Polynomial-Time Classical Simulation of Noisy IQP Circuits with Constant Depth

TL;DR

This work shows that noisy IQP circuits become classically simulatable in polynomial time once their depth crosses a constant threshold $d_c=O(p^{-1}\log(kp^{-1}))$, independent of circuit architecture or anticoncentration assumptions. The authors leverage a graph-percolation perspective: noise effectively removes entangling edges, partitioning the circuit into small, independent subcircuits that can be simulated efficiently by state-vector methods. They also establish a matching hardness result below the threshold, yielding a sharp depth-based boundary under standard complexity assumptions, and discuss consequences for fault-tolerance and QAOA. Overall, the paper provides a concrete, percolation-based framework explaining how noise can erode quantum advantage in IQP-type circuits and suggests directions for analyzing other noisy quantum tasks.

Abstract

Sampling from the output distributions of quantum computations comprising only commuting gates, known as instantaneous quantum polynomial (IQP) computations, is believed to be intractable for classical computers, and hence this task has become a leading candidate for testing the capabilities of quantum devices. Here we demonstrate that for an arbitrary IQP circuit undergoing dephasing or depolarizing noise, whose depth is greater than a critical $O(1)$ threshold, the output distribution can be efficiently sampled by a classical computer. Unlike other simulation algorithms for quantum supremacy tasks, we do not require assumptions on the circuit's architecture, on anti-concentration properties, nor do we require $Ω(\log(n))$ circuit depth. We take advantage of the fact that IQP circuits have deep sections of diagonal gates, which allows the noise to build up predictably and induce a large-scale breakdown of entanglement within the circuit. Our results suggest that quantum supremacy experiments based on IQP circuits may be more susceptible to classical simulation than previously thought.

Figures (5)

• Figure 1: A generic IQP circuit on a 1D architecture involving $d$ layers of diagonal gates
• Figure 2: A plot of the depth thresholds for approximate sampling $(d^*)$ and exact sampling $(d_c)$. We also include the depth threshold for hardness of exact sampling, which we obtain in \ref{['Sec: Tightness']}
• Figure 3: The graph on the left shows the interaction graph of a particular circuit, where the qubits are vertices and two vertices are joined by an edge if there is an entangling gate acting between them in the circuit. On the right, the red vertices indicate qubits which have been hit with noise. The effect of the noise is essentially to remove all interactions with this qubit from the larger circuit. The circuit can then effectively be simulated by considering only the connected components which are much smaller.
• Figure 4: In (a), we show an example of $\tilde{C}$ on 5 qubits, and the list of characters $b = (b_1,b_2,b_3,b_4,b_5)$ which represents the initial state. In (b), we show the circuit $\tilde{C}_2$ constructed after steps 1 and 2. Certain noise channels, indicated in red, have been sampled as $\mathcal{N}_2 \circ \mathcal{N}_{0,0,1/2}$ (step 1), and after this, each $\mathcal{N}_{0,0,1/2}$ channel has been removed and its corresponding initial state in $b$ randomized, as $b_3=0,b_4=1$ (step 2). In (c), we show an intermediate layer of step 3, where the $\mathcal{D}'$ channels, indicated in blue, represent the replacements made in step 3a while $b_3 = 1$ due to step 3b. In (d), we show the final circuit $\tilde{C}_3$ and indicate the disjoint subsets $V_1,V_2,V_3,V_4$ into which it is partitioned for independent state vector simulation (steps 4 and 5).
• Figure 5: Using our analysis, the phase transition occurs at $d^* \approx 33$ for $p=0.05$ and $d^* = 117$ for $p=0.02$. In the figures on the left, we plot the size of the largest subcircuit ($\max_i |V_i|$) on a logarithmic scale against the depth of the circuit for different values of $n$. Before percolation occurs, the size of the largest subcircuit should decay exponentially with depth, because the number of non-decohered qubits decays as $\max_i |V_i| \leq (1-2p)^dn$: this is observed in the early linear portion of the plot. After the phase transition, the size should decay further to $\log(n)$ which is observed in the later portion of the plot (note that the y-axis values for each n, which were previously evenly spaced, get closer together). To directly observe the change in scaling with $n$, in the figures on the right, we plot the size of the largest circuit against the number of qubits on a log-log scale for different values of $d$. We see that depths below the phase transition show linear growth of the largest subcircuit with $n$ while depths above the phase transition show logarithmic growth of the largest subcircuit with $n$.

Theorems & Definitions (38)

• Theorem 1
• Corollary 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 8
• proof
• Definition 9: Noisy Circuit Sampling Distribution
• Lemma 10