Quantum Gates Between Distant Qubits via Spin-Independent Scattering

We show how the spin independent scattering of two initially distant qubits, say, in distinct traps or in remote sites of a lattice, can be used to implement an entangling quantum gate between them. The scattering takes place under 1D confinement for which we consider two different scenarios: a 1D wave-guide and a tight-binding lattice. We consider models with contact-like interaction between two fermionic or two bosonic particles. A qubit is encoded in two distinct spins (or other internal) states of each particle. Our scheme enables the implementation of a gate between two qubits which are initially too far to interact directly, and provides an alternative to photonic mediators for the scaling of quantum computers. Fundamentally, an interesting feature is that"identical particles"(e.g., two atoms of the same species) and the 1D confinement, are both necessary for the action of the gate. Finally, we discuss the feasibility of our scheme, the degree of control required to initialize the wave-packets momenta, and show how the quality of the gate is affected by momentum distributions and initial distance. In a lattice, the control of quasi-momenta is naturally provided by few local edge impurities in the lattice potential.

Recent progress in the control of the motion of neutral atoms in restricted geometries such as traps [1,2], 1D optical lattices [3,4] and wave-guides [5] has been astounding. Naturally, the question arises as to whether they can be used in a similar manner as photons are used, i.e., as "flying qubits" for logic as well as for connecting well separated registers in quantum information processing. Quantum logic between flying qubits exploits their indistinguishability and assume them to be mutually non-interacting -hence the names "linear optics" [6] and "free electron" [7] quantum computation. In fact, for such an approach to be viable one has to engineer circumstances so that the effect of the inter-qubit interactions can be ignored [8]. On the other hand, in the context of photonic qubits, it is known that effective interactions, engineered using atomic or other media, may enhance the efficacy of processing information [9][10][11][12][13]. One is thereby motivated to seek similarly efficient quantum information processing (QIP) with material flying qubits which have the advantage of naturally interacting with each other. Further motivation stems from the fact that for non-interacting mobile fermions, additional "whichway" detection is necessary for quantum computation [7] and even for generating entanglement [14], which are not necessarily easy. Thus, if interactions do exist between flying qubits of a given species, one should aim to exploit these for QIP.
While it is known that both spin-dependent [15] and spin-independent [16,17] scattering can entangle, it is highly non-trivial to obtain a useful quantum gate. The amplitudes of reflection and transmission in scattering generally depend on the internal states of the particles involved which makes it difficult to ensure that a unitary operation i.e., a quantum gate acts exclusively on the limited logical (e.g. internal/spin) space that encodes the qubits. Only recently, for non-identical (one static and one mobile) particles, it has been shown that a quantum gate can be engineered from a spin dependent scattering combined with an extra potential [18]. We will show here that one can accomplish a quantum gate merely from the spin independent elastic scattering of two identical particles. This crucially exploits quantum indistinguishability to label qubits by their momenta, as well as the equality of the incoming pair and outgoing pair of momenta in one dimension (1D). In our scheme the quantum gate is only dictated by the Scattering matrix or S-matrix acting on the initial state of the two free moving qubits. This is thus an example of minimal control QIP where nothing other than the initial momenta of the qubits is controlled. Not only will it enable QIP beyond the paradigm of linear optics with material flying qubits, but also potentially connect well separated registers of static qubits. One static qubit from each register should be out-coupled to momenta states in matter wave-guides and made to scatter from each other. The resulting quantum gate will connect separated quantum registers. This may be simpler than interfacing static qubits with photons.
While quantum gates exploiting the mutual interactions of two material flying qubits has not been considered yet, the corresponding situation for static qubits has been widely studied (e.g., Refs. [19][20][21][22][23][24][25]). However, these methods typically require a precise control of the interaction time of the qubits or between them and a mediating bus (e.g., Refs. [26][27][28]). On the other hand, our method exploits a much lower control process, namely the scattering of flying qubits. Still static qubits offer the natural candidate for information storage. Motivated by this, and also by the high degree of control reached in current optical lattice experiments [3,4], as a second result of this Letter we consider a lattice implementation of our gate. This experimental proposal is particularly compelling also because the qubit can be made either static of mobile depending on the tunable potential barrier on different lattice sites, thus avoiding to seek some mechanism to couple static and mobile particles and allowing arXiv:1412.3582v1 [quant-ph] 11 Dec 2014 for both storage and computation with the same physical setup.
Our study interfaces QIP and quantum indistinguishability with two other areas, namely the Bethe-Ansatz exact solution of many-body models [29] and the 1D confinement of atoms achieved in recent experiments [30][31][32][34][35][36][37][38] Quantum gate between flying qubits:-A two qubit entangling gate is important as it enables universal quantum computation when combined with arbitrary one qubit rotations [39]. We consider the spin independent interaction to be a contact interaction between point-like non-relativistic particles. For two two spinless bosons on a line (1D) the Hamiltonian with a delta-function interaction is [29] where x 1 and x 2 are the coordinates of the two particles. The above model is called the Lieb-Liniger model and has an interesting feature which we will actively exploit. This is the fact that the momenta are individually conserved during scattering. If the incoming particles have momenta p 1 and p 2 , then the outgoing particles also have momenta p 1 and p 2 [40], as shown in Fig.1(a). Thus the scattering matrix is diagonal in the basis of momenta pairs and, is, in fact, only a phase which accumulates on scattering. The scattering matrix extracted from these wavefunctions is given, for incident particles with momenta p 2 >p 1 , by [29] S(p 2 , p 1 )= p2−p1−ic p2−p1+ic . The phase accumulated on scattering is −i ln S(p 2 , p 1 ). Note that, as expected, for non-interacting bosons (c→0), their exchange causes no phase change, while when c→∞ (impenetrable bosons equivalent to free fermions) and have a −1 factor multiplying on exchange.
We consider the case of colliding particles having some internal degrees of freedom in which a qubit can be encoded ( Fig.1(a)). The collision is assumed to have the form of a spin independent contact (delta) interaction of point-like particles as in Eq.(1). We first consider bosons with two relevant states |↑ and |↓ of some internal degree of freedom (could be any two spin states of a spin-1 boson, for example). For symmetric states of the internal degrees of freedom, the external degrees of freedom also have to be symmetric and have the same scattering matrix as spinless bosons. On the other hand, for antisymmetric spin states, the spatial wave function of the two particles is fermionic so that the amplitude for x 1 =x 2 (the chance of a contact delta interaction) is zero implying that they do not scatter from each other. The above observations lead to the S-matrix (for p 2 >p 1 ) where Π 12 is the SWAP (permutation) operator acting on the internal (spin) degree of freedom (Π 12 (|u 1 |v 2 )=|v 1 |u 2 , where |u 1 and |v 2 are arbitrary spin states of the particles, so that Π 12 is a 4×4 matrix). We also consider the case where qubit states are spin states of a spin-1/2 particle (say, electrons or fermionic atoms). This is the conventional encoding in many quantum computation schemes. In this case the S−matrix was computed by C. N. Yang [41] to be (for We consider a frame in which two qubits are moving towards each other so that they eventually collide. Let us call the qubit with momentum towards the right as qubit A, while the qubit with momentum towards the left is called qubit B. Each qubit is in a definite momenta state, whose magnitudes are p A and p B respectively [42]. Thus p 2 =p A and p 1 =−p B . The evolution of the 4 possible qubit states due to the scattering is thereby given by where p A+B =p A +p B , e iφ B = p A+B −ic p A+B +ic and e iφ F =1. Unless either p A+B or c vanishes, the above is manifestly an entangling gate, as is evident from the fact that the right hand sides of the last two lines of Eq. (4) is an entangled state. This gate is most entangling (i.e., the most useful in context of quantum computation, equivalent in usefulness to the well known Controlled NOT or CNOT gate) when p A+B ≈c, as then the right hand sides of the last two lines of Eq(4) correspond to maximally entangled states e −i π Error estimates:-The amplitudes in Eqs.(4) depend only on the ratio of p A+B /c and thereby any spread δp A+B of the relative momenta of the incoming particles only affects the amplitudes as δp A+B /c. As a relevant example we consider two Gaussian wavepackets in the internal state |↑ A |↓ B where p A+B is centered around c(1+δ) with a relative variance ηc. After the scattering the generated entanglement between the internal degrees of freedom is measured by the concurrence [43] and f (z)= √ πe z 2 erfc(z). From the asymptotic expansion [44] zf (z)≈1−z −2 /2 one obtains that C slowly decays as a function of (δ, η), and that the case δ 0 is less prone to errors when η increases. Errors can thereby be arbitrarily reduced in principle by choosing particles with higher c. This is opposite to the usual paradigm of gates based on "timed" interactions, where for a given timing error δt, stronger interactions enhance the error (while weaker interactions make gates both slower and susceptible to decoherence).
Implementation via flying qubits:-One of the most promising implementation of our gates is with neutral bosonic/fermionic atoms. The delta function interaction we use is, in fact, very realistic and realizable for neutral atoms under strong 1D confinement [45]. 87 Rb atoms have already been strongly confined to 1D atomic waveguides leading to delta interactions [30]. For 87 Rb, with a 3D scattering length a 50Å an axial (for 1D) confinement of ω ⊥ 100kHz gives (using e.g. Refs. [45,46]) c 10 6 m −1 . Velocities of atoms in 1D waveguides (c.f. atom lasers [34]) can be mm s −1 , which translates to p A+B 10 6 m −1 (in units of wavenumbers). Thereby, p A+B ≈c for optimal gates is achievable in current technology [47]. Deviation from the 1D effective δ-potential are expected when the condition p A+B √ µω ⊥ is not satisfied (µ being the atomic mass). In that case the scattering matrix has still the form Eq.(4) where c shows a (weak) dependence on p A+B [45]. The optimal gate is then found by solving p A+B =c(p A+B ) c−ζ 3/2 (µω ⊥ / ) −3/2 (c p A+B /4) 2 , being ζ the Riemann zeta function. State independent waveguides for two spin states has been met [35] in magnetic waveguides (trivially possible in optical waveguides/hollow fibers). Our gate will be an extension of recent collision experiments between different spin species [48] with pairs of atoms at a time. Launching exactly two atoms towards each other in 1D should be feasible with microtrap arrays [1,2,49] or in atom chips [31,32] and is also a key assumption in many works [8,33]. For example, our gates can be made with the technique of Ref. [33] whereby atoms are trapped initially in potential dips inside a larger well and let to roll towards each other in a harmonic potential to acquire their momenta (note that our gate scheme is completely different from Ref. [33], where the atomic motion is guided by internal states). The initial position of the particles x B =−x A =x 0 can be tailored so that their relative momentum has minimum variance ∆p A+B when the particles reach the collision point (x=0). As ω z ω ⊥ (being ω z the frequency of the longitudinal harmonic confinement) the collision does not feel the longitudinal potential, so it is approximated by Eq.(4). As shown previously, the generated entanglement is very high (C 1) provided that η∼∆p A+B /p A+B ∼∆x 0 /x 0 1. A different approach (depicted is Fig. 1b) consists in suddenly moving the local trapping potentials so that the particles in A and B move towards their new potential minima. As in the previous case, wave-packets with well-defined and tunable momenta are created by switching off the potential when they reach the minima where their momentum uncertainties are minimal. Lastly, static atomic qubits (c.f. [25,50]) in well separated traps may be Raman outcoupled [24] imparting them momenta towards each other in a 1D waveguide -this will link distinct registers.
Ballistic electrons inside carbon nano-tubes or semiconductor nanowires, or edge states may be another implementation [51,52]. As electrons are typically screened in a solid, a contact interaction should describe their scattering. Note that the Hubbard model with solely on site interactions models a variety of electronic systems. In a low energy limit, in which the electronic de Broglie wavelength is far larger than the lattice spacing of the Hubbard model, the Hamiltonian reduces to that of Eq.(1).
Quantum gates between distant stationary qubits:-A discrete variant of the system with a δ-interaction is the Bose-Hubbard Hamiltonian [29,53]: where α={↑, ↓} labels two internal degrees of freedom of the particles. We call N the length of the chain. In a lattice setup, free-space evolution is replaced with particle hopping. Particle collisions lead to a scattering matrix which, for uniform couplings J j =J, U αβ j =U αβ , is given by Eq.(2) with the substitutions [53,54] p j → sin p j , c → U αβ /J.
We consider different initial internal degrees of freedom and set U αα j =0. A maximally entangling gate is therefore realized when sin p 1 − sin p 2 ≈2U , being U =U ↑↓ /(2J). In particular, p 1 = sin −1 U when p 1 =−p 2 .
The Bose-Hubbard Hamiltonian (5) naturally models cold bosonic atoms in optical lattice [55]. Owing to single atom addressing techniques [56] 87 Rb atoms in different lattice sites can be initialized in either two distinguishable hyperfine internal states |↓ ≡|F =1, m F =−1 and |↑ ≡|F =2, m F =−2 . The coupling constants U αβ j depend on the strength g αβ of the effective interaction between cold atoms [57]. These parameters are usually experimentally measured [58,59] and can be tuned by Feshbach resonances [60]. Spin-exchange collisions are highly suppressed due to the little difference (less than 5%) between singlet and triplet scattering length of 87 Rb [55]. The one-dimensional regime is obtained by increasing the harmonic lattice transverse confinement (ω ⊥ /2π 18 kHz see [37,61] for typical values). We obtain the 1D pseudo-potential coupling constants g αβ from the 3D measured values [59] following [45], respectively g ↑↑ =1.14×10 −37 J m, g ↑↓ =1.12×10 −37 J m, g ↓↓ =1.09×10 −37 J m. The internal spin state and the position of particles are detected by fluorescence microscopy techniques [56,62]. The parameters U αβ j and J j can be physically controlled in optical lattice systems locally varying the depth of the optical potential [63]. Arbitrary optical potential landscapes are generated directly projecting a light pattern by using holographic masks or micromirror device [4,62]. In particular, U αβ √ 2π (g αβ /λ) (V 0 /E R ) 1/4 and J j where λ is the laser wavelength, V 0 is the lattice depth and E R is the recoil energy [57]. For flying qubits we considered a fixed c and we tuned p j to obtain the desired gate. In a lattice, on the other hand, U αβ can be controlled precisely, while the creation of a wave-packet requires the control and initialization of many-sites. This kind of control can be avoided by initially placing two particles at the two distant boundaries of the lattice (particle A on the left and particle B on the right) and locally tuning the coupling J 0 between the boundaries and the rest of the chain [64] (all the other couplings are uniform J j =J). An optimal choice of J 0 /J has a twofold effect: firstly it generates two wavepackets whose momentum distribution is Lorentzian, narrow around p A =−p B ±π/2, respectively, and with a width dependent on J 0 ; secondly it generates a quasidispersionless evolution, allowing an almost perfect reconstruction of the wave-packets after the transmission (occurring in a time ≈N/J) to the opposite end. In this scheme (Fig. 1c), the particles starts from opposite loca-tions, interact close to the center of the chain and then reach the opposite end where the wave-function is again almost completely localized, allowing a proper particle addressing. Since p A/B is fixed, a high amount of entanglement is generated when U =| sin p A/B |=1. For 87 Rb we found that the latter condition is satisfied, e.g., when V 0 /E R 2.2, giving also J/h 240Hz.
In this scheme there are two error sources. The first is due to the transmission quality, though it is above 85% even for long chains [64]. The second one is due to the finite width of the Lorentzian momentum profile around |p|=π/2 which, in turn, yields slightly different gates for different momentum components. To quantify the amount of those errors we evaluate numerically the join probability amplitude A αβ ij (t) to have respectively particle A in sites i and particle B in j as function of the inter-particle interactions U . The indices α, β refer to the initial internal state of particles A and B,t is the transfer time, and the initial condition is A αβ 1N (0)=1. We find that for distinguishable particles A ↑↓ 1N /A ↑↓ N 1 (t)=−iU/U opt in agreement with the prediction, apart from U opt which models finite size effects: e.g. U opt =0.95 (U opt =0.97) for N =25 (N =51) and U opt →1 for long chains. For indistinguishable particles we obtain that A αα 11 /A αα 1N (t) is zero for α=↑, ↓ and for any value of U ↑↓ . Therefore, apart from a global damping factor due to the non-perfect wavepacket reconstruction, the resulting transformation is in agreement with the gate (4). The entanglement generation between the boundaries is evaluated via the concurrence [43] C 1N =2|A ↑↓ 1N A ↑↓ N 1 * |. From the asymptotic analysis [64] we find that C 1N =f 4 1N 2U/Uopt (U/Uopt) 2 +1 where f 1N is the transmission probability from site 1 to site N at the transmission time. Surprisingly, C 1N does not directly depend on the width of the wave-packet in momentum space: its maximum value depends only on the transfer quality f 1N . For example, for N =25, f 1N =0.97 so C opt 1N =0.88; for N =51, f 1N =0.95 so C opt 1N =0.81.
Concluding remarks: -We have described a mechanism for useful gates between flying qubits and hopping qubits in a lattice using the low control method of scattering. 1D matter waveguides, identical particles and a contact interaction are key requirements for the gate. Exactly solvable models enable the study [29], while experiments and results on 1D gases [1, 2, 30-32, 34, 35, 37, 38] and optical lattices [3,56] suggest implementations with flying and static atomic qubits. The scheme may both enhance linear optics-like QIP and connect separated atomic qubit registers.
Acknowledgements: -VK acknowledges financial support by NSF Grant No. DMS-1205422. SB, LB and EC are currently supported by the ERC grant PACOMANE-DIA.