We propose a novel physical realization of a quantum computer. The qubits are electric dipole moments of ultracold diatomic molecules, oriented along or against an external electric field. Individual molecules are held in a 1-D trap array, with an electric field gradient allowing spectroscopic addressing of each site. Bits are coupled via the electric dipole-dipole interaction. Using technologies similar to those already demonstrated, this design can plausibly lead to a quantum computer with > ∼ 10 4 qubits, which can perform ∼ 10 5 CNOT gates in the anticipated decoherence time of ∼ 5 s.PACS numbers: 03.67. Lx, 33.80.Ps, 33.55.Be We describe a new technical approach to the design of a quantum computer (QC). The basic QC architecture is shown in Fig. 1. The qubits consist of the electric dipole moments of diatomic molecules, oriented along or against an external electric field. Bits are coupled by the electric dipole-dipole interaction. Individual molecules are held in a 1-D trap array, with an electric field gradient allowing spectroscopic addressing of each site. Loading with ultracold molecules makes it possible to use a weak trapping potential, which should allow long decoherence times for the system. This design bears various features in common with other recent proposals which employ electric dipole couplings [1,2,3]. However, the technical parameters of our design appear very favorable, and apparently only incremental improvements of demonstrated techniques are required in order to build a QC of unprecedented size.We describe the molecular qubits as permanent electric dipoles oriented along (|0 ) or against (|1 ) an external electric field ( E ext ). (This model reproduces the exact behavior well in a certain regime.) Lattice sites are equally spaced in the x-direction and each contains one molecule, prepared initially in its ground state |0 . The external field is perpendicular to the trap axis and consists of a constant bias field plus a linear gradient:, where H 0 is the internal energy of a bit, d a is the electric dipole moment of bit a, and E a = E ext (x a ) + E int (x a ) is the total electric field at x a . The internal field E int is created by the electric dipole moments of neighboring bits:The scheme for gate operations is as outlined for the electric dipole moments of quantum dots in Ref. [1]. Transitions between qubit states can be driven by electric resonance, either directly in the microwave region or indirectly by an optical stimulated Raman process. Resonant drive pulses are tuned to frequency ν a = ν 0 + d ef f E a /h, where hν 0 is the difference in internal energies between states |0 and |1 in zero field; the effective dipole momentis the dipole moment in state |0 (|1 ); and h is Planck's constant. Pulses of sufficient temporal length to resolve the energy splitting due to E int can be used for CNOT gates; shorter pulses suffice for one-bit rotations. Final-state readout can be accomplished by state-selective, resonant multiphoton ionization [4] and imaging detection of the resulti...