#variationalquantumalgorithms

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Berry Phase Calculation with Variational Quantum Algorithms

Berry Phase Calculation

A new adaptive variational quantum technique for Berry Phase Calculation in topological systems is presented in this study. The authors use cyclic adiabatic evolution to reduce quantum circuit depth while maintaining accuracy. Dimerized Fermi–Hubbard chains are used to identify topological phase transitions in both interacting and noninteracting regimes. The algorithm shows outstanding resilience, operating effectively even with large nonadiabatic effects or numerical restrictions. With near-term quantum technology, this work provides an efficient foundation for simulating topological materials and complex geometric phases.

Iowa State University and Ames researchers National Laboratory has devised a quantum computing method that dramatically improves Berry phase computation, a critical geometric component of quantum states. APL Quantum reported this discovery, which could be used in near-term quantum technologies to characterize topological phases of matter.

The Berry Phase Challenge

In condensed matter physics, the Berry phase is a key indicator of quantum Hall effect, topological insulators, and superconductors. Traditional computer methods struggle to calculate geometric phases in strongly linked systems. The exponential increase of computing cost or the fermionic sign problem during sampling can plague classical techniques.

Quantum computing inherently captures quantum correlations, but Trotterization of adiabatic state evolution requires deep quantum circuits. Deep circuits on Noisy Intermediate-Scale Quantum (NISQ) devices are difficult to implement because to quick decoherence and noise growth.

Adaptive Circuits: Dynamic Solution

Mootz and Yao developed an adaptive variational quantum method that dynamically creates efficient quantum circuits to overcome these limitations. The Adaptive Variational Quantum Dynamics Simulation (AVQDS) method generates the circuit on the fly without a fixed ansatz or circuit depth.

Process starts with Adaptive Variational To prepare the system's ground state, use AVQITE. Cycles of adiabatic development occur when the system is ready. The method selects unitaries from a defined operator pool and appends them to the circuit only when needed to keep the McLachlan distance, a measure of state development deviation, below a threshold. This “pseudo-Trotter” method keeps the circuit small without sacrificing precision.

Successful SSHH Model Benchmarking

Their method was proven using the dimerized Fermi-Hubbard chain Su–Schrieffer–Heeger–Hubbard (SSHH) model, a flexible platform for electron interaction research. Eight qubits were encoded for four-site chain benchmarking.

The study examined noninteracting and strongly connected regimes. It produced precise simulations with circuit depths of roughly 106 layers in the noninteracting example. By adding electron correlations (the interaction regime), the complexity increased to 279 layers.

Amazingly, AVQDS saved more resources than conventional methods. The researchers estimated that first-order Trotterization would need 970 to 7,200 times more CNOT gates to match their adaptive technique's precision.

Unmatched Durability

The algorithm's stability across several parameters is a key study finding. The approach reliably computed the Berry phase even with a short simulation length (T) or large time step size.

This robustness comes from a unique “perfect error cancellation” mechanism. The Berry phase is obtained by merging a quantum-circuit-based and calculated global phase. Despite strong nonadiabaticity, these two components compensate for each other, yielding a very accurate phase.

They isolated the Berry phase with time-reversal symmetry. A forward development and time-reversed return remove the dynamical phase contribution, leaving the geometric Berry phase.

Road Ahead

This adaptive technique's success suggests it could improve topological material simulations that traditional computers cannot handle. By using compact state representations and Quantum Processing Units (QPUs), AVQDS may soon be used for larger systems and more complicated models, including interacting topological superconductors.

Future work will combine these algorithms with error mitigation methods and expand the framework to quantum embedding frameworks to simulate bulk systems with fewer quantum resources. “Our findings underscore the potential of AVQDS for efficient quantum simulations of topological materials,” the scientists said, noting that geometric phase analysis is essential to understanding strongly correlated quantum events.

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Hamiltonian Expressibility: Variational Quantum Algorithms

Variational Quantum Algorithms Use Hamiltonian Expressibility to Unlock Quantum Potential.

Variable Quantum Algorithms

The rapidly emerging field of quantum computing may use Variational Quantum Algorithms (VQAs) to harness the power of near-term, noisy quantum devices, also known as Noisy Intermediate-Scale Quantum (NISQ) devices. Fusing quantum circuits with classical optimisation, these methods solve complex problems through parameter refinement.

The Variational Quantum Eigensolver (VQE) is designed to find a Hamiltonian operator's ground state. Quantum chemistry uses VQE to determine molecular ground-state energies and combinatorial optimisation using ground-state searches.

The Main Challenge: Selecting a Quantum Circuit In Variational Quantum Algorithms, choosing an effective parametric quantum circuit, or ansatz, is crucial. The ansatz controls quantum computation and affects the quality of the answer and the algorithm's trainability. A good ansatz should work with current quantum hardware, have a short circuit depth to reduce noise, and resist barren plateaus.

The Barren Plateau Issue

Variational Quantum Algorithms struggle with barren plateaus, which are cost function gradients that decrease exponentially with qubit count. This flattening of the optimisation landscape makes it difficult for classical optimisers to recognise meaningful parameter updates, especially for larger quantum systems, preventing convergence.

Even if a circuit needs exploratory power to find the ideal solution, too much power might lead to barren plateaus.

Introduce Hamiltonian Expressibility

Hamiltonian expressibility may help solve the barren plateau problem and ansatz selection. This metric assesses how evenly a circuit explores the problem's energy landscape as defined by a Hamiltonian. The idea is that a circuit's exploration capability should boost its chances of finding good answers. However, its potential to improve solutions has not been completely realised.

Researchers have a method to evaluate Hamiltonian expressibility for a given ansatz and Hamiltonian. VQE solves optimisation issues. A correlation analysis between the two metrics will determine if Hamiltonian expressibility improves Machine Learning for ansatz design.

Key Results: Ideal and Quiet

The study found several noteworthy things under ideal or extremely low noise conditions:

Ansatz Depth and Expressivity

Until saturation, a circuit's Hamiltonian expressibility improves with layer depth. Once saturation is attained, the expressibility value tends to swing inside a “maximally expressive zone” and more layers are no longer needed.

Problem-Specific Expressibility

Hamiltonian expressibility targets issues. Circuits often have higher expressibility than non-diagonal Hamiltonians for problems like Maximum Cut, Minimum Vertex Cover, Maximum Clique, and Random Diagonal. Because diagonal Hamiltonians have basis states as their best solutions, their energy space is naturally “narrower” and easier to explore.

Small-Scale Solution Quality Association

Small-scale problems (like four-qubit ones) are linked.

High Expressibility for Superposition Solutions: Ansätze with high Hamiltonian expressibility perform better for superposition-state and non-diagonal Hamiltonian problems. Classes like Heisenberg Superposition State and Random Non-Diagonal problems showed strong negative associations, meaning lower expressibility values lead to higher accuracy.

Basis-State Solutions Have Limited Expressibility: Circuits with low expressibility are better for problems having basis states, such as diagonal Hamiltonians. Extreme expressibility may backfire in some instances.

The link between expressibility and solution quality weakens as qubits increase, such as to eight. This evidence supports theoretical findings that barren plateaus, which can hinder trainability, have high expressibility.

Key Findings: Noise

Realistic quantum noise sources like decoherence and gate failures substantially modify the relationship between expressibility and solution quality, especially for small-scale settings (4 qubits).

Diagonal Hamiltonians and basis-state issues benefit more from low Hamiltonian expressibility than noiseless problems. This is because simpler, less expressive circuits are shallower and less noisy.

This relationship becomes more complicated for issues with superposition-state solutions and non-diagonal Hamiltonians. An intermediate level of expressibility may yield the best results for some superposition-state circumstances, such as the Heisenberg XXZ model, even if more expressive circuits are more noisy due to their complexity.

This requires balancing a circuit's exploration and noise resistance.

The study demonstrated a bell-shaped pattern in solution quality for some non-diagonal problems under noise: expressibility grows, peaks at an intermediate level, and then falls. This bell-shaped pattern appears when noise levels climb.

Conclusion and Future Plans

The results show that Hamiltonian expressibility can guide VQE protocol ansatz selection for small-scale challenges in ideal and noisy contexts. Expressibility metrics-based circuit selection processes are better for systems with more complex solutions (superposition states).

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Barren Plateaus Quantum With Dissipative Computation & Noise

Barren Plateaus Quantum

Despite its rapid progress, quantum computation has faced “barren plateaus.” These risky parts of the computational landscape hinder the scalability of  variational quantum algorithms (VQAs), making optimisation harder as the quantum system grows. Noise exacerbates this issue. According to groundbreaking research by Elias Zapusek, Ivan Rojkov, and Florentin Reiter of ETH Zürich and the Fraunhofer Institute, dissipative quantum algorithms may help.

The Barren Plateaus Challenge

A barren plateau occurs when VQA cost function gradients fall exponentially with system growth. As quantum computers get bigger, the signal needed to properly train or optimise quantum algorithms becomes vanishingly small, requiring an impractical, exponentially huge number of measurement shots to detect any significant change. It is rare to find large, useful gradients focused about zero, which is often the case.

Several factors cause barren plateaus in conventional VQAs, including:

Overly expressive ansatz that approaches random quantum circuits can flatten loss landscapes and reduce gradient magnitudes.

Cost functions globally: Even shallow, layered ansätze can experience exponential gradient degradation when the cost function comprises multiqubit observations.

High entanglement between quantum system components can also generate bare plateaus.

Noise-induced barren plateaus (NIBPs) in quantum systems worsen this issue. As circuit depth increases, gradients and expectation values converge exponentially to a noise-induced fixed point, flattening the cost landscape deterministically. This eliminates smart initialisation benefits and prevents optimisation.

Dissipative engineering: cooling and entropy extraction

The latest work suggests dissipative quantum algorithms can overcome these constraints. This is discussed in “Scaling Quantum Algorithms via Dissipation: Avoiding Barren Plateaus.” Dissipative quantum algorithms use non-unitary dynamics and tailored dissipation in their circuit architecture, unlike typical VQAs, which use solely unitary dynamics. Their clever system includes:

For engineered cooling, supplementary qubits are reset frequently.

Entropy, a measure of disorder or uncertainty in the quantum state, is actively extracted by periodic resetting. This differs substantially from standard methods.

Actively eliminating entropy helps these dissipative circuits overcome unitary and noise-induced barren plateaus. This approach maintains gradient magnitudes, enabling scalable and noise-resilient optimisation in conditions where standard VQAs would fail.

Validity and Efficiency

Researcher analytical conditions make dissipative circuits trainable even with realistic noise levels. Their deployment on error-prone quantum devices, both current and future, depends on theoretical backing.

Numerous numerical simulations support these theoretical predictions. The simulations prove that dissipative circuits are efficient when unitary algorithms hit empty plateaus. While preparing toric code ground states, a classic topologically ordered state, numerical evidence shows that dissipative learners lack NIBPs. The dissipative technique kept stable, trainable gradients, while unitary circuits preparing such states experienced exponential suppression of gradients and expectation values with system scale due to noise.

The work solves the barren plateau problem and highlights dissipative circuits' efficiency. They can eliminate the processing load of step-by-step layer-wise simulations that only approximate system evolution by directly determining a system's steady state, which is a more accurate final configuration approximation.

Implications for Future Quantum Computing

This revolutionary technique allows VQAs to construct quantum algorithms that are fundamentally more robust to hardware faults in the real world while addressing their most pressing scalability difficulties. The method can be applied to more complex quantum systems with additional study on hardware implementation, circuit layout, noise qualities, and entropy extraction rates.

The study also teaches that not all dissipative structures are noise-resistant. Although conceptually appealing, several purely dissipative universal quantum processing approaches fail when noise is present because they simply repeat unitary protocols in a dissipative environment without actively leveraging dissipation to overcome noise limitations. The main difference is active entropy extraction.

This work positions dissipative quantum algorithms as a top possibility for scalable and reliable quantum computing on noisy, near-term devices. Dissipative circuits can also remove unnecessary information and improve generalisation in quantum machine learning. More research should examine the complicated relationships between noise and intended dissipation.

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Quantum Feature Maps And Classical Data In Quantum Space

Quantum Feature Maps

Iterative quantum feature mappings (IQFMs) are a quantum machine learning (QML) innovation that introduces a hybrid classical-quantum architecture that solves some of the primary issues with implementing quantum models on conventional hardware. Fujitsu Research researchers Nasa Matsumoto, Quoc Hoan Tran, Koki Chinzei, Yasuhiro Endo, and Hirotaka Oshima designed this framework. Quantum News published “Quantum-enhanced Machine Learning Boosts Performance with Iterative Feature Maps” on June 25, 2025, detailing their research's benefits and results.

IQFMs build on Quantum Feature Maps. QFMs convert conventional data into quantum states using quantum computers' growing Hilbert space. The transformation makes QML models universal approximators of continuous functions and may speed up classification problems exponentially. Classical machine learning (ML) methods that map input data into new feature spaces improve separability.

Practitioners face various challenges when using deep QFMs. Current quantum technology is limited and susceptible to circuit noise. The conventional variational quantum algorithms (VQAs) used to train these models often experience computational bottlenecks, especially when obtaining precise gradient estimation, which requires substantial quantum resources and can lead to local minima or “barren plateaus” in the optimisation landscape.

IQFMs generate deep learning systems by iteratively integrating shallow quantum feature maps (QFMs) with classically derived augmentation weights to address these difficulties. This hybrid design intentionally reduces learning quantum resources.

Hybrid Architecture: IQFMs connect shallow QFMs via measurement outputs, which are processed by classical augmentation, in place of deep quantum circuits. This form promotes expressiveness and flexibility for near-term quantum computers.

A key feature of IQFMs is that they optimise just the classical augmentation weights between QFMs, not the quantum circuits' variational characteristics. This technique avoids a major shortcoming of standard QML algorithms and reduces quantum computational runtime by offloading gradient estimates to classical processors. The classical augmentation parameters (𝑾_l) can be taught, while the quantum circuit parameters (𝜽_l) are often fixed to random values.

IQFMs use contrastive learning, an important representation learning strategy. The model is trained to produce similar representations with similar inputs and divergent representations with unrelated inputs. By focussing on key data similarities and differences, contrastive learning stabilises feature extraction across noisy quantum circuits and reduces variability caused by hardware faults or quantum measurements in IQFMs.

Using supervised contrastive-learning, a “anchor” feature vector is produced for an input. A “negative” sample (one with a different label) is processed to yield a representation farther from the anchor than a “positive” sample (one with the same label). This is done by minimising a contrastive loss function.

Layer-wise Training: IQFMs supplement contrastive learning with layer-by-layer training. Instead of optimising every parameter, which would demand a lot of quantum resources, this method trains classical augmentation weights for each QFM layer. It also avoids the "barren plateaus" phenomenon in VQAs where gradients disappear in long quantum circuits, reducing computational complexity.

QFM block characteristics are extracted via quantum measurements in many bases. The approach involves transferring classical features into a quantum state using an embedding circuit (𝒰_𝚿), entangling and mixing data with a preprocessing circuit (P_l), and adjusting the measurement basis with a parameterised circuit (Ω_l). The process creates a feature vector (𝒈_l) based on measurement operator expectations. To improve the feature set, measurements in bases other than the computational (Pauli-Z) basis concatenate feature vectors. This multi-basis strategy improves classification and prevents classical simulation of some quantum correlations.

Flexibility and Performance: IQFMs can categorise classical and quantum data.

Quantum Data Classification: IQFMs consistently outperform Quantum Convolutional Neural Networks (QCNN) in quantum phase recognition tasks (Task A and Task B, which categorise Hamiltonians' ground states into discrete quantum phases). This suggests that random measurement bases and typical post-processing are effective without QFM circuit optimisation.

Robustness to Noise: IQFMs beat QCNN in the presence of statistical errors from a small number of measurement shots and physical RX noise (random data rotations). At higher noise levels, IQFMs outperformed QCNN in accuracy. Visualisations reveal that contrastive learning strengthened discriminative representations by forming more coherent feature space clusters.

Classification: On the Fashion-MNIST test, IQFMs fared comparably to classical neural networks with similar designs. Classical data was processed using modular IQFMs to handle large datasets. Multiple QFMs process classical data simultaneously in this configuration. This modular design, which only allows subcircuits on quantum devices, allows large-scale tasks on near-term quantum devices with constrained qubits.

Future Outlook: IQFMs are a promising step towards quantum-enhanced machine learning's full potential, making them ideal for practical applications with limited computational resources and uneven data quality. The researchers note that IQFMs' architecture avoids the incompatibility of back-propagation with quantum circuits, which lack intermediate states for gradient computation, and that Direct Feedback Alignment (DFA) may be investigated as a training method.