Quantifying the magic of quantum channels

Abstract

Keywords: resource theory of magic quantum channels, completely positive-Wigner-preserving quantum operations, thauma of a quantum channel, distillable magic, classical simulation of noisy quantum circuits Abstract 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.states also motivates the resource theory of magic states [15][16][17][18][19][20], where the free operations are the SOs and the free states are the stabilizer states (abbreviated as 'Stab'). On the other hand, since a key step of fault-tolerant quantum computing is to implement non-SOs, a natural and fundamental problem is to quantify the nonstabilizerness or 'magic' of quantum operations. As we are at the stage of noisy intermediate-scale quantum (NISQ) technology, a resource theory of magic for noisy quantum operations is desirable both to exploit the power and to identify the limitations of NISQ devices in fault-tolerant quantum computation (FTQC). Overview of resultsIn this paper, we develop a framework for the resource theory of magic quantum channels, based on qudit systems with odd prime dimension d. Related work on this topic has appeared recently [21], but the set of free operations that we take in our resource theory is larger, given by the completely positive-Wigner-preserving (CPWP) perations as we detail below. We note here that d-level FTQC based on qudits with prime d is of considerable interest for both theoretical and practical purposes [22][23][24][25][26].Our paper is structured as follows.• In section 2, we first review the stabilizer formalism [4] and the discrete Wigner function [27][28][29]. We further review various magic m...

Application of the Resource Theory of Channels to Communication Scenarios

“…The resource theoretic framework has been applied to various kinds of quantities [3][4][5][6][7][8], as well as employed to extract common features shared by a wide class of resources [9][10][11][12][13][14][15][16][17][18][19][20][21]. Recently, the framework has been extended beyond the consideration of static resources attributed to quantum states to dynamic resources attributed to quantum measurements and channels, and it has been under active investigation [15][16][17][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].…”

“…We introduce resource theory for communication, a resource theory of channels relevant to communication via quantum channels, in which we choose the set of constant channels, the useless channels for communication, as free resources. Unlike many of the resource theories of channels with underlying state theories [15,27,29,[31][32][33][34][35], our setting is not equipped with any theory of state, making the consideration of the resource theories of channels crucial. We introduce generalized robustness for communication and another equivalent quantity, max-relative entropy for communication, as resource quantifiers, and discuss resource transformations under free operations, for which we consider the maximal set of superchannels that map constant channels to constant channels.…”

Contextual bound states for qudit magic state distillation

“…We have yet to understand the implication of our result for contextuality and Wigner negativity as a resource [14,27,[37][38][39]. In particular, what is the computational power of a Clifford computer with access to a large, but finite, supply of states which exhibit a small violation of a single non-contextuality inequality?…”

Geometric Rényi Divergence and its Applications in Quantum Channel Capacities

“…The capacities in other communication scenarios are still under investigation. Some recent works (e.g [3,4]) also extend the use of quantum channels to generate quantum resources such as magic state, a key ingredient for fault-tolerant quantum computation. The capability of a channel to generate such resource is thus characterized by its corresponding generation capacity.…”