Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Complexity of quantum circuits via sensitivity, magic, and coherence

Kaifeng Bu, Roy J. Garcia, Arthur Jaffe, Dax Enshan Koh, Lu Li

TL;DR

This work investigates quantum circuit complexity through three quantum resources: sensitivity (influence), magic, and coherence. It provides a unified framework showing that the circuit cost $Cost(U)$ is lower bounded by a maximum of terms involving these resources, specifically $Cost(U) \ge c\max\{CiS[U], M[U]/d^2, C_r(U)/\log d\}$, and it characterizes stable (zero-sensitivity) unitaries as those generated by local gates and swaps, with matchgates emerging as Gaussian-stable gates. The authors develop a quantum Fourier entropy–influence relation to connect magic to sensitivity and demonstrate how scrambling, via average OTOCs, ties to influence, while cohering power yields another independent bound on cost. Together, these results illuminate how non-Gaussianity, scrambling, and coherence constrain quantum computational resources and provide pathways to quantify speedups or classical simulability. The framework has potential implications for understanding quantum advantage, black-hole information dynamics, and stability in quantum machine learning.

Abstract

Quantum circuit complexity-a measure of the minimum number of gates needed to implement a given unitary transformation-is a fundamental concept in quantum computation, with widespread applications ranging from determining the running time of quantum algorithms to understanding the physics of black holes. In this work, we study the complexity of quantum circuits using the notions of sensitivity, average sensitivity (also called influence), magic, and coherence. We characterize the set of unitaries with vanishing sensitivity and show that it coincides with the family of matchgates. Since matchgates are tractable quantum circuits, we have proved that sensitivity is necessary for a quantum speedup. As magic is another measure to quantify quantum advantage, it is interesting to understand the relation between magic and sensitivity. We do this by introducing a quantum version of the Fourier entropy-influence relation. Our results are pivotal for understanding the role of sensitivity, magic, and coherence in quantum computation.

Complexity of quantum circuits via sensitivity, magic, and coherence

TL;DR

This work investigates quantum circuit complexity through three quantum resources: sensitivity (influence), magic, and coherence. It provides a unified framework showing that the circuit cost is lower bounded by a maximum of terms involving these resources, specifically , and it characterizes stable (zero-sensitivity) unitaries as those generated by local gates and swaps, with matchgates emerging as Gaussian-stable gates. The authors develop a quantum Fourier entropy–influence relation to connect magic to sensitivity and demonstrate how scrambling, via average OTOCs, ties to influence, while cohering power yields another independent bound on cost. Together, these results illuminate how non-Gaussianity, scrambling, and coherence constrain quantum computational resources and provide pathways to quantify speedups or classical simulability. The framework has potential implications for understanding quantum advantage, black-hole information dynamics, and stability in quantum machine learning.

Abstract

Quantum circuit complexity-a measure of the minimum number of gates needed to implement a given unitary transformation-is a fundamental concept in quantum computation, with widespread applications ranging from determining the running time of quantum algorithms to understanding the physics of black holes. In this work, we study the complexity of quantum circuits using the notions of sensitivity, average sensitivity (also called influence), magic, and coherence. We characterize the set of unitaries with vanishing sensitivity and show that it coincides with the family of matchgates. Since matchgates are tractable quantum circuits, we have proved that sensitivity is necessary for a quantum speedup. As magic is another measure to quantify quantum advantage, it is interesting to understand the relation between magic and sensitivity. We do this by introducing a quantum version of the Fourier entropy-influence relation. Our results are pivotal for understanding the role of sensitivity, magic, and coherence in quantum computation.
Paper Structure (19 sections, 39 theorems, 188 equations, 2 figures)

This paper contains 19 sections, 39 theorems, 188 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Sensitivity and circuit complexity
  4. Influence
  5. Circuit sensitivity and complexity
  6. Stable unitaries
  7. Matchgates are Gaussian stable gates
  8. Quantifying scrambling by influence on average case
  9. Quantum Fourier entropy and influence
  10. Quantum Fourier entropy-influence relation and conjecture
  11. Magic entropy-circuit sensitivity relation
  12. Magic and circuit complexity
  13. A lower bound on circuit cost from magic-influence relation
  14. A lower bound on circuit cost by magic power
  15. Coherence and circuit complexity
  16. ...and 4 more sections

Key Result

Theorem 2

The circuit cost of a quantum circuit $U\in \mathrm{SU}(d^n)$ is lower bounded as follows: where $c$ is a universal constant independent of $d$ and $n$. The quantities $\mathrm{CiS}[U]$ ($\mathcal{M}[U]$, $\mathcal{C}_r(U)$, respectively), defined formally in eq:circuit_sensitivity (eq:magic_power, eq:cohering_power, respectively), quantify the sensitivity (magic, coherence, respectively)

Figures (2)

  • Figure 1: A Venn diagram illustrating the overlap between the stable gate set and the Gaussian stable gate set on $n$-qubit systems, as explained in the text.
  • Figure 2: A plot of the function $g(x)=x(\log x)^2$ for $x\in[0,1]$, where the logarithm is taken to be of base 2. The maximum value of $g(x)$ is $g(\mathrm{e}^{-2}) \approx 1.1267$, which occurs at $x = \mathrm{e}^{-2} \approx 0.135$. The function $g(x)$ vanishes at both $x=0$ and $x=1$. In addition, it is increasing on $[0,\mathrm{e}^{-2}]$ and decreasing on $[\mathrm{e}^{-2},1]$.

Theorems & Definitions (94)

  • Definition 1: Nielsen et al. nielsen2006quantum
  • Theorem 2: Results on Circuit Complexity
  • Theorem 3: Matchgates via Sensitivity
  • Theorem 4: Quantum Fourier Entropy-Influence Relation
  • Definition 5: Montanaro and Osborne montanaro2010quantum
  • Definition 6: Circuit Sensitivity
  • Lemma 7
  • proof
  • Proposition 8: Small Total Circuit Sensitivity
  • proof
  • ...and 84 more