Abstract In the stabilizer formalism of fault-tolerant quantum computation, stabilizer states serve as classical objects, while magic states (non-stabilizer states) are a kind of quantum resource (called magic resource) for promoting stabilizer circuits to universal quantum computation. In this framework, the T-gate is widely used as a non-Clifford gate which generates magic resource from stabilizer states. A natural question arises as whether the T-gate is in some sense optimal for generating magic resource. We address this issue by employing an intuitive and computable quantifier of magic based on characteristic functions (Weyl transforms) of quantum states. We demonstrate that the qubit T-gate, as well as its qutrit extension, the qutrit T-gate, are indeed optimal for generating magic resource among the class of diagonal unitary operators. Moreover, up to Clifford equivalence, the T-gate is essentially the only gate having such an optimal property. This reveals some intrinsic optimal features of the T-gate. We further compare the T-gate with general unitary gates for generating magic resource.
In the stabilizer formalism of quantum error correction and fault-tolerant quantum computation, stabilizer states, as common eigenstates (with the eigenvalue 1) of operators in maximal Abelian subgroups of the Pauli group, play the role of classical states, while magic (non-stabilizer) states are quantum resources enabling universal quantum computation when injected into stabilizer circuits.The natural issue arises as how to characterize and quantify the ability of quantum channels (including quantum gates) in generating magic resource (non-stabilzerness) from stabilizer states.In this work, we introduce two intuitive quantities characterizing magic-resource generating power of quantum channels and reveal their basic properties.We evaluate these quantifies for several prototypical quantum channels and quantum gates.In particular, we demonstrate that the qubit T -gate (i.e., π/8-gate) and its qutrit generalization are optimal in generating magic resource in a class of diagonal unitaries.This highlights the significance of the T -gate and its extensions from a resource-theoretic perspective and provides a theoretical support of their wide applications in stabilizer quantum computation.
Abstract In a remarkable work [Phys. Rev. A 86 022316 (2012)], Howard and Vala introduced a qudit version of the qubit T-gate (i.e., π/8-gate) for any prime dimensional system. This non-Clifford gate is a key ingredient of the paradigm ‘Clifford +T’, which are widely employed in the stabilizer formalism of universal and fault-tolerant quantum computation. Considering the applications and significance of the T-gate, it is desirable to characterize it from various angles. Here we prove that in any prime dimensional system, the Howard-Vala T-gate is optimal, among all diagonal gates, for generating magic resources from stabilizer states when the magic is quantified via the L 1-norm of characteristic functions (Fourier transforms) of quantum states. The quadratic Gaussian sum in number theory plays a key role in establishing this optimality. This highlights an extreme feature of the Howard-Vala T-gate. We further reveal an intrinsic relation between the Howard-Vala T-gate and the Watson-Campbell-Anwar-Browne T-gate [Phys. Rev. A 92 022312 (2015)] for any prime dimensional system.
Quantum gate teleportation is a key technique in fault-tolerant quantum computation that uses resource states to implement logical gates. Here, we develop a theory of magic gate teleportation (MGT), a form of quantum gate teleportation that uses non-stabilizer resource states to implement non-Clifford unitary gates on arbitrary input states without revealing information about those states. This setting is motivated by the canonical approach to implementing logical non-Clifford gates in fault-tolerant quantum computation. We uncover a hidden structure within MGT---after backpropagating the Pauli measurements, MGT protocols can be viewed as encoding the input state into a stabilizer code heralded by the measurement outcomes, followed by a logical non-Clifford gate. Using this structure, we construct MGT protocols for any resource state obtained by applying commuting Pauli rotations to a stabilizer state, and provide an efficient algorithm for synthesizing their circuit implementations. Conversely, we prove that directly useful single-qubit resource states are necessarily Clifford-equivalent to diagonal states, and establish the same constraint for multi-qubit resource states used in a broad class of MGT protocols. In particular, the output state distilled from the
[[5,1,3]] protocol is not directly useful for any MGT protocols, though it can be probabilistically converted into directly useful states. Finally, we identify conditions under which the feedforward operators can be implemented by Pauli operators, shedding light on the paradigm of algorithmic fault tolerance and simplifying the feedforward operations needed for quantum computing.
Abstract Magic is a precious resource necessary for achieving universal fault-tolerant quantum computation. Therefore, it is of vital importance to study the detection and quantification of the magic resource encompassed in quantum states and quantum gates both theoretically and experimentally. In this work, we adopt the quantum Jensen–Shannon divergence to quantify the magic resource of quantum states and quantum gates. On the one hand, we determine the magic resource of a pure state as the minimal and average distance between this state and the set of pure stabilizer states via the quantum Jensen–Shannon divergence, respectively, and extend them to the general mixed states through the method of convex roof construction. We investigate the basic properties of these two magic quantifiers and utilize them to evaluate the magic resource for some typical qubit and qutrit states. By comparing the magic quantifier via the quantum Jensen–Shannon divergence with the min-relative entropy of magic and the stabilizer α-Rényi entropies, we find that the min-relative entropy of magic provides both an upper bound and a lower bound for the magic quantifier via the quantum Jensen-Shannon divergence, and the stabilizer α-Rényi entropies provide a series of lower bounds for the magic quantifier via the quantum Jensen–Shannon divergence. On the other hand, based on the magic quantifier via the quantum Jensen–Shannon divergence for quantum states, we further propose two quantifiers for the magic-resource-generating power of quantum gates and demonstrate that the T-gate is the optimal diagonal unitary gate in creating magic resource for both qubit and qutrit systems in the sense of Clifford equivalence.
Motivated by their necessity for most fault-tolerant quantum computation schemes, we formulate a resource theory for magic states. First, we show that robustness of magic is a well-behaved magic monotone that operationally quantifies the classical simulation overhead for a Gottesman-Knill-type scheme using ancillary magic states. Our framework subsequently finds immediate application in the task of synthesizing non-Clifford gates using magic states. When magic states are interspersed with Clifford gates, Pauli measurements, and stabilizer ancillas-the most general synthesis scenario-then the class of synthesizable unitaries is hard to characterize. Our techniques can place nontrivial lower bounds on the number of magic states required for implementing a given target unitary. Guided by these results, we have found new and optimal examples of such synthesis.
Recent advances in quantum hardware are bringing fault-tolerant quantum computing (FTQC) closer to reality. In the early stage of FTQC, however, the numbers of available logical qubits and high-fidelity
T gates remain limited, making it crucial to optimize the quantum resource usage. In this work, we aim to study the simulation cost of general quantum states under the constraint that only
k T gates can be used, alongside an unlimited number of Clifford gates. Inspired by the notion of robustness of magic (RoM) which quantifies the cost of quantum-circuit simulation using stabilizer states (
k=0), we introduce its generalization, which we call Clifford+
kT robustness, treating Clifford+
kT states as free resources. We explore theoretical properties of Clifford+
kT robustness and in particular derive a lower bound that reveals the (in)efficiency of quantum-circuit simulation using Clifford+
kT states. Through numerical computations, we also evaluate Clifford+
kT robustness for key resource states for universal quantum computation, such as tensor products of the magic states. Our results allow to assess the sampling-cost reduction achieved by the use of Clifford+
kT states instead of stabilizer states, providing practical guidance for efficient resource usage in the early-FTQC era.