Polynomial-time Algorithm Achieves Certified Fidelity Bounds for Many-body Mixed States
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Scientists have long struggled with the computational cost of verifying the fidelity of complex quantum states. Yuhan Liu from the Max Planck Institute of Quantum Optics, alongside Yijian Zou from the Perimeter Institute for Theoretical Physics, and their colleagues, now present a groundbreaking polynomial-time algorithm to certify fidelity for many-body mixed states, a significant leap forward in quantum verification. Their research elegantly maps fidelity estimation onto variational optimisation of quantum circuits, enabling systematic improvements in lower bounds and, crucially, providing certified upper bounds via trace distance calculations. This innovative approach demonstrates exponential precision gains over existing methods and unlocks the potential to accurately determine universal scaling behaviour, even extracting previously unknown critical exponents from systems like critical mixed states and approximate quantum codes , paving the way for scalable verification of increasingly complex many-body systems. Fidelity Bounds via Variational Quantum Circuits Scientists have developed a novel polynomial-time algorithm for computing certified lower and upper bounds on the fidelity between many-body mixed states, a significant advancement overcoming limitations in quantifying the similarity between quantum states. This breakthrough addresses a longstanding challenge in quantum verification, where traditional methods often require exponentially scaling computational resources, particularly when dealing with high-rank mixed states like those found in thermal or dissipative systems. The research team mapped the complex fidelity estimation problem onto a variational optimization of sequential quantum circuits, enabling systematic improvement of lower bounds simply by increasing the circuit depth. Complementarily, they derived certified upper bounds on fidelity using variational lower bounds on the trace distance, all within the same framework. This innovative approach leverages the natural representation of many-body states as Matrix Product Density Operators (MPDOs), allowing for efficient computation without full diagonalization of exponentially large matrices. The method employs Uhlmann’s theorem, connecting fidelity to the overlap between purifications of density matrices, and translates this into an optimization problem solvable with variational quantum circuits. Experiments demonstrate the power of this technique through two compelling examples: fidelity correlators in critical mixed states and codeword distinguishability in an approximate quantum error-correcting code. Remarkably, the variational lower bound accurately captures the universal scaling behaviour of fidelity with a size-consistent relative error of less than 1%, facilitating the extraction of previously unknown critical exponents. The study reveals an exponential improvement in precision compared to existing moment-based bounds, establishing a scalable framework for verifying many-body quantum systems. In analysing fidelity correlators for critical mixed states, the algorithm certified a notably slower decay of these correlators under dephasing noise for systems up to 32 spins, indicating a distinct universality class. Furthermore, the research successfully evaluated the distinguishability of low-energy eigenstates in critical systems experiencing extensive decoherence, providing a diagnostic tool for assessing the error decodability of the approximate quantum error-correcting code. The sequential circuit ansatz, a key component of the algorithm, utilizes ancilla degrees of freedom and allows for efficient computation while maintaining precision. This work not only provides a powerful tool for quantum simulation verification but also opens new avenues for understanding mixed-state phases of matter and characterizing their information-theoretic properties, paving the way for more robust and reliable quantum technologies. The ability to accurately compute fidelity in these complex systems is crucial for advancing the field of quantum information science and unlocking the full potential of quantum computation. Fidelity Bounds via Variational Quantum Circuits Scientists have developed a polynomial-time algorithm for computing certified lower and upper bounds on the fidelity between matrix product density operators (MPDOs). This work addresses a significant challenge in quantifying the quality of quantum state preparation, particularly for inherently high-rank mixed states where traditional methods fail. The research team mapped the fidelity estimation problem onto a variational optimization of sequential quantum circuits, leveraging Uhlmann’s theorem to establish a novel approach. By systematically increasing the circuit depth, they achieved a progressive refinement of the lower bounds on fidelity, offering a scalable solution for complex quantum systems. Experiments employed a framework where fidelity estimation is recast as an optimisation problem solved on a quantum computer. The core innovation lies in representing mixed states as MPDOs, exploiting their natural suitability for tensor network methods and circumventing the exponential computational cost of full diagonalization. Researchers constructed sequential quantum circuits designed to maximise a cost function directly related to the fidelity, effectively searching for the optimal circuit configuration that yields the tightest lower bound. Complementary to the lower bound calculation, the study also obtained certified upper bounds on fidelity by utilising variational lower bounds on the trace distance within the same framework.
The team demonstrated the power of this method through two compelling examples: fidelity correlators in critical mixed states and codeword distinguishability in an approximate quantum error-correcting code. Remarkably, the variational lower bound accurately tracked the universal scaling behaviour of the fidelity with a size-consistent relative error of approximately 0.01, allowing for the extraction of previously unknown critical exponents. This precision represents an exponential improvement over existing moment-based bounds, enabling a more detailed characterisation of mixed-state phases of matter. Furthermore, the study revealed that dephasing noise induces a notably slower decay of fidelity correlators for systems up to N = 32 spins, indicating a distinct universality class for the decohered state. Analysis of the approximate – code showed that the lower bounds accurately capture system-size scaling, distinguishing between correctable and uncorrectable noise, a crucial step towards robust quantum error correction. This scalable framework establishes a powerful tool for verifying many-body quantum systems and advancing our understanding of complex quantum phenomena. Fidelity Bounds from Variational Quantum Circuits Scientists have developed a polynomial-time algorithm for computing certified lower and upper bounds for the fidelity between matrix product density operators (MPDOs), overcoming a significant challenge in many-body quantum system verification. The research team mapped the fidelity estimation problem to a variational optimization of sequential quantum circuits, enabling systematic improvement of lower bounds by increasing circuit depth, a crucial advancement for assessing complex quantum states. Complementarily, they obtained certified upper bounds on fidelity by utilising variational lower bounds on the trace distance, providing a robust and comprehensive approach to fidelity assessment. Experiments revealed that the variational lower bound accurately tracks the universal scaling behaviour of fidelity with a size-consistent relative error, allowing for the extraction of previously unknown critical exponents in mixed states. This breakthrough delivers an exponential improvement in precision over existing moment-based bounds, offering a scalable framework for verifying many-body quantum systems, a feat previously hampered by exponential computational requirements.
The team demonstrated the power of this method through two key examples: fidelity correlators in critical mixed states and codeword distinguishability in an approximate quantum – code. Measurements confirm that, when analysing fidelity correlators, dephasing noise induces a notably slower decay of these correlators for system sizes up to N = 32 spins, exhibiting distinct critical exponents compared to the noiseless case. This indicates a different universality class for the decohered state, providing valuable insights into the behaviour of quantum systems under realistic conditions. Tests prove that the algorithm accurately captures the system-size scaling of fidelity, distinguishing between correctable and uncorrectable noise in the context of quantum error correction, a vital step towards building fault-tolerant quantum computers. Data shows that the optimal circuit is achievable with finite depth in analytically tractable limiting cases, further validating the efficiency and practicality of the proposed method. The work establishes a scalable framework for the verification of many-body quantum systems, offering a powerful tool for exploring mixed-state phases of matter and advancing the field of quantum simulation. This innovative approach unlocks the potential to analyse systems previously inaccessible due to computational limitations, paving the way for deeper understanding and control of complex quantum phenomena. Fidelity Bounds via Optimised Sequential Circuits are presented Scientists have developed a polynomial-time algorithm for computing certified lower and upper bounds on the fidelity between many-body mixed states represented as matrix product density operators. This method overcomes the exponential scaling issues of exact diagonalization by optimising fidelity over sequential circuits with ancilla, enabling calculations for systems up to size N = 32. The researchers demonstrated the algorithm’s effectiveness through applications to fidelity correlators in decohered critical states and codeword distinguishability within the CFT code. Notably, the variational lower bound accurately captures the universal scaling behaviour of fidelity, with a size-consistent relative error of approximately 5.5% for N = 14. This precision represents a significant improvement over existing moment-based bounds and allows for the extraction of previously unknown critical exponents, such as a change in critical exponent for the fidelity correlator under dephasing. The authors acknowledge that while the lower bound is quantitatively distinct from the exact fidelity, it faithfully represents the underlying universal physics. Future research could focus on understanding the scaling of the variational error with circuit depth, potentially revealing insights into the complexity of optimal purification. Further extensions to higher-dimensional tensor networks and adaptation to other information-theoretic quantities, like α-Renyi relative entropies, are also promising avenues for exploration, as is implementation on quantum hardware via variational quantum algorithms. 👉 More information 🗞 Polynomial-time certification of fidelity for many-body mixed states and mixed-state universality classes 🧠 ArXiv: https://arxiv.org/abs/2601.13333 Tags: