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Classical simulation of a quantum circuit with noisy magic inputs

Jiwon Heo, Sojeong Park, Changhun Oh

TL;DR

We study how noise acting solely on injected magic states affects the classical simulability of otherwise efficiently simulable quantum circuits built from resourceless primitives. By framing the problem as sampling from a noisy output distribution with a fixed resourceless backbone and an ensemble representation of noisy magic inputs, we derive explicit noise thresholds under which a polynomial-time classical sampler exists. The proposed three-step algorithm—ensemble construction, truncation of high-magic samples, and low-rank stabilizer decomposition—yields tractable simulation costs in both qubit Clifford-based and fermionic matchgate-based settings, with concrete thresholds for dephasing and particle loss. Numerical results illustrate the regimes where the resourceless rank remains small and where the method remains efficient, informing the practical boundary between quantum advantage and classical simulability. The work thereby clarifies how realistic noise on magic resources impacts the potential for near-term quantum devices to demonstrate quantum advantage.

Abstract

Magic states are essential for universal quantum computation and are widely viewed as a key source of quantum advantage, yet in realistic devices they are inevitably noisy. In this work, we characterize how noise on injected magic resources changes the classical simulability of quantum circuits and when it induces a transition from classically intractable behavior to efficient classical simulation. We adopt a resource-centric noise model in which only the injected magic components are noisy, while the baseline states, operations, and measurements belong to an efficiently simulable family. Within this setting, we develop an approximate classical sampling algorithm with controlled error and prove explicit noise-dependent conditions under which the algorithm runs in polynomial time. Our framework applies to both qubit circuits with Clifford baselines and fermionic circuits with matchgate baselines, covering representative noise channels such as dephasing and particle loss. We complement the analysis with numerical estimates of the simulation cost, providing concrete thresholds and runtime scaling across practically relevant parameter regimes.

Classical simulation of a quantum circuit with noisy magic inputs

TL;DR

We study how noise acting solely on injected magic states affects the classical simulability of otherwise efficiently simulable quantum circuits built from resourceless primitives. By framing the problem as sampling from a noisy output distribution with a fixed resourceless backbone and an ensemble representation of noisy magic inputs, we derive explicit noise thresholds under which a polynomial-time classical sampler exists. The proposed three-step algorithm—ensemble construction, truncation of high-magic samples, and low-rank stabilizer decomposition—yields tractable simulation costs in both qubit Clifford-based and fermionic matchgate-based settings, with concrete thresholds for dephasing and particle loss. Numerical results illustrate the regimes where the resourceless rank remains small and where the method remains efficient, informing the practical boundary between quantum advantage and classical simulability. The work thereby clarifies how realistic noise on magic resources impacts the potential for near-term quantum devices to demonstrate quantum advantage.

Abstract

Magic states are essential for universal quantum computation and are widely viewed as a key source of quantum advantage, yet in realistic devices they are inevitably noisy. In this work, we characterize how noise on injected magic resources changes the classical simulability of quantum circuits and when it induces a transition from classically intractable behavior to efficient classical simulation. We adopt a resource-centric noise model in which only the injected magic components are noisy, while the baseline states, operations, and measurements belong to an efficiently simulable family. Within this setting, we develop an approximate classical sampling algorithm with controlled error and prove explicit noise-dependent conditions under which the algorithm runs in polynomial time. Our framework applies to both qubit circuits with Clifford baselines and fermionic circuits with matchgate baselines, covering representative noise channels such as dephasing and particle loss. We complement the analysis with numerical estimates of the simulation cost, providing concrete thresholds and runtime scaling across practically relevant parameter regimes.
Paper Structure (23 sections, 20 theorems, 81 equations, 5 figures)

This paper contains 23 sections, 20 theorems, 81 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Our contributions
  4. General framework
  5. Problem description
  6. Algorithm description
  7. Ensemble conditions
  8. Sampling procedure and truncation method
  9. Finding a low-rank superposition of resourceless states
  10. Qubit system
  11. Problem description
  12. Classical simulation
  13. Fermionic system
  14. Problem description
  15. Classical simulation
  16. ...and 8 more sections

Key Result

Lemma 1

Let $\hat{\rho}$ be a density matrix described as an ensemble $\{p_i, |\psi_i\rangle\}$ and $\delta_1>0$ be given. Let $k$ be the smallest integer such that $k\geq tp$ and where $D(a\| p)$ is the Kullback-Leibler divergence of a binomial distribution, defined as Consider a truncation procedure described as Then, the trace distance between $\hat{\rho}^{\otimes t}$ and the density operator genera

Figures (5)

  • Figure 1: A qubit-based example of quantum circuits that permitted universal quantum computation with noisy magic components, which is our desired circuit. White boxes are Clifford operations, and measurements are the computational basis measurement (a) A $n$-qubit circuit with $t$$T$ gates, whereas these suffer from physical noise. (b) Injecting $|T\rangle$ with adaptive measurement; however, it also suffers from physical noise. We mention that we also permit adaptive measurements in the injected circuit.
  • Figure 2: Classical simulable regions in each case (Blue boxes). For the dephasing noise cases, we only represent when the noise rate is in $[0, 1/2]$, because of symmetry. (a) In this case, the noisy magic states can be represented as a convex summation of stabilizer states when the noise rate is in $[(1-\tan(\pi/8))/2, 1/2]$, so that this interval is in the classical simulable region. For the remaining interval, the area of the classical simulable region is $O(\log n /t)$. (b) The area of the classical simulable region is $O(\sqrt{\log n /t})$. (c) The area of the classical simulable region is $O((\log n /t)^{1/4})$.
  • Figure 3: Blueprint of our problem and algorithm. The goal of our strategy is to find a proper low-rank resourceless superposition state. Since the state $|0^{\otimes n}\rangle$ does not affect the resourceless rank of the output state, we present $\hat{\rho}^{\otimes t}$ only in the figure.
  • Figure 4: Our desired circuit in the fermionic system. We injects $t$ noisy $4$-mode magic states, $|\psi_4\rangle = (|0011\rangle+|1100\rangle)/\sqrt{2}$. The white boxes represent Gaussian gates. As in the qubit case, we allow adaptive measurements in the injected circuit.
  • Figure 5: The resourceless rank of the output state (log scale) over the noise rate $p$. We set $\delta=0.01$ and investigate the various numbers of qubits (or modes). Each dashed line indicates the number of state coefficients that can be stored for a given amount of data storage. (a) Qubit case with the dephasing noise. (b) Fermion case with the particle loss. (c) Fermion case with the dephasing noise.

Theorems & Definitions (32)

  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 1
  • Lemma 3
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Theorem 4
  • Theorem 5
  • ...and 22 more