Quantifying the magic of quantum channels

To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum "magic" or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension $d$, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.Comment: 44 pages, 7 figures; v2 close to published versio

Quantifying magic for multi-qubit operations

The development of a framework for quantifying ‘non-stabilizerness’ of quantum operations is motivated by the magic state model of fault-tolerant quantum computation and by the need to estimate classical simulation cost for noisy intermediate-scale quantum (NISQ) devices. The robustness of magic was recently proposed as a well-behaved magic monotone for multi-qubit states and quantifies the simulation overhead of circuits composed of Clifford + T gates, or circuits using other gates from the Clifford hierarchy. Here we present a general theory of the ‘non-stabilizerness’ of quantum operations rather than states, which are useful for classical simulation of more general circuits. We introduce two magic monotones, called channel robustness and magic capacity, which are well-defined for general n-qubit channels and treat all stabilizer-preserving CPTP maps as free operations. We present two complementary Monte Carlo-type classical simulation algorithms with sample complexity given by these quantities and provide examples of channels where the complexity of our algorithms is exponentially better than previously known simulators. We present additional techniques that ease the difficulty of calculating our monotones for special classes of channels

Benchmarking one-shot distillation in general quantum resource theories

We study the one-shot distillation of general quantum resources, providing a unified quantitative description of the maximal fidelity achievable in this task, and revealing similarities shared by broad classes of resources. We establish fundamental quantitative and qualitative limitations on resource distillation applicable to all convex resource theories. We show that every convex quantum resource theory admits a meaningful notion of a pure maximally resourceful state which maximizes several monotones of operational relevance and finds use in distillation. We endow the generalized robustness measure with an operational meaning as an exact quantifier of performance in distilling such maximal states in many classes of resources including bi- and multipartite entanglement, multi-level coherence, as well as the whole family of affine resource theories, which encompasses important examples such as asymmetry, coherence, and thermodynamics.Comment: 8+5 pages, 1 figure. v3: fixed (inconsequential) error in Lemma 1

Quantifying the resource content of quantum channels: An operational approach

We propose a general method to operationally quantify the resourcefulness of quantum channels via channel discrimination, an important information processing task. A main result is that the maximum success probability of distinguishing a given channel from the set of free channels by free probe states is exactly characterized by the resource generating power, i.e. the maximum amount of resource produced by the action of the channel, given by the trace distance to the set of free states. We apply this framework to the resource theory of quantum coherence, as an informative example. The general results can also be easily applied to other resource theories such as entanglement, magic states, and asymmetry.Comment: v2. 9 pages, new references are added v1. 8 pages, no figure

Optimized Surface Code Communication in Superconducting Quantum Computers

Quantum computing (QC) is at the cusp of a revolution. Machines with 100 quantum bits (qubits) are anticipated to be operational by 2020 [googlemachine,gambetta2015building], and several-hundred-qubit machines are around the corner. Machines of this scale have the capacity to demonstrate quantum supremacy, the tipping point where QC is faster than the fastest classical alternative for a particular problem. Because error correction techniques will be central to QC and will be the most expensive component of quantum computation, choosing the lowest-overhead error correction scheme is critical to overall QC success. This paper evaluates two established quantum error correction codes---planar and double-defect surface codes---using a set of compilation, scheduling and network simulation tools. In considering scalable methods for optimizing both codes, we do so in the context of a full microarchitectural and compiler analysis. Contrary to previous predictions, we find that the simpler planar codes are sometimes more favorable for implementation on superconducting quantum computers, especially under conditions of high communication congestion.Comment: 14 pages, 9 figures, The 50th Annual IEEE/ACM International Symposium on Microarchitectur

Advancing classical simulators by measuring the magic of quantum computation

Stabiliser operations and state preparations are efficiently simulable by classical computers. Stabiliser circuits play a key role in quantum error correction and fault-tolerance, and can be promoted to universal quantum computation by the addition of "magic" resource states or non-Clifford gates. It is believed that classically simulating stabiliser circuits supplemented by magic must incur a performance overhead scaling exponentially with the amount of magic. Early simulation methods were limited to circuits with very few Clifford gates, but the need to simulate larger quantum circuits has motivated the development of new methods with reduced overhead. A common theme is that algorithm performance can often be linked to quantifiers of computational resource known as magic monotones. Previous methods have typically been restricted to specific types of circuit, such as unitary or gadgetised circuits. In this thesis we develop a framework for quantifying the resourcefulness of general qubit quantum circuits, and present improved classical simulation methods. We first introduce a family of magic state monotones that reveal a previously unknown formal connection between stabiliser rank and quasiprobability methods. We extend this family by presenting channel monotones that measure the magic of general qubit quantum operations. Next, we introduce a suite of classical algorithms for simulating quantum circuits, which improve on and extend previous methods. Each classical simulator has performance quantified by a related resource measure. We extend the stabiliser rank simulation method to admit mixed states and noisy operations, and refine a previously known sparsification method to yield improved performance. We present a generalisation of quasiprobability sampling techniques with significantly reduced exponential scaling. Finally, we evaluate the simulation cost per use for practically relevant quantum operations, and illustrate how to use our framework to realistically estimate resource costs for particular ideal or noisy quantum circuit instances