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On the Hardness of Measuring Magic

Roy J. Garcia, Gaurav Bhole, Kaifeng Bu, Liyuan Chen, Haribabu Arthanari, Arthur Jaffe

TL;DR

This paper studies the feasibility of measuring quantum magic on large devices and introduces a scalable magic monotone called Pauli instability. It defines I(U) = -log( E_{P1,P2 in Q^{⊗ n}} |OTOC(U,P1,P2)| ) based on the OTOC, and shows that approximating it with I_N(U) = -log( (1/N) Σ_i |OTOC(U,P1^i,P2^i)| ) incurs a Pauli sample complexity N = e^{2 I(U)} f(η,δ), with f(η,δ) = ln(1/δ) / (2 (1 − e^{g η})^2) and g = sign(I(U) − I_N(U)). The main results establish faithfulness, invariance, additivity, and T gate scaling for Pauli instability, and give corollaries that efficient, accurate measurement is possible only when I(U) = O(log n), while in the linear-n regime measurement becomes intractable; an OTOC sample complexity bound M = ln(1/δ) / (γ^2 OTOC^2) further quantifies the measurement effort. The work is supported by numerical simulations and IBM Eagle experiments that illustrate the growth of magic with the number of T gates and reveal noise-induced biases, leading to a conjecture that measuring any reliable magic monotone becomes intractable in large-scale, noisy devices. Overall, the study links chaos and scrambling to the practical limitations of diagnosing quantum advantage via direct magic measurements and motivates developing robust, noise-tolerant protocols for future experiments.

Abstract

Quantum computers promise to solve computational problems significantly faster than classical computers. These 'speed-ups' are achieved by utilizing a resource known as magic. Measuring the amount of magic 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 can be accurately measured on large-scale quantum computers has remained an open problem. To address this question, we introduce Pauli instability as a measure of magic and experimentally measure it on the IBM Eagle quantum processor. We prove that measuring large (i.e., extensive) quantities of magic is intractable. Our results suggest that one may only measure magic when a quantum computer does not provide a speed-up. We support our conclusions with both theoretical and experimental evidence. Our work illustrates the capabilities and limitations of quantum technology in measuring one of the most important resources in quantum computation.

On the Hardness of Measuring Magic

TL;DR

This paper studies the feasibility of measuring quantum magic on large devices and introduces a scalable magic monotone called Pauli instability. It defines I(U) = -log( E_{P1,P2 in Q^{⊗ n}} |OTOC(U,P1,P2)| ) based on the OTOC, and shows that approximating it with I_N(U) = -log( (1/N) Σ_i |OTOC(U,P1^i,P2^i)| ) incurs a Pauli sample complexity N = e^{2 I(U)} f(η,δ), with f(η,δ) = ln(1/δ) / (2 (1 − e^{g η})^2) and g = sign(I(U) − I_N(U)). The main results establish faithfulness, invariance, additivity, and T gate scaling for Pauli instability, and give corollaries that efficient, accurate measurement is possible only when I(U) = O(log n), while in the linear-n regime measurement becomes intractable; an OTOC sample complexity bound M = ln(1/δ) / (γ^2 OTOC^2) further quantifies the measurement effort. The work is supported by numerical simulations and IBM Eagle experiments that illustrate the growth of magic with the number of T gates and reveal noise-induced biases, leading to a conjecture that measuring any reliable magic monotone becomes intractable in large-scale, noisy devices. Overall, the study links chaos and scrambling to the practical limitations of diagnosing quantum advantage via direct magic measurements and motivates developing robust, noise-tolerant protocols for future experiments.

Abstract

Quantum computers promise to solve computational problems significantly faster than classical computers. These 'speed-ups' are achieved by utilizing a resource known as magic. Measuring the amount of magic 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 can be accurately measured on large-scale quantum computers has remained an open problem. To address this question, we introduce Pauli instability as a measure of magic and experimentally measure it on the IBM Eagle quantum processor. We prove that measuring large (i.e., extensive) quantities of magic is intractable. Our results suggest that one may only measure magic when a quantum computer does not provide a speed-up. We support our conclusions with both theoretical and experimental evidence. Our work illustrates the capabilities and limitations of quantum technology in measuring one of the most important resources in quantum computation.
Paper Structure (6 sections, 4 theorems, 4 equations, 2 figures)

This paper contains 6 sections, 4 theorems, 4 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Simulations and Experiments
  5. Conclusion
  6. Acknowledgements

Key Result

Theorem 1

Let $\delta, \eta>0$. Then $\left| \mathbb{I}_N(U)-\mathbb{I}(U) \right|<\eta$ with probability at least $1-\delta$ when the Pauli sample complexity is Here, $f(\eta,\delta)=\frac{\ln(1/\delta)}{2(1-e^{g\eta})^2}$ and $g=\mathrm{sign}(\mathbb{I}(U)-\mathbb{I}_N(U))$.

Figures (2)

  • Figure 1: (a) Numerical simulations (blue points) of $\mathbb{I}_N$ for a unitary $U_k$ as in (c, top), which is a single layer of $k$ T gates. Black points are the exact values of the $\mathbb{I}$: $\mathbb{I}(U_k)=k\log(4/3)$. The system size is $n=10$, the Pauli sample complexity is 500 and the OTOC sample complexity is 500. (b) Experimental measurement of $\mathbb{I}_N$ for $U_k$. The system size is $n=5$, the Pauli sample complexity is 500, and the OTOC sample complexity is 500. (d) Experimental measurement of $\mathbb{I}_N$ for the unitary $V_k$ in (c, bottom), which contains $k$ layers. Layer $i$ is composed of a layer of H gates, two layers of staggered CNOT gates, a layer of S gates, and a single T gate applied to the $i$-th qubit. The system size is $n=4$, the Pauli sample complexity is $500$ and the OTOC sample complexity is $500$. In all plots, each data point is computed by independently measuring $\mathbb{I}$ (or $\mathbb{I}_N$) 5 times and averaging. The OTOC is measured using the circuit in Fig. \ref{['Fig:Circuit']}.
  • Figure 2: Quantum circuit to measure the OTOC for a unitary $U$ and Pauli strings $P_1$ and $P_2$. 'Ref' denotes $n$ reference qubits and 'Sys' denotes $n$ system qubits. The control qubit is in the state $\ket{+}_{\mathcal{C}}=(\ket{0}_{\mathcal{C}}+\ket{1}_{\mathcal{C}})/\sqrt{2}$. The dotted box denotes a measurement in the $X$ basis. The circuit outputs the expectation value $\langle X_\mathcal{C} \rangle=\mathrm{OTOC}(U, P_1, P_2)$, where $\mathcal{C}$ denotes the control qubit.

Theorems & Definitions (6)

  • Definition 1
  • Theorem 1: Pauli sample complexity
  • Corollary 1
  • Conjecture 1
  • Proposition 1: OTOC sample complexity
  • Corollary 2