Scales of Time Where the Quantum Discord Allows an Efficient Execution of the DQC 1 Algorithm

The power of one qubit deterministic quantum processor (DQC1) (Knill and Laflamme (1998)) generates a nonclassical correlation known as quantum discord. The DQC1 algorithm executes in an efficient way with a characteristic time given by τ = Tr[U n ]/2 , whereU n is an n qubit unitary gate. For pure states, quantum discord means entanglement while for mixed states such a quantity is more than entanglement. Quantumdiscord can be thought of as themutual information between two systems.Within the quantum discord approach the role of time in an efficient evaluation of τ is discussed. It is found that the smaller the value of t/T is, where t is the time of execution of the DQC1 algorithm and T is the scale of time where the nonclassical correlations prevail, the more efficient the calculation of τ is. A Mösbauer nucleus might be a good processor of the DQC1 algorithm while a nuclear spin chain would not be efficient for the calculation of τ.

The relation between nonclassical correlation and entanglement has been extensively studied in the past and it is not clear yet what are their common elements [1][2][3][4].The systems used in the literature are diverse and they consist either of two qubits [1], four qubits [3], stochastic systems of qubits [2], or a few quantum dots [4].In [5] a study on multipartite entanglement was done within the framework of the generalized Schmidt-correlated states.In such a work entanglement measures of negativity, concurrence, and relative entropy were investigated and concluded that valuable information can be extracted from such quantities.However, in spite of such considerable efforts, a complete understanding of entanglement in quantum information processing (QIP) is an open question.On the other hand, there are suggestions within the context of macroscopic quantum correlations that new tools are required to define and quantify entanglement, beyond the traditional microscopic point of view [2].In such a direction it has been suggested that mixed-states quantum computation leads to exponential speedup over the best classical algorithms [6,7].In particular, in [6] the power of one qubit deterministic quantum processor was introduced, known also as DQC1 which executes with a characteristic time  = Tr[  ]/2  , where   is an  qubit unitary gate which means it is efficient.In the DQC1 algorithm the mixed states play a fundamental role due to the fact that they are coupled to a single control qubit with a given degree of impurity [8].It has been proposed that DQC1 generates nonclassical correlations, from which entanglement is a particular case [9].Such correlations present interesting computational advantages over the traditional approaches [10].A notable nonclassical correlation of such a kind is the so-called quantum discord which is related both to the information on a classical system and its accessibility without perturbing the state for independent observers [11].For pure states, quantum discord means entanglement but for mixed states is more than entanglement [10].A key quantity for defining the measure of correlations between two systems S and M is the mutual information where if the classical systems states  and  are described by a probability distribution (, ), one has that () = − ∑    log   is the Shannon entropy.If  and  are quantum systems described by a combined density matrix   , then () = − Tr( log ) is the von Neumann entropy.On the other hand ( | ) is the conditional entropy which is an average of Shannon entropies for , conditioned on the alternatives for .Meanwhile, for quantum systems such a quantity is an average of von Neumann entropies.By the way, the quantity H( | ) represents measurement-independent conditional information.
One advantage of DQC1 is that this evaluates efficiently the normalized trace of a unitary matrix [10].The respective quantum circuit performing the above consists of a control qubit with the following degree of impurity: where the purity is defined as At this stage, it is worth emphasizing that within the several different approaches essayed so far for evaluating the normalized trace of a unitary matrix, there is scarce information about the time of execution of such algorithm.In other words, the common approaches are within the static point of view where the time of execution of such algorithm does not elapse.By the above reason, the purpose of the present paper is to discuss the role of time in the efficient evaluation of  = Tr(  )/2  .In order to do the above, we are assuming that the time evolution is accounted in an approximated way by the changes in the control qubit.That is, the control qubit reflects the changes on the  target qubits when these are subjected to a unitary gate   .Another assumption that we will make is that the changes in time of the control qubit are manifested through changes in the purity.
According to the static approach of the algorithm under discussion, the control qubit is initially subjected to a Hadamard operation.Furthermore, with the control qubit there is a collection of  qubits in the completely mixed state,   /2  .These  target qubits are subjected to a unitary gate   .Such a circuit evaluates in a time  = Tr(  )/2  more efficiently in the quantum approach.Classically, the calculation of  has not been performed yet and it is believed to be difficult.Quantum mechanically the value of  is achieved in an approximated way by measuring the expectation values of the Pauli operators for example ⟨⟩ and ⟨⟩ [8].In the limit where  ≫ 1 and || are small (i.e., () ≃ 1) the static DQC1 discord is given by [8,10] where  2 (⋅) is the binary Shannon entropy, which is given by where The quantity || has an approximated polynomial overhead which goes as 1/ 2 .It is worth mentioning that accurate bounds can be imposed on ||.Indeed, it is well known that complex eigenvalues of a unitary operator are of modulus one [12].The latter together with the Cauchy-Schwarz inequality implies that where  = 0 for the particular case of   is equal to one of the generators of the group SU().The above is due to the fact that, by definition, the generators of SU() are traceless [13].In Figure 1 we have plotted the static quantum discord D of (4) as a function of the impurity  and || in the ranges 0 ≤  ≤ 1 and 0 ≤ || ≤ 1.Let us observe from Figure 1 that D( = 1) = 0 and that D( = 0) = 1.6; that is the larger the impurity the less the quantum discord.It is worth mentioning that quantum correlations can be characterized by the quantum discord.This has been confirmed in two experiments to date: measurement of the discord in a twoqubit optics setup using full-state tomography [10] and a four-qubit NMR implementation [14].Let us discuss a simple example that illustrates the difference between the traditional static approach where the time does not elapse and the timedependent approach introduced in the present work.Take for instance a single-qubit unitary operator  1 =  −r⋅ →  /2 which represents a three-dimensional rotation by  about the unitary vector r = (  ,   ,   ), →  = (  ,   ,   ) being the vector whose components are the Pauli matrices.The trace of  1 can be evaluated easily yielding tr  1 = cos(/2) [15].It is clear that from the value of tr  1 neither any information about the time required for executing the calculation of the trace of  1 nor the relation that this keeps with the scale of time where the nonclassical correlations prevail can be extracted.The above is the problem of the nonrealistic, traditional static approach where the time does not elapse for executing efficiently  = Tr(  )/2  .
Concerning the quantity  in ( 4) and ( 5) it is important to pinpoint here that in [8] a given constant value of  was employed through all of the execution of the algorithm.Such an assumption limits the knowledge about the execution time of the algorithm which must be less than , the minimum time required to execute one quantum gate in the different quantum computer architectures.In Table 1 we list the respective values of  corresponding to several different quantum systems.In the present paper the condition of  equal to a constant will be relaxed.In fact, we are assuming that such a quantity depends on time in the following way: In the particular case where the nonclassical correlation means entanglement, the quantity  of ( 7) would be the minimum time required to execute one quantum gate.In general terms,  indicates the scale of time where the regime of the nonclassical correlations of the DQC1 system prevails.
A justification of the above equation is that if the time of preparation of the system elapses within the interval of time 0 ≤  ≤ /2, the quantity  grows until this one reaches its maximal value of one.As it is expected, as the time elapses to values /2 <  ≤ , decoherence makes that  decreases monotonically with time.That is, for  > /2, the nonclassical correlations deteriorate as a consequence of the interaction with the environment.From (7) it follows that the purity of the control qubit fluctuates with the time.This implies that the normalized trace  will be also a time-dependent quantity.However, the unitary character of   is not lost although certainly its eigenvalues should be time dependent.In any case, the normalized trace of the unitary matrix   satisfies the bounds given by ( 6).We point out that a key quantity for an efficient calculation of the trace  by making use of the advantages of the nonclassical correlations is precisely the time interval  within which the nonclassical correlations prevail.Due to the fact that within the present approach both the quantity || are small and  ≪ 1, in the present paper the following approximation is proposed: where () is given by (7).The assumption of ( 8) is well justified, whereas in [16] it has been shown that both  and the quantum discord of (4) do not depend on .Furthermore, it is reasonable that the trace of the unitary matrix does depend on the impurity () of the control qubit of the DQC1 model.The above is precisely the central idea of the DQC1 model where the mixture of the control qubit regulates the efficient calculation of .It is expected that the greater the quantum discord is the more efficient the calculation of  = Tr  1 is.To substitute ( 7) and ( 8) into ( 4) and ( 5) we obtain the DQC1 discord D as a function of time which is plotted in Figure 2. Let us observe from Figure 2 that for / = 0 one has that C(/ = 0) = 1.65343 while for / = 10 −3 the quantum discord is C(/ = 10 −3 ) = 1.65341.The above means that as the time elapses the nonclassical correlations become smaller making less efficient the calculation of  = Tr  1 .The latter is a consequence of the presence of decoherence.On the other hand, we observe from Figure 2 that the smaller the value of / the more efficient the calculation of  providing that 1/2 ≤ / ≤ 1.From Figure 2 and Table 1 it is observed that the most appropriate device for an efficient calculation of  should be the Mösbauer nucleus.However, the electrons-GaAs, electrons-Au, and trapped ions-In devices might also be good for an efficient calculation of .On the other hand, from such a figure we can see that quantum dots and nuclear spin chains technologies are not appropriate for an efficient calculation of .

Advances in Mathematical Physics
As a conclusion, we have related the quantum discord with the scales of time where the nonclassical correlations prevail for several different quantum devices.From Figure 1 one can conclude that, for values of the purity of (3) close to unit, the nonclassical correlations become important.Within the DQC1 model the nonclassical correlations are given by the quantum discord whose value is given by (4) providing that || is small.The above might indicate that the larger the value of the quantum discord the smaller the value of ||.In other words, the nonclassical correlations are important for an efficient calculation of  = Tr  1 .This fact is accounted by (7) and ( 8) that reflect that when  becomes close to unit (i.e., a larger quantum discord), the value of  is small.Finally, we have found that a Mösbauer nucleus might be a good processor of the DQC1 algorithm while a nuclear spin chains quantum device would not be efficient for the calculation of .

Figure 2 :
Figure2: Quantum discord of (4) as a function of the / where  is the time and  is the scale of time where the nonclassical correlations prevail.The interval considered is 0 ≤ / ≤ 10 −3 and the value of the impurity is given by(7).

Table 1 :
[17]es of the minimum time required to execute one quantum gate for several different devices.The employed data is the same of[17].The time  is approximately the scale of time where the nonclassical correlations prevail.