Selective writing and read-out of a register of static qubits

We propose a setup comprising an arbitrarily large array of static qubits (SQs), which interact with a flying qubit (FQ). The SQs work as a quantum register, which can be written or read-out by means of the FQ through quantum state transfer (QST). The entire system, including the FQ's motional degrees of freedom, behaves quantum mechanically. We demonstrate a strategy allowing for selective QST between the FQ and a single SQ chosen from the register. This is achieved through a perfect mirror located beyond the SQs and suitable modulation of the inter-SQ distances.


Introduction
A prominent paradigm in quantum information processing (QIP) [1] is to employ flying qubits (FQs) and static qubits (SQs) as carriers and registers of quantum information, respectively [2]. Key to such idea is the ability to write and read-out the information content of a SQ by means of a FQ. By this, here we mean that efficient quantum state transfer (QST) between these two types of qubits must be possible on demand. In this picture, control over memory allocation appears a desirable if not indispensable requirement. For instance, one can envisage the situation where only one or a few SQs are available, e.g. because the remaining ones are encoding some information to save. On the other hand, one may need to carry away only the information saved in certain specific SQs. Alternatively, only a restricted area of the register of SQs may be interfaced with some external processing network where one would like to eventually convey information or from which output data are to be received. In such cases, the ability of selecting the exact location where the information content of the FQ should be uploaded or downloaded is demanded. Ideally, according to the schematics in Fig. 1, one would like the FQ to reach the specific target SQ, then fully transfer its quantum state to this and eventually fly away. Evidently, this picture is implicitly based on the assumption that, firstly, the motional degrees of freedom of the FQ are in fact fully classical and, secondly, that these can be accurately controlled. Despite its simplicity, although interesting research along this line is being carried out mostly through so called surface acoustic waves (see e.g. Ref. [3] and references therein), such an approach calls for a very high level of control.
If we set within a fully quantum framework, the most natural situation to envisage is the one where the FQ, besides bearing an internal spin, moves in a quantum mechanical way and hence propagates as a wavelike object. Such a circumstance substantially complicates the dynamics in that, besides the complex spin-spin interactions, intricate wavelike effects such as multiple reflections between the many SQs occur either. This appears an adverse environment to accomplish selective QST: while ideally one would like to focus the FQ's wave packet right on the target SQ, the former is expected to spread throughout the SQs' register. Thereby, not only it is non-trivial what strategy would enable selective QST but even the mere possibility that this could occur can be questioned.
In this work, we consider a paradigmatic Hamiltonian memory read-out model where the FQ propagates along a 1D line comprising a collection of (fixed) spatiallyseparated non-interacting SQs and couples to them via a contact-type spin-spin Heisenberg interactions (see Fig. 2). We start with a single SQ and prove that a unitary swap between the itinerant and static spins is unattainable. The insertion of a perfect mirror along the 1D line, however, makes it possible. At the same time, since the transmission channel is suppressed there is no uncertainty over the final path followed by the FQ. Next, we find that even for a pair of SQs this can be achieved with either of the two SQs through an ad hoc setting of distances and coupling strengths. Surprisingly enough, this means that Feynman paths entering multiple reflections can combine so as to effectively decouple one SQ while enabling at the same time a unitary swap involving the other one. Even more surprisingly, the working principle behind this phenomenon is such that it is naturally generalized to the case of an arbitrarily large register of SQs, as we rigorously prove. arbitrary N (c) SQs. The FQ f incomes from the left with a given wave vector k, undergoes multiple scattering between the SQs and the perfect mirror and eventually moves away from the register with the same k.

Read-out of a single static memory qubit
Consider the case where a single memory static qubit SQ 1 lies on the x-axis close to position x = 0. To read-out the quantum information stored in SQ 1 (or write it there) a FQ f is injected along the axis with momentum k, say from the left-hand side.
We model the f -SQ 1 interaction as a contact-type spin-dependent scattering potential having the Heisenberg coupling form. The system Hamiltonian can thus be expressed asĤ =p 2 /2 +V , wherep is the momentum operator of f (its mass being set equal to one for simplicity) and is the coupling potential with associated strength G ‡. Here, x is the spatial coordinate of f whileσ f andσ 1 are the spin operators of qubits f and SQ 1 , respectively, i.e.σ = (σ x , σ y , σ z ) withσ β=x,y,z having eigenvalues ±1/2 (we set = 1 throughout). We ask whether or not, when f will emerge from the scattering process, the internal degree of freedom (i.e. the spin) of the two qubits have been exchanged according to the mapping where ρ f 1 is the (joint) input spin state of f and SQ 1 , whileŴ ij is the usual swap two-qubit unitary operator exchanging the states of qubits i and j [1]. While there are in fact counterexamples [4,5] showing that this is impossible §, we give next the general proof that such swap operation cannot occur. For this purpose, let us define where for each qubit, either flying or static, |↑ and |↓ stand for the eigenstates ofσ z with eigenvalues 1/2 and −1/2, respectively (from now on, we omit particle subscripts whenever unnecessary). State |Ψ − is the well-known singlet, while the triplet subspace is spanned by {|↑↑ , |Ψ + , |↓↓ }. Using the identitŷ σ f ·σ 1 = (Ŝ 2 f 1 −σ 2 f −σ 2 1 )/2, whereŜ f 1 =σ f +σ 1 , the interaction Hamiltonian can be written asV = (G/2)(Ŝ 2 f 1 − 3/2)δ(x), entailing [Ĥ,Ŝ 2 f 1 ] = 0 [5,6,7]. Within the singlet (triplet) subspace the effective interaction is thus spinless and readsV s = −(3G/4)δ(x) [V t = (G/4)δ(x)]: the problem is reduced to a scattering from a (spin-independent) δ-barrier. For a δ-potential step Γδ(x) and a particle incoming with momentum k, the reflection and transmission probability amplitudes r (0) (γ) and t (0) (γ), respectively, are found through a textbook calculation as where we have introduced the rescaled parameter γ = Γ/k. These functions allow to calculate the reflection coefficient for the singlet and triplet sectors as ‡ The assumption of the δ-shaped potential is a standard one, and for the present setup it relies on the usually met condition that the FQ's wavelength is significantly larger than the characteristic SQ size. § In Ref. [5], it was proven that, given the initial spin state |↑↓ f 1 , the scattering process between f and SQ 1 can never lead to σ 1z = 1/2. Owing to conservation ofσ f z +σ 1z , this is equivalent to state that the transformation |↑↓ f 1 → |↓↑ f 1 is unattainable.
where we have set g = G/k. Evidently, |r t | = |r s | for any G = 0. This is the very reason which forbids one from using the above scattering process for implementing any unitary gate on the spin degree of freedom of f and SQ 1 , hence, in particular, the swap gate (2) enabling perfect writing/read-out of SQ 1 . Observe in fact that, once the orbital degree of freedom of the FQ are traced out, the final spin state ρ ′ f 1 of the joint system f -SQ 1 can be related to the initial one ρ f 1 (in general mixed) through the completely positive, trace-preserving map [1] where the first contribution refers to the f -wave component emerging from the right of the 1D line (transmission channel), while the second to the one emerging from the left (reflection channel). The Kraus operators [1,10]T f 1 andR f 1 describing these two complementary events are provided, respectively, by the transmission and reflection operators of the model, namelŷ whereΠ (s) f 1 are the projector operators associated with the singlet and triplet subspaces, respectively, of the f -SQ 1 system. Notice that in the computational basis yields the probability amplitude that, given the initial joint spin state |α ′ f α ′ 1 , f is reflected back and the final spin state is |α f α 1 [8,9] (an analogous statement holds forT f 1 ). Via the identities (4) and (5) one can easily verify that Eq. (7) immediately entails the proper Furthermore, expressed in this form it is now easy to see why the mapping (6) is never unitary: in fact for this to happen,R f 1 andT f 1 should be mutually proportional, i.e. r s(t) = ξt s(t) . This is impossible since it requires r s /t s = r t /t t , which can be fulfilled only provided that r s = r t (conflicting with |r s | = |r t | proven above).
A strategy to get around this hindrance is to insert a perfect mirror at x = 0 beyond the SQ located at x = x 1 at a distance d 1 as sketched in Fig. 2(a) (this is inspired by Ref. [9], where, however, a somewhat different system was addressed). First of all, such modified geometry suppresses the transmission channel eliminating the uncertainty in the direction along which f propagates after interacting with SQ 1 . Specifically, in the presence of the perfect mirror we haveT where now the reflection matrixR More interestingly, Eq. (8) allows for the perfect swap gate (2) to be implemented. To see this, observe that since the squared total spin is still a conserved quantity as in the no-mirror case, the problem reduces to a spinless particle scattering from a spinless barrier Γδ(x − x 1 ) and a perfect mirror which, via a simple textbook calculation, gives the reflection amplitude (recall that γ = Γ/k). Therefore, a reasoning fully analogous to the previous case leads toR Observe thatR (m) f 1 is unitary because r (m) (γ) has unit modulus. To work out the conditions for realizing an f -SQ 1 swap gate (2), we use the fact that this unitary can be written asŴ t . This identity is fulfilled provided that g and kd 1 are related to each other according to the function which is plotted in Fig. 3(a). Interestingly,g(kd 1 ) ≥ 1 means that g must exceed the threshold g th = 1 to ensure occurrence of the swap. To summarize, in the presence of a single SQ and for a given spin-spin coupling strength, for any 0 < kd 1 < π [see Fig. 3(a)] there always exists a corresponding coupling constant G ≥ k ensuring the occurrence of the f -SQ 1 swap . Conversely, as long as G is strictly larger than k, there are always two distinct values of kd 1 enabling the perfect swap between f and SQ 1 . Before concluding this section, we point out that, based on the form of r This situation is indeed equivalent to moving the mirror to SQ 1 's location: the chance for the FQ to be found at such position then vanishes and its spin is thus unable to couple to the SQs. More in general, the property that two objects whose optical separation is an integer multiple of π behave as if they were at the same place will be exploited repeatedly in this work.

Two static qubits
In addition to SQ 1 and the perfect mirror, the setup now comprises a further SQ, dubbed SQ 2 , located on the left of 1 at a distance d 2 from it as shown in Fig. 2(b). Hence, the spin-spin coupling term inĤ now readŝ where . We aim to implement either an f -SQ 1 or an f -SQ 2 swap operation, i.e. either the unitaryŴ f 1 ⊗Î 2 orÎ 1 ⊗Ŵ f 2 , respectively (note that in any case we require one of the two SQs to be unaffected). Analogously to the single-SQ case, the mirror suppresses the transmission channel and thereby one can define a unitary reflection operatorR f 12 within the 8-dimensional (8D) overall spin space that fully describes the interaction process output. In the spirit of scattering matrices combination via sum over different Feynman paths [13], the scattering operatorR f 12 results from a superposition of all possible paths, the first of which are sketched in Fig. 4. The overall sum is obtained in terms of a geometric series aŝ where although not shown by our notation, despite it involves qubits f and SQ 1(2) , each reflection or transmission operator on the right-hand side is intended as the extension to the present 8D spin space. Also, note thatR (m) f 1 is a function of kd 1 . The present setup ensures QST between f -SQ 1 and f -SQ 2 , respectively, in the regimes f -SQ 1 QST : kd 2 = h(kd 1 ), g =g(kd 1 ), (16) f -SQ 2 QST : kd 1 = nπ, g =g(kd 2 ), where n = 1, 2, . . ., while h(kd 1 ) = π − arg[r (m) s (g)]/2 is a periodic function of period π plotted in Fig. 3(b). Condition (17) is easily understood: we have already discussed (see the previous section) that when kd 1 = nπ the optical distance between SQ 1 and the mirror is effectively zero, hence it is as if the mirror lied at x = x 1 so as to inhibit the f -SQ 1 coupling. We are thus left basically with the same setup as the one in the previous section, which shows that if condition g =g(kd 2 ) is fulfilled [cf. Eq. (13)] then To prove Eq. (16), which is key to the central findings in this paper, it is convenient to introduce the coupled spin basis arising from the coupling ofσ f ,σ 1 andσ 2 . We defineŜ f i =σ f +σ i (i = 1, 2) and the total spinŜ =σ f + i=1,2σ i . It is then straightforward to check that Eq. (14) can be expressed aŝ t |1 1|, where we have introduced the concise notation |s f 1 = |s f 1 ; s = 1/2, m . As forR f 2 =T f 2 −Î f 2 , one has to solve an effective scattering problem in a 2D spin space in the presence of the spin-dependent potential barrier (G/2)(Ŝ 2 , where s 2 is the quantum number associated withσ 2 2 and we have introduced the discrete function q j = j(j + 1) (here, although s 2 = 1/2, we leave such quantum number unspecified for reasons that will become clear later on). Such task can be carried out easily, as we show in the Appendix. Next, by requiring condition (13), which ensures thatR with (for compactness of notation the dependance ofg on kd 1 is not shown).
Since for the 1D blocks s = 3/2, as mentioned, s f 1 can only take value 1 and the same occurs for s f 2 as is easily seen. Hence, s f 1 = s f 2 = 1 and the interaction Hamiltonian is given byV = = −e −2ikd 2 immediately yields r (3/2) = −1 (matching the value found for r 11 as it must be given that they both correspond to s f 1 = 1). This demonstrates that, up to an irrelevant global phase factor, the f -SQ 1 swap indeed occurs under condition (16). It is important to stress that this result is independent of the value taken by r t . In other words, the same result is achieved by replacing (G/4)δ(x − x 2 ) with Γδ(x − x 2 ) with arbitrary Γ.

Arbitrary number of static qubits
We now address the case where an arbitrary number N of SQs are present, the νth one lying at x = x ν in a way that d ν = x ν−1 − x ν is the distance between the νth and (ν − 1)th ones [see Fig. 2(c)]. Hence, noŵ Again, we aim at implementing a selective swap between f and SQ ν (ν = 1, . . . , N). Selective QST is achieved for where n i can be any positive integer. Regime (25) is immediately explained since it entails that |x N −1 |, namely the distance between SQ N −1 and the mirror, is a multiple integer of π, hence the mirror behaves as if it lied at x = x N −1 . All the static qubits from SQ 1 to SQ N −1 are thus decoupled from f . We in fact retrieve the case of one SQ Strictly speaking, the solution is kd 2 = nπ − arg[r (m) s (g)]/2 for n = 1, 2, . . . (n integer). All these solutions are physically equivalent. Lower values of n, i.e. n ≤ 0, are to be discarded since they would make kd 2 negative. at a distance d N from the mirror, where QST is ensured by condition (13) (with the replacement d 1 → d N ).
The case in Eq. (24) is explained as follows. The mirror is effectively positioned at x = x ν−1 since each kd i≤ν−1 is a multiple integer of π. On the other hand, kd i>ν+1 = n i π holds as well: the static qubits indexed by i such that ν + 1 ≤ i ≤ N behave as if they were all located at x = x ν+1 . Thereby, effectivelyV = G N i=ν+1 (σ f · σ i )δ(x − x ν+1 ) + G(σ f ·σ ν )δ(x − x ν ) (subject to a hard-wall boundary condition at x = x ν−1 ). Letσ eff = N i=ν+1σ i be the total spin of the N − ν SQs effectively located at x = x ν+1 and s eff the quantum number associated withσ 2 eff . For N − ν even, s eff = 0, 1, . . . , (N − ν)/2, while for N − ν odd s eff = 1/2, 3/2, . . . , (N − ν)/2. As, clearly, s eff is a good quantum number, in each subspace of fixed s eff an effective static spin-s eff particle lies at x = x ν+1 ¶. By coupling this spin to f and SQ ν , we find that the total quantum number can take values s = s eff −1, s eff , s eff +1 (we can assume s eff ≥ 1 since the case s eff = 1/2 has been analyzed in the previous section). Among these, only s = s eff is degenerate since in the corresponding eigenspace eitherŜ 2 f ν orŜ 2 f e = (σ f +σ eff ) 2 can take two possible values, i.e. s f ν = 0, 1 and s f e = s eff ± 1/2 (s f e is the quantum number associated withŜ 2 f e ). The reflection matrix for the system is thus block-diagonal, where each block corresponding to either s = s eff − 1 or s = s eff + 1 is 1D, while a block corresponding to s = s eff is 2D. In the latter case, the corresponding reflection amplitudes in the basis {|s f 1 ; s, m s = |s f 1 } can then be worked out in full analogy with the s = 1/2 subspace in the case of two SQs (see the previous section). Hence, they are given by Eqs. = −e −2ikd ν+1 then ensures that in each case the corresponding overall reflection amplitude equals −1 (see the comment at the end of the previous section). A swap operation between f and SQ ν is therefore implemented.

Working conditions
Based on the above findings, in particular Eq. (24), the following working conditions to achieve selective writing/read-out of the static register can be devised. Firstly, one fixes once for all the desired coupling strength g = g 0 [provided that it exceeds the threshold value g th = 1, equivalent to G = k; see Fig. 3(a)]. Next, we choose one of the two different distances (in unit of k −1 ) that correspond to g = g 0 according to the functioñ g(kd) [see Fig. 3(a)]. Let us call such a distance d a , which therefore fulfillsg(kd a ) ≡ g 0 . ¶ Unlike a very spin-s eff particle, in our case a given value of s eff can exhibit degeneracies (e.g. for N = 3 the value s eff = 1/2 is two-fold degenerate). Yet, such degeneracies do not play any role here and can in fact be ignored.
A further distance d b = h(kd a )/k [cf. Fig. 3(b)] is then univocally identified. All the nearest-neighbour distances are set equal to an integer multiple of π (in unit of k −1 ) but the νth and (ν + 1)th ones, which are set to d a and d b , respectively. In a practical implementation, such tunable setting of nearest-neighbor distances could be achieved by fabricating the setup in such a way that the FQ can propagate along three possible paths instead of a single one (similarly to the geometry of the well-known Aharonov-Bohm rings). If the paths have different lengths, the actual path followed by the FQ can be chosen by means of tunable beam splitters, in fact setting the effective SQ-SQ distance.
In practice, unavoidable static disorder will affect the ideal pattern of nearestneighbor SQ distances. Through a proof-of-principle resilience analysis we have assessed that, by assuming Gaussian noise and in the case of a single SQ, an uncertainty in its position of order of about 10% yields a process fidelity above the 95%-threshold. This witnesses an excellent level of tolerance, in line with similar tests [9,15]. Preliminary studies for the cases of two and three SQs have been carried out as well, confirming comparable performances. A comprehensive conclusive characterization of the effects of static disorder in the case of an arbitrary number of SQs, though, requires a rather involved analysis and thus goes beyond the scopes of this paper.

Conclusions
We have considered a typical scenario envisaged in distributed quantum information, where writing and read-out of a register of SQs is performed through a FQ. In a fully quantum theory, the motional degrees of freedom (MDOFs) of the FQ should be treated as quantum, which is expected to substantially complicate the dynamics. By taking a paradigmatic Hamiltonian, we have discovered that, as long as the f -SQ coupling is above a certain threshold value (i.e. G ≥ k with k being the input momentum of the FQ), for an arbitrary number of SQs selective QST can be achieved on demand by tuning only two SQ distances.
Throughout, as is customary in scattering-based theories, we have assumed to deal with a perfectly monochromatic plane wave for the flying qubit. In practice, clearly, this is a narrow-bandwidth wavepacket centered at a carrier wave vector k 0 . A detailed resilience study of the performances of our protocol in such conditions is beyond the scope of the present paper. Yet, similarly to Refs. [9,15,16], it is reasonable to expect the gate fidelity to be only mildly affected owing to the smoothness of functions g(kd) and h(kd) (cf. Fig. 3). In our model, we assumed a Heisenberg-type spin-spin interaction. As already stressed, our attitude here was to take this well-known coupling as a paradigmatic model to show the possibility that selective writing/read-out is in principle achievable. However, there exist setups where the Heisenberg-type coupling occurs so as to make them potential candidates for realizing our protocol. For instance (see also Refs. [19]) this is the case of an electron propagating along a semiconducting carbon nanotube [20] and scattered from single-electron quantum dots or molecular spin systems featuring unpaired electrons, such as Sc@C82 [21]. Alternatively, one can envisage a photon propagating in a 1D waveguide to embody the FQ in a way that its spin is encoded in the polarization DOFs. A three-level Λ-type atom could then work as the static qubit, where the {|↑ , |↓ } basis is encoded in the ground doublet, while each transition to the excited state requires orthogonal photonic polarizations; see Refs. [22,23]. Although similar, the corresponding (pseudo) spin-spin coupling, yet, is not equivalent to a Heisenberg-type one. We found some numerical evidence that this alternative coupling model could work as well, at least in the few-SQ case. An analytical treatment, however, is quite involved and thus no definite answer can be given. This is connected to the question whether some specific symmetry is a necessary prerequisite for such remarkable effects to take place (in passing, note that the Heisenberg model conserves the squared total spin, which was crucial to carry out our proofs). All these issues are the focus of ongoing investigations.
It is worth mentioning that in a recent work [24], Ping et al. proposed a protocol for imprinting the quantum state of a "writing" FQ on an array of SQs and retrieving it through a "reading" FQ at a next stage [24]. There, information is intentionally encoded over the entire register, which has some advantages, while MDOFs are in fact treated as classical. Significantly enough, here we have shown that the inclusion of quantum MDOFs can allow for control over local encoding/decoding. In line with other works [16], such apparent complication appears instead a powerful resource to carry out refined QIP tasks.