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arXiv:2311.17991 · Physical review. D/Physical review. D.

Sachdev-Ye-Kitaev model on a noisy quantum computer

Muhammad Asaduzzaman, Raghav G. Jha, Bharath Sambasivam

Published May 1, 2024• 26 citations
arXivHTMLPDF

Abstract

We study the Sachdev-Ye-Kitaev (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 $\mathcal{O}({N}^{5}{J}^{2}{t}^{2}/\ensuremath{\epsilon})$ where $J$ is the dimensionful coupling parameter, $t$ is the evolution time, and $\ensuremath{\epsilon}$ is the desired precision. With this improved resource requirement, we perform the time evolution for $N=6$, 8 with maximum two-qubit circuit depth and gate count 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 vacuum return probability after time $t$ and out-of-time order correlators which is a standard observable of quantifying the chaotic nature of quantum systems.

DOI
10.1103/physrevd.109.105002

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We study the Sachdev-Ye-Kitaev (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 NNN Majorana fermions is O(N5J2t2/\ensuremathϵ)\mathcal{O}({N}^{5}{J}^{2}{t}^{2}/\ensuremath{\epsilon})O(N5J2t2/\ensuremathϵ) where JJJ is the dimensionful coupling parameter, ttt is the evolution time, and \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ is the desired precision. With this improved resource requirement, we perform the time evolution for N=6N=6N=6, 8 with maximum two-qubit circuit depth and gate count 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 vacuum return probability after time ttt and out-of-time order correlators which is a standard observable of quantifying the chaotic nature of quantum systems.
The Sachdev–Ye–Kitaev (SYK) model is a prominent model of strongly interacting fermions that serves as a toy model of quantum gravity and black hole physics. In this work, we study the Trotter error and gate complexity of the quantum simulation of the SYK model using Lie–Trotter–Suzuki formulas. Building on recent results by Chen and Brandão \cite{Anthony} — in particular their uniform smoothing technique for random matrix polynomials — we derive bounds on the first- and higher-order Trotter error of the SYK model, and subsequently find near-optimal gate complexities for simulating these models using Lie–Trotter–Suzuki formulas. For the k -local SYK model on n Majorana fermions, at time t , our gate complexity estimates for the first-order Lie–Trotter–Suzuki formula scales with O ~ ( n k + 5 2 t 2 ) for even k and O ~ ( n k + 3 t 2 ) for odd k , and the gate complexity of simulations using higher-order formulas scales with O ~ ( n k + 1 2 t ) for even k and O ~ ( n k + 1 t ) for odd k . Given that the SYK model has Θ ( n k ) terms, these estimates are close to optimal. These gate complexities can be further improved upon in the context of simulating the time evolution of an arbitrary fixed input state | ψ ⟩ , leading to a O ( n 2 ) -reduction in gate complexity for first-order formulas and O ( n ) -reduction for higher-order formulas.We also apply our techniques to the sparse SYK model, which is a simplified variant of the SYK model obtained by deleting all but a Θ ( n ) fraction of the terms in a uniformly i.i.d. manner. We find the average (over the random term removal) gate complexity for simulating this model using higher-order formulas scales with
Abstract The Sachdev-Ye-Kitaev (SYK) model describes a strongly correlated quantum system that shows a strong signature of quantum chaos. Due to its chaotic nature, the simulation of real-time dynamics becomes quickly intractable by means of classical numerics, and thus, quantum simulation is deemed to be an attractive alternative. Nevertheless, quantum simulations of the SYK model on noisy quantum processors are severely limited by the complexity of its Hamiltonian. In this work, we simulate the real-time dynamics of a sparsified version of the SYK model with 24 Majorana fermions on a trapped-ion quantum processor. We adopt a randomized quantum algorithm, TETRIS, and develop an error mitigation technique tailored to the algorithm. Leveraging the hardware’s high-fidelity quantum operations and all-to-all connectivity of the qubits, we successfully calculate the Loschmidt amplitude for sufficiently long times so that its decay is observed. Based on the experimental and further numerical results, we assess the future possibility of larger-scale simulations of the SYK model by estimating the required quantum resources. Moreover, we present a scalable mirror-circuit benchmark based on the randomized SYK Hamiltonian and the TETRIS algorithm, which we argue provides a better estimate of the decay of fidelity for local observables than standard mirror-circuits.
We show that one can quantum simulate the dynamics of a Sachdev-Ye-Kitaev model with NNN Majorana modes for time ttt to precision $\ensuremath{\epsilon}$ with gate complexity $O({N}^{7/2}t+{N}^{5/2}t\phantom{\rule{0.16em}{0ex}}\mathrm{polylog}(N/\ensuremath{\epsilon}))$. In addition to scaling sublinearly in the number of Hamiltonian terms, this gate complexity represents an exponential improvement in $1/\ensuremath{\epsilon}$ and large polynomial improvement in NNN and ttt over prior state-of-the-art algorithms which scale as $O({N}^{10}{t}^{2}/\ensuremath{\epsilon})$. Our approach involves a variant of the qubitization technique in which we encode the Hamiltonian HHH as an asymmetric projection of a signal oracle UUU onto two different signal states prepared by state oracles, $A|0\ensuremath{\rangle}\ensuremath{\mapsto}|A\ensuremath{\rangle}$ and $B|0\ensuremath{\rangle}\ensuremath{\mapsto}|B\ensuremath{\rangle}$, such that $H=\ensuremath{\langle}B|U|A\ensuremath{\rangle}$. Our strategy for applying this method to the Sachdev-Ye-Kitaev model involves realizing BBB using only Hadamard gates and realizing AAA as a random quantum circuit.
We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two q^\hat{q}q^​-body Sachdev-Ye-Kitaev (SYK) models with general complex couplings, one the complex conjugate of the other, living in off-diagonal blocks of a larger WSYK Hamiltonian. In the limit of large number NNN of Majoranas and q^∝N\hat{q} \propto \sqrt Nq^​∝N​, we employ diagrammatic and combinatorial techniques to show that the spectral density of the model can be approximated by the weight function of the Al-Salam-Chihara Q-Laguerre polynomials. For q^=2,3,4\hat{q} = 2, 3, 4q^​=2,3,4, the analytical spectral density, with Q=Q(q^,N)Q=Q(\hat{q},N)Q=Q(q^​,N) computed analytically, agrees well with numerical results obtained by exact diagonalization. The spectrum is positive with a hard edge at zero energy and, for odd q^\hat{q}q^​, low-energy excitations grow as a stretched exponential, though the functional form is different from that of the supersymmetric SYK model. The microscopic spectral density and level statistics close to the hard edge are very well approximated by that of an ensemble of random matrices belonging to non-standard universality classes, which indicates quantum chaotic dynamics.
Quantum computers are expected to be vital for exploring complex dynamics in many-body quantum systems. Thus, validating established results on current quantum computers is essential for evaluating their future utility. Hence, we investigate the entanglement entropy of the Sachdev-Ye-Kitaev (SYK) model, a paradigmatic model of quantum chaos, many-body physics, and holographic duality, in current IBM’s superconducting quantum computers. We implement optimized swap-based many-body interference protocol and randomized measurement protocol tailored for IBM quantum computers’ limited qubit connectivity. Additionally, we employ quantum multi-programming that parallelizes circuit execution to improve the results obtained by the randomized measurement protocol. Finally, by incorporating the quantum error mitigation techniques into our implementation of the entropy measurement protocols on IBM quantum hardware, we show that the current noisy quantum computer can yield results aligned with theoretical expectations, therefore affirming its capability to explore chaotic quantum dynamics in complex quantum systems. • Measurement of the real-time dynamics of the entanglement entropy (second Rényi entropy) in the SYK Model using IBM’s superconducting quantum computers. • Implementation of the optimized SWAP-based many-body interference protocol designed for limited qubit connectivity to measure the entanglement entropy using IBM’s superconducting quantum computers. • Using quantum multi-programming to parallelize circuit execution to improve the randomized measurement protocol in determining the entanglement entropy.
Abstract The Sachdev–Ye–Kitaev (SYK) model, known for its strong quantum correlations and chaotic behavior, serves as a key platform for quantum gravity studies. However, variationally preparing thermal states on near-term quantum processors for large systems ( N > 12, where N is the number of Majorana fermions) presents a significant challenge due to the rapid growth in the complexity of parameterized quantum circuits. This paper addresses this challenge by integrating reinforcement learning (RL) with convolutional neural networks, employing an iterative approach to optimize the quantum circuit and its parameters. The refinement process is guided by a composite reward signal derived from entropy and the expectation values of the SYK Hamiltonian. This approach reduces the number of controlled-NOT gates by two orders of magnitude for systems N ⩾ 12 compared to traditional methods like first-order Trotterization. We demonstrate the effectiveness of the RL framework in both noiseless and noisy quantum hardware environments, maintaining high accuracy in thermal state preparation. This work advances a scalable, RL-based framework with applications for quantum gravity studies and out-of-time-ordered thermal correlators computation in quantum many-body systems on near-term quantum hardware. The code is available at solving_SYK_with_RL repository.