Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes

3 January 2013

Earl T. Campbell, Hussain Anwar, Dan E. Browne

Phys. Rev. X 2, 041021 (2012)

Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes

A quantum computer exploits the nonclassical aspects of quantum mechanics, but its extreme sensitivity to noise makes fault-tolerant techniques a must for it to operate reliably. A key component in high-threshold fault-tolerance schemes is the preparation of magic states, quantum states in a superposition of classical states, that are required to exploit quantum effects. However, the slightest of experimental imperfections results in the preparation of flawed magic states, unsuitable for immediate use in quantum computers.

Fortunately, many copies of flawed magic states can be distilled down to a smaller number of suitable magic states. This process of magic-state distillation, however, forms a bottleneck in implementations of the already proposed fault-tolerant techniques in terms of the impractically high overheads in resources such as memory and running time. Unless overcome, such overhead costs could consign quantum computers to history books as a theoretically possible, but practically infeasible, technology. Together with former UCL researcher Earl Campbell, Hussain Anwar and Dan Browne, members of the AMOPP group, propose in this paper that a shift of magic-state distillation from the current qubit paradigm (involving two-level systems) to a qudit one (involving d-level systems) may provide a way to overcome the overhead bottleneck.

Their starting point was curiosity: Can we achieve magic-state distillation in more complex systems? Quantum computing is usually conceived in terms of qubits, but this need not be the case. In fact, qudits can be the basis for a quantum computer, and we designed methods of magic-state distillation for all prime numbers d. To our surprise, by exploiting new insights into number-theoretic properties of qudit computers, some of our protocols offer substantial improvements over their better-known qubit cousins. In particular, five levels appear as the optimal case where our protocol outperforms all previously discovered protocols in all figures of merit. Notably, the potential resource savings in device memory could easily be on the order of a millionfold.

This work opens the door for the development of qudit-based fault-tolerance schemes based on magic-state distillation. These schemes may offer significant advantages over their qubit counterparts and hence bring the development of a scalable quantum computer closer to our grasp.