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actly π (other errors are still possible, as discussed be- low). This attractive feature of our gate relies on the capability of using the presence and absence of a fermion as a qubit [18, 19], which is in turn enabled by the inter- conversion between bosonic photons and fermionic spin waves [2]. This interconversion is achieved in two steps. First, photon storage and retrieval techniques [12–15] are used to interconvert between photons and bosonic spin waves, and, second, hard-core interactions between spin waves effectively endow them with fermionic character. The first step requires strong collectively enhanced cou- pling between photons and atoms [12–15], which can be achieved in recently demonstrated waveguides that trans- versely confine both atoms and photons to dimensions on the order of a wavelength [8–10]. The second step is achieved by means of strong atom-atom interactions [11, 16]. Our proposal thus bears a similarity to those of Refs. [5, 6], which also rely on strong interatomic inter- actions, but contrasts with methods where atom-photon interactions alone suffice [2, 7]. At the same time, our proposal is fundamentally different from Refs. [5, 6] in that it relies on the robustness and conceptual simplicity of fermionic exchange for achieving the nonlinear phase. 

Before proceeding, we note that fermionic exchange in spin chains, including their cold atom realization [19], has been suggested for use as an entangling gate in Refs. [19, 20]. The advantage of our approach is that we require neither spatially inhomogeneous coupling nor single-site addressiblity and that we extend the gate from spins to photons via photon storage techniques [12–15]. 

Details of the protocol. —We begin with a bosonic two- component single-band Hubbard model in a 1D lattice, with the transverse motion frozen. There are N sites with nearest-neighbor spacing a , yielding a total length Na . The tunneling amplitudes are t$_{α}$ for species α ( α = g , s ), while the s-wave interaction energies are U$_{αα}$ for two α atoms and U$_{sg}$ for an s atom with a g atom [11]. As- suming one atom per each of N sites in the weak tun- neling limit t$_{g}$, t$_{s}$ ≪ U$_{gg}$, U$_{ss}$, U$_{sg}$ , ignoring edge effects, and dropping the term ∑ j S z j (which would contribute a simple linear phase [21]), the superexchange XXZ Hamil- tonian is ( ℏ = 1 throughout the Letter) [11, 16] 

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where V = 2 t 2 $_{g}$+ t 2 s U$_{sg}$ − 4 t 2 g U$_{gg}$ − 4 t 2 s U$_{ss}$ and J = 2 t$_{g}$t$_{s}$ U$_{sg}$ . Here ⃗ S$_{j}$ = ⃗ σ$_{j}$/ 2, where ⃗ σ$_{j}$ are Pauli operators in the { s, g } basis and S ± j = S x j ± iS y $_{j}$. It is convenient to use the Jordan-Wigner transformation [17] to map the spins onto fermions with creation operators c † $_{j}$: S + j = c † j exp ( iπ ∑ j − 1 k $_{=1}$n$_{k}$ ) and S z j = n$_{j}$ − 1 $_{2}$, where n$_{j}$ = c † $_{j}$c$_{j}$ . Then, dropping the term 

∑ j n$_{j}$ , which would contribute only a linear phase [21], 

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With V = 0, Eq. (1) is the XX Hamiltonian, while Eq. (2) then represents free fermions. We can obtain V = 0, for example, by taking t$_{g}$ = t$_{s}$ = t and U$_{gg}$ = U$_{ss}$ = 2 U$_{sg}$ = U , so that J = 4 t $^{2}$/U [19]. When all three scattering lengths are similar in size, as e.g. in some ground-state alkali atoms, U$_{sg}$ can be reduced by shifting the s and g lattices relative to each other [22]. 

The free fermion ( V = 0) case can be diagonalized, ignoring boundaries, by going to k -space: 

where k takes N values in the first Brillouin zone from − π to π at intervals of 2 π/N , c † k = (1 $_{/}$√$_{N}$$_{)}$ ∑ j e $^{ikj}$c † $_{j}$, and the dispersion ϵ ( k ) = − 2 J cos( k ) is shown in Fig. 1(c). With all atoms in g (denoted as vac ) and tunnel- 

| 〉 | 〉 ing turned off, two counterpropagating photonic modes, labeled as R and L in Fig. 1(a), are incident on our 1D chain of atoms near resonance with the g - e transition [see Fig. 1(b)]. The wave packets carry photonic qubits, in which | 0 〉 ( | 1 〉 ) corresponds to the absence (presence) of a single photon in the mode. The goal is to implement a two-qubit phase gate, which puts a π phase on the state | 1 〉| 1 〉 , leaving the other three basis states ( | 0 〉| 0 〉 , | 0 〉| 1 〉 , and | 1 〉| 0 〉 ) unchanged [21]. We use two control fields Ω (one for R and one for L ) near resonance with the s - e transition [see Fig. 1(b)] applied at different angles from the side of the chain [see Fig. 1(a)] to store [12–15] the two photons into the s - g coherence over two spatially sep- arated regions. As shown below, by appropriately choos- ing the angle of incidence of the control fields, we can arrange for the R and L spin waves to be centered in k -space around k = π/ 2 and k = − π/ 2, respectively, as shown in Fig. 1(c). This ensures that the two spin waves will move through each other with maximum speed [largest possible difference of first derivatives of ϵ ( k )] and with minimal distortion [vanishing second derivative of ϵ ( k )]. Photonic band gap effects coming from the peri- odicity of atomic positions [23] can be avoided if light of different wavelengths is used for the lattice and the photons, which is typically the case in experiments [24]. 

The XX Hamiltonian is then turned on. Since H is diagonal in the Fourier basis [Eq. (3)], it is convenient to define, for any function q ( j ) on the sites j , its Fourier transform ˜ q ( k ) = (1 $_{/}$√$_{N}$$_{)}$ ∑ j q ( j ) e − ikj together with the usual inverse relation q ( j ) = (1 $_{/}$√$_{N}$$_{)}$ ∑ k ˜ q ( k ) e $^{ikj}$. Us- ing Eq. (3) and S + j | vac 〉 = c † j | vac 〉 , it is then easy to check that the R spin wave alone evolves in time τ as ∑ j R ( j, τ ) S j + | vac 〉 , where ˜ R ( k, τ ) = ˜ R ( k, 0) e − iτϵ ( k $^{)}$. We now assume that R ( j, 0) is centered around j = N/ 4 

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