Vadym Kliuchnikov

Asymptotically optimal approximation of single qubit unitaries by Clifford and T circuits using a constant number of ancillary qubits.

Decomposing unitaries into a sequence of elementary operations is at the core of quantum computing. Information theoretic arguments show that approximating a random unitary with precision ε requires Ω(log(1/ε)) gates. Prior to our work, the state of the art in approximating a single qubit unitary included the Solovay-Kitaev algorithm that requires O(log(3+δ)(1/ε)) gates and does not use ancillae and the phase kickback approach that requires O(log(2)(1/ε)loglog(1/ε)) gates but uses O(log(2)(1/ε)) ancillae. Both algorithms feature upper bounds that are far from the information theoretic lower bound. In this Letter, we report an algorithm that saturates the lower bound, and as such it guarantees asymptotic optimality. In particular, we present an algorithm for building a circuit that approximates single qubit unitaries with precision ε using O(log(1/ε)) Clifford and T gates and employing up to two ancillary qubits. We connect the unitary approximation problem to the problem of constructing solutions corresponding to Lagranges four-square theorem, and thereby develop an algorithm for computing an approximating circuit using an average of O(log(2)(1/ε)loglog(1/ε)) operations with integers.

Practical Approximation of Single-Qubit Unitaries by Single-Qubit Quantum Clifford and T Circuits

We present an algorithm, along with its implementation that finds T-optimal approximations of single-qubit Z-rotations using quantum circuits consisting of Clifford and T gates. Our algorithm is capable of handling errors in approximation down to size 10-15, resulting in the optimal single-qubit circuit designs required for implementation of scalable quantum algorithms. Our implementation along with the experimental results are available in the public domain.

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).

Floating point representations in quantum circuit synthesis

We provide a non-deterministic quantum protocol that approximates the single qubit rotations Rx(2ϕ21ϕ22) using Rx(2ϕ1) and Rx(2ϕ2) and a constant number of Clifford and T operations. We then use this method to construct a ‘floating point’ implementation of a small rotation wherein we use the aforementioned method to construct the exponent part of the rotation and also to combine it with a mantissa. This causes the cost of the synthesis to depend more strongly on the relative (rather than absolute) precision required. We analyze the mean and variance of the T-count required to use our techniques and provide new lower bounds for the T-count for ancilla free synthesis of small single-qubit axial rotations. We further show that our techniques can use ancillas to beat these lower bounds with high probability. We also discuss the T-depth of our method and see that the vast majority of the cost of the resultant circuits can be shifted to parallel computation paths.

Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets

We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov, and Mosca [Quantum Inf. Comput. 13(7,8), 607–630 (2013)]. Their algorithm takes as input an exactly synthesizable single-qubit unitary—one which can be expressed without error as a product of Clifford and T gates—and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer n, we consider the “Clifford-cyclotomic” gate set consisting of the Clifford group plus a z-rotation by π/n. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of π/n z-rotations. For the Clifford+T case n = 4, the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring Z[e^(i π/n), 1/2]. We prove that this characterization holds for a handful of other small values of n but the fraction of positive even integers for which it fails to hold is 100%.

Q#: Enabling Scalable Quantum Computing and Development with a High-level DSL

Quantum computing exploits quantum phenomena such as superposition and entanglement to realize a form of parallelism that is not available to traditional computing. It offers the potential of significant computational speed-ups in quantum chemistry, materials science, cryptography, and machine learning. The dominant approach to programming quantum computers is to provide an existing high-level language with libraries that allow for the expression of quantum programs. This approach can permit computations that are meaningless in a quantum context; prohibits succint expression of interaction between classical and quantum logic; and does not provide important constructs that are required for quantum programming. We present Q#, a quantum-focused domain-specific language explicitly designed to correctly, clearly and completely express quantum algorithms. Q# provides a type system; a tightly constrained environment to safely interleave classical and quantum computations; specialized syntax; symbolic code manipulation to automatically generate correct transformations of quantum operations; and powerful functional constructs which aid composition.

Efficient topological compilation for a weakly integral anyonic model

A class of anyonic models for universal quantum computation based on weakly-integral anyons has been recently proposed. While universal set of gates cannot be obtained in this context by anyon braiding alone, designing a certain type of sector charge measurement provides universality. In this paper we develop a compilation algorithm to approximate arbitrary

Optimization of Clifford circuits

n

Fast and efficient exact synthesis of single-qubit unitaries generated by clifford and T gates

-qutrit unitaries with asymptotically efficient circuits over the metaplectic anyon model. One flavor of our algorithm produces efficient circuits with upper complexity bound asymptotically in

An algorithm for the T-count

O(3^{2\,n} \, \log{1/\varepsilon})