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arXiv:2611.03207

Spin SYK-dot chains: entanglement bound and resource estimates towards a quantum advantage

Sayantan Dey, Abel Carreras, Siddhartha Patra

quant-phcond-mat.str-elSubmitted Nov 2, 2026• Announced Nov 3, 2026
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Abstract

Ground states of Sachdev-Ye-Kitaev (SYK)-type models are volume-law entangled, which makes them costly both for tensor networks and for quantum hardware simulations. Thus, it remains unclear where quantum hardware could outperform classical simulation of such models. In this work, we study a spin SYK-dot chain, in which q dense SYK dots of p qubits are joined by single ZZ bonds of strength K. The entanglement across any cut is then bounded by floor(p/2) ln 2 plus the inter-dot entanglement, and the two-qubit depth of one Trotter step is set by p alone. A resource estimate shows that a quantum advantage for the ground state could appear only for dots of about 15 to 21 qubits or more. There, a single Trotter step already requires two-qubit error rates of order 10^{-6}, beyond the reach of noisy intermediate-scale quantum (NISQ) hardware. We test the assumptions of this estimate with an imaginary-time matrix product state (MPS) that applies each dot as one block. It agrees with exact diagonalisation at (p, q) = (6, 2). At (5, 2), a variational circuit with the same dot-plus-link layout comes within 3.6% of the exact energy, and noisy simulations show that its fidelity decays as assumed in the resource estimate. At (8, 5), i.e., N = 40 qubits, the energy converges at bond dimension chi = 768, far below the exact value 2^20, and the inter-dot entanglement stays small and is set mainly by K. Out-of-time-order correlators across the links decay on a time scale that shrinks rapidly with |K|. Thus, the dot size p and the inter-dot coupling K together set the entanglement of the chain, and with it where a quantum advantage could appear.

Comments
17 pages, 13 figures

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Ground states of Sachdev-Ye-Kitaev (SYK)-type models are volume-law entangled, which makes them costly both for tensor networks and for quantum hardware simulations. Thus, it remains unclear where quantum hardware could outperform classical simulation of such models. In this work, we study a spin SYK-dot chain, in which q dense SYK dots of p qubits are joined by single ZZ bonds of strength K. The entanglement across any cut is then bounded by floor(p/2) ln 2 plus the inter-dot entanglement, and the two-qubit depth of one Trotter step is set by p alone. A resource estimate shows that a quantum advantage for the ground state could appear only for dots of about 15 to 21 qubits or more. There, a single Trotter step already requires two-qubit error rates of order 10^{-6}, beyond the reach of noisy intermediate-scale quantum (NISQ) hardware. We test the assumptions of this estimate with an imaginary-time matrix product state (MPS) that applies each dot as one block. It agrees with exact diagonalisation at (p, q) = (6, 2). At (5, 2), a variational circuit with the same dot-plus-link layout comes within 3.6% of the exact energy, and noisy simulations show that its fidelity decays as assumed in the resource estimate. At (8, 5), i.e., N = 40 qubits, the energy converges at bond dimension chi = 768, far below the exact value 2^20, and the inter-dot entanglement stays small and is set mainly by K. Out-of-time-order correlators across the links decay on a time scale that shrinks rapidly with |K|. Thus, the dot size p and the inter-dot coupling K together set the entanglement of the chain, and with it where a quantum advantage could appear.
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We establish tight connections between entanglement entropy and the approximation error in Trotter-Suzuki product formulas for Hamiltonian simulation. Product formulas remain the workhorse of quantum simulation on near-term devices, yet standard error analyses yield worst-case bounds that can vastly overestimate the resources required for structured problems. For systems governed by geometrically local Hamiltonians with maximum entanglement entropy SmaxS_\text{max}Smax​ across all bipartitions, we prove that the first-order Trotter error scales as O(t2Smaxpolylog⁡(n)/r)\mathcal{O}(t^2 S_\text{max} \operatorname{polylog}(n)/r)O(t2Smax​polylog(n)/r) rather than the worst-case O(t2n/r)\mathcal{O}(t^2 n/r)O(t2n/r), where nnn is the system size and rrr is the number of Trotter steps. This yields improvements of Ω~(n2)\tildeΩ(n^2)Ω~(n2) for one-dimensional area-law systems and Ω~(n3/2)\tildeΩ(n^{3/2})Ω~(n3/2) for two-dimensional systems. We extend these bounds to higher-order Suzuki formulas, where the improvement factor involves 2pS∗/22^{pS^*/2}2pS∗/2 for the ppp-th order formula. We further establish a separation result demonstrating that volume-law entangled systems fundamentally require Ω~(n)\tildeΩ(n)Ω~(n) more Trotter steps than area-law systems to achieve the same precision. This separation is tight up to logarithmic factors. Our analysis combines Lieb-Robinson bounds for locality, tensor network representations for entanglement structure, and novel commutator-entropy inequalities that bound the expectation value of nested commutators by the Schmidt rank of the state. These results have immediate applications to quantum chemistry, condensed matter simulation, and resource estimation for fault-tolerant quantum computing.
We aim to rigorously address how local noise affects classical simulability of quantum dynamics benchmarked by tensor-network methods. Using operator entanglement entropy (OEE), we prove the following: (1) For single-qubit depolarizing noise on arbitrary circuits, tensor networks with poly(n)\mathrm{poly}(n)poly(n) bond dimension suffice for fixed absolute Hilbert-Schmidt error after $\order{1}$ depth, while relative error demands $\order{\log n}$ depth; and this bound is optimal. (2) For single-qubit depolarizing noise on 1D local circuits, the existence of whole-trajectory error-bounded matrix product operator (MPO) of poly(n)\mathrm{poly}(n)poly(n) bond dimension at all depths. (3) For general single-qubit noise in 1D brickwall circuits, random two-design gates with contraction coefficient c<1/3c<1/3c<1/3 yield an $\order{1}$ OEE plateau with probability 1−Te−Ω(n)1-Te^{-Ω(n)}1−Te−Ω(n), while arbitrary gates with c<1/48c<1/48c<1/48 give $\order{\log n}$ OEE in the worst case. (4) In higher dimensions, these bounds yield uniform-in-depth poly(n)\mathrm{poly}(n)poly(n) average boundary-bond dimensions for projected entangled pair operators~(PEPO) across every cut -- under depolarizing noise at either absolute or relative accuracy, and under general noise with strong contraction at absolute accuracy. Our results establish a rigorous connection between certain noise models, circuit types, and their classical simulability.
The Sachdev-Ye-Kitaev (SYK) model has been identified as a promising candidate to run on early fault-tolerant quantum computers due to the relatively modest resources required to probe non-trivial physics (namely holographic duality and AdS/CFT correspondence). As such, it is crucial that the details of how to run such a simulation are well understood. Using PsiQuantum's Construct platform, we implement and analyze three different approaches to simulate the SYK model: Trotterization, qDRIFT, and asymmetric qubitization with Quantum Signal Processing. We provide an open-source library containing implementations for SYK simulation using all three methods, which we use to obtain quantum resource estimates for qubit and T gate count as functions of the number of Majorana modes and precision. We find that while qDRIFT and Trotterization benefit from a lower qubit count, the large number of T gates required lead to asymmetric qubitization being advantageous in most cases. This reinforces previous theoretical considerations. We intend both the implementations and the estimates to be useful for researchers to continue to study the SYK model and understand how the techniques and resources vary.
The time evolution of quantum many-body systems is one of the most important processes for benchmarking quantum simulators. The most curious feature of such dynamics is the growth of quantum entanglement to an amount proportional to the system size (volume law) even when interactions are local. This phenomenon has great ramifications for fundamental aspects, while its optimisation clearly has an impact on technology (e.g., for on-chip quantum networking). Here we use an integrated photonic chip with a circuit-based approach to simulate the dynamics of a spin chain and maximise the entanglement generation. The resulting entanglement is certified by constructing a second chip, which measures the entanglement between multiple distant pairs of simulated spins, as well as the block entanglement entropy. This is the first photonic simulation and optimisation of the extensive growth of entanglement in a spin chain, and opens up the use of photonic circuits for optimising quantum devices.
The ferromagnetic state of an Ising chain can represent a twofold degenerate subspace or equivalently a logical qubit that is protected from excitations by an energy gap. Here, we study a braiding-like exchange operation through the movement of the ferromagnetic state in the qubit subspace, which resembles that of the localized edge modes in a Kitaev chain. The system consists of two Ising chains in a one-dimensional geometry where the operation is simulated through the adiabatic time evolution of the ground state. The time evolution is implemented via the Suzuki-Trotter expansion on basic single- and two-qubit quantum gates using IBM's Aer QASM simulator. The fidelity of the system is investigated as a function of the evolution and system parameters to obtain optimum efficiency and accuracy for different system sizes. Various aspects of the implementation, including the circuit depth, Trotterization error, and quantum gate errors pertaining to noisy intermediate-scale quantum (NISQ) hardware, are discussed as well. We demonstrate that the quantum gate errors, i.e., bit-flip, phase errors, are the dominating factor in determining the fidelity of the system as the Trotter error and the adiabatic condition are less restrictive even for modest values of Trotter time steps. We reach an optimum fidelity >99>99%>99 on systems of up to 11 sites per Ising chain. We show that running such simulations is far beyond the reach of current NISQ hardware with the most efficient implementation of a single braiding-like operation for a fidelity above 9090%90 requiring a circuit depth of the order of $\ensuremath{\sim}{10}^{3}$ restricting the individual gate errors to be less than $\ensuremath{\sim}{10}^{\ensuremath{-}6}$.