Quantum communication with a spin chain

In the development of large-scale quantum computation, one of the essential tasks is the transmission of quantum information between distant quantum registers and processors. Although photons are the best option for long distance communication, the short- to mid-range communication between processors could benefit from alternative approaches. In particular, new communication schemes could avoid problems that arise when converting stationary qubits into flying qubits, and vice versa.

One way to transmit quantum information is to realize a quantum channel with a spin chain, i.e., a linear string of N magnets or equivalent systems, as first suggested by Bose.¹ Information encoded in the first spin at one end of the chain automatically travels to the other end, thanks to the natural evolution of an interacting spin system. Figure 1 shows a linear chain of spins initially all pointing down (like in a ferromagnetic domain), except for the very first spin at the sender's site. This spin excitation contains the information that the sender wants transmitted to the receiver, whose site is located at the opposite end of the chain. No action from the outside is required, except the ability to switch on and off the interactions that connect both the sender's spin and the receiver's spin to the rest of the chain. Besides connecting remote quantum processors, a spin chain is also able to share entanglement, a peculiar feature of quantum systems at the heart of quantum technologies, such as teleportation.²Moreover, such a spin channel can interface directly to a quantum computer because both can be made with the same kind of physical system.

Figure 1. A linear chain of spins that initially are all down except the first one at the sender's site. (top) A uniformly filled chain. (bottom) The so-called double-hole configuration, where the nearest-neighboring spins of sender and receiver have been removed.

Over the last years, research efforts devoted to improving the performance of spin chains have increased the fidelity³and communication speed⁴ in these systems. The variety of approaches being pursued often rely on particular geometries, ancillary systems, dynamical control, or careful design of the interactions between the system components.

In our work,^5,6 we devised a simple way to attain perfect state transfer with a ferromagnetic spin chain, characterized by a realistic long-range interaction. The guiding principle is to reduce the N-spin system to an effective two-spin system by slightly detaching sender and receiver from the rest of the chain. In more formal terms, our procedure increases the energy separation between the sender-receiver subspace and the rest of the chain. The system eigenvectors, corresponding to the two lowest energy eigenvalues, exhibit a more pronounced localization towards the ends of the chain. As a result, the excitation tends to localize in the sender-receiver subspace with little or even negligible mixing with the other spins in the chain.⁵ Quite surprisingly, this protocol allows the ferromagnetic system to generate maximally entangled states between distant parties. At a given time, the two spins at the opposite ends of the chain may be correlated in a way that has no classical counterpart.⁶

We can achieve the effective two-spin system by leaving an empty site next to sender and receiver, as illustrated in the bottom of Figure 1. We refer to this case as the double-hole configuration. Another possibility is to have a uniformly filled string of spins, but then isolate the spins at sites 2 and N−1 so they do not participate in the system dynamics. This level of control is possible, for example, in linear ion traps. Thanks to the single particle addressability, one can put the ions, sitting at sites 2 and N−1, out of resonance with respect to the driving field used to establish the spin-spin interaction.

In mathematical terms, we consider a linear chain of interacting spins described by the isotropic XYZ Heisenberg model

where S_i is the total spin operator and S_i^z its component along the z-axis. We assume that the interaction strength J_{i, j} decreases with the third power of the distance between spins (dipolar coupling). The ability of the quantum channel to faithfully transmit the information from sender to receiver is measured by the fidelity

where

is the probability amplitude of propagating the excitation from the sender site s to the receiver r, as a function of time t. Perfect transmission, f_{s, r}(t)=1, leads to the maximum value of the fidelity, F(t)=1. In Figure 2, we plot the fidelity as a function of time (in dimensionless units defined previously⁵) for a chain consisting of 50 sites. The red curve refers to the complete chain with uniform filling depicted in the top panel of Figure 1, whereas the blue curve represents the double-hole configuration from the bottom panel of Figure 1. The double-hole configuration performs much better in terms of fidelity and transmission rate, achieving perfect transmission and being almost three times faster than the complete chain. Moreover, the fidelity of the double-hole configuration is a smooth function of time, which greatly relaxes the requirements on experimental time resolution for performing the communication protocol.

Figure 2. Plot of the fidelity, F, as a function of the time, t, for a spin chain consisting of 50 sites. Both the complete chain (red) and the double-hole configuration (blue) are represented.

It may seem counterintuitive that a chain with missing links would outperform a complete chain, but the double-hole configuration effectively uses only the ends of the chain near the sender and receiver. Therefore, it reduces the information loss through the middle of the chain. Our scheme's performance is, most notably, independent of the system size. This is demonstrated in Figure 3, where we have varied the length of the chain from a few to 100 spins. While the performance of the complete chain decreases with increasing distance between sender and receiver, the maximum fidelity in the double-hole scheme is consistently one. The same size-independence holds true for the entanglement generation between sender and receiver.⁶

Figure 3. Plot of the maximum fidelity, F_max, as a function of the chain length, N. Red circles correspond to the complete chain, whereas blue diamonds correspond to the double-hole configuration.

In summary, spin chains are a promising candidate for quantum information transfer and entanglement generation between different registers and processors of a quantum computer. Our approach is fully scalable and requires minimal control and design from the experimental point of view. For this reason it should also be relatively robust against noise, technical imperfections, fluctuations of the relevant parameters and other potential sources of decoherence. The next step will be an experimental implementation based on, for instance, strings of trapped electrons⁷ or ions.⁸The realization of spin chains would provide not only a test of communication protocols, but also an efficient way to simulate quantum systems.