[Paper Review] Computational quantum-classical boundary of complex and noisy quantum systems

Keisuke Fujii, Shuhei Tamate|arXiv (Cornell University)|Jun 26, 2014

Quantum Computing Algorithms and Architecture52 references3 citations

TL;DR

This paper establishes a computational quantum-classical (CQC) boundary for noisy commuting quantum circuits by analyzing classical simulatability under single-qubit depolarizing noise. It shows that when noise exceeds a threshold of 14.6%, quantum circuits become classically simulatable via Gottesman-Knill and separable state criteria, while below this rate, they remain quantumly hard to simulate—providing a sharp boundary for experimental verification of quantum advantage in noisy regimes.

ABSTRACT

It is often said that the transition from quantum to classical worlds is caused by decoherence originated from an interaction between a system of interest and its surrounding environment. Here we establish a computational quantum-classical boundary from the viewpoint of classical simulatability of a quantum system under decoherence. Specifically, we consider commuting quantum circuits being subject to decoherence. Or equivalently, we can regard them as measurement-based quantum computation on decohered weighted graph states. To show intractability of classical simulation above the boundary, we utilize the postselection argument introduced by M. J. Bremner, R. Jozsa, and D. J. Shepherd [Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 465, 1413 (2009).] and crucially strengthen its statement by taking noise effect into account. Classical simulatability below the boundary is also shown constructively by using both separable criteria in a projected-entangled-pair-state picture and the Gottesman-Knill theorem for mixed state Clifford circuits. We found that when each qubit is subject to a single-qubit complete-positive-trace-preserving noise, the computational quantum-classical boundary is sharply given by the noise rate required for the distillability of a magic state. The obtained quantum-classical boundary of noisy quantum dynamics reveals a complexity landscape of controlled quantum systems. This paves a way to an experimentally feasible verification of quantum mechanics in a high complexity limit beyond classically simulatable region.

Motivation & Objective

• To define a computational quantum-classical (CQC) boundary based on classical simulatability of noisy quantum systems.
• To determine the noise threshold at which quantum advantage in commuting quantum circuits transitions to classical simulatability.
• To provide a physically feasible method for verifying quantumness without full state tomography.
• To extend the postselection argument to noisy quantum systems, ensuring robustness against realistic noise models.
• To establish a sharp boundary using both intractability (quantum side) and constructible classical simulation (classical side).

Proposed method

• Uses noisy commuting quantum circuits with single-qubit complete-positive-trace-preserving (CPTP) noise as a model system.
• Applies the Gottesman-Knill theorem to mixed-state Clifford circuits to show classical simulatability on the classical side.
• Employs a projected-entangled-pair-state (PEPS) picture with separable criteria to constructively demonstrate classical simulatability.
• Extends the postselection argument of Bremner et al. to noisy settings, proving quantum intractability under noise.
• Derives the joint probability distribution for successful projections using Bayesian updating of local states on entangled bonds.
• Uses brute-force counting of self-avoiding walks on the RHG lattice to evaluate low-weight error accumulation in magic state injection.

Experimental results

Research questions

• RQ1 What noise level separates classically simulatable from quantumly hard-to-simulate commuting quantum circuits?
• RQ2 Can the postselection argument be extended to noisy quantum systems to preserve quantum computational hardness?
• RQ3 How does noise affect the distillability of magic states in the context of quantum advantage?
• RQ4 Can quantumness be verified experimentally without full tomography, using single-shot measurements?
• RQ5 What is the role of topological protection in error accumulation during magic state injection?

Key findings

• The CQC boundary is sharply defined at a noise rate of 14.6% for single-qubit CPTP noise in non-constant-depth commuting quantum circuits.
• For depth-four circuits, the CQC boundary is bounded between 13.4% and 14.6%, confirming robustness of the threshold.
• Classical simulatability on the classical side is constructively proven using the Gottesman-Knill theorem and separable state criteria in the PEPS picture.
• Quantum intractability on the quantum side is established via a noise-robust extension of the postselection argument, valid even under additive and multiplicative error models.
• Low-weight errors in magic state injection are quantified via self-avoiding walk counting, showing significant error accumulation from primal and dual 1-chains on the RHG lattice.
• The results enable single-shot experimental verification of quantumness without tomography, relying only on measurement statistics and physical assumptions.