  Recent advances in laser interactions with coherent free electrons have
enabled to shape the electron's quantum state. Each electron becomes a
superposition of energy levels on an infinite quantized ladder, shown to
contain up to thousands of energy levels. We propose to utilize the quantum
nature of such laser-driven free electrons as a "synthetic Hilbert space" in
which we construct and control qudits (quantum digits). The question that
motivates our work is what qudit states can be accessed using electron-laser
interactions, and whether it is possible to implement any arbitrary quantum
gate. We find how to encode and manipulate free-electron qudit states, focusing
on dimensions which are powers of 2, where the qudit represents multiple qubits
implemented on the same single electron - algebraically separated, but
physically joined. As an example, we prove the possibility to fully control a
4-dimenisonal qudit, and reveal the steps required for full control over any
arbitrary dimension. Our work enriches the range of applications of free
electrons in microscopy and spectroscopy, offering a new platform for
continuous-variable quantum information.
