Quantum image processing?

Transcription

1 Quantu iage processing? Mario Mastriani DLQS LLC, 443 NW 63RD Drive, Coconut Creek, FL 3373, USA Abstract This paper presents a nuber of probles concerning the practical (real) ipleentation of the techniques known as Quantu Iage Processing. The ost serious proble is the recovery of the outcoes after the quantu easureent, which will be deonstrated in this work that is equivalent to a noise easureent, and it is not considered in the literature on the subject. It is noteworthy that this is due to several factors: ) a classical algorith that uses Dirac s notation and then it is coded in MATLAB does not constitute a quantu algorith, ) the literature ephasizes the internal representation of the iage but says nothing about the classical-to-quantu and quantu-to-classical interfaces and how these are affected by decoherence, 3) the literature does not ention how to ipleent in a practical way (at the laboratory) these proposals internal representations, 4) given that Quantu Iage Processing works with generic qubits this requires easureents in all axes of the Bloch sphere, logically, and 5) aong others. In return, the technique known as Quantu Boolean Iage Processing is entioned, which works with coputational basis states (CBS), exclusively. This ethodology allows us to avoid the proble of quantu easureent, which alters the results of the easured except in the case of CBS. Said so far is extended to quantu algoriths outside iage processing too. Keywords Quantu algoriths Quantu Boolean Iage Processing - Quantu/Classical Interfaces Quantu iage processing - Quantu easureent - Quantu signal processing.

2 Introduction Quantu coputation and quantu inforation is the study of the inforation processing tasks that can be accoplished using quantu echanical systes. Like any siple but profound ideas it was a long tie before anybody thought of doing inforation processing using quantu echanical systes []. Quantu coputation is the field that investigates the coputational power and other properties of coputers based on quantu-echanical principles. An iportant objective is to find quantu algoriths that are significantly faster than any classical algorith solving the sae proble. The field started in the early 98s with suggestions for analog quantu coputers by Paul Benioff [] and Richard Feynan [3, 4], and reached ore digital ground when in 985 David Deutsch defined the universal quantu Turing achine [5]. The following years saw only sparse activity, notably the developent of the first algoriths by Deutsch and Jozsa [6] and by Sion [7], and the developent of quantu coplexity theory by Bernstein and Vazirani [8]. However, interest in the field increased treendously after Peter Shor s very surprising discovery of efficient quantu algoriths (or siulations on a quantu coputer) for the probles of integer factorization and discrete logariths in 994 [9]. Quantu Inforation Processing (QuIn) - The ain concepts related to Quantu Inforation Processing ay be grouped in the next topics: quantu bit (qubit, which is the eleental quantu inforation unity), Bloch s Sphere (geoetric environent for qubit representation), Hilbert s Space (which generalizes the notion of Euclidean space), Schrödinger s Equation (which is a partial differential equation that describes how the quantu state of a physical syste changes with tie.), Unitary Operators, Quantu Circuits (in quantu inforation theory, a quantu circuit is a odel for quantu coputation in which a coputation is a sequence of quantu gates, which are reversible transforations on a quantu echanical analog of an n-bit register. This analogous structure is referred to as an n-qubit register.), Quantu Gates (in quantu coputing and specifically the quantu circuit odel of coputation, a quantu gate or quantu logic gate is a basic quantu circuit operating on a sall nuber of qubits), and Quantu Algoriths (in quantu coputing, a quantu algorith is an algorith which runs on a realistic odel of quantu coputation, the ost coonly used odel being the quantu circuit odel of coputation) [, -]. Nowadays, other concepts copleent our knowledge about QuIn, they are: Quantu Signal Processing (QuSP) - The ain idea is to take a classical signal, saple it, quantify it (for exaple, between and 55), use a classical-to-quantu interface, give an internal representation to that signal, ake a processing to that quantu signal (denoising, copression, aong others), easure the result, use a quantu-to-classical interface and subsequently detect the classical outcoe signal. Interestingly, and as will be seen later, the quantu iage processing has aroused ore interest than QuSP. In the words of its creator: any new classes of signal processing algoriths have been developed by eulating the behavior of physical systes. There are also any exaples in the signal processing literature in which new classes of algoriths have been developed by artificially iposing physical constraints on ipleentations that are not inherently subject to these constraints. Therefore, Quantu Signal Processing (QuSP) is a signal processing fraework [3, 4] that is aied at developing new or odifying existing signal processing algoriths by borrowing fro the principles of quantu echanics and soe of its interesting axios and constraints. However, in contrast to such fields as quantu coputing and quantu inforation theory, it does not inherently depend on the physics associated with quantu echanics. Consequently, in developing the QuSP fraework we are free to ipose quantu echanical constraints that we find useful and to avoid those that are not. This fraework provides a unifying conceptual structure for a variety of traditional processing techniques and a precise atheatical setting for developing generalizations and extensions of algoriths, leading to a potentially useful paradig for signal processing with applications in areas including frae theory, quantization and sapling ethods, detection, paraeter estiation, covariance shaping, and ultiuser wireless counication systes. The truth is that to date, papers on this discipline are less than half a dozen, and its practical use is practically nil. Moreover, although it is an interesting idea, developed so far, does not withstand further coent.

3 Quantu Iage Processing (QuIP) - it is a young discipline and it is in training now, however, it s uch ore developed than QuSP. QuIP starts in 997. That year, Vlasov proposed a ethod of using quantu coputation to recognize so-called orthogonal iages [5]. Five years later, in, Schutzhold described a quantu algorith that searches specific patterns in binary iages [6]. A year later, in October 3, Beach, Loont, and Cohen fro Cybernet Systes Corporation, (an organization with a close cooperative relationship with the US Defense Departent) deonstrated the possibility that quantu algoriths (such as Grover s algorith) can be used in iage processing. In that paper, they describe a ethod which uses a quantu algorith to detect the posture of certain targets. Their study iplies that quantu iage processing ay, in future, play a valuable role during wartie [7]. Later, we can found the works of Venegas-Andraca [8], where he proposes quantu iage representations such as Qubit Lattice [9, ]; in fact, this is the first doctoral thesis in the specialty, The history continues with the quantu iage representation via the Real Ket [] of Latorre Sentís, with a special interest in iage copression in a quantu context. A new stage begins with the proposal of Le et al. [], for a flexible representation of quantu iages to provide a representation for iages on quantu coputers in the for of a noralized state which captures inforation about colors and their corresponding positions in the iages. History continues up to date by different authors and their innovative internal representation techniques of the iage [3-44]. Very siilar to the case of QuSP, the idea in back of QuIP is to take a classic iage (captured by a digital caera or photon counter) and place it in a quantu achine through a classical-to-quantu interface, give soe internal representation to the iage using the procedures entioned above, perfor processing on it (denoising, copression, aong others), easure the results, restore the iage through another interface but this tie quantu-classical, y ready. The contribution of a quantu achine over a classic achine when it coes to process iages it is that the forer has uch ore power of processing. This last advantage can handle iages and algoriths of a high coputational cost, which would be unanageable in a classic achine in a practical sense. The proble of this discipline lies in its genetic, given that QuIP is the daughter of Quantu Inforation Processing and Digital Iage Processing, thus, we fall into the old dilea of teaching, i.e.: to teach Latin to Peter, we should know ore about Latin and ore about Peter? The answer is siple: we should know very well of both, but the ission becoes ipossible. In other words, what is acceptable in Quantu Inforation Processing, and (at the sae tie) inadissible in Digital Iage Processing? The entioned proble begins with the quantu easureent, then, if after processing the iage within the quantu coputer, we want to retrieve the result by toography of quantu states, we will encounter a serious obstacle, this is: if we ake a toography of quantu states in Quantu Inforation Processing (even, this can be extended to any ethod of quantu easureent after the toography) with an error of 6% in our knowledge of the state, this constitutes an excellent easure of such state, but on the other hand, and this tie fro the standpoint of Digital Iage Processing [45-48], an error of 6 % in each pixel of the outcoe iage constitutes a disaster, since this error becoes unanageable and exaggerated noise. So overwheling is the aforeentioned disaster that the recovered iage loses its visual intelligibility, i.e., its look and feel, and orphology, due to the destruction of edges and textures. This speaks clearly (and for this purpose, one need only read the papers of QuIP cited above) that these works are based on coputer siulations in classical achines, exclusively (in ost cases in MATLAB [49]), and do not represent test in a laboratory of Quantu Physics. In fact, if these field trials were held, the result would be the aforeentioned. We just have to go to the lab and try with a single pixel of an iage, then extrapolate the results to the entire iage and therefore the inconvenience will be explicit. On the other hand, today there are obvious difficulties to treat a full iage inside a quantu achine, however, there is no difficulty for a single pixel, since that pixel represents a single qubit, and this can be tested in any laboratory in the world, without probles. Therefore, there are no excuses.

4 Definitely, the proble lies in the hostile relationship between the internal representation of the iage (inside quantu achine), the outcoe easureent, and the recovery of the iage outside of quantu achine. Therefore, the only technique of QuIP that survives is QuBoIP [5]. This is because it works with CBS, exclusively, and the quantu easureent does not affect the value of states. However, it is iportant to clarify that both, i.e., traditional techniques QuIP and QuBoIP share a coon eney, and this is the decoherence [, 5]. Prolegoenous to the probles of Quantu Iage Processing. Qubits and Bloch s sphere The bit is the fundaental concept of classical coputation and classical inforation. Quantu coputation and quantu inforation are built upon an analogous concept, the quantu bit, or qubit for short. In this section we introduce the properties of single and ultiple qubits, coparing and contrasting their properties to those of classical bits []. The difference between bits and qubits is that a qubit can be in a state other than or []. It is also possible to for linear cobinations of states, often called superpositions:, () where is called wave function,, with the states polarization states of light. Besides, a colun vector is called a ket vector and are understood as different T, where, ( ) T eans * * transpose of ( ), while a row vector is called a bra vector, where, ( ) * eans coplex conjugate of ( ). The nubers and are coplex nubers, although for any purposes not uch is lost by thinking of the as real nubers. Put another way, the state of a qubit is a vector in a two-diensional coplex vector space. The special states and are known as Coputational Basis States (CBS), and for an orthonoral basis for this vector space, being and One picture useful in thinking about qubits is the following geoetric representation. Because, we ay rewrite Eq.() as i i i e cos e sin e cos cos i sin sin () where,. We can ignore the factor of effects [], and for that reason we can effectively write i e out the front, because it has no observable i cos e sin (3) The nubers and define a point on the unit three-diensional sphere, as shown in Fig.. Quantu echanics is atheatically forulated in Hilbert space or projective Hilbert space. The space of pure states of a quantu syste is given by the one-diensional subspaces of the corresponding Hilbert space (or the "points" of the projective Hilbert space). In a two-diensional Hilbert space this is siply the coplex projective line, which is a geoetrical sphere. This sphere is often called the Bloch s sphere; it provides a useful eans of visualizing the state of a single qubit, and often serves as an excellent testbed for ideas about quantu coputation and quantu inforation.

5 Fig. Bloch s Sphere. Many of the operations on single qubits which can be seen in [] are neatly described within the Bloch s sphere picture. However, it ust be kept in ind that this intuition is liited because there is no siple generalization of the Bloch s sphere known for ultiple qubits []. In the general case, with n qubits there would be n possible states []. Except in the case where and unit sphere in is one of the ket vectors or the representation is unique. The paraeters, re-interpreted as spherical coordinates, specify a point a sincos sin sin cos on the (according to Eq.). 3 Fig. Details of the poles, as well as an exaple of parallel and several qubit states on the sphere.

6 Figure highlights all coponents (details) concerning the Bloch s sphere, naely Spin down = and = = = qubit basis state = North Pole Spin up = = = = qubit basis state = South Pole Both poles play a fundaental role in the developent of the quantu coputing []. Besides, a very iportant concept to the affections of the developent quantu inforation processing, in general, i.e., the notion of latitude (parallel) on the Bloch s sphere is hinted. Such parallel as shown in green in Fig., where we can see the coplete coexistence of poles, parallels and eridians on the sphere, including coputational basis states. The poles and the parallels for the geoetric bases of criteria and logic needed to, ipleent any quantu gate or circuit.. Schrödinger s equation and unitary operators A quantu state can be transfored into another state by a unitary operator, sybolized as U (U : H H on a Hilbert space H, being called an unitary operator if it satisfies U U UU I, where is the adjoint of ( ), and I is the identity atrix), which is required to preserve inner products: If we transfor U and U, then UU. In particular, unitary operators preserve lengths: and to UU, if is on the Bloch s sphere (i.e., it is a pure state). (4) On the other hand, the unitary operator satisfies the following differential equation known as the Schrödinger equation [, -]: d U t dt i Hˆ U t (5) where Ĥ represents the Hailtonian atrix of the Schrödinger equation, i, and is the reduced Planck constant, i.e., h/. Multiplying both sides of Eq.(5) by and setting t U t (6) yields d dt i Hˆ (7) t t Equation (6) represents the quantu algorith eployed in Quantu Inforation Processing. Besides, the solution to the Schrödinger equation is given by the atrix exponential of the Hailtonian atrix: i Hˆ t e (if Hailtonian is not tie dependent) (8) U t and

7 i t Ĥ dt e (if Hailtonian is tie dependent) (9) U t Thus the probability aplitudes evolve across tie according to the following equation: i Hˆ t t e (if Hailtonian is not tie dependent) () or i t Ĥ dt t e (if Hailtonian is tie dependent) () The Eq.() is the ain piece in building circuits, gates and quantu algoriths, being U who represents such eleents []. Finally, the discrete version of Eq.(7) is ihˆ, () n n for a tie dependent (or not) Hailtonian, being n the discrete tie..3 Quantu Circuits, Gates, and Algoriths; Reversibility and Quantu Measureent As we can see in Fig.3, and reeber Eq.(6), the quantu algorith (identical case to circuits and gates) viewed as a transfer (or apping input-to-output) has two types on output: a) the result of algorith (circuit of gate), i.e., b) part of the input, i.e., (underlined critical need in quantu coputing []. n ), in order to ipart reversibility to the circuit, which is a Besides, we can see clearly a odule for easuring with their respective output, i.e., (or, n p post-easureent), and a nuber of eleents needed for the physical ipleentation of the quantu algorith (circuit or gate), naely: control, ancilla and trash []. n In Fig.3 as well as in the rest of the (unlike []) a single fine line represents a wire carrying qubit or N qubits (qudit), interchangeably, while a single thick line represents a wire carrying or N classical bits, interchangeably too. However, the entioned concept of reversibility is closely related to energy consuption, and hence to the Landauer s Principle []. On the other hand, coputational coplexity studies the aount of tie and space required to solve a coputational proble. Another iportant coputational resource is energy. In [], the authors show the energy requireents for coputation. Surprisingly, it turns out that coputation, both classical and quantu, can in principle be done without expending any energy! Energy consuption in coputation turns out to be deeply linked to the reversibility of the coputation. In other words, it is inexcusable the need of the presence to the output of quantu gate []. On the other hand, in quantu echanics, easureent is a non-trivial and highly counter-intuitive process. Firstly, because easureent outcoes are inherently probabilistic, i.e. regardless of the carefulness in the preparation of a easureent procedure, the possible outcoes of such easureent will be distributed according to a certain probability distribution []. Secondly, once the easureent has been perfored [], we can see that the very act of easuring alters the easureent, since there is a disturbance, and this is due n

8 Fig. 3 Module to easuring, quantu algorith and the eleents needs to its physical ipleentation. to the fact that a quantu syste is unavoidably altered due to the interaction with the easureent apparatus [5]. Consequently, for an arbitrary quantu syste, pre-easureent and post-easureent quantu states are different in general []. Postulate. Quantu easureents are described by a set of easureent operators, index labels the different easureent outcoes, which act on the state space of the syste being easured. Measureent outcoes correspond to values of observables, such as position, energy and oentu, which are Heritian operators [] corresponding to physically easurable quantities. Let be the state of the quantu syste iediately before the easureent. Then, the probability that result occurs is given by p( ) Mˆ Mˆ (3) where (4) and the post-easureent quantu state is Mˆ Mˆ p (5) Operators ust satisfy the copleteness relation of Eq.(6), because that guarantees that probabilities will su to one; see Eq.(7) []: ˆ ˆ M M I (6) ˆ ˆ M M p( ) (7)

9 as well as, orthogonality and positive-definite condition, respectively, Mˆ M ˆ, if i j (8) i j Mˆ M ˆ,. (9) On the other hand, Mˆ M ˆ,, () n with, I () Let us work out a siple exaple. Assue we have a polarized photon with associated polarization orientations horizontal and vertical. The horizontal polarization direction is denoted by and the vertical polarization direction is denoted by. Thus, an arbitrary initial state for our photon can be described by the quantu state (reebering Subsection.), where and are coplex nubers constrained by the noralization condition spanning., and, is the coputational basis Now, we construct two easureent operators and and two easureent outcoes,. Then, the full observable used for easureent in this experient is According to Postulate, the probabilities of obtaining outcoe and or outcoe. are given by. Corresponding post-easureent quantu states are as follows: if outcoe = p( ) ; if outcoe = p p then p( ), then, see Eq.(9) and (3), respectively. Besides, p p,, () which is equivalent to,. (3) On the other hand, M M. (4) n n n Now, considering that,, (5a), (5b), (5c)

10 and, (6a), (6b), (6c) and, if in Eq.(5) we replace by, and considering Eq.(5c), for the generic ψ, we will have p ˆ M ˆ M, (7) and, by, and considering Eq.(6c), we will have p ˆ M ˆ M. (8) Equations (7) and (8) represent the geoetric projections of they are the quantu easureent []. ψ onto the coputational basis,, i.e., Finally, aking the sae but regarding to coputational basis,, and using Eq.(5a) and (6c), we have, (9) p Mˆ M ˆ and. (3) p Mˆ M ˆ Equations (9) and (3) deonstrate that only CBS are not disturbed by quantu easureent []. Finally, using Equations (5b) and (6a), respectively, we have the cross projections, that is to say, onto z axis, and onto x and y axis (see Fig., Bloch s sphere), Mˆ Mˆ, (3) p and Mˆ Mˆ. (3) p

11 3 Quantu Iage Processing and its ipleentation probles Fro the above in Section on this subject, it is clear that between the literature [5-44] there are serious confusions about: - What are classical-to-quantu and quantu-to-classical interfaces? How to ipleent the physically and really? See [5-57]. - What returns a real quantu coputer (non-adiabatic), in the general case? i.e., states, density atrix of such states, or what? The answer is siple, and as we can see in the last Subsection, we obtain probabilities and post easureent quantu states [58]. However, this obviousness is not clear in the papers dealing algoriths for Quantu Iage Processing (QuIP) [44, 59-7]. In fact, they try classical algorith with Dirac notation, and after such algorith are coded in MATLAB, in the best. Without further clarification concerning a real ipleentation (in laboratory) of the above algoriths, it appears that the quantu coputers eployed are ideal. A look at the appropriate literature helps clarify the situation [, 7]. - What the eaning of an appropriate internal representation of the iage? See [9-8, 3-36, 73, 74]. - What are the consequence of decoherence [57, 75-84] on the different algorith for QuIP? - The ost of papers lacks of coputational cost of treated quantu algorith - The ost of papers did not clarify which type of quantu easureent [58, 85-93] used (weak [94-99], weak and strong, or siply a toography [-]), if they used any. - In relation to the previous point, the ost of papers lacks of an estiate of the error range of the quantu easureent used [58]. - What are the consequences of quantu easureent on the outcoes? The latter is the ost serious proble, which we try to address it in detail here. On the other hand, it is noteworthy that the sae proble extends to Quantu Signal Processing (QuSP) [3, 4]. What is the quantu easureent equivalent in QuIP and QuSP? According to Eq.(5) of Subsection.3, we will have f p ˆ M ˆ M I II III f f f f 3!!! 3! (33) where the second row of this equation represents its decoposition in a Taylor s series. Quickly, we can see that Eq.(3) is an odd syetric function (see the first row of this equation), therefore the ters of even degree disappear, I III V VII f f 3 f 5 f 7 (34) p! 3! 5! 7! Besides, at this point, we have two practical alternatives based on CBS, i.e.: (Mac Laurin s series), and. In the first case, we have, f p Mˆ ˆ M I III V VII f f 3 f 5 f 7! 3! 5! 7! (35)

12 with f (36) because is a CBS (see Eq.9), and the Quantu Measureent does not alter the outcoe. On the other hand, doing successive derivatives of the first row of Eq.(3), then, replacing account Equations () and (5a), we will have sybolically, by, and, taking into f I Mˆ Mˆ (37a) f Mˆ (37b) I f Mˆ Mˆ II 3 ˆ M ˆ M (38a) f (38b) II III 3 ˆ M ˆ M f 3 (39a) f Mˆ (39b) III Mˆ Mˆ IV 5 ˆ M ˆ M f (4a) f 3 (4b) IV 4 3 V 5 ˆ M ˆ M f 5 (4a) f 3 Mˆ (4b) V Mˆ Mˆ VI 7 ˆ M ˆ M f (4a) f 3 5 (4b) VI 6

13 3 5 VII 7 ˆ M ˆ M f 7 (43a) f 3 5 Mˆ (43b) VII Mˆ Mˆ VIII 9 ˆ M ˆ M f (44a) f (44b) VIII IX 9 ˆ M ˆ M f 9 (45a) f Mˆ (45b) IX 9 Now, replacing Equations (37b), (39b), (4b), (43b), (45b) into the forth row of Eq.(35), we have Mˆ Mˆ 3 Mˆ p 5! 3! 5! Mˆ 7 Mˆ 9 7! 9! (46) Obviously, if (outcoe = ), then p Quantu Measureent does not alter the outcoe. On the other hand, if an odd syetric function (this is identical to the previous case), thus, it will be (satisfying Eq.9), i.e., it is a CBS case where and considering Eq.(33) is f p Mˆ ˆ M I III V f f 3 f 5! 3! 5! (47) with f (48) because is a CBS (see Eq.3), and the Quantu Measureent does not alter the outcoe. On the other hand, doing successive derivatives of the first row of Eq.(47), then, replacing by, and, taking into account Equations () and (6b), and with siilar considerations with respect to the previous case, we will have sybolically and finally,

14 3 5 3 Mˆ Mˆ 3 Mˆ p 5! 3! 5! Mˆ 7 Mˆ 9 7! 9! (49) Obviously, if (outcoe = ), then p here too, Quantu Measureent does not alter the outcoe. Now, regrouping ters of Eq.(46), and reeber Equations (5c), we have (satisfying Eq.3), i.e., it another CBS case where, n n p,qm,qm (5) where, n,qm is a residue owing to the higher degree coponents. In fact, both Equation 46 and the 49, the ters of degree greater than result fro the power (with exponent, and divided by the corresponding factorials and powers of, therefore, they have a rather attenuated ipact on the ter of degree equal to. Besides, fro Stochastic Processes [55] we know that is a noise due to the quantu easureent, in fact, its subscript eans just that, while, 3) for coponents with odule / n,qm represents the easureent atrix. That is to say, the very act of easuring introduces noise, since, it is a disturbance. On the other hand, fro Eq.(7) we can estiate the ean value (considering the errors of easureent in this projection) of such residue, that is to say, n,qm (5) The precision involved in this estiation depends on the ethod of eployed quantu easureent [53]. In the sae way, if we have grouped ters of Eq.(49), and reeber Equations (6b), we have n n p,qm,qm (5) and through a reasoning identical to the previous case, and taking in account Eq.(8), we coe to an estiation of the ean value of n,qm with siilar considerations to those established for the previous case, regarding to an analysis with focusing in Stochastic Processes. Then, n,qm (53) Equations (5) and (5) tell us that when we easure (in each of its projections, see Equations 7 and 8), we are introducing a noise easureent, which is inherent to the type of easureent, however, always exists and it cannot be overlooked, and this noise will disappear only when we easure CBS,. In other words, when we easure (generic case), we do not have the value of each projection direct, clean and clearly, i.e., the values obtained are corrupted by noise.

15 In case of generic qubits, i.e., Eq. (), the question is: what is possible without disturbing partially known quantu states? The answer is: nothing [4, 5]. Besides, optial, reliable estiation of quantu states is only theoretical [6]. Now, if we express Eq.(6) in a ore general discrete for, we will obtain U (54) t t,t t being Ut,t the quantu algorith for QuIP. We can see this for each pixel in Fig.4. Fig. 4 Equivalence between quantu easureent and easureent noise. In fact, in Fig.4 we can see the classical iage I, and the post-easureent outcoe t p, where ClQu eans part of classical-to-quantu interface, and QuCl eans part of quantu-to-classical interface. Figure 4 shows clearly that represented by the Eq.(9), that is to say, quantu easureent becoing a noise. Besides, is the recovered iage after the quantu iage processing, i.e., it is a classical iage. t p Now, let's look at a couple of very conspicuous cases. Case # - the quantu algorith does nothing: this case can be seen on Fig.5. Here quantu easureent introduces noise anyway, then, the first conclusion that we arrived is:, however, t t Fig. 5 The quantu algorith does nothing.

16 how is possible to store iages in a quantu coputer? It is obvious that trying to recover these iages we introduce noise to the. Then, we need a classical filter to iage denoising at the output, where eans estiated I, see Fig.6. Obviously, it is absurd. However, there is uch literature on this topic [8, 9, 3, 7-], which occurs on that istake. How is possible? Î Fig. 6 Classical iage denoising filtering after quantu easureent. In reality, after quantu-to-classical interface. Case # - the quantu algorith filters iages: this case can be seen on Fig.7. Here where U t,t U, t t,t t represents the quantu algorith for iage denoising [, 3]. In Fig.7 we can see that at the entrance a noise adds to the iage (iage and noise are classic). This noise is known as noise of state, and this can be generated in two ways: a) It brings the classic original iage that ust be filtered b) It is the result of a suboptial preparation of the qubits Reebering Eq.(5) and considering that t = t, and t = t + t, we can represent the intervention of this noise as follows: Fig. 7 Quantu iage denoising algorith for the filtering of classical noise. U n (55) tt t,t t t state Equation (53) is known as equation of state, which, together with the equations (48) and (5) constitute the general odel of any linear (or linearizable) syste affected siultaneously by state and easureent noise at the sae tie, where n is the noise of state, while state Ut,t is the atrix of state [4-]. t Under these conditions, we could only estiate optially the state if we knew the statistics of both noise. Obviously, this is not the case in quantu easureent [6]. Therefore, in this case we also need a filter to the output (i.e., in the classic environent, and as in the previous case, see Fig.6), that is to say, re-filter but in the classic world, where it was the original classic noise. Then, why do we use a quantu coputer? Ergo, why do we use Quantu Iage Processing? Absurd. Suing-up, we cannot dispense with the external filter in any case. The quantu easureent can put ore noise than the original classic noise, at ost of the sae order, or worse yet, we don't know.

17 The sae is true in Quantu Iage Segentation [, 3], and Quantu Iage Encryption [37-43, 4-3], and any another intervention of Quantu Iage Processing. In fact, the foregoing is extended to any quantu algorith [], including Quantu Signal Processing [3, 4]. Although it is incredible, no paper on quantu algoriths (with generic qubits) perfors an exhaustive analysis of the robustness of the proposed algorith. On the other hand, returning to Section I, if we have and uncertainty in the quantu easureent equal to QM (e.g., 6 %), then, we will have a spread of x QM (e.g., %). It is obvious that this has a nasty ipact on both the visual intelligibility of the iage as well as in the quality of the definition of their edges, in its texture, etc. In after words, this is unacceptable fro the point of view of Digital Iage Processing [45-48]. All this can be seen clearly in Figures 8 (Agus in Miai), 9 (Angelina) and (Lena). The three figures show original iage, the resultant noisy iage by fault of Quantu Measureent, a esh with the noise of Quantu Measureent, and, the noise by Quantu Measureent in a colorap scale. Fig. 8 Agus in Miai: top-left is original iage, top-right is noisy iage because of Quantu Measureent, down-left is a esh with the noise of Quantu Measureent, and, down-right is the noise by Quantu Measureent in a colorap scale. This obliges us to filter in the classical world (outside of the quantu coputer) and then we ust to restore the edges through asks of enhanceent [45-48]. That is to say, we have ore activity outside quantu coputer than inside quantu coputer. Is this logical? Is this practical? Justifies this the use of a quantu coputer to process iages? All these probles are a direct result of underestiating the quantu easureent. If the proble of quantu easureent would be solved, in such a way that it could be overlooked (or at best underestiate) as indeed happens in "whole" literature on Quantu Iage Processing [5-44,59-7,73,74,7-3], only then we could replace the notation of the classic algoriths of Digital Iage Processing with Dirac's notation, and ready! But this is not the case. The only act of easure collapses the wave function since the easureent is a disturbance that anifests as an equivalent noise of easureent. The ipact of such noise can be saw in a quantitative way in the next Sections which include a set of conspicuous siulations.

18 Fig. 9 Angelina: top-left is original iage, top-right is noisy iage because of Quantu Measureent, down-left is a esh with the noise of Quantu Measureent, and, down-right is the noise by Quantu Measureent in a colorap scale. Fig. Lena: top-left is original iage, top-right is noisy iage because of Quantu Measureent, down-left is a esh with the noise of Quantu Measureent, and, down-right is the noise by Quantu Measureent in a colorap scale. A direct consequence of this is that we cannot store infinite inforation in the infinite decials of both projections of a generic qubit. See Eq.(3). In reality, we could. What we cannot do is to know this inforation because the quantu easureent would alter the stored values.

19 4 Prolegoena to experients and siulations In this section, we present a series of tools which are necessaries for an iportant set of coparative siulations. We consider such siulations are essential for the purpose of establishing the central idea of this paper. The entioned tools are: - Quantu Boolean Iage Processing (QuBoIP) [5] - Flexible Representation of Quantu Iages (FRQI) [] - Novel Enhanced Quantu Representation (NEQR) [3] - Quantu State Toography (QuST) [-,3] In the Section 5 (which will be fully dedicated to siulations), the first three tools will be copared between the, for which we take as a reference the values estiated in QuST, i.e., we will calculate the easureent errors of QuBoIP, FRQI and NEQR against such estiated values. Further, we are going to do a coparative analysis of different aspects related to perforance of entioned techniques. Other techniques for the internal representation of the iage are Qubit Lattice [9], Entangled Iage [], Real Ket [], Multi-channel representation of quantu iages (MCQI) [63], Noral arbitrary superposition state (NASS) [3], Noral arbitrary quantu superposition state (NAQSS) [8], Geoetric Quantu Iage Representation (GQIR) [69], Quantu representation for color digital iages (QUALPI) [34], QSMC represents M colors and QSNC represents the coordinates of N pixels in an iage (QSMC&QSNC) [9], Color quantu iage based on phase transfor (CQIPT) [73], Iproveent of FRQI (IFRQI) [33], Geoetric transforations on quantu iages (GTQI) [34], Iproved novel enhanced quantu representation (INEQR) [35], and Multi-channel representation for quantu iages (MCRQI) [36]. This techniques also have been tested and have not given better results than QuBoIP. Therefore, in Section 5 we will present a set of interesting and coparative siulation based on FRQI, NEQR, and QUBoIP, whereby, the central idea of this work is set sufficiently highlighted. Quantu Boolean Iage Processing (QuBoIP): this technology is presented as a viable alternative to the proble of practical ipleentation of algoriths for quantu iage processing (QuIP) [5]. The algoriths grouped under this technology does not suffer the ravages of the quantu easureent. However, QuBoIP and QuIP share a coon eney: the decoherence [57, 75-84]. Of all fors, and as it is logical to assue, decoherence affects a lot ore to the QuIP that the QuBoIP, given that QuBoIP works with CBS, it is ore easy to operate with threshold logic, until the ultiate collapse. On the other hand, and as we can see in previous sections, there is a direct and autoatic correspondence between, and,. Such correspondence (and in that order) will be the classical-to-quantu interface. In the sae way, but in reverse order, there is a direct and autoatic correspondence between, and,. In this correspondence, but in that order, we know it as a quantu-to-classical interface. Unlike of QuIP, in QuBoIP the easureent is not a proble, since it does not alter the outcoe easure. Therefore, in QuBoIP is not necessary a filter (and ore) after quantu easureent [5], as in QuIP. In Table I we can see these stateents, where left colun represents the state before easureent, while right colun represents the state after that, for CBS and generic state (see Subsection.), where is the easureent operator (see Eq.4), and is the posteasureent quantu state in its generic for []. p TABLE I MEASUREMENT OUTCOME WITH CBS AND GENERIC STATE. Before quantu easureent After quantu easureent Mˆ Mˆ p

20 That is to say, here too, it is only necessary to consider the poles of Bloch s sphere when we work with CBS (see poles of Figures and ), i.e., when we working with one qubit only, which is all that is used in this technology. Besides, while in the case of CBS before and after quantu easureents are atching situations, for generic qubits (third row of Table I) the case is copletely different [, 58]. Fig. Quantu entangleent in QuBoIP between second and third coluns of each case: first row for Agus in Miai, second row for Angelina, and third row for Lena. The first colun is coposed by the original iages, while, the second colun is constituted by the MSB of gray version of each iage. Then, the second and third colun are entangled between the. Table I is the cornerstone of this ethodology called quantu Boolean iage processing in general, and quantu Boolean iage denoising in particular. It will allow us (aong others): a) to build ore robust interfaces with respect to quantu easureent noise, decoherence, etc., b) to ignore the proble of quantu easureent [, 58], which was above entioned, c) a design ore close to the hardware,

21 d) a lower coputational and eory cost [5], working only with the ost significant bit (MSB) and its quantu counterpart, in siulations of quantu coputing on classical systes like GPGPU [37], FPGA [38], given that in a real and future quantu coputer becoes irrelevant an evaluation and consideration regarding to coputational cost, because, this technology proises to introduce a huge quantity of quantu processors (olecules or particles) in an incredibly sall nanoetric space, allowing us face great coputational challenges, which are unacceptable today for the classic coputers, such as the probles of the type NP coplete. e) to export this criterion beyond the QuIP, e.g., to Quantu Signal Processing [3, 4], and f) aong others. Working with CBS frees us fro the influence that superposition have norally when we work with generic cubits. In other words, we are in a very special borderline case, i.e., it is siilar to classical Boolean [5]. In fact, there is no difference between quantu Boolean and pure Boolean (classical) regarding to their results in a filtering context. The quantu Boolean version has the sae coputational cost of classical Boolean version. Besides, for this algorith, we will seek that be CBS (all the tie), because it s easier its state treatent in presence of quantu decoherence. On the other hand, the quantu decoherence tie depends on the ipleentation technology of the quantu coputer and not the quantu algorith itself []. However, it is easier a collapse of the wave function with generic qubits that with CBS, the state is ore stable in a quantu Boolean context than in a generic quantu context, in special after easureent of the state. Besides, QuBoIP allow us work in a ore practical way with the entangleent. Figure show us quantu entangleent in QuBoIP between second and third coluns of each case: first row for Agus in Miai, second row for Angelina, and third row for Lena. The first colun is coposed by the original iages, while, the second colun is constituted by the MSB of gray version of each iage. Then, the second and third colun are entangled between the. This is easily verifiable in the laboratory for each ost significant qubit of the iage. For everything that we said, we can see that QuIP can be ipleented in a practical way through techniques QuBoIP [5]. Flexible Representation of Quantu Iages (FRQI): As we can see in [3], every pixel is apped into a basis state of a four-diensional qubit sequence. FRQI was proposed by Le []. This quantu iagerepresentation odel can be expressed as in (56) for a n n iage. In FRQI, the position inforation of every pixel is stored in a basis state of a two-diensional qubit sequence, and the gray-scale inforation is stored as the probability aplitude of a single qubit which is entangled with the qubit sequence. Figure shows a 4 4 FRQI quantu iage. n n I cos YX sinyx YX (56) n Y X Qubit Lattice and Entangled Iage are siilar to classical digital iage processing, and therefore it is easy to find the quantu equivalents for classical iage-processing operations, yet without uch perforance iproveent. The Real Ket and FRQI odels use the superposition of a qubit sequence to store iages, and therefore they can process inforation for all pixels siultaneously. Copared with Real Ket, FRQI aintains a D pixel atrix, and therefore it is the ost suitable and flexible odel for designing quantu iage-processing algoriths aong all the existing quantu iage representations. Many studies on quantu iage processing have been carried out based on FRQI [3,64] discussed siple geoetric and color operations based on FRQI and proved the great perforance iproveent of these quantu operations copared with classical iage operations expanded FRQI using three color qubits to store the full-color inforation of an RGBα iage designed quantu waterarking algoriths based on FRQI, obtaining good perforance. In general, all previous studies based on FRQI have focused on certain

22 Fig. A 4 4 FRQI quantu iage. θ YX is the phase of the color qubit which denotes the gray-scale value of pixel (Y, X). siple operations and algoriths in digital iage processing. The ain reason for this is that the FRQI odel has soe drawbacks:. The tie coplexity of quantu iage preparation for FRQI is too high. For a n n iage, the procedure costs O( 4n ), which is quadratic in the iage size.. To siplify preparation of an FRQI quantu iage, the Boolean expression iniization ethod is used to perfor iage copression. However, it has been found that the FRQI copression ratio depends on the gray-scale distribution of the digital iage; especially when the iage is highly disordered, the FRQI copression ratio is very low, even approaching zero. 3. Because FRQI stores the gray-scale inforation of the iage pixels as the probability aplitude of a single qubit, it is ipossible to obtain accurate probability aplitude for this qubit through finite quantu easureents. In other words, accurate iage retrieval is an ipossible task for the FRQI odel. 4. FRQI stores the digital iage into the superposition of a qubit sequence according to the position inforation of different pixels. Therefore, all operations which are suitable for FRQI should perfor the sae operations for pixels at different positions. Other operations cannot be accoodated. Besides, fro [64] we can see an exaple of a x FRQI quantu iage with its corresponding state is presented in Fig.3. Fig. 3 A siple FRQI iage and its quantu state.

23 To effectively capture all the inforation about the colour and position of a n x n -sized FRQI iage, a total of n+ qubits are required as shown in the generalised circuit of the FRQI representation in Fig.4. Fig. 4 Generalised circuit showing how inforation in an FRQI quantu iage state is encoded. These disadvantages of FRQI have becoe liitations on the exploration of coplex quantu iageprocessing algoriths. Hence, it is of significant iportance to iprove existing quantu iage odels and to find better ways to represent quantu iages for the future developent of quantu iage processing. Novel Enhanced Quantu Representation (NEQR): As we can see in [3], the advantages of FRQI result fro using the superposition of a qubit sequence to store the position inforation of all the pixels, so that all can be operated on siultaneously. However, the ain reason for the drawbacks of this odel is that FRQI uses only a single qubit to store the gray-scale inforation for each pixel. To iprove FRQI, the NEQR odel uses two entangled qubit sequences to store the gray-scale and position inforation, and stores the whole iage in the superposition of the two qubit sequences. Suppose the gray range of iage is q q q, binary sequence CYX CYX C YX C encodes the gray-scale value YX f(y,x) of the corresponding pixel (Y,X) as in (57): q q k q f (Y,X) CYX CYX CYX C YX,C YX,, f (Y,X), (57) The representative expression of a quantu iage for a n n iage can be written as in (58): n n n n q I f Y,X Y X C Y X i n (58) YX n i Y X Y X Figure 5 shows a 4 4 NEQR iage exaple. Copared with the exaple of FRQI in Fig., the obvious difference is that NEQR utilizes the basis state of qubit sequence to represent the gray scale of pixels instead of probability aplitude of a single qubit in FRQI. Different gray scales can be distinguished in NEQR because different basis states of qubit sequence are orthogonal. All the drastic iproveents of NEQR in the following discussion are depended on this peculiar change. Figure 6 illustrates a gray-scale iage and its representative expression in NEQR. In this figure, because the gray scale ranges between and 55, eight qubits are needed in NEQR to store the gray-scale inforation for the pixels. Therefore, NEQR needs q+n qubits to represent a n n iage with gray range q.

24 Fig. 5 A 4 4 NEQR quantu iage. f (Y, X) denotes the gray-scale value of pixel (Y, X), which is stored as the basis of a qubit sequence. state f Y,X Fig. 6 A exaple iage and its representative expression in NEQR. Quantu State Toography (QuST): As we can see in [-,3], this technique is very iportant in the context of Quantu Measureent proble for Quantu Inforation Processing in general, and Quantu Iage Processing in particular. Besides, fro [39] we know that for pure states like Equations, and 3, the density atrix is: (59) However, it is possible to define a statistical ixture of pure states (called a ixed state) using the density operator, which is defined as: Pi i i (6) i where P i is the probability of obtaining state i. The su can in general extend arbitrarily far, but any single qubit density operator can be diagonalized (in the linear algebraic sense) and represented as the probabilistic su of just two states. For a pure state, P = and P =. The density operator is Heritian and ust only have positive eigenvalues since these eigenvalues correspond to the probabilities for obtaining i in the eigenvector basis. In addition, the probabilities P i ust su to one. Since the states are noralized, this condition can be concisely written as Tr() =. The purity of the state is a useful quantity that can be defined as i

25 P Pi (6) i We will see later that any density operator can be written as a point within the Bloch sphere, with the pure states lying on the surface. The goal of state toography is to produce an estiate of an unknown quantu state. Typically, in a realworld scenario an experientalist is given a certain nuber of copies of an unknown state (or equivalently, a liited tie in which she ust perfor easureents). The experientalist then perfors easureents on her copies of the unknown state and fro these easureents she produces an estiate. This situation is depicted in Fig.7. The nuber of easureents the experientalist ust perfor is dependent on the size of the quantu state being easured. In [39] they stated that for an n-qubit state there are n - free paraeters that describe the state. For exaple, looking at Eq.(6), i z x y i x y z (6) it is obvious that in order to deterine a single qubit state one ust easure the expectation values of the three Pauli operators [39] i x, y, x. i (63) while, the purity of the state is siply: P x y z r (64) where r is the radius of the Bloch vector. In our geoetric picture of the single qubit, this corresponds to easuring the Bloch vector's projection onto three orthogonal axes. It is often the case that a paraetrization that yields paraeters which are directly easurable is not possible. In this case, each easureent ade will yield a linear cobination of paraeters. After a sufficient nuber of easureents is ade, this syste of equations - which is linear in the paraeters - can be inverted. This technique, aptly naed linear inversion, is a for of toography. Fig. 7 A depiction of state toography. N copies of an unknown state are easured, and fro the outcoes of these easureents an estiate for the unknown density atrix is obtained. We begin by writing a general density atrix in ters of a inial set of unknown paraeters. For exaple, this can be done for a two qubits by writing: t t it t it t it t it t t it t it t it t it t t it t7 it3 t9 it5 t it6 t (65)

26 In experients with photons, usually a rate of detected photons is easured. I will show in [39] how these photon rates are converted to probabilities in the lab, but for now we will assue that soe absolute rate N is known, so the rates are converted to probabilities (technically these are called observed frequencies since one cannot 'observe' a probability) as i i i ij j N j i i N. Then for each easureent setting, we have: Tr O M t (66) where the easureent operator O is said to be a projector onto the state if it takes the for: O (67) being Tr( ) the trace of ( ). On the other hand, Eq.(66) along with the equation t + t + t 3 + t 4 = (for noralization) can be written as Mt (68) One need only invert the atrix M to solve this syste of linear equations. There is a proble with linear inversion, however. The for of Eq.(65) does not constrain sufficiently. In an experient, each of the expectation values can only be estiated with a finite nuber of copies of the unknown state. The estiates of the expectation values are all constrained to be within the eigenvalue spectru of the operators that they represent (there is no way to obtain values outside of the eigenvalue spectru), but the su of the squares of the expectation values are not constrained at all. As we can see in [39], the su of the squares of the expectation values of the Pauli operators is the radius of the state vector. Physical states are represented by state vectors with a radius less than one. If each of the expectation values can only be estiated to within soe finite accuracy, it is possible that Eq.(65) will yield an unphysical density atrix. What, if anything, can be concluded fro this unphysical density atrix? The etric of accuracy of toography, fidelity, does not yield eaningful results for unphysical density atrices (for exaple, fidelities greater than one can be obtained). In the next section, I will show how to confine the estiate produced by toography to a physical state in a eaningful way. In general, we represent our quantu state as a density atrix which is a Heritian atrix X of size q x q. The quantu toography easureent process collects easureents * i A n, A Tr O (69) i * where n is a ter that represents noise, and O is a Heritian atrix that represents an observable. For i experiental reasons, is the tensor product of Pauli atrices [39]. Note the siilarity between Equations * O i 5, 5 and 69, that is to say, the odel of noise associated to quantu easureent. 5 Experients and siulations Of all the current ethods of Quantu State Toography (QuST) [-,3,39-43], that is to say, Maxiu Likehood, Estiation via Backprojections, Linear Regression Estiation, Copressed Sensing, and Kalan Filter Approach, we will use in here the last one, because, it is ore robust in the presence of noise [43]. On the other hand, in this Section, it is vital iportance the reference that gives us the QuST which it is estiated later than the quantu process which uses soe internal representation of the iage such as FRQI, NEQR, and QuBoIP (or siply: QuBo). The use of the QuST will give us a realistic notion of how the quantu easureent noise affects the quality of the outcoe according to the internal representation of the selected iage.

27 Fig. 8 Quantu state toography for qubits, after a quantu iage process which involves a particular internal iage representation, naely, top-left: real-part for FRQI, top-right: iaginary-part for FRQI, iddle-left: real-part for NEQR, iddle-right: iaginary-part for NEQR, botto-left: real-part for QuBo, and finally, botto-right: iaginarypart for QuBo. In real part, and coordinates,, and,, both should give an exact. On the other hand, NEQR values were noralized to, to equate the 3 techniques. Figure 8 shows us real and iaginary part of QuST after the use of FRQI, NEQR and QuBo. The other techniques for internal representation of the iage (naed in Section 4) are not representative, because they are in the range between FRQI and NEQR, and all are worse outcoe than QuBo.

28 Fig. 9 With the sae approach and the sae order of the previous figure, the graph shows the easured values in gray and the easureent errors in red, that is to say, fro left to right: real and iaginary part, and, fro top to botto: FRQI, NEQR, and QuBo.