Limitations of Noisy Geometrically Local Quantum Circuits

TL;DR

This work analyzes the sampling complexity of output distributions from noisy geometrically local quantum circuits under depolarizing noise. Using an information-theoretic framework based on relative entropy decay, coarse-graining into sublattices, and a Pauli-based inclusion-exclusion decomposition, it proves that beyond a critical depth $d^* = \Theta(p^{-1}\log(p^{-1}n))$ one can approximate the circuit’s output in quasipolynomial time, and beyond $d^* = \Theta(p^{-1}\log(p^{-1}))$ via percolated approximations. The authors conjecture a further $\Theta(1)$-depth simulability arising from a percolation-induced loss of long-range entanglement and propose a candidate algorithm whose accuracy hinges on an approximate Markov property of the output distribution. These results deepen our understanding of how locality and noise constrain quantum advantage, with practical implications for NISQ-era experiments and fundamental implications for decoherence and quantum-classical boundaries.

Abstract

It has been known for almost 30 years that quantum circuits with interspersed depolarizing noise converge to the uniform distribution at $ω(\log n)$ depth, where $n$ is the number of qubits, making them classically simulable. We show that under the realistic constraint of geometric locality, this bound is loose: these circuits become classically simulable at even shallower depths. Unlike prior work in this regime, we consider sampling from worst-case circuits and noise of any constant strength. First, we prove that the output distribution of any noisy geometrically local quantum circuit can be approximately sampled from in quasipolynomial time, when its depth exceeds a fixed $Θ(\log n)$ critical threshold which depends on the noise strength. This scaling in $n$ was previously only obtained for noisy random quantum circuits (Aharonov et. al, STOC 2023). We further conjecture that our bound is still loose and that a $Θ(1)$-depth threshold suffices for simulability due to a percolation effect. To support this, we provide analytical evidence together with a candidate efficient algorithm. Our results rely on new information-theoretic properties of the output states of noisy shallow quantum circuits, which may be of broad interest. On a fundamental level, we demonstrate that unitary quantum processes in constant dimensions are more fragile to noise than previously understood.

Figures (4)

• Figure 1: We consider the complexity of sampling from the output distribution of noisy gometrically local quantum circuits on $n$ qubits, of depth $d$, with depolarizing noise strength $p$. We denote what is currently known about these circuits in black, depict our contributions in blue, and highlight a few existing results that require additional assumptions in pink.
• Figure 2: Figure (a) shows how convergence of the entire state to within a constant relative entropy distance of $\sigma$ requires noise on all qubits, and further $O(\log n)$ layers of this noise. Figure (b) shows how convergence to $\sigma_{A} \otimes \rho_{\overline{A}}$ only requires noise within the lightcone of $A$, and further this means only $O(\log(|L(A)|))$ layers are required.
• Figure 3: In Fig (a), we show how non-adjacent sublattices independently converge towards the maximally mixed state due to our specific choice of coarse-graining. Fig (b) depicts what our relative entropy bounds tell us about the state after constant depth. This can be compared with \ref{['fig:2']}(a): while $O(\log n)$ depth is required for the full state to be within constant relative entropy distance to the maximally mixed state, we show that after $O(1)$ depth, every single sublattice is within a constant relative entropy distance to the maximally mixed state on that sublattice.
• Figure 4: In Fig. (a), we depict how $\rho$ can be trivially decomposed into $\sigma_{J_i} \otimes \rho_{J \backslash J_i}$ and a residual term $\rho - \sigma_{J_i} \otimes \rho_{J \backslash J_i} =\mathcal{M}_{J_i}(\rho)$. We use a completely blue box to depict $\sigma_{J_i}$ since it is a maximally mixed state. We use a thinner box with shading above it to denote $\mathcal{M}_{J_i}(\rho)$, since it is not a state, but rather a trace-zero matrix of small, $O(\delta)$, trace norm. In Fig. (b), we show how this decomposition can be applied recursively for multiple sublattices, and we show how each residual term is exponentially suppressed (note the scaling in $\delta$), which allows them to be truncated without incurring too much error.

Theorems & Definitions (42)

• theorem 1.1: Informal
• conjecture 1.2: Informal
• lemma 3.1: Relative Entropy Convergence on Subsets
• lemma 3.2: Relative Entropy Decay in Circuits
• definition 3.3: Coarse-Graining
• corollary 3.4: Relative Entropy Decay in Geo. Local Circuits
• definition 4.1: Inclusion-Exclusion Map
• lemma 4.2: Bounds on Inclusion-Exclusion Terms
• definition 4.3: Sparse Approximations
• theorem 4.4: Convergence to sparse approximations