Quantum supremacy and random circuits

TL;DR

The paper tackles quantum supremacy by proving average-case $\#P$-Hardness for estimating output amplitudes of random quantum circuits under Random Circuit Sampling (RCS). It introduces the Cayley path to interpolate between worst-case circuits and Haar-random average-case instances, enabling a formal reduction from worst-case hardness to average-case hardness. A key technical advance is extending the Berlekamp–Welch algorithm to rational functions, together with a TVD analysis showing that Cayley-path circuits remain close to Haar distributions, which supports robust hardness bounds even in the presence of additive errors. The results yield concrete robustness: additive-error hardness bounds $\epsilon=2^{-\Omega(m^2)}$ for depth-appropriate regimes, strengthening the case for quantum supremacy and offering a rigorous criterion against classical simulations that align with current experimental scales.

Abstract

As Moore's law reaches its limits, quantum computers are emerging with the promise of dramatically outperforming classical computers. We have witnessed the advent of quantum processors with over $50$ quantum bits (qubits), which are expected to be beyond the reach of classical simulation. Quantum supremacy is the event at which the old Extended Church-Turing Thesis is overturned: A quantum computer performs a task that is practically impossible for any classical (super)computer. The demonstration requires both a solid theoretical guarantee and an experimental realization. The lead candidate is Random Circuit Sampling (RCS), which is the task of sampling from the output distribution of random quantum circuits. Google recently announced a $53-$qubit experimental demonstration of RCS. Soon after, classical algorithms appeared that challenge the supremacy of random circuits by estimating their outputs. How hard is it to classically simulate the output of random quantum circuits? We prove that estimating the output probabilities of random quantum circuits is formidably hard ($\#P$-Hard) for any classical computer. This makes RCS the strongest candidate for demonstrating quantum supremacy relative to all other proposals. The robustness to the estimation error that we prove may serve as a new hardness criterion for the performance of classical algorithms. To achieve this, we introduce the Cayley path interpolation between any two gates of a quantum computation and convolve recent advances in quantum complexity and information with probability and random matrices. Furthermore, we apply algebraic geometry to generalize the well-known Berlekamp-Welch algorithm that is widely used in coding theory and cryptography. Our results imply that there is an exponential hardness barrier for the classical simulation of most quantum circuits.

Figures (7)

• Figure 1: Left: The architecture $\mathcal{A}$ is the blue print. Right: Circuit $C$ with architecture $\mathcal{A}$
• Figure 2: Schematics of the Cayley path on the unitary group induced by $C_{k}(\theta)\equiv C_{k}f(\theta h_{k})$.
• Figure 3: The worst- to average-case hardness reduction for a circuit with architecture $\mathcal{A}$
• Figure 4: Plot of the Cayley function in the complex plane (Eq. \ref{['eq:f_x']}). The arrow shows how the function fills the unit circle as $x$ increases from $x=-\infty$. The non-uniform spacing is due to the finite step size in $x$ and aggregation of points at infinity.
• Figure 5: Schematics of Definition \ref{['def:(Deformed-Haar)']}: The scrambling of the circuit $C$ to $C(\theta)$.

Theorems & Definitions (21)

• Remark 1
• Lemma 1
• proof
• proof
• Remark 2
• Theorem
• Lemma 2
• proof
• Conjecture 1
• Lemma 3