Abstract Magic states are the resource that allows quantum computers to attain an advantage over classical computers. This resource consists in the deviation from a property called stabilizerness which in turn implies that stabilizer circuits can be efficiently simulated on a classical computer. Without magic, no quantum computer can do anything that a classical computer cannot do. Given the importance of magic for quantum computation, it would be useful to have a method for measuring the amount of magic in a quantum state. In this work, we propose and experimentally demonstrate a protocol for measuring magic based on randomized measurements. Our experiments are carried out on two IBM Quantum Falcon processors. This protocol can provide a characterization of the effectiveness of a quantum hardware in producing states that cannot be effectively simulated on a classical computer. We show how from these measurements one can construct realistic noise models affecting the hardware.
Abstract Magic states are the resource that allows quantum computers to attain an advantage over classical computers. This resource consists in the deviation from a property called stabilizerness which in turn implies that stabilizer circuits can be efficiently simulated on a classical computer. Without magic, no quantum computer can do anything that a classical computer cannot do. Given the importance of magic for quantum computation, it would be useful to have a method for measuring the amount of magic in a quantum state. In this work, we propose and experimentally demonstrate a protocol for measuring magic based on randomized measurements. Our experiments are carried out on two IBM Quantum Falcon processors. This protocol can provide a characterization of the effectiveness of a quantum hardware in producing states that cannot be effectively simulated on a classical computer. We show how from these measurements one can construct realistic noise models affecting the hardware.
Nonstabilizerness 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 show the transition of classically simulable stabilizer states into intractable quantum states on the IonQ quantum computer.For applications, we efficiently distinguish stabilizer and nonstabilizer states with low measurement cost even in the presence of experimental noise.Further, we propose a variational quantum algorithm to maximize our measure via the shift rule.Our algorithm can be free of barren plateaus even for highly expressible variational circuits.Finally, we experimentally demonstrate a Bell-measurement protocol for the stabilizer Rényi entropy as well as the Wallach-Meyer entanglement measure.Our results pave the way to understanding the nonclassical power of quantum computers, quantum simulators, and quantum many-body systems.
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.
Quantum computers promise to solve computational problems significantly faster than classical computers. These “speedups” are achieved by utilizing a resource known as magic resource. Measuring the amount of magic resource used by a device allows us to quantify its potential computational power. Without this property, quantum computers are no faster than classical computers. Whether magic resource can be accurately measured on large-scale quantum computers has remained an open problem. To address this question, we introduce as a measure of magic resource and experimentally measure it on the IBM Eagle quantum processor. We show that sufficiently small quantities of magic resource can be measured efficiently. We also prove that measuring large (i.e., extensive) quantities of magic resource with our measure is intractable. Our results lead us to surmise that magic resource may be difficult to measure when a quantum computer provides a speedup. We find that one of the most sought-after properties in quantum computation is only partially accessible to experiments. We support our conclusions with both theoretical and experimental evidence. Our work illustrates the capabilities and limitations encountered when studying the power of quantum computers.
Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical magic state preparation protocols under the standard circuit-level noise model. When applied to protocols based on code switching, magic state cultivation, and magic state distillation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the non-stabilizerness, e.g., stabilizer rank or Pauli rank, of the target encoded magic state. The efficiency of our simulation method is rooted in a curious fact: every circuit-level Pauli error in these protocols propagates to a Clifford error at the end. This property is satisfied by a large family of protocols, including those that repeatedly measure a transversal Clifford that squares to a Pauli. We provide a proof-of-principle numerical simulation that prepares a magic state using such logical Clifford measurements. Our work enables practical simulation of logical magic state preparation protocols without resorting to approximations or resource-intensive state-vector simulations.
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.
Non-local magic is the non-stabilizerness that no local unitary operation can erase. It captures the joint action of entanglement and magic underlying quantum advantage, and it has never been measured on quantum hardware. Here we report its first experimental demonstration, on a superconducting quantum processing unit, through two independent routes: an optimal local-erasure protocol and a direct, state-agnostic measurement of subsystem purity. The two agree with each other and with theory. Exploiting direct access to the device, we construct a noise model with no free parameters that identifies readout error and a depolarizing controlled-Z channel as the dominant mechanisms, and we show that local and non-local magic can be addressed separately, erasing local magic in situ while preserving the non-local part. Non-local magic provides a hardware benchmark beyond standard gate-fidelity protocols and points toward more reliable pre-fault tolerant devices.The same tools underlie a purity-estimation protocol with exponential speedup and the decoding of Hawking radiation in a black-hole toy model.