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arXiv:2607.19070

How to improve the discrimination power of classically simulable measurements?

Yiran Wang, Yongming Li

quant-phmath-phSubmitted Jul 21, 2026• Announced Jul 22, 2026• 0 citations
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Abstract

Classically simulable measurements (CSMs) constitute an important class of restricted measurements in the odd-prime-dimensional magic resource theory, referring to those measurements with positive discrete Wigner functions. Since their discrimination power is weaker than that of global measurements, it is necessary to study how to improve the discrimination power of CSMs. In this paper, we consider three methods to improve the discrimination power of CSMs, including adding magic resources, using quantum catalysts, and using quantum memories. Specifically, we relate measurements with positive discrete Wigner functions to completely positive Wigner-preserving measurement channels, thereby transforming the problem of improving the discrimination power of CSMs into the problem of determining how many magic resources are required to simulate quantum channels using free operations. Based on this, we derive the lower and upper bounds of the simulation cost. Moreover, we provide a concrete example for which these bounds coincide and prove that consumable magic resources can enhance the discrimination power of CSMs. Finally, we establish a no-go theorem, which shows that for discriminating a pair of states with positive Wigner functions, neither finite-dimensional quantum catalysts nor finite-dimensional quantum memories can improve the optimal success probability of discrimination using CSMs.

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Classically simulable measurements (CSMs) constitute an important class of restricted measurements in the odd-prime-dimensional magic resource theory, referring to those measurements with positive discrete Wigner functions. Since their discrimination power is weaker than that of global measurements, it is necessary to study how to improve the discrimination power of CSMs. In this paper, we consider three methods to improve the discrimination power of CSMs, including adding magic resources, using quantum catalysts, and using quantum memories. Specifically, we relate measurements with positive discrete Wigner functions to completely positive Wigner-preserving measurement channels, thereby transforming the problem of improving the discrimination power of CSMs into the problem of determining how many magic resources are required to simulate quantum channels using free operations. Based on this, we derive the lower and upper bounds of the simulation cost. Moreover, we provide a concrete example for which these bounds coincide and prove that consumable magic resources can enhance the discrimination power of CSMs. Finally, we establish a no-go theorem, which shows that for discriminating a pair of states with positive Wigner functions, neither finite-dimensional quantum catalysts nor finite-dimensional quantum memories can improve the optimal success probability of discrimination using CSMs.
Abstract In this paper, we mainly study the indistinguishability of classically simulable measurements (measurements that can be effectively simulated on classical computers) in the tasks of quantum channel discrimination. First, we investigate the limitations of classically simulable measurements under unambiguous and perfect discrimination strategies. For unambiguous discrimination, we establish a conditional limitation result for channel-induced output-state discrimination scenarios, while for perfect discrimination we derive a sufficient condition under which classically simulable measurements fail to perfectly distinguish quantum channels. When considering the minimum-error discrimination strategy, we formulate the corresponding bilinear optimization problem and discuss its formal Lagrange dual formulation. Subsequently, we give the lower bound of the minimum-error probability obtained by using classically simulable measurements in the minimum-error discrimination task under the adaptive protocol. Furthermore, we give two measures of the nonstabilizerness (magic) of measurements. We prove that one measure we proposed can be used as an exact quantifier for the advantage in some tasks of quantum state discrimination. Finally, we investigate the relationship between classically simulable measurements and the data hiding ratio.
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.
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.
An important problem in quantum computation is to characterize the resources required for a computational speedup over classical computation. Veitch et al. showed that one necessary condition for a computational speedup in the model of quantum computation with magic states is that the discrete Wigner function representing the input state of the quantum circuit must take negative values. The amount of negativity in the discrete Wigner function quantifies the complexity of classical simulation of a quantum computation with simulation being efficient if the Wigner function is nonnegative everywhere. In this sense, negativity of the Wigner quasiprobability representation serves as an indicator of quantumness from a computational perspective. However, this result only holds for systems of qudits where the local Hilbert space dimension is odd. The first main result discussed in this thesis relates to a discrete Wigner function suitable for describing quantum computation with magic states defined for any local Hilbert space dimension. When the local Hilbert space dimension is odd it subsumes the standard discrete Wigner function. When the local Hilbert space dimension is even, as a result of state-independent proofs of contextuality among multiqubit (or multi-even-dimensional-qudit) Pauli observables, the phase space over which the Wigner function is defined becomes much larger. However, for systems of qubits, the properties required for simulation of quantum computation with magic states remain. This simulation method effectively extends the result described above for odd-dimensional qudits to qubits. Although in the even-dimensional case the phase space is much larger, points in multiqubit phase space can be characterized and classified and some structure can be imparted on the phase space. The second main result discussed here is a hidden variable model for quantum computation with magic states on qubits. This model is similar in structure to quasiprobability representations like the discrete Wigner function, but unlike those representations the model is capable of representing all elements of any quantum computation---states, operations, and measurements---using only classical probabilities. No negativity is required. This calls into question the role of negativity in quasiprobability representations as an indicator of quantumness for models of quantum computation.
According to the celebrated Gottesman-Knill theorem, magic resource (nonstabilizerness) is necessary to drive universal quantum computation. Thus characterization and quantification of magic resource for quantum states and quantum channels are significant issues. In this work, we propose a method to construct quantifiers of magic resource for quantum channels from those for quantum states by employing the channel-state duality to convert quantum channels to the corresponding Choi states. We focus on a specific quantifier of magic resource based on characteristic functions (Fourier transforms) of quantum states. This quantifier is well defined for all dimensions and easy to calculate, as opposed to those defined via discrete Wigner functions. It satisfies some desirable properties and provides an upper bound to the mana (a kind of logarithmic negativity of discrete Wigner functions) of quantum channels. Using this quantity, we evaluate the amount of magic resource for some typical quantum channels and compare them with some existing quantifiers of generating-power of magic resource. For the dephasing channels in prime dimensional systems, our quantifier of magic resource essentially characterizes the average generating power of magic resource, while, for general channels, they are quite different and some challenging issues arise.
In the realm of fault-tolerant quantum computing, stabilizer operations play a pivotal role, characterized by their remarkable efficiency in classical simulation. This efficiency sets them apart from nonstabilizer operations within the quantum computational theory. In this Letter, we investigate the limitations of classically simulable measurements in distinguishing quantum states. We demonstrate that any pure magic state and its orthogonal complement of odd prime dimensions cannot be unambiguously distinguished by stabilizer operations, regardless of how many copies of the states are supplied. We also reveal intrinsic similarities and distinctions between the quantum resource theories of magic states and entanglement in quantum state discrimination. The results emphasize the inherent limitations of classically simulable measurements and contribute to a deeper understanding of the quantum-classical boundary.