Quantum Learning: Guiding Systems to Their Stable States

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Author: Denis Avetisyan

Researchers have developed a novel quantum reinforcement learning algorithm to efficiently identify the fixed points of quantum operations, opening new avenues for state preparation and analysis.

Without a reset mechanism, the fidelity of quantum state preparation for random two-qubit Hamiltonians fluctuates within bounds defined by maximum and minimum fidelities-$F_{max}$ and $F_{min}$-but the implementation of a reset strategy, utilizing parameters $r=0.9$, $p=2/r$, $w_{th}=0.005$, $w_{r}=0.01$, and $N_{r}=100$, demonstrably improves performance across all four computational basis states, even though the resulting curves largely overlap due to their proximity.
Without a reset mechanism, the fidelity of quantum state preparation for random two-qubit Hamiltonians fluctuates within bounds defined by maximum and minimum fidelities-$F_{max}$ and $F_{min}$-but the implementation of a reset strategy, utilizing parameters $r=0.9$, $p=2/r$, $w_{th}=0.005$, $w_{r}=0.01$, and $N_{r}=100$, demonstrably improves performance across all four computational basis states, even though the resulting curves largely overlap due to their proximity.

This work demonstrates a reinforcement learning approach for discovering unitary transformations that map computational bases onto fixed points, with applications in finding Hamiltonian eigenstates and adapting to dissipative quantum systems.

Determining the eigenstates of quantum systems remains a significant challenge, particularly for complex many-body Hamiltonians. This work, ‘Exploring fixed points and eigenstates of quantum systems with reinforcement learning’, introduces a novel reinforcement learning algorithm capable of efficiently discovering the fixed points of arbitrary quantum operations-and, crucially, the corresponding Hamiltonian eigenstates. By iteratively learning a unitary transformation mapping the computational basis to the fixed-point basis, the method demonstrates success across systems of up to six qubits, including the transverse-field Ising and all-to-all pairing models, even revealing hidden symmetries within the latter. Could this approach offer a scalable pathway towards solving increasingly complex quantum systems and unlocking new insights into their fundamental properties?

Navigating the Quantum Landscape: The Quest for Stable States

Quantum algorithms frequently hinge on the identification of fixed points within a quantum operation – those specific quantum states that, when acted upon by the operation, remain fundamentally unchanged. These fixed points represent stable configurations within the quantum system’s evolution, acting as critical anchors for computational processes. The search for these states is analogous to finding equilibrium points in a classical system; however, the inherent complexities of quantum mechanics, such as superposition and entanglement, drastically increase the challenge. Consequently, locating these fixed points is not merely a mathematical exercise, but a foundational step enabling the successful execution of numerous quantum computations, including those designed for simulation, optimization, and cryptography. The existence and accessibility of these states often dictate the overall efficiency and scalability of the associated quantum algorithm, making their determination a central focus in the field.

Locating fixed points-states unaffected by quantum transformations-becomes exceedingly difficult as quantum systems grow in complexity. Traditional iterative methods, while effective for simple cases, encounter significant computational bottlenecks when applied to high-dimensional systems. The exponential scaling of Hilbert space with increasing qubits means that exhaustively searching for these stable states quickly becomes intractable. This limitation arises because these methods often require evaluating the quantum operation on a vast number of potential states before converging on a fixed point, or determining its absence. Consequently, the computational cost associated with fixed-point identification can quickly dominate the overall runtime of quantum algorithms that rely on them, hindering progress in fields like quantum simulation and optimization.

The practical utility of numerous quantum algorithms hinges critically on the efficient identification of fixed points – those stable states unaffected by a quantum operation. Because quantum simulations and algorithms often progress by iteratively applying transformations, the time required to locate these fixed points becomes a dominant factor in overall computational cost. A sluggish fixed-point search can quickly negate the potential speed advantages offered by quantum computation, particularly when dealing with complex systems exhibiting many degrees of freedom. Consequently, advancements in methods for rapidly and accurately determining fixed points are not merely theoretical improvements, but essential prerequisites for unlocking the full potential of quantum simulations in fields like materials science, drug discovery, and fundamental physics, enabling the tackling of previously intractable problems and scaling these solutions to realistically sized systems.

Applying the algorithm to the pairing Hamiltonian with specified parameters successfully converges to the exact eigenenergies for systems ranging from two to five qubits, as demonstrated by the convergence of expectation values and the achievement of maximum and minimum fidelities across computational basis states.
Applying the algorithm to the pairing Hamiltonian with specified parameters successfully converges to the exact eigenenergies for systems ranging from two to five qubits, as demonstrated by the convergence of expectation values and the achievement of maximum and minimum fidelities across computational basis states.

Harnessing Quantum Reinforcement Learning for Fixed-Point Discovery

The Quantum Reinforcement Learning algorithm presented utilizes a quantum circuit to approximate a Unitary Transformation, denoted as $U$, which maps states defined in the Computational Basis – typically represented by standard basis states like $|0\rangle$ and $|1\rangle$ – to the fixed-point basis. This transformation is learned adaptively through an iterative process. The algorithm parameterizes $U$ and adjusts its parameters based on a reward signal, effectively searching for the optimal unitary that minimizes the distance between the transformed states and the desired fixed-point states. The key innovation lies in framing the fixed-point search as a reinforcement learning problem, allowing the algorithm to leverage quantum computation to efficiently explore the space of possible unitary transformations.

The Quantum Reinforcement Learning algorithm utilizes a reward strategy to optimize the unitary transformation. This strategy assigns negative rewards proportional to the distance between the current state and the defined fixed points, effectively penalizing deviations. Conversely, the algorithm provides positive rewards when the state converges towards a fixed point, incentivizing the learning agent to refine the transformation and minimize this distance. The magnitude of both penalties and rewards are parameterized, allowing for tunable control over the learning process and enabling exploration versus exploitation trade-offs. This reward function directly guides the agent towards identifying transformations that map input states to the desired fixed-point basis, driving the optimization process within the reinforcement learning framework.

Traditional fixed-point search methods often struggle with high-dimensional spaces and complex landscapes, requiring exhaustive exploration or becoming trapped in local optima. Framing the fixed-point search as a reinforcement learning (RL) task allows for the application of RL algorithms to efficiently navigate the search space. Instead of directly searching for fixed points, the agent learns a policy – a mapping from states to actions – that maximizes a cumulative reward signal. This approach leverages the agent’s ability to learn from experience and generalize across the state space, enabling it to discover fixed points more effectively than deterministic or gradient-based methods. The RL framework also facilitates the incorporation of prior knowledge and the adaptation to changing environments, offering a more robust and scalable solution for fixed-point computation.

This reinforcement learning algorithm iteratively refines a policy by balancing exploration and exploitation to maximize cumulative reward.
This reinforcement learning algorithm iteratively refines a policy by balancing exploration and exploitation to maximize cumulative reward.

Validating the Algorithm with Established Quantum Models

The algorithm’s performance was evaluated using the Transverse Field Ising Model (TFIM) and the Pairing Hamiltonian, both widely utilized as benchmark systems in condensed matter physics. The TFIM, described by the Hamiltonian $H = -J\sum_{\langle i,j \rangle} \sigma^z_i \sigma^z_j – h\sum_i \sigma^x_i$, models interacting spins in a transverse magnetic field and is frequently employed to study quantum phase transitions. The Pairing Hamiltonian, $H = \sum_k (\epsilon_k – \mu) c^\dagger_k c_k + \sum_{k,k’} V_{k,k’} c^\dagger_k c^\dagger_{k’} c_{k’} c_k$, represents a simplified model of electron pairing in superconductivity. Utilizing these models allows for direct comparison with existing quantum algorithms and provides a standardized assessment of the algorithm’s capabilities in simulating many-body quantum systems.

The algorithm successfully identified Eigenstates for both the Transverse Field Ising Model (TFIM) and the Pairing Hamiltonian. Performance was benchmarked by calculating the fidelity of the resulting Eigenstates against known solutions, achieving values between 0.968 and 0.996 for TFIM instances with 2 to 4 qubits. This high level of accuracy was maintained for the Pairing Hamiltonian, extending to systems with up to 6 qubits. These fidelity scores indicate a strong correlation between the algorithm’s output and the expected Eigenstates of the target Hamiltonians, validating its effectiveness in characterizing both ground and excited states.

Implementation of a Symmetry Restriction significantly improved algorithmic performance by reducing the computational complexity of the state space. This restriction leverages known symmetries within the Hamiltonian – specifically, the conservation of particle number parity for the Pairing Hamiltonian and spin-flip symmetry for the Transverse Field Ising Model ($TFIM$) – to constrain the search to relevant Hilbert subspaces. By effectively diminishing the size of the search space, convergence rates were accelerated and computational resources were conserved, allowing for the efficient calculation of Eigenstates for larger qubit systems compared to unrestricted searches. This approach did not compromise accuracy, maintaining high fidelity results across both benchmark models.

Symmetry-restricted reinforcement learning successfully converged to high-fidelity, low-energy states for both five- and six-qubit systems across all Hamming-weight sectors, as demonstrated by mean fidelity and expectation value analysis.
Symmetry-restricted reinforcement learning successfully converged to high-fidelity, low-energy states for both five- and six-qubit systems across all Hamming-weight sectors, as demonstrated by mean fidelity and expectation value analysis.

Ensuring Robustness: Refining Solutions with Post-Selection

Recognizing that initial solutions generated through reinforcement learning might contain inaccuracies stemming from computational noise or instability, a post-selection process was implemented to rigorously assess their reliability. This process centers on the meticulous examination of energy fluctuations within each proposed solution; solutions exhibiting significant variation in energy-indicating potential instability or non-physical behavior-are systematically filtered. The magnitude of these fluctuations serves as a critical metric, allowing for the identification and rejection of unreliable solutions before they can influence subsequent analysis. By prioritizing stability, this post-selection methodology effectively refines the dataset, ensuring that only physically plausible and consistently accurate fixed points are retained for further investigation and ultimately contribute to the overall robustness of the quantum simulations.

The identification of stable, physically relevant solutions within complex quantum simulations hinges on discerning genuine fixed points from transient fluctuations. To achieve this, a post-selection process was implemented, rigorously evaluating potential solutions based on their energy fluctuations. Solutions exceeding a threshold of 0.02 were systematically filtered, as high fluctuations indicate instability and a departure from true fixed points. This careful curation ensures that only solutions representing minimal energy states – and therefore, the most likely and physically meaningful configurations – are retained for further analysis. By prioritizing stability, the methodology significantly enhances the reliability of the simulation results, enabling the accurate determination of ground states and other critical properties of the quantum system under investigation.

The synergistic application of reinforcement learning and post-selection techniques demonstrably elevates the fidelity of quantum simulations. By initially employing reinforcement learning to navigate the complex landscape of potential solutions, researchers can efficiently identify promising candidates. However, recognizing the inherent susceptibility to minor inaccuracies, a subsequent post-selection process-focused on filtering solutions exhibiting excessive energy fluctuations-acts as a crucial validation step. This dual approach not only refines the accuracy of identified fixed points, ensuring they represent genuinely stable quantum states, but also establishes a robust methodology applicable to increasingly complex simulations previously intractable with conventional methods. The result is a pathway towards reliable quantum modeling, unlocking the potential for advancements in materials science, drug discovery, and fundamental physics.

Post-selecting states based on a fluctuation threshold of 0.02 significantly alters the estimated energies for the N=5 pairing Hamiltonian, as demonstrated by comparing the results to those obtained when all final states are included.
Post-selecting states based on a fluctuation threshold of 0.02 significantly alters the estimated energies for the N=5 pairing Hamiltonian, as demonstrated by comparing the results to those obtained when all final states are included.

The pursuit of efficient quantum state preparation, as detailed in this research, echoes a fundamental principle of responsible technological development. This work demonstrates a quantum reinforcement learning algorithm capable of navigating complex quantum systems to locate Hamiltonian eigenstates-a feat requiring careful consideration of encoded values. As Richard Feynman once stated, “The first principle is that you must not fool yourself – and you are the easiest person to fool.” This sentiment highlights the necessity of rigorous validation and transparency in algorithms, ensuring that the values guiding the search for these states align with desired outcomes and do not inadvertently perpetuate biases or limitations. The adaptability of the algorithm to dissipative systems and symmetry restrictions further emphasizes the importance of building tools that are both powerful and ethically grounded, acknowledging that technology without care for people is techno-centrism. Ensuring fairness is part of the engineering discipline.

Where to Next?

The demonstrated capacity to navigate quantum Hilbert spaces via iteratively refined unitary transformations presents a compelling, if subtly disquieting, prospect. This work effectively formalizes a procedure for finding fixed points – and, by extension, eigenstates – but sidesteps the deeper question of what constitutes a meaningful search. The algorithm excels at mapping computational bases onto solutions, but the choice of initial basis, the reward function, and the very definition of “success” remain largely external to the learning process. It creates the world through algorithms, often unaware of the encoded assumptions.

Future investigations should address the limitations imposed by symmetry restrictions. While beneficial for accelerating convergence, these constraints necessarily bias the search, potentially obscuring alternative, equally valid solutions. More fundamentally, extending this framework to dissipative systems, where fixed points represent stable states rather than energy eigenstates, necessitates a re-evaluation of the reward structure and a consideration of non-unitary dynamics. The true test will be its application to systems where analytic solutions are unavailable – not merely to confirm what is already known, but to genuinely discover novel quantum phenomena.

Transparency is minimal morality, not optional. As quantum machine learning algorithms proliferate, a critical examination of their inherent biases and underlying value systems becomes paramount. The efficiency gained through automation should not come at the expense of interpretability or control. The algorithm doesn’t understand the physics; it merely finds patterns. And the patterns it finds reflect the values of its creators.

Original article: https://arxiv.org/pdf/2511.17491.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-11-24 12:36