A practical quantum computer must not merely store information, but also process it. To prevent errors introduced by noise from multiplying and spreading, a fault-tolerant computational architecture is required. Current experiments are taking the first steps toward noise-resilient logical qubits. But to convert these quantum devices from memories to processors, it is necessary to specify how a universal set of gates is performed on them. The leading proposals for doing so, such as magic-state distillation and colour-code techniques, have high resource demands. Alternative schemes, such as those that use high-dimensional quantum codes in a modular architecture, have potential benefits, but need to be explored further.

Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of stabilizer rank, which for a pure state ψ is defined to be the smallest integer χ such that ψ is a superposition of χ stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of χ ∼ 10 6 stabilizer states and sample from the full n-bit output distribution, improving on previous simulations which used ∼ 10 3 stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64 T gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components. CONTENTS

Motivated by their necessity for most fault-tolerant quantum computation schemes, we formulate a resource theory for magic states. We first show that robustness of magic is a well-behaved magic monotone that operationally quantifies the classical simulation overhead for a Gottesman-Knill type scheme using ancillary magic states. Our framework subsequently finds immediate application in the task of synthesizing non-Clifford gates using magic states. When magic states are interspersed with Clifford gates, Pauli measurements and stabilizer ancillas-the most general synthesis scenariothen the class of synthesizable unitaries is hard to characterize. Our techniques can place non-trivial lower bounds on the number of magic states required for implementing a given target unitary. Guided by these results we have found new and optimal examples of such synthesis.

The dynamics of a quantum system can be simulated using a quantum computer by breaking down the unitary into a quantum circuit of one and two qubit gates. The most established methods are the Trotter-Suzuki decompositions, for which rigorous bounds on the circuit size depend on the number of terms L in the system Hamiltonian and the size of the largest term in the Hamiltonian Λ. Consequently, Trotter-Suzuki is only practical for sparse Hamiltonians. Trotter-Suzuki is a deterministic compiler but it was recently shown that randomised compiling offers lower overheads. Here we present and analyse a randomised compiler for Hamiltonian simulation where gate probabilities are proportional to the strength of a corresponding term in the Hamiltonian. This approach requires a circuit size independent of L and Λ, but instead depending on λ the absolute sum of Hamiltonian strengths (the 1 norm). Therefore, it is especially suited to electronic structure Hamiltonians relevant to quantum chemistry. Considering propane, carbon dioxide and ethane, we observe speed-ups compared to standard Trotter-Suzuki of between 306× and 1591× for physically significant simulation times at precision 10 −3 . Performing phase estimation at chemical accuracy, we report that the savings are similar.Quantum computers could be used to mimic the dynamics of other quantum systems, providing a computational method to understand physical systems beyond the reach of classical supercomputers. A quantum computation is broken down into a discrete sequence of elementary one and two qubit gates. To simulate the continuous unitary evolution of the Schrödinger equation, an approximation must be made into a finite sequence of discrete gates. The precision of this approximation can be improved by using more gates. The standard approaches are the Trotter and higher order Suzuki decompositions [1][2][3]. In addition to simulating dynamics, we are often interested in learning the energy spectra of Hamiltonians. Assuming a good ansatz for the ground state, we can combine quantum simulation with phase estimation to find the energy of the ground state [4] and excited states [5][6][7]. For a molecule with unknown electronic configuration, this is called the electronic structure problem [8,9] and it is crucially important in chemistry and material science. However, electronic structure Hamiltonians contain a very large number of terms and unfortunately the gate count of Trotter-Suzkui increases with the number of terms. While the scaling is formally efficient, the required number of gates is impractically large. An alternative to Trotter-Suzkui without this scaling problem would therefore have significant applications.A recurrent theme in the literature is that stochastic noise can be less harmful than coherent noise [10,11], which hints that randomisation might be useful for washing out coherent errors in circuit design. Poulin et al [12] showed that randomness is especially useful in simulation of time-dependent Hamiltonians as it allows us to average out rapid Hamilto...

We propose families of protocols for magic state distillation -important components of fault tolerance schemes -for systems of odd prime dimension. Our protocols utilize quantum Reed-Muller codes with transversal non-Clifford gates. We find that, in higher dimensions, small and effective codes can be used that have no direct analogue in qubit (two-dimensional) systems. We present several concrete protocols, including schemes for three-dimensional (qutrit) and five-dimensional (ququint) systems. The five-dimensional protocol is, by many measures, the best magic state distillation scheme yet discovered. It excels both in terms of error threshold with respect to depolarising noise (36.3%) and the efficiency measure know as "yield", where, for a large region of parameters, it outperforms its qubit counterpart by many orders of magnitude. PACS numbers: 03.67.Pp,03.67.LxThe central challenge of implementing scalable quantum computing is to protect quantum systems against noise and decoherence while retaining the capacity to perform computation. Quantum error correction and fault tolerant techniques provide a solution to this problem, and a variety of constructions for fault tolerant quantum computation have been proposed [1][2][3][4]. In all these schemes, a delicate balance must be maintained between coherently manipulating the encoded system while preserving the protected subspace and prohibiting the proliferation of errors. For example, for schemes built on stabilizer codes [5] transversal gates have the desired properties, while in topological systems, topologically protected braiding operations [2] provide the logical gates. While much work in quantum computation has focussed upon qubits (twolevel systems), it is known that for any prime d, effective codes exist for storing d-level quantum systems [5][6][7]. Thus qudit systems are also candidates for scalable fault tolerant quantum computation.In many approaches, the protected unitary gates are a subset of the so-called Clifford group. The stabilizer operations (comprising Clifford unitaries as well as preparation and measurements in the computational basis) are known to be efficiently classically simulatable [5,6,8], and on their own are not universal for quantum computation. Furthermore, several theorems have shown [9-12] that, in general, there is a tension between providing protection against generic noise and achieving universal quantum computing.Despite these obstacles, fault tolerant universal quantum computing is possible [1]. One particularly successful approach, known as state-injection, is to achieve universality by augmenting the fault tolerant operations with a supply of many copies of a suitable ancillary resource state. While methods for direct preparation of sufficiently noise-free protected resource states have been proposed [1], a particularly elegant solution can be provided by distillation techniques, where many noisy copies of a resource state can be distilled to arbitrary fidelity by using only error-protected operations, * Electronic address...