Abstract
Magic state distillation (MSD) is a purification protocol that plays an important role in fault-tolerant quantum computation. Repeated iteration of the steps of an MSD protocol generates pure single non-stabilizer states, or magic states, from multiple copies of a mixed resource state using stabilizer operations only. Thus mixed resource states promote the stabilizer operations to full universality. MSD was introduced for qubit-based quantum computation, but little has been known concerning MSD in higher-dimensional qudit-based computation. Here, we describe a general approach for studying MSD in higher dimensions. We use it to investigate the features of a qutrit MSD protocol based on the five-qutrit stabilizer code. We show that this protocol distils non-stabilizer magic states, and identify two types of states that are attractors of this iteration map. Finally, we show how these states may be converted, via stabilizer circuits alone, into a state suitable for state-injected implementation of a non-Clifford phase gate, enabling non-Clifford unitary computation.
Highlights
Quantum computers hold the promise of solving certain computational tasks at an exponentially faster rate than is currently believed to be possible with classical computers [1]
In this paper we have demonstrated that magic state distillation protocols do exist for higher dimensional systems
We have shown that the five qutrit code [[5, 1, 3]]3 is capable of distilling the qutrit Hadamard eigenstates, implying that, in analogy to the qubit case, there exist H-type qutrit magic states
Summary
Quantum computers hold the promise of solving certain computational tasks at an exponentially faster rate than is currently believed to be possible with classical computers [1]. Stabilizer operations consist of a family of unitary circuits known as the Clifford group, preparation of 0⟩ state and measurement in the computation basis. We demonstrate that magic state distillation can be achieved in a qutrit system, and find both similarities and differences with previously known MSD protocols for qubits. They are distilled up to an error threshold of 23.3% This family is generic in that all quantum states can be mapped into the Hadamard plane by random application of the Hadamard unitaries, a process known as Hadamard-twirling. The second family contain four eigenstates of the qutrit Hadamardsquared operator, but lies within a degenerate eigenspace of this operator, and so is not uniquely defined by it Under depolarizing noise, they are distilled up to an error threshold of 34.5%.
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