Three-Dimensional Color Code Thresholds via Statistical-Mechanical Mapping.

Three-dimensional (3D) color codes have advantages for fault-tolerant quantum computing, such as protected quantum gates with relatively low overhead and robustness against imperfect measurement of error syndromes. Here we investigate the storage threshold error rates for bit-flip and phase-flip noise in the 3D color code (3DCC) on the body-centered cubic lattice, assuming perfect syndrome measurements. In particular, by exploiting a connection between error correction and statistical mechanics, we estimate the threshold for 1D stringlike and 2D sheetlike logical operators to be p_{3DCC}^{(1)}≃1.9% and p_{3DCC}^{(2)}≃27.6%. We obtain these results by using parallel tempering Monte Carlo simulations to study the disorder-temperature phase diagrams of two new 3D statistical-mechanical models: the four- and six-body random coupling Ising models.

Elucidating reaction mechanisms on quantum computers.

With rapid recent advances in quantum technology, we are close to the threshold of quantum devices whose computational powers can exceed those of classical supercomputers. Here, we show that a quantum computer can be used to elucidate reaction mechanisms in complex chemical systems, using the open problem of biological nitrogen fixation in nitrogenase as an example. We discuss how quantum computers can augment classical computer simulations used to probe these reaction mechanisms, to significantly increase their accuracy and enable hitherto intractable simulations. Our resource estimates show that, even when taking into account the substantial overhead of quantum error correction, and the need to compile into discrete gate sets, the necessary computations can be performed in reasonable time on small quantum computers. Our results demonstrate that quantum computers will be able to tackle important problems in chemistry without requiring exorbitant resources.

Efficient synthesis of universal repeat-until-success quantum circuits.

Recently it was shown that the resources required to implement unitary operations on a quantum computer can be reduced by using probabilistic quantum circuits called repeat-until-success (RUS) circuits. However, the previously best-known algorithm to synthesize a RUS circuit for a given target unitary requires exponential classical runtime. We present a probabilistically polynomial-time algorithm to synthesize a RUS circuit to approximate any given single-qubit unitary to precision ϵ over the Clifford+T basis. Surprisingly, the T count of the synthesized RUS circuit surpasses the theoretical lower bound of 3 log_{2}(1/ϵ) that holds for purely unitary single-qubit circuit decomposition. By taking advantage of measurement and an ancilla qubit, RUS circuits achieve an expected T count of 1.15 log_{2}(1/ϵ) for single-qubit z rotations. Our method leverages the fact that the set of unitaries implementable by RUS protocols has a higher density in the space of all unitaries compared to the density of purely unitary implementations.

Asymptotically optimal topological quantum compiling.

We address the problem of compiling quantum operations into braid representations for non-Abelian quasiparticles described by the Fibonacci anyon model. We classify the single-qubit unitaries that can be represented exactly by Fibonacci anyon braids and use the classification to develop a probabilistically polynomial algorithm that approximates any given single-qubit unitary to a desired precision by an asymptotically depth-optimal braid pattern. We extend our algorithm in two directions: to produce braids that allow only single-strand movement, called weaves, and to produce depth-optimal approximations of two-qubit gates. Our compiled braid patterns have depths that are 20 to 1000 times shorter than those output by prior state-of-the-art methods, for precisions ranging between 10(-10) and 10(-30).

Resource-optimal single-qubit quantum circuits.

Determining the optimal implementation of a quantum gate is critical for designing a quantum computer. We consider the crucial task of efficiently decomposing a general single-qubit quantum gate into a sequence of fault-tolerant quantum operations. For a given single-qubit circuit, we construct an optimal gate sequence consisting of fault-tolerant Hadamard (H) and π/8 rotations (T). Our scheme is based on a novel canonical form for single-qubit quantum circuits and the corresponding rules for exactly reducing a general single-qubit circuit to our canonical form. The result is optimal in the number of T gates. We demonstrate that a precomputed epsilon net of canonical circuits in combination with our scheme lowers the depth of approximation circuits by up to 3 orders of magnitude compared to previously reported results.