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.
In this paper, we extend the resource theory of magic to the channel case by considering completely stabilizer preserving operations (CSPOs) as free. We introduce and characterize the set of CSPO preserving and completely CSPO preserving superchannels. We quantify the magic of quantum channels by extending the generalized robustness and the min relative entropy of magic from the state to the channel domain and show that they bound the single-shot dynamical magic cost and distillation. We also provide analytical conditions for qubit interconversion under CSPOs and show that it is a linear progr…
Quantum resource theories are a powerful framework to characterize and quantify relevant quantum phenomena and identify processes that optimize their use for different tasks. Here, we define a resource measure for magic, the sought-after property in most fault-tolerant quantum computers. In contrast to previous literature, our formulation is based on bosonic codes, well-studied tools in continuous-variable quantum computation. Particularly, we use the Gottesman-Kitaev-Preskill code to represent multi-qubit states and consider the resource theory for the Wigner negativity. Our techniques are us…
The development of a framework for quantifying "non-stabiliserness" 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-stabiliserness" of quantum operations r…
Consumption of magic states promotes the stabilizer model of computation to universal quantum computation. Here, we propose three different classical algorithms for simulating such universal quantum circuits, and characterize them by establishing precise connections with a family of magic monotones. Our first simulator introduces a new class of quasiprobability distributions and connects its runtime to a generalized notion of negativity. We prove that this algorithm has significantly improved exponential scaling compared to all prior quasiprobability simulators for qubits. Our second simulator…
We study the effect of quantum memory in magic squares game when played in quantum domain. We consider different noisy quantum channels and analyze their influence on the magic squares quantum pseudo-telepathy game. We show that the probability of success can be used to distinguish the quantum channels. It is seen that the mean success probability decreases with increase of quantum noise. Where as the mean success probability increases with increase of quantum memory. It is also seen that the behaviour of amplitude damping and phase damping channels is similar. On the other hand, the behaviour…
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 …
We study the possibility of reversing an action of a quantum channel. Our principal objective is to find a specific channel that reverses as accurately as possible an action of a given quantum channel. To achieve this goal we use semidefinite programming. We show the benefits of our method using the quantum pseudo-telepathy Magic Square game with noise. Our strategy is to move the pseudo-telepathy region to higher noise values. We show that it is possible to reverse the action of a noise channel using semidefinite programming.
In the article we analyse how noisiness of quantum channels can influence the magic squares quantum pseudo-telepathy game. We show that the probability of success can be used to determine characteristics of quantum channels. Therefore the game deserves more careful study aiming at its implementation.
We present a scheme for magic state distillation using punctured polar codes. Our results build on some recent work by Bardet et al. (ISIT, 2016) who discovered that polar codes can be described algebraically as decreasing monomial codes. Using this powerful framework, we construct tri-orthogonal quantum codes (Bravyi et al., PRA, 2012) that can be used to distill magic states for the $T$ gate. An advantage of these codes is that they permit the use of the successive cancellation decoder whose time complexity scales as $O(N\log(N))$. We supplement this with numerical simulations for the erasur…
We present a simple and precise protocol for standard quantum teleportation of N-qubit state, considering the most general resource q-channel and Bell states. We find condition on these states for perfect teleportation and give explicitly the unitary transformation required to be done by Bob for achieving perfect teleportation. We discuss connection of our simple theory with the complicated related work on this subject and with character matrix, transformation, judgment and kernel operators defined in this context. We also prove that the magic basis discussed by Hill and Wootters [Phys. Rev. L…
In this work we aim at quantifying quantum channel output similarity. In order to achieve this, we introduce the notion of quantum channel superfidelity, which gives us an upper bound on the quantum channel fidelity. This quantity is expressed in a clear form using the Kraus representation of a quantum channel. As examples, we show potential applications of this quantity in the quantum control field.
Quantifying Nonstabilizerness of Codeword-Stabilized Codes
MIIT Key Laboratory of Complex-field Intelligent Exploration, School of Optics and Photonics, Beijing Institute of Technology（工业和信息化部复杂场智能探测重点实验室，光电学院，北京理工大学）
We present an alternative framework for quantifying the coherence of quantum channels, which contains three conditions: the faithfulness, nonincreasing under sets of all the incoherent superchannels and the additivity. Based on the characteristics of the coherence of quantum channels and the additivity of independent channels, our framework provides an easier way to certify whether a function is a bona fide coherence measure of quantum channels. Detailed example is given to illustrate the effectiveness and advantages of our framework.
Spreading of Magic Resource under Unitary Clifford Dynamics
King’s College London（伦敦国王学院）；University of Cambridge（剑桥大学）；Trinity College Dublin（都柏林三一学院）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）
Non-stabilizerness or magic resource characterizes the amount of non-Clifford operations needed to prepare quantum states. It is a crucial resource for quantum computing and a necessary condition for quantum advantage. However, quantifying magic resource beyond a few qubits has been a major challenge. Here, we introduce efficient measures of magic resource for pure quantum states with a sampling cost that is independent of the number of qubits. Our method uses Bell measurements over two copies of a state, which we implement in experiment together with a cost-free error mitigation scheme. We sh…
The ideal realization of quantum teleportation relies on having access to a maximally entangled state; however, in practice, such an ideal state is typically not available and one can instead only realize an approximate teleportation. With this in mind, we present a method to quantify the performance of approximate teleportation when using an arbitrary resource state. More specifically, after framing the task of approximate teleportation as an optimization of a simulation error over one-way local operations and classical communication (LOCC) channels, we establish a semi-definite relaxation of…
Magic states can be used as a resource to circumvent the restrictions due to stabilizer-preserving operations, and magic-state conversion has not been studied in the single-copy regime thus far. Here we solve the question of whether a stabilizer-preserving quantum operation exists that can convert between two given magic states in the single-shot regime. We first phrase this question as a feasibility problem for a semi-definite program (SDP), which provides a procedure for constructing a stabilizer-preserving quantum operation (free channel) if it exists. Then we employ a variant of the Farkas…
Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels -- that is, not completely positive -- can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simu…
In this paper we introduce a way to quantify the noise level associated to a given quantum transformation. The key mechanism lying at the heart of the proposal is "noise addition": in other words we compute the amount of extra noise we need to add to the system, through convex combination with a reference noisy map or by reiterative applications of the original map, before the resulting transformation becomes entanglement-breaking. We also introduce the notion of entanglement-breaking channels of order n (i.e. maps which become entanglement-breaking after n iterations), and the associated noti…
The coupled-channels density-matrix technique for nuclear reaction dynamics, which is based on the Liouville-von Neumann equation with Lindblad dissipative terms, is developed with the inclusion of full angular momentum couplings. It allows a quantitative study of the role and importance of quantum decoherence in nuclear scattering. Formulae of asymptotic observables that can reveal effects of quantum decoherence are given. A method for extracting energy-resolved scattering information from the time-dependent density matrix is introduced. As an example, model calculations are carried out for t…
Quantum channels breaking entanglement, incompatibility, or nonlocality are defined as such because they are not useful for entanglement-based, one-sided device-independent, or device-independent quantum information processing, respectively. Here, we show that such breaking channels are related to complementary tests of macrorealism i.e., temporal separability, channel unsteerability, temporal unsteerability, and the temporal Bell inequality. To demonstrate this we first define a steerability-breaking channel, which is conceptually similar to entanglement and nonlocality-breaking channels and …
We introduce a non-Markovianity measure for continuous variable open quantum systems based on the idea put forward in H.-P. Breuer et al., Phys. Rev. Lett.\textbf{103}, 210401 (2009), i.e., by quantifying the flow of information from the environment back to the open system. Instead of the trace distance we use here the fidelity to assess distinguishability of quantum states. We employ our measure to evaluate non-Markovianity of two paradigmatic Gaussian channels: the purely damping channel and the quantum Brownian motion channel with Ohmic environment. We consider different classes of Gaussian…
Federal University of Rio Grande do Norte（北里奥格兰德联邦大学）；International Institute of Physics, Federal University of Rio Grande do Norte（北里奥格兰德联邦大学国际物理研究所）；Center for Macroscopic Quantum States bigQ, Department of Physics, Technical University of Denmark（丹麦技术大学宏观量子态中心bigQ）
Entirely nonlocal quantum magic without entanglement
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Beijing Key Laboratory of Fault-Tolerant Quantum Computing, Beijing Academy of Quantum Information Sciences（北京容错量子计算北京市重点实验室，北京量子信息科学研究院）
Nonstabilizerness, also known as magic, quantifies the number of non-Clifford operations needed in order to prepare a quantum state. As typical measures either involve minimization procedures or a computational cost exponential in the number of qubits $N$, it is notoriously hard to characterize for many-body states. In this work, we show that nonstabilizerness, as quantified by the recently introduced Stabilizer Rényi Entropies (SREs), can be computed efficiently for matrix product states (MPSs). Specifically, given an MPS of bond dimension $χ$ and integer Rényi index $n>1$, we show that the S…
Classical simulation of a quantum circuit with noisy magic inputs
Graduate School of Quantum Science and Technology, Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院量子科学与技术研究生院）；Department of Physics, Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院物理系）
The Gaussian states are essential ingredients in many tasks of quantum information processing. The presence of the noises imposes limitations on achieving these quantum protocols. Therefore, examining the evolution of quantum entanglement and quantum correlations under the coherence of Gaussian states in noisy channels is of paramount importance. In this paper, we propose and analyze a scheme that aims to specify and examine the dynamic evolution of the quantum correlations in two-modes Gaussian states submitted to the influence of the Gaussian thermal environment. We describe the time evoluti…
We quantify the dynamical quantum resource in the resource non-increasing (RNI) framework, namely, the free dynamical resource is defined by the channels that cannot increase the static resourcefulness of any input state. We present two kinds of approaches to quantifying the dynamical resource, the distance measures and the maximal increasing static resource (MISR). As a demonstration, we quantify the dynamical total coherence with our presented measures. It is shown that the distance based measures have good operational interpretation through quantum processing tasks and can be numerically ca…
Experimental demonstration of non-local magic in a superconducting quantum processor
Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学）；Scuola Superiore Meridionale（南方高等学院）；Istituto Nazionale di Fisica Nucleare (INFN)（意大利国家核物理研究所）
Based on the nonincreasing property of quantum coherence via skew information under incoherent completely positive and trace-preserving maps, we propose a non-Markovianity measure for open quantum processes. As applications, by applying the proposed measure to some typical noisy channels, we find that it is equivalent to the three previous measures of non-Markovianity for phase damping and amplitude damping channels, i.e., the measures based on the quantum trace distance, dynamical divisibility, and quantum mutual information. For the random unitary channel, it is equivalent to the non-Markovi…
