Scalable quantum computation via local control of only two qubits

We apply quantum control techniques to a long spin chain by acting only on two qubits at one of its ends, thereby implementing universal quantum computation by a combination of quantum gates on these qubits and indirect swap operations across the chain. It is shown that the control sequences can be computed and implemented efficiently. We discuss the application of these ideas to physical systems such as superconducting qubits in which full control of long chains is challenging.

Figure 1

Our approach for universal quantum computation on a chain of $N$ spins. By modulating the magnetic field $B_1(t)$ on qubit $1$, we induce information transfer and swap gates on the chain (dashed lines). The states of the qubits from the uncontrolled register can be brought to the controlled part. There, the gates from a quantum algorithm are performed by local operations. Afterward, the (modified) states are swapped back into their original positions.

Figure 2

Fourier transform of the optimized function $B_1^{*}(t)$ that produces the physical swap between $n=1$ and $n=29$ for the chain $N=30$ with a fidelity of $F=1-{10}^{-4}$ in a total time $T_N=870.$ The inset shows the effect of a high-frequency cutoff on the fidelity; frequency components above $2J$ can be neglected.

Figure 3

The swap operation time $T_N$ versus the chain length $N$: the red continuous line is $(N-1){}^2$ while the black dots are the lengths for which we numerically verified this scaling with a fidelity $F>1-{10}^{-4}$.