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.
Highlights
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
We develop a framework for the resource theory of magic quantum channels, based on qudit systems with odd prime dimension d
We apply our magic measures for quantum channels in order to evaluate the magic cost of quantum channels, and we explore further applications in quantum gate synthesis
Summary
One of the main obstacles to physical realizations of quantum computation is decoherence that occurs during the execution of quantum algorithms. The addition of non-stabilizer quantum resources, such as non-stabilizer operations, can lead to universal quantum computation [12]. One solution for the above scenario is to implement a non-stabilizer operation via state injection [100] of so-called “magic states,” which are costly to prepare via magic state distillation [12] (see [11, 19, 21, 45, 46, 53, 56]). Since a key step of fault-tolerant quantum computing is to implement non-stabilizer operations, a natural and fundamental problem is to quantify the non-stabilizerness 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
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