Sachdev-Ye-Kitaev model on a noisy quantum computer (2311.17991v4)

Abstract: We study the SYK model -- an important toy model for quantum gravity on IBM's superconducting qubit quantum computers. By using a graph-coloring algorithm to minimize the number of commuting clusters of terms in the qubitized Hamiltonian, we find the gate complexity of the time evolution using the first-order product formula for $N$ Majorana fermions is $\mathcal{O}(N^5 J^{2}t^2/\epsilon)$ where $J$ is the dimensionful coupling parameter, $t$ is the evolution time, and $\epsilon$ is the desired precision. With this improved resource requirement, we perform the time evolution for $N=6, 8$ with maximum two-qubit circuit depth of 343. We perform different error mitigation schemes on the noisy hardware results and find good agreement with the exact diagonalization results on classical computers and noiseless simulators. In particular, we compute return probability after time $t$ and out-of-time order correlators (OTOC) which is a standard observable of quantifying the chaotic nature of quantum systems.

• The paper’s main contribution is reducing circuit complexity from O(N^10) to O(N^5) by applying a graph-coloring algorithm.
• The study demonstrates quantum simulations of the SYK model for N=6 and N=8 Majorana fermions using a Lie-Trotter-Suzuki approach to optimize Trotter steps.
• Error mitigation techniques such as Pauli twirling and randomized compiling are employed to align noisy quantum outputs with exact diagonalization results.

Analysis of the Sachdev-Ye-Kitaev Model on Noisy Quantum Computers

This paper presents an exploration of the Sachdev-Ye-Kitaev (SYK) model through IBM's superconducting qubit quantum computers, focusing on the optimization of quantum circuit complexity and error mitigation. The SYK model, with its significant implications in quantum gravity and chaos theory, serves as an ideal testbed for demonstrating the potential and challenges of current quantum computing hardware.

The paper's primary contribution is the enhancement of circuit complexity for simulating the SYK model. The authors demonstrate a significant improvement by reducing the complexity scaling from $\mathcal{O}(N^{10})$ to $\mathcal{O}(N^{5})$ when mapping the Hamiltonian of the SYK model to qubits using a graph-coloring algorithm. This improvement is achieved without sacrificing the fidelity of the simulation, which is critical for handling larger instances of the SYK model.

For practical simulations, the authors execute quantum circuits on IBM processors, demonstrating time evolution calculations for the SYK model with $N = 6$ and $N = 8$ Majorana fermions. They implemented a first-order Lie-Trotter-Suzuki product formula while optimizing the number of Trotter steps to reduce error accumulation. They found the necessary ECR gate depth for these computations was 343 for maximum depth, showcasing the hardware's capability in handling complex quantum simulations but also highlighting the substantial resources required even for relatively small systems.

From an analytical standpoint, the research underscores the critical role of error mitigation in quantum simulations. Basic methods such as Pauli twirling/randomized compiling and self-mitigation are employed to address gate noise and decoherence, demonstrating their effectiveness in aligning quantum computer outputs with exact diagonalization results.

In terms of physical analysis, the researchers compute observables like the vacuum return probability and the out-of-time order correlators (OTOC), which are essential metrics for studying the chaotic nature of quantum systems. The paper showcases the ability to replicate expected behaviors such as the ramp feature in return probabilities when averaged over disorder, a characteristic feature of quantum chaotic systems.

The implications of this paper are twofold. Practically, it solidifies the position of superconducting qubit architectures as viable, albeit currently limited, platforms for simulating complex quantum models like SYK. By improving circuit complexity, the research provides a framework that can be scaled with technological advances in coherence times, gate fidelities, and connectivity. Theoretically, the work paves the way for further exploration of quantum chaotic behaviors and the verification of holographic principles in quantum systems, with the eventual goal of bridging gaps between theoretical predictions and empirical observations.

Looking forward, this research invites future exploration into more complex SYK variants with higher fermion numbers and interaction terms. Improved quantum hardware with advanced connectivity, such as trapped-ion systems, may provide additional pathways for simulation with potentially lower gate errors, enabling insights into the holographic duality and strongly coupled field theories. Additionally, further developments in error mitigation and circuit optimization could expand the practical limits of such quantum simulations, allowing for broader applications in quantum many-body physics and beyond.