Quantum analog-to-digital and digital-to-analog conversion

An algorithm is proposed which transfers the quantum information of a wave function (analog signal) into a register of qubits (digital signal) such that $n$ qubits describe the amplitudes and phases of $2^n$ points of a sufficiently smooth wave function. We assume that the continuous degree of freedom couples to one or more qubits of a quantum register via a Jaynes-Cummings Hamiltonian and that we have universal quantum computation capabilities on the register as well as the possibility to perform bang-bang control on the qubits. The transfer of information is mainly based on the application of the quantum phase estimation algorithm in both directions. Here, the running time increases exponentially with the number of qubits. We pose it as an open question which interactions would allow polynomial running time. One example would be interactions which enable squeezing operations.

Figure 1

The function $\mid g\left(x\right){\mid }^2$ for $n=6$.