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arXiv:2607.01653 · Physical review. D/Physical review. D.

Preparing a Thermofield Double State with Feedback Quantum Algorithms

Guilherme E. L. Pexe, Lucas A. M. Rattighieri, Felipe F. Fanchini, Dario Rosa, Amilson R. Fritsch

hep-thquant-phSubmitted Jul 2, 2026• Announced Jul 3, 2026• 0 citations
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Abstract

The efficient preparation of correlated thermal states, such as the Thermofield Double (TFD) state, is a fundamental prerequisite for simulating quantum gravity models and many-body thermodynamics on quantum processors. In this work, we investigate the ground state preparation of the Two Coupled Sachdev-Ye-Kitaev model, known as the Maldacena-Qi model, which is dual to a traversable wormhole in $AdS_2$, utilizing feedback-based quantum algorithms. We demonstrate that the standard feedback-based quantum algorithm (FALQON) and its time-rescaled variant (TR-FALQON) face severe kinetic limitations in this system, failing to converge to the highly entangled ground state when initialized in trivial product states. To overcome these barriers, we propose the hybrid ITE-TR-FALQON protocol, which integrates the imaginary-time evolution present in imaginary-time-enhanced FALQON (ITE-FALQON) with the time-rescaling mechanism. Our numerical results indicate that the introduction of non-unitary dynamics is strictly necessary to break symmetry traps and filter out excited states, while time-rescaling drastically accelerates algorithm convergence. The proposed method achieves fidelities close to unity and reproduces the von Neumann and Rényi entropy spectra of the exact TFD state with high precision.

Comments
10 pages, 4 figures

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arXiv:2102.05687•Physical review. B./Physical review. B•Jul 22, 2021•19 citations
The efficient preparation of correlated thermal states, such as the Thermofield Double (TFD) state, is a fundamental prerequisite for simulating quantum gravity models and many-body thermodynamics on quantum processors. In this work, we investigate the ground state preparation of the Two Coupled Sachdev-Ye-Kitaev model, known as the Maldacena-Qi model, which is dual to a traversable wormhole in AdS2AdS_2AdS2​, utilizing feedback-based quantum algorithms. We demonstrate that the standard feedback-based quantum algorithm (FALQON) and its time-rescaled variant (TR-FALQON) face severe kinetic limitations in this system, failing to converge to the highly entangled ground state when initialized in trivial product states. To overcome these barriers, we propose the hybrid ITE-TR-FALQON protocol, which integrates the imaginary-time evolution present in imaginary-time-enhanced FALQON (ITE-FALQON) with the time-rescaling mechanism. Our numerical results indicate that the introduction of non-unitary dynamics is strictly necessary to break symmetry traps and filter out excited states, while time-rescaling drastically accelerates algorithm convergence. The proposed method achieves fidelities close to unity and reproduces the von Neumann and Rényi entropy spectra of the exact TFD state with high precision.
A major promise of quantum computers is the controlled preparation of many-body quantum states beyond the reach of efficient classical computation. Among the most important targets are thermal mixed states and their thermofield double (TFD) purifications, which play central roles in quantum many-body physics and quantum gravity. For target systems with a bounded energy spectrum that obey the eigenstate thermalization hypothesis (ETH), we present a parent Hamiltonian built from two copies of the target Hamiltonian and ultra-local couplings between the copies, which we argue is gapped with a ground state that approximates a TFD state of the target Hamiltonian. By adiabatically evolving down from strong coupling, we can thus prepare a high-fidelity TFD state. We study two variants of the parent Hamiltonian using numerical methods in two classes of models: mixed field Ising models in one and two dimensions and non-local "spin Sachdev-Ye-Kitaev'' models. In the simpler variant, the parent Hamiltonian ground state has high overlap with a TFD for system sizes accessible to near-term quantum devices. However, the global overlap decays exponentially with the number of qubits, with a small error per degree of freedom. The second variant introduces an additional penalty term which can be tuned to reduce or remove the decay of the overlap with system size. Together with a general ETH-based analysis, these results suggest a broadly applicable method for TFD preparation that is not limited to particular temperatures or geometric locality.
Preparing ground states of strongly correlated quantum systems is a central goal in quantum simulation and optimization. The feedback-based quantum algorithm (FALQON) provides an attractive alternative to variational methods with a fully quantum feedback rule, but it fails in the presence of spectral degeneracies, where the feedback signal collapses and the evolution cannot reach the ground state. Using the Fermi-Hubbard model on lattices up to 3x3, we show that this breakdown appears at half-filling on the 2x2 lattice and extends to both half-filled and doped configurations on the 3x3 lattice. We then introduce an imaginary-time-enhanced FALQON (iiiFALQON) scheme, which inserts short imaginary-time evolution steps into the feedback loop. This proposed method suppresses excited-state components, escapes degenerate subspaces, restores monotonic energy descent and improves ground-state convergence across the tested fillings. We further benchmark its performance against existing methods, showing improved results for iFALQON, demonstrating the potential of hybrid real-imaginary-time feedback dynamics for ground-state preparation in strongly correlated quantum systems.
We propose quantum algorithms for preparing thermal states via the simulation of the thermofield double states. The key idea is to leverage double-bracket quantum algorithms to implement imaginary-time evolution on thermofield double states, whose reduced state realizes the Gibbs state. Our method, termed double-bracket thermofield double (DB-TFD), introduces two variants. The first, the vanilla DB-TFD algorithm, directly implements imaginary-time evolution using double-bracket quantum imaginary-time evolution. The second, poly DB-TFD, employs double-bracket quantum signal processing to approximate the imaginary-time evolution operator via a polynomial transformation. We demonstrate that the complexity of the poly DB-TFD algorithm scales exponentially with the inverse temperature in a broad practical regime. This scaling is consistent with existing methods, and numerical simulations support the corresponding theoretical bound. We further demonstrate the utility of DB-TFD in quantum Boltzmann machines for generative modeling, achieving improved performance compared with variational imaginary-time evolution approaches. These results establish DB-TFD as a promising route for thermal state preparation in the near-term and early-fault-tolerant regimes.
Variational preparation of thermofield double (TFD) states can require deep quantum circuits, particularly for interacting many-body systems. Reducing these circuits while retaining high fidelity is therefore crucial for TFD-state preparation on noisy quantum processors. We study this problem by applying the multi-angle quantum approximate optimization algorithm (ma-QAOA) to TFD-state preparation and introducing two top-down sequential angle-pruning algorithms. Starting from the optimized initial ma-QAOA circuit, both algorithms sequentially remove Pauli-string evolutions with small optimized angles and reoptimize the remaining parameters after each removal. We apply these algorithms to Gaussian and binary Sachdev--Ye--Kitaev (SYK) models in both dense and sparse cases. We find that ma-QAOA prepares the target TFD states with high fidelity and that sequential small-angle pruning retains high fidelity while reducing the circuit depth, particularly at low temperature. Moreover, using the post-reoptimization cost in sequential small-angle pruning further improves the fidelity. For the binary sparse N=10N=10N=10 SYK model at β=10β=10β=10, 88.8%88.8\%88.8%--92.1%92.1\%92.1% of the nonlocal Pauli-string evolutions are removed while retaining an average fidelity of approximately 95%95\%95%. Finally, we propose extensions of the sequential pruning algorithms toward quantum--classical hybrid implementation.
We analyze a simple and efficient experimental protocol to cool the Sachdev-Ye-Kitaev (SYK) model to low temperatures. The protocol utilizes local couplings between two copies of an SYK model to create a gapped adiabatic path, between a high temperature product state and a low temperature thermofield double state. By smoothly varying the coupling strength between these two limits, one efficiently cools the SYK model. We support these predictions-and demonstrate fast cooling to the low-temperature gravitational regime-via exact numerical solutions to the large-N equations of motion that govern the ground state and dynamical properties of the coupled system. Finally, we present a theoretical framework based upon eigenstate thermalization that provides a microscopic explanation for the efficacy of the cooling protocol; intriguingly, this suggests that the protocol may be applicable for cooling strongly-interacting quantum Hamiltonians more broadly.
A pair of identical Sachdev-Ye-Kitaev (SYK) models with bilinear coupling forms a quantum dual of a traversable wormhole, with a ground state close to the so called thermofield double state. We use adiabaticity arguments and numerical simulations to show that the character of the ground state remains unchanged when interaction strengths in the two SYK models become imbalanced. Analysis of thermodynamic and dynamical quantities highlights the key signatures of the wormhole phase in the imbalanced case. Further adiabatic evolution naturally leads to the ``maximally imbalanced'' limit where fermions in a single SYK model are each coupled to a free Majorana zero mode. This limit is interesting because it could be more easily realized using various setups proposed to implement the SYK model in a laboratory. We find, based on numerical studies of the quantum mechanical model, that this limiting case is marginal in that it retains some of the characteristic signatures of the wormhole physics, such as the spectral gap and revival dynamics, but lacks others so that it likely does not represent the full-fledged wormhole dual. We discuss how this scenario could be implemented in the proposed realization of the SYK physics in quantum wires of finite length coupled to a disordered quantum dot.