Earl T. Campbell

Qutrit magic state distillation

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.

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.

Enhanced fault-tolerant quantum computing in d-level systems.

Error-correcting codes protect quantum information and form the basis of fault-tolerant quantum computing. Leading proposals for fault-tolerant quantum computation require codes with an exceedingly rare property, a transversal non-Clifford gate. Codes with the desired property are presented for d-level qudit systems with prime d. The codes use n=d-1 qudits and can detect up to ∼d/3 errors. We quantify the performance of these codes for one approach to quantum computation known as magic-state distillation. Unlike prior work, we find performance is always enhanced by increasing d.

Measurement-based entanglement under conditions of extreme photon loss.

The act of measuring optical emissions from two remote qubits can entangle them. By demanding that a photon from each qubit reaches the detectors, one can ensure that no photon was lost. But retaining both photons is rare when loss rates are high, as in Moehring et al. where 30 successes occurred per 10(9) attempts. We describe a means to exploit the low grade entanglement heralded by the detection of a lone photon: A subsequent perfect operation is quickly achieved by consuming this noisy resource. We require only two qubits per node, and can tolerate both path length variation and loss asymmetry. The impact of photon loss upon the failure rate is then linear; realistic high-loss devices can gain orders of magnitude in performance and thus support quantum computing.

Cellular-automaton decoders for topological quantum memories

A new error correction method for quantum computing memories is based on local computing elements. Michael Herold from the Freie Universitat Berlin in Germany, with colleagues in Germany, Denmark and the UK, sought to address the challenge of maintaining information stored in topological quantum memories. Without a stable memory, delicate quantum states can decay quickly, introducing errors in stored information. Error correction is an important process in stabilizing topological memories, but was previously conceived as a system-wide process. The proposed practical error correction mechanism relies on parallel cellular operations within the topological quantum memory, so that the local operations replace the need for a complex system-wide scheme. The concept has the further benefit of being compatible with classical hardware, and it is easily scalable.

Application of a Resource Theory for Magic States to Fault-Tolerant Quantum Computing

Motivated by their necessity for most fault-tolerant quantum computation schemes, we formulate a resource theory for magic states. First, we 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 scenario-then the class of synthesizable unitaries is hard to characterize. Our techniques can place nontrivial 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.

Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost

The standard approach to fault-tolerant quantum computation is to store information in a quantum error correction code, such as the surface code, and process information using a strategy that can be summarized as distill then synthesize. In the distill step, one performs several rounds of distillation to create high-fidelity logical qubits in a magic state. Each such magic state provides one good T gate. In the synthesize step, one seeks the optimal decomposition of an algorithm into a sequence of many T gates interleaved with Clifford gates. This gate-synthesis problem is well understood for multiqubit gates that do not use any Hadamards. We present an in-depth analysis of a unified framework that realizes one round of distillation and multiqubit gate synthesis in a single step. We call these synthillation protocols, and show they lead to a large reduction in resource overheads. This is because synthillation can implement a general class of circuits using the same number of T states as gate synthesis, yet with the benefit of quadratic error suppression. This general class includes all circuits primarily dominated by control-control-Z gates, such as adders and modular exponentiation routines used in Shor’s algorithm. Therefore, synthillation removes the need for a costly round of magic state distillation. We also present several additional results on the multiqubit gate-synthesis problem. We provide an efficient algorithm for synthesizing unitaries with the same worst-case resource scaling as optimal solutions. For the special case of synthesizing controlled unitaries, our techniques are not just efficient but exactly optimal. We observe that the gate-synthesis cost, measured by T count, is often strictly subadditive. Numerous explicit applications of our techniques are also presented.

Roads towards fault-tolerant universal quantum computation

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.

Fast decoders for qudit topological codes

Qudit toric codes are a natural higher-dimensional generalization of the wellstudied qubit toric code. However, standard methods for error correction of the qubit toric code are not applicable to them. Novel decoders are needed. In this paper we introduce two renormalization group decoders for qudit codes and analyse their error correction thresholds and efficiency. The first decoder is a generalization of a ‘hard-decisions’ decoder due to Bravyi and Haah (arXiv:1112.3252). We modify this decoder to overcome a percolation effect which limits its threshold performance for many-level quantum systems. The second decoder is a generalization of a ‘soft-decisions’ decoder due to Poulin and Duclos-Cianci (2010 Phys. Rev. Lett. 104 050504), with a small cell size to optimize the efficiency of implementation in the high dimensional case. In each case, we estimate thresholds for the uncorrelated bit-flip error model and provide a comparative analysis of the performance of both these approaches to error correction of qudit toric codes.

An efficient magic state approach to small angle rotations

Standard error correction techniques only provide a quantum memory and need extra gadgets to perform computation. Central to quantum algorithms are small angle rotations, which can be fault-tolerantly implemented given a supply of an unconventional species of magic state. We present a low-cost distillation routine for preparing these small angle magic states. Our protocol builds on the work of Duclos-Cianci and Poulin [Phys. Rev. A, 91, 042315 (2015)] by compressing their circuit. Additionally, we present a method of diluting magic states that reduces costs associated with very small angle rotations. We quantify performance by the expected number of noisy magic states consumed per rotation, and compare with other protocols. For modest size angles, our protocols offer a factor 24 improvement over the best known gate synthesis protocols and a factor 2 over the Duclos-Cianci and Poulin protocol. For very small angle rotations, the dilution protocol dramatically reduces costs, giving several orders magnitude improvement over competitors. There also exists an intermediary regime of small, but not very small, angles where our approach gives a marginal improvement over gate synthesis. We discuss how different performance metrics may alter these conclusions.