Universal Quantum Computation with the Exchange Interaction

Experimental implementations of quantum computer architectures are now being investigated in many different physical settings. The full set of requirements that must be met to make quantum computing a reality in the laboratory is daunting, involving capabilities well beyond the present state of the art. Theory can aid experiment by exposing new alternative routes for achieving quantum computation by simpler means. In this report we indicate one route to simplification that would apply to many recent solid-state approaches, using quantum dots, and using donor-atom nuclear spins or electron spins. In all these approaches, the basic two-qubit quantum gate is generated by a tunable Heisenberg interaction (the Hamiltonian is $H_{ij}=J(t){\vec S}_i\cdot{\vec S}_j$ between spins i and j); we show here that, at the expense of increasing the number of devices and the number of computational steps by constant factors, the Heisenberg interaction alone suffices to exactly implement any quantum computer circuit. An explicit assessment of the tradeoffs involved is given.