Quantum Computing with an Always-On Heisenberg Interaction

Many promising schemes for quantum computing (QC) involve switching “on” and “off” a physical coupling between qubits. This may prove extremely difficult to achieve experimentally. Here we show that systems with a constant Heisenberg coupling can be employed for QC if we actively “tune” the transition energies of individual qubits. Moreover, we can collectively tune the qubits to obtain an exceptionally simple scheme: computations are controlled via a single “switch” of only six settings. Our schemes are applicable to a wide range of physical implementations, from excitons and spins in quantum dots through to bulk magnets.

Figure 1

Strategies for implementing QC on a 1D chain. Blocks represent individual spins and qubits are denoted by letters $T,U,\dots ,Z$. Fixed Zeeman energies are denoted by $A$ and $B$, tunable Zeeman energies by ${ϵ}_i$. Here we assume an independent mechanism exists for single qubit gates. If these are much faster than $1/J$, then one simply adopts the trivial scheme shown in (a). Qubits are placed on adjacent cells and will suffer continuous phase gates with their neighbors, but techniques developed for NMR QC [10] can be employed to actively negate this via fast rotations. However, universal single qubit gates in solid state systems are usually slow [4] compared to $1/J$. In this case we would choose to place qubits only on alternate spins (b), with intervening spins in definite classical states. The Ising interaction is then entirely negated, and two-qubit operations are achieved by “tuning” a spin’s Zeeman energy, as described in the text. Part (c) displays data from a numerical analysis of the process: a gate on $X$ and $Y$ is achieved by tuning ${ϵ}_2$ in the nine spin section shown. Plots show worst case defects over the 16 possible basis states. Phase noise (not shown) was always smaller.

Figure 2

Strategies for implementing QC when no independent mechanism for single qubit gates exists; these are now synthesised via Zeeman tuning alone. This requires an encoding (a) of two spins per qubit, with a third acting as a barrier. All gates can be implemented purely by tuning the Zeeman energy of the qubit that initially has energy $ϵ=B$. For single qubit gates: If we abruptly tune this energy to $ϵ=A$, perfectly matching the energy of its neighbor, then the logical qubit represented by this pair will experience [11] a continuous rotation given by $\exp (ikt{\widehat{\sigma }}_x)$ as shown in (b)(i). Tuning to an energy $ϵ=A+\delta$ will produce a rotation about an axis in the $z\mathrm{\text{−}}x$ plane, the axis being determined by the ratio $J$ to $\delta$, as depicted in (b)(ii). Such rotations can synthesize any one-qubit gate. For a two-qubit gate we employ the process shown in (c): we set the energy $ϵ$ of our tunable spin to a value near $C$. This effectively allows qubit $X$ to “spread” onto the barrier spin, where it experiences a conditional phase gate due to the proximity $Y$. Part (d) shows an architecture for the case where Zeeman energies cannot be tuned independently for nearby spins. QC can still be achieved, by collectively tuning one of the two subsets with energies ${ϵ}_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}$ and ${ϵ}_{\mathrm{o}\mathrm{d}\mathrm{d}}$.

Figure 3

Left: Cartoon emphasizing that, in our 3rd architecture, the entire computation on all qubits can be implemented simply by switching between six settings. In practice such switching would be of course be performed by a conventional computer (not manually), and moreover one would require additional setting(s) for measurement. Right: in certain physical systems, the switch can be equivalent to adjusting just a single global parameter.