Characterizing Quantum Supremacy in Near-Term Devices

TL;DR

The paper investigates quantum supremacy in the near term by framing sampling from output distributions of universal random quantum circuits as a concrete computational task. It introduces cross-entropy difference as a practical benchmark that links experimental data to the ideal circuit distribution and shows that chaotic circuit outputs follow Porter-Thomas statistics, making classical simulation exponentially hard. Through complexity-theoretic arguments and a detailed Ising-model partition-function mapping, it argues that efficient classical sampling is unlikely under widely held conjectures, while numerical evidence up to 42 qubits demonstrates Porter-Thomas-like behavior and measurable fidelity insights via cross entropy. The work provides a concrete, scalable framework for validating quantum supremacy in noisy, intermediate-scale devices and a path to extrapolate to larger quantum systems.

Abstract

A critical question for the field of quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of state-of-the-art classical computers, achieving so-called quantum supremacy. We study the task of sampling from the output distributions of (pseudo-)random quantum circuits, a natural task for benchmarking quantum computers. Crucially, sampling this distribution classically requires a direct numerical simulation of the circuit, with computational cost exponential in the number of qubits. This requirement is typical of chaotic systems. We extend previous results in computational complexity to argue more formally that this sampling task must take exponential time in a classical computer. We study the convergence to the chaotic regime using extensive supercomputer simulations, modeling circuits with up to 42 qubits - the largest quantum circuits simulated to date for a computational task that approaches quantum supremacy. We argue that while chaotic states are extremely sensitive to errors, quantum supremacy can be achieved in the near-term with approximately fifty superconducting qubits. We introduce cross entropy as a useful benchmark of quantum circuits which approximates the circuit fidelity. We show that the cross entropy can be efficiently measured when circuit simulations are available. Beyond the classically tractable regime, the cross entropy can be extrapolated and compared with theoretical estimates of circuit fidelity to define a practical quantum supremacy test.

Figures (15)

• Figure 1: Example of a random quantum circuit in a 1D array of qubits. Vertical lines correspond to controlled-phase (${\rm CZ}$) gates (see Sec. \ref{['sec:pt']}).
• Figure 2: Distribution function of rescaled probabilities $Np$ to observe individual bit-strings as an output of a typical random circuit. Blue curve ($r=0$) shows the distribution of $\{N p_U(x_j)\}$ obtained from numerical simulations of the ideal random circuit (see Sec. \ref{['sec:pt']}) . This distribution is very close to the Porter-Thomas form ${\rm Pr}(Np) = e^{-N p}$ shown with blue dots. Curves with different colors show the distributions of probabilities obtained for different Pauli error rates $r$. The dashed line at $Np = 1$ corresponds to the uniform distribution $\delta(p-1/N)$. These numerics are obtained from simulations of a planar circuit with $5 \times 4$ qubits and gate depth of 40 ($n=20$ and $N = 2^{20}$).
• Figure 3: The blue line shows the probabilities $p_U(x_j)$ of bit-strings $x_j$ sorted in ascending order. The red line shows the corresponding probabilities after adding a Pauli error ($X$ or $Z$) in a single location in the circuit, using the same ordering. The circuit used has $5 \times 4$ qubits and depth 40 (see Sec. \ref{['sec:pt']}). We average over all possible error locations. The average over errors gives almost the uniform distribution. The small residual correlation (slight upper curvature seen in the red line) is analyzed numerically in App. \ref{['app:correlations']}.
• Figure 4: The circuit fidelity $\alpha$ as a function of the number of qubits. Different colors correspond to different Pauli error rates $r_2 = r_{\rm init} = r_{\rm mes} = r$ and $r_1 = r/10$. Circular markers correspond to the numerically simulated fidelities, Eq. \ref{['eq:6']}. Square markers correspond to the average cross entropy difference among 10 instances, Eq. \ref{['eq:5']}. The circuit depth in these simulations is 40 (see Sec. \ref{['sec:pt']}). The red line, at 48 qubits, is a reasonable estimate of the largest size that can be simulated with state-of-the-art classical supercomputers in practice. Using state-of-the-art superconducting circuits we expect $\alpha \gtrsim 0.1$ (blue line) for a $7 \times 7$ circuit. Error bars correspond to the standard deviation among instances.
• Figure 5: Probability distribution of $\log(Np_U(x))$ where bit-strings $x$ are sampled from a circuit of fidelity $\alpha$. The continuous step histograms are obtained from numerical simulations with different Pauli error rates $r_2 = r_{\rm init} = r_{\rm mes} = r$ and $r_1 = r/10$. The values of $r$ are $r=0$ for $\alpha=1$ (blue), $r=0.005$ for $\alpha=0.43$ (red), $r=0.01$ for $\alpha=0.18$ (green) and uniform sampling of bit-strings for $\alpha=0$. The value of $\alpha$ is estimated using Eq. \ref{['eq:6']}. The superimposed dashed lines correspond to the theoretical distribution of Eq. \ref{['eq:pal']}. We chose a circuit of $5 \times 4$ qubits and depth 40 (see Sec. \ref{['sec:pt']}).