Quantum Networks for Generating Arbitrary Quantum States

Many results in quantum information theory require the generation of specific quantum states, such as EPR pairs, or the implementation of specific quantum measurements, such as a von Neumann measurement in a Fourier transformed basis. Some states and measurements can be efficiently implemented using standard quantum computational primitives such as preparing a qubit in the state |0〉 and applying a sequence of quantum gates (from a finite set). EPR pairs can be prepared from the state |0〉|0〉 using a Hadamard gate and a controlled-NOT gate. A von Neumann measurement in the Fourier basis can be efficiently realized by applying an inverse quantum Fourier transform and performing a von Neumann measurement in the standard computational basis, i.e. {|0〉, |1〉}. However many states and basis changes cannot be efficiently realized. This paper focusses on the generation of quantum states. For example, in [HMPEPC98], their improved frequency standard experiment requires the preparation of specific symmetric states on n qubits, where n is a parameter (number of ions). The algorithm we describe here will efficiently prepare the required symmetric state. This short paper will focus on the algorithm for generating the state, and will ignore issues related to errors and decoherence (for which the theory of fault-tolerant error-correction, or other stabilization methods, will apply). We do not have space to elaborate on the details of precision, but simple calculations that require O(log( ǫ )) extra space and polylog( ǫ ) elementary operations allow us to generate any state with fidelity at least 1− ǫ.