirac, “Stor- [17] D. Deutsch, A. Barenco, and A. Ekert, “Univer- ing quantum dynamics in quantum states: sality in quantum computation,” Proc. R. Soc. stochastic programmable gate for U(1) oper- London Ser. A 449, 669–677 (1995). ations,” Phys. Rev. A , 047905 (2002); 88 [18] A. Barenco, C. H. Bennett, R. Cleve, D. P. Di- quant-ph/0102037, (2001). Vincenzo, N. Margolus, P. W. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, “Elementary [5] D. Deutsch, “Quantum computational net- gates for quantum computation,” Phys. Rev. A works,” Proc. R. Soc. London Ser. A 425, 73– 52, 3457–3467 (1995). 90 (1989). [19] G. P. Berman, G. D. Doolen, R. Mainieri, [6] S. Lloyd and L. Viola, “Control of open quan- andV.I.Tsifrinovich,Introduction to Quantum tum systems dynamics,” quant-ph/0008101 Computers (World Scientiﬁc, Singapore 1998). (2000). [20] D. P. DiVincenzo, “Two-bit gates are universal [7] S. G. Schirmer, A. D. Greentree, V. Ramakr- for quantum computation,” Phys. Rev. A 51, ishna, and H. Rabitz, “Constructive control 1015 – 1022 (1995). of quantum systems using factorization of unitary operators,” Jour. Phys. A 35, 8315–8339; [21] S. Lloyd, “Universal quantum simulators,” Sci- quant-ph/0211042 (2002). ence 273, 1073 (1996).

6 [22] A. Yu. Vlasov, “Cliﬀord algebras and universal sets of quantum gates,” quant-ph/0010071 (2000); Phys. Rev. A 63, 054302 (2001). [23] A. Yu. Vlasov, “Noncommutative tori and universal sets of non-binary quantum gates,” quant-ph/0012009 (2000); Jour. Math. Phys. 43, 2959 – 2964 (2002). [24] A. Peres and D. R. Terno, “Hybrid classical– quantum dynamics,” Phys. Rev. A 63, 022101 (2001); quant-ph/0008068 (2000). [25] S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. A 82, 1784–1787 (1999); quant-ph/981082, (1998). [26] S. Lloyd, “Hybrid quantum computing,” quant-ph/0008057, (2000). [27] A. Yu. Vlasov, “Algebras and universal quantum computations with higher dimensional systems,” (Proc. QI2002) quant-ph/0210049 (2002). [28] A. Yu. Vlasov, “Quantum theory of computation and relativistic physics,” (Proc. PhysComp96) quant-ph/9701027 (1996). [29] A. Peres and D. Terno, “Quantum information and relativity,” Rev. Mod. Phys. 75 [in press]; quant-ph/0212023 (2003). [30] D. Terno, “Quantum information and relativity theory,” PhD Thesis (Technion, 2003).