Magic State Distillation Research Articles

Transversality Versus Universality for Additive Quantum Codes

Logic gates can be performed on data encoded in quantum code blocks such that errors introduced by faulty gates can be corrected. The important class of transversal gates acts bitwise between corresponding qubits of code blocks and thus limits error propagation. If any quantum gate could be implemented using transversal gates, the set would be universal. We study the structure of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF</i> (4)-additive quantum codes and prove that no universal set of transversal logic gates exists for these codes. This result is in stark contrast with the classical case, where universal transversal gate sets exist, and strongly supports the idea that additional quantum techniques, based, for example, on quantum teleportation or magic state distillation, are necessary to achieve universal fault-tolerant quantum computation on additive codes.

Noise thresholds for higher-dimensional systems using the discrete Wigner function

For a quantum computer acting on $d$-dimensional systems, we analyze the computational power of circuits wherein stabilizer operations are perfect, and we allow access to imperfect nonstabilizer states or operations. If the noise rate affecting the nonstabilizer resource is sufficiently high, then these states and operations can become simulable in the sense of the Gottesman-Knill theorem, reducing the overall power of the circuit to no better than classical. In this paper we find the depolarizing noise rate at which this happens and consequently the most robust nonstabilizer states and non-Clifford gates. In doing so, we make use of the discrete Wigner function and derive facets of the so-called qudit Clifford polytope, i.e., the inequalities defining the convex hull of all qudit Clifford gates. Our results for robust states are provably optimal. For robust gates we find a critical noise rate that, as dimension increases, rapidly approaches the the theoretical optimum of 100%. Some connections with the question of qudit magic state distillation are discussed.

Universal quantum computation in a semiconductor quantum wire network

Universal quantum computation (UQC) using Majorana fermions on a 2D topological superconducting (TS) medium remains an outstanding open problem. This is because the quantum gate set that can be generated by braiding of the Majorana fermions does not include \emph{any} two-qubit gate and also the single-qubit $\pi/8$ phase gate. In principle, it is possible to create these crucial extra gates using quantum interference of Majorana fermion currents. However, it is not clear if the motion of the various order parameter defects (vortices, domain walls, \emph{etc.}), to which the Majorana fermions are bound in a TS medium, can be quantum coherent. We show that these obstacles can be overcome using a semiconductor quantum wire network in the vicinity of an $s$-wave superconductor, by constructing topologically protected two-qubit gates and any arbitrary single-qubit phase gate in a topologically unprotected manner, which can be error corrected using magic state distillation. Thus our strategy, using a judicious combination of topologically protected and unprotected gate operations, realizes UQC on a quantum wire network with a remarkably high error threshold of $0.14$ as compared to $10^{-3}$ to $10^{-4}$ in ordinary unprotected quantum computation.

Implementing Arbitrary Phase Gates with Ising Anyons

Ising-type non-Abelian anyons are likely to occur in a number of physical systems, including quantum Hall systems, where recent experiments support their existence. In general, non-Abelian anyons may be utilized to provide a topologically error-protected medium for quantum information processing. However, the topologically protected operations that may be obtained by braiding and measuring topological charge of Ising anyons are precisely the Clifford gates, which are not computationally universal. The Clifford gate set can be made universal by supplementing it with single-qubit pi/8-phase gates. We propose a method of implementing arbitrary single-qubit phase gates for Ising anyons by running a current of anyons with interfering paths around computational anyons.

Bound States for Magic State Distillation in Fault-Tolerant Quantum Computation

Magic state distillation is an important primitive in fault-tolerant quantum computation. The magic states are pure nonstabilizer states which can be distilled from certain mixed nonstabilizer states via Clifford group operations alone. Because of the Gottesman-Knill theorem, mixtures of Pauli eigenstates are not expected to be magic state distillable, but it has been an open question whether all mixed states outside this set may be distilled. In this Letter we show that, when resources are finitely limited, nondistillable states exist outside the stabilizer octahedron. In analogy with the bound entangled states, which arise in entanglement theory, we call such states bound states for magic state distillation.