Fault-Tolerant Quantum Computation via Exchange Interactions

Quantum computation can be performed by encoding logical qubits into the states of two or more physical qubits, and control of effective exchange interactions and possibly a global magnetic field. This “encoded universality” paradigm offers potential simplifications in quantum computer design since it does away with the need to control physical qubits individually. Here we show how encoded universality schemes can be combined with fault-tolerant quantum error correction, thus establishing the scalability of such schemes.

Figure 1

Analytically derived circuits for the $\sqrt{{\mathrm{S}\mathrm{W}\mathrm{A}\mathrm{P}}^{\prime }}$ operation. Time flows from left to right. Data-physical qubits are numbered 1, 2, while 3–6 are ancilla-physical qubits. Ancilla qubits 5 and 6 are added to allow parallel operations within each LCU, which reduces the time overhead by a factor of 2. (a) The circles represent a possible arrangement of qubits so that all are nearest neighbors throughout the pulse sequence. (b) $\sqrt{{\mathrm{S}\mathrm{W}\mathrm{A}\mathrm{P}}^{\prime }}$ in the $XY$ model: an angle $\varphi$ under an arrow connecting qubits $i,j$ represents the pulse $\exp (-i\varphi {\overline{X}}_{ij})$. (c) $\sqrt{{\mathrm{S}\mathrm{W}\mathrm{A}\mathrm{P}}^{\prime }}$ in the Heisenberg or $XXZ$ model: an angle $\varphi$ under a solid (checkered) arrow connecting qubits $i,j$ represents the pulse $\exp (-i\varphi Z_i{Z}_j)$ [$\exp (-i\varphi {\overline{X}}_{ij})$]. These pulses are, in turn, realized with the pulse sequence in (d), where the angle $2\varphi$ in (d) corresponds to the angles $\pm \pi /4,\pm \pi /2$ used in circuit (c). (d) Refocusing the Ising terms: each arrow represents a Heisenberg or $XXZ$ exchange interaction between corresponding qubits. The choice $+{\tau }_E$ [$-{\tau }_E$] for the central pulse selects $\exp (-i2\varphi Z_i{Z}_j)$, with $\varphi =\frac{1}{\hslash }{\int }^{{\tau }_E}{J}_{ij}^z(t)dt$ [$\exp (-i2\varphi {\overline{X}}_{ij})$, with $\varphi =\frac{2}{\hslash }{\int }^{{\tau }_E}{J}_{ij}(t)dt$] used in circuit (c). Qubit indices are $i,j=1,2,3,4$ and $k=5,6$. The vertical bars represent global magnetic fields $\exp (-i\sum _l{\theta }_l{Z}_l)$, where ${\theta }_l=({\mu }_B{g}_l/\hslash ){\int }^{{\tau }_B}{B}_l^{z}f(t)dt$, in which $B_l^z$ is the $z$ component of the global magnetic field at spin $l$ and $f(t)$ is the global time dependence of the magnetic field. The pulse duration ${\tau }_B$ (indicated above the vertical bars) is adjusted such that ${\theta }_k-{\theta }_i=\pi /2$.