# Does Preparing the Maldacena-Qi Ground State Require Non-Unitary Dynamics?

Yes — and that conclusion carries consequences well beyond one paper. Researchers from the Instituto de Física de São Carlos, Universidade de São Paulo, UNESP (São Paulo State University), and affiliated institutions have demonstrated that standard feedback-based quantum algorithms — specifically FALQON and TR-FALQON — cannot converge to the highly [entangled](https://quantumintel.tech/glossary/entanglement) ground state of the two-coupled Sachdev-Ye-Kitaev (SYK) model, also known as the Maldacena-Qi model, when initialized in trivial product states. The failure is not a bug in the algorithms; it is a structural feature of the model's energy landscape. The team's central finding, stated explicitly in their numerical results, is that "the introduction of non-unitary dynamics is strictly necessary to break symmetry traps and filter out excited states." To overcome this, they developed ITE-TR-FALQON — a [hybrid quantum-classical](https://quantumintel.tech/glossary/hybrid-quantum-classical) protocol that integrates imaginary-time evolution (ITE) with time-rescaling. Their approach achieves fidelities close to unity and reproduces both the von Neumann and Rényi entropy spectra of the exact Thermofield Double (TFD) state with high precision. For the broader [NISQ](https://quantumintel.tech/glossary/nisq)-era hardware community, this means that any platform targeting wormhole-analog simulations or TFD-based benchmarks will need a credible path to implementing non-unitary operations — a non-trivial engineering demand.

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## What Is the Maldacena-Qi Model and Why Does It Matter for Quantum Hardware?

The Maldacena-Qi model is the two-coupled SYK model — two SYK systems connected by a coupling term — and it is holographically dual to a traversable wormhole in AdS₂ spacetime. That duality makes it one of the most direct computational handles on quantum gravity that currently exists. The ground state of this model is, to good approximation, the Thermofield Double state: a maximally entangled state across two subsystems that encodes thermal correlations at a specific inverse temperature.

The TFD state is more than a curiosity from holography. It has concrete applications in quantum information: it serves as a basis for certain quantum error correction codes, and it provides a controlled benchmark for testing scrambling dynamics and information propagation in quantum processors. Efficient, high-fidelity TFD preparation is therefore both a physics milestone and a hardware stress test. Any processor that can prepare a high-fidelity TFD is simultaneously validating its [gate fidelity](https://quantumintel.tech/glossary/gate-fidelity), [coherence time](https://quantumintel.tech/glossary/coherence-time), and mid-circuit measurement capabilities under a workload that has genuine theoretical grounding.

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## Why FALQON and TR-FALQON Fail Here

FALQON (Feedback-based ALgorithm for Quantum OptimizatioN) and its time-rescaled variant TR-FALQON are designed to iteratively lower a system's energy through feedback-driven unitary evolution. On simpler many-body systems they perform well. The Maldacena-Qi model is a different category of problem.

When initialized in a trivial product state — the natural starting point for any near-term quantum device — the feedback-based algorithms get trapped. The source material is specific on the mechanism: the algorithms face "severe kinetic limitations," stalling in symmetry traps and excited-state manifolds. Critically, the researchers found that increasing computational resources does not resolve the problem. The algorithms remain stalled regardless. This is not a scaling issue; it is a topological one. The energy landscape of the coupled SYK model contains barriers that purely unitary, feedback-driven dynamics cannot cross from a product-state initialization.

This finding has a sharp implication for the variational quantum eigensolver (VQE) and QAOA communities more broadly: for systems with sufficiently complex energy landscapes, unitary ansatz circuits initialized at simple product states may be structurally incapable of reaching the correct ground state, irrespective of optimizer choice or circuit depth.

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## The ITE-TR-FALQON Protocol: What It Does Differently

The team led by Guilherme E. L. Pexe, alongside Lucas A. M. Rattighieri, Felipe F. Fanchini, Dario Rosa, and Amilson R. Fritsch, developed ITE-TR-FALQON by fusing two mechanisms:

**Imaginary-time evolution (ITE):** Unlike real-time unitary evolution, ITE is inherently non-unitary. It filters the quantum state toward lower-energy components by evolving under e^(−τH) rather than e^(−itH). In the limit of large imaginary time τ, a generic state projects onto the ground state — provided the overlap is nonzero. This is the operation that breaks the symmetry traps and eliminates contamination from excited states.

**Time-rescaling:** This component, inherited from TR-FALQON, drastically accelerates algorithm convergence. It effectively rescales the time parameter to navigate the feedback loop more efficiently, reducing the number of iterations needed before the state approaches the ground manifold.

The combined protocol achieves fidelities close to unity with the exact TFD state and reproduces both von Neumann and Rényi entropy spectra with high precision. Those entropy metrics are particularly important: the Rényi entropy spectrum encodes the full entanglement structure of the state, not merely its average entanglement. Matching it means the prepared state genuinely captures the thermal correlations of the Maldacena-Qi ground state, not merely a superficially similar approximation.

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## Hardware Implications: Who Can Actually Run This?

Here is where the analysis diverges from the paper's immediate scope. ITE is non-unitary, which means it cannot be directly implemented as a standard quantum circuit on hardware that only supports unitary gates and projective measurement. Implementing ITE on real quantum hardware typically requires one of several strategies:

- **Ancilla-based dilation:** Embed the non-unitary operation in a larger unitary acting on system plus ancilla qubits, followed by post-selection on the ancilla. This is probabilistic and resource-intensive.
- **Quantum imaginary-time evolution (QITE) algorithms:** Approximate ITE using sequences of local unitary operations. These require additional circuit depth and repeated measurement cycles.
- **Mid-circuit measurement and classical feedback:** Some trapped-ion and superconducting platforms now support this natively, enabling measurement-based non-unitary operations without ancilla dilation.

Platforms with high-fidelity mid-circuit measurement and reset — currently a differentiating feature of trapped-ion systems and a growing capability on superconducting architectures — are the most natural candidates for implementing ITE-TR-FALQON on hardware. The coherence time demands are also significant: ITE-based preparation of a highly entangled many-body state is not a shallow circuit task.

This is not a near-term plug-and-play result. It is a research protocol establishing what is *necessary*, and the hardware community will need to treat the non-unitary requirement as a genuine engineering constraint rather than an algorithmic detail.

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## Broader Trajectory: What This Tells Us About Quantum Gravity Simulation

The Maldacena-Qi model is not the only quantum gravity analog being targeted for quantum hardware. The Google Quantum AI group's 2022 wormhole-signaling experiment (on a superconducting processor) used a heavily truncated and classically pretrained version of the SYK model — precisely because full ground-state preparation of the genuine model was not tractable. The Brazilian team's work makes the preparation problem explicit and proposes a resolution, which is a meaningful step toward closing that gap.

More broadly, this paper joins a growing body of evidence that the NISQ paradigm's reliance on purely unitary, classically initialized variational circuits has structural limits. Thermofield Double states, topological ground states, and other highly entangled many-body targets appear to consistently require either non-unitary dynamics, carefully engineered initial states, or both. The fault-tolerant era, with access to mid-circuit measurement, [logical qubit](https://quantumintel.tech/glossary/logical-qubit) operations, and deeper circuits, may be where these simulations become tractable at scale.

For enterprise buyers and research groups evaluating quantum platforms: prioritize vendors with documented mid-circuit measurement fidelity and reset capabilities if quantum gravity simulation or TFD-based benchmarking is on your roadmap.

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## Key Takeaways

- **Non-unitary dynamics are not optional** for preparing the Maldacena-Qi (two-coupled SYK) ground state from a product state — the researchers' own numerical results state this explicitly.
- **FALQON and TR-FALQON both fail** on this model due to severe kinetic limitations and symmetry traps; adding computational resources does not resolve the stalling.
- **ITE-TR-FALQON** — integrating imaginary-time evolution with time-rescaling — achieves fidelities close to unity and accurately reproduces the von Neumann and Rényi entropy spectra of the Thermofield Double state.
- **The Thermofield Double state** benchmarks scrambling dynamics, informs QEC code construction, and is dual to a traversable wormhole in AdS₂ — making high-fidelity preparation both theoretically and practically significant.
- **Hardware platforms** with mid-circuit measurement and reset capabilities are the natural implementation target; purely unitary gate-based processors face a structural mismatch with ITE requirements.
- **Research institutions** involved: Instituto de Física de São Carlos, Universidade de São Paulo, UNESP (São Paulo State University), and affiliated institutions.
- **For the variational algorithm community**, this is a data point that complex energy landscapes can be topologically inaccessible to unitary feedback-based methods regardless of optimizer or circuit depth.

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## Frequently Asked Questions

**What is the Maldacena-Qi model in simple terms?**
It is two copies of the Sachdev-Ye-Kitaev (SYK) model — a maximally chaotic quantum many-body system — coupled together. The coupled system is mathematically equivalent (dual) to a traversable wormhole in two-dimensional anti-de Sitter spacetime, making it one of the most direct quantum computing handles on quantum gravity.

**What is the Thermofield Double state and why is it hard to prepare?**
The Thermofield Double is a maximally entangled state across two subsystems encoding thermal correlations. It is the approximate ground state of the Maldacena-Qi model. It is hard to prepare because standard unitary feedback algorithms get kinetically trapped when starting from simple, unentangled product states — the kind of states quantum hardware naturally initializes in.

**What does imaginary-time evolution do that real-time evolution cannot?**
Real-time evolution under a Hamiltonian H is unitary (e^(−itH)) and preserves the overlap structure of the initial state. Imaginary-time evolution (e^(−τH)) is non-unitary and contracts the state toward lower-energy components, effectively filtering out excited states. This is why ITE can escape symmetry traps that unitary algorithms cannot.

**Can current quantum hardware implement ITE-TR-FALQON?**
Not directly as a standard circuit. ITE requires non-unitary operations, which on gate-based hardware means ancilla-based dilation with post-selection, QITE approximation schemes with additional circuit depth, or mid-circuit measurement-and-feedback approaches. Platforms with high-fidelity mid-circuit measurement (some trapped-ion and superconducting systems) are the most viable current candidates.

**Does this result affect variational quantum algorithms more broadly?**
Yes, as an important cautionary data point. The finding that FALQON and TR-FALQON cannot reach the correct ground state regardless of added computational resources suggests that purely unitary variational methods have structural limitations on sufficiently complex many-body landscapes — a concern that extends to VQE and QAOA applied to strongly correlated systems.