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Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits

Although single-qubit re-uploading circuits can represent the same Fourier series as multi-qubit circuits with an equivalent encoding budget, they suffer from a "phase-locking" phenomenon that severely restricts their tangent space dimension, leading to significantly harder first-order training and lower convergence rates compared to wider multi-qubit controls.

Original authors: Michael Poppel, David Bucher, Maximilian Zorn, Markus Baumann, Sebastian Wölckert, Claudia Linnhoff-Popien, Philipp Altmann, Jonas Stein

Published 2026-10-16

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CC BY 4.0

Original authors: Michael Poppel, David Bucher, Maximilian Zorn, Markus Baumann, Sebastian Wölckert, Claudia Linnhoff-Popien, Philipp Altmann, Jonas Stein

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the emerging field of quantum computing, researchers are learning to harness the strange laws of physics that govern the subatomic world to solve problems too complex for today's supercomputers. A central tool in this effort is the variational quantum circuit, a type of algorithm designed to learn patterns from data much like a neural network, but using quantum bits, or qubits, instead of classical ones. These circuits work by encoding information into the angles of rotation applied to qubits, effectively turning the machine into a sophisticated mathematical engine that can approximate complex curves. For simple, one-dimensional data, scientists have long known that they can trade the number of qubits they use for the number of steps they take. In other words, a machine with a single qubit that runs for a long time can, in theory, produce the exact same results as a machine with many qubits that runs for a short time. This flexibility suggests that building smaller, simpler quantum devices might be just as powerful as building larger, more complex ones, provided the smaller ones are deep enough.

However, a new study by researchers at the University of Munich and Aqarios GmbH reveals that this theoretical equivalence breaks down when it comes to the actual process of training these machines. While a single-qubit circuit can represent the same mathematical functions as a two-qubit circuit, it is significantly harder to teach the single-qubit version to find the correct solution. The researchers discovered that the single-qubit machine suffers from a structural limitation they call "phase-locking." Because the state of a single qubit is confined to the surface of a sphere, every adjustment made to its parameters moves the system along a single, shared direction. This constraint means the machine has no extra room to maneuver; it is forced to walk a tightrope where every step must be perfectly aligned to change the output. In contrast, a two-qubit machine has a much larger state space, allowing its parameters to move in multiple independent directions simultaneously. This extra freedom provides a safety net, making it far easier for the training algorithm to navigate the complex landscape of errors and find the optimal solution.

To test this, the team set up a controlled experiment comparing three different circuit designs. They used a "serial" circuit with one qubit and a long chain of operations, and compared it against two-qubit circuits that had the same total capacity for encoding information. They trained these machines on a variety of mathematical targets, ranging from smooth, predictable curves to complex, real-world data sets like monthly temperature records and daily stock prices. The results were stark. When using standard training methods that rely on calculating gradients, the single-qubit circuit failed to reach the best possible solution in the vast majority of attempts, especially as the complexity of the problem increased. In one test with a high encoding budget, the single-qubit circuit succeeded in only three out of one hundred attempts, while the two-qubit control succeeded in ninety-nine. Even when the researchers matched the number of adjustable parameters to ensure a fair comparison, the single-qubit machine consistently lagged behind, often producing predictions that were significantly less accurate than those of its two-qubit counterpart.

The researchers explored whether this difficulty was simply due to the training algorithm getting stuck in a bad starting position or struggling with a poorly conditioned mathematical landscape. They found that while the single-qubit circuit did often encounter difficult mathematical conditions during training, the problem went deeper than just bad luck or a poor start. When they switched to a more powerful, second-order training method that is less sensitive to these mathematical hurdles, the gap between the two designs narrowed but did not disappear. The single-qubit machine still failed to converge in many cases where the two-qubit machine succeeded. This suggests that the limitation is not just a temporary obstacle that a better algorithm can overcome, but a fundamental structural disadvantage. The single-qubit machine simply lacks the extra degrees of freedom required to explore the solution space effectively, leaving it with no spare directions to correct its course when it veers off track.

This finding has important implications for the future of quantum machine learning. It suggests that while building smaller, single-qubit devices might seem like an attractive shortcut to avoid the engineering challenges of managing multiple qubits, it may come at a steep cost in performance. The study indicates that for one-dimensional data, the extra complexity of adding a second qubit is not just a luxury but a necessity for reliable training. The two-qubit system, with its ability to move in multiple directions at once, provides the robustness needed to consistently find the best solution. While the researchers did not test every possible scenario, their simulations on real-world data sets confirmed that the single-qubit approach often leads to higher errors in prediction. Ultimately, the work demonstrates that in the quantum realm, having more room to move is not just about having more space; it is about having the freedom to find the right path in a landscape that would otherwise be impossible to navigate.

Technical Summary: Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits

Problem Statement
Variational Quantum Circuits (VQCs) with angle encoding implement truncated Fourier series. For one-dimensional inputs, it is theoretically established that a single-qubit "serial" circuit (depth $L=E$ , width $N=1$ ) possesses the same expressivity as a multi-qubit circuit with the same encoding budget $E = NL$ . Both can represent every real trigonometric polynomial of degree $\le E$ with values in $[-1, 1]$ . While numerical observations suggest that fewer qubits may lead to training plateaus on complex targets, the structural reason for this disparity and its quantitative cost in first-order optimization remained unproven. This paper investigates whether the exchange of width for depth in single-qubit re-uploading circuits incurs a hidden cost in trainability, despite identical spectral capabilities.

Methodology
The authors compare three circuit architectures with identical encoding budgets $E$ :

Serial Circuit: One qubit, $L=E$ layers.
Ancilla Control: Two qubits, encoding on one and reading out the other, $L=E$ layers. This shares the serial circuit's model class and chain length but operates in a four-dimensional state space.
Parallel Circuit: Two qubits, both encoded, $L=E/2$ layers. This serves as a reference to separate chain length from the number of tangent components.

The study employs a controlled experimental design:

Exact Loss: Training on random trigonometric polynomial targets where the optimal loss is theoretically zero. This isolates optimization dynamics from sampling noise.
Real-World Data: Training on three time series (Nottingham temperatures, US gasoline prices, IBM closing prices) and Poisson-kernel targets with matched parameter counts.
Optimizers: The study utilizes Gradient Descent (with Armijo line search), Levenberg–Marquardt (a second-order method), and Adam.
Metrics: Convergence rates (share of runs reaching a loss threshold within a fixed step budget) and held-out prediction error ( $1-R^2$ ).

Key Theoretical Contributions
The paper establishes a structural difference termed phase-locking:

Phase-Locking: In a single-qubit circuit, the state resides on the Bloch sphere. At any input $x$ , all parameter derivatives move the state along a single complex direction (a vector orthogonal to the state). Consequently, all output derivatives share a common complex factor (the "envelope" $\mu(x)$ ).
Tangent Direction Count:
- The serial circuit has at most $2E + 2$ tangent directions (real dimensions) across all inputs. Since it must control $2E + 1$ Fourier coefficients, it has at most one spare direction (the readout rotation), which leaves the output unchanged.
- The two-qubit ancilla circuit generically has at least $6E + 6$ tangent directions, providing at least $4E - 2$ spare directions.
Full Row Rank: Despite the limitation in tangent directions, the coefficient Jacobian of the serial circuit retains full row rank almost everywhere, confirming that no Fourier directions are structurally lost; the limitation is purely in the number of available directions to traverse the loss landscape.

Experimental Results

Exact Loss (Flat Targets): Under Gradient Descent, the serial circuit converges significantly less often than the ancilla control as $E$ increases. At $E=32$ , the serial circuit converges in only 3 runs (Gradient Descent), 53 (Adam), and 83 (Levenberg–Marquardt) out of 100, whereas the control converges in 99–100 runs. The gap widens with $E$ .
Conditioning: The serial circuit's Jacobian remains poorly conditioned more frequently during training. While Levenberg–Marquardt (which is robust to poor conditioning) narrows the convergence gap, it does not eliminate it entirely at high $E$ .
Real-World Data: At matched parameter counts, the serial circuit reaches the least-squares optimum in at most 9 runs under Adam, compared to at least 98 for the control. The held-out error of the serial circuit is up to 1.5 times that of the control under Adam. Under Levenberg–Marquardt, the typical serial run predicts as well as the control, but a significant fraction of runs still fail to reach the optimum.
Failure Analysis: Runs that fail under first-order methods often end at points where the Jacobian loses rank or singular values drop significantly. These failures are not attributed to poor initialization, as Levenberg–Marquardt starting from the same points frequently converges.

Significance and Claims
The paper claims that while width can be exchanged for depth without losing expressivity (the ability to represent functions), this exchange is not free in first-order training.

Structural Constraint: The "phase-locking" phenomenon in single-qubit circuits restricts the tangent space to a single complex direction per input, leaving the circuit with no spare tangent directions to maneuver around the loss landscape.
Optimization Cost: This structural rigidity makes the serial circuit harder to optimize with first-order methods compared to two-qubit circuits of the same spectrum, which possess significantly more tangent directions.
Practical Implication: In practical settings using common first-order optimizers (like Adam), the single-qubit architecture may fail to reach the optimal solution, leading to higher generalization error. While second-order methods (Levenberg–Marquardt) can mitigate this, they do not fully resolve the issue at large encoding budgets.

The authors explicitly state they do not claim a causal link between the theoretical tangent count and the observed training cost, but rather that the structure provides a candidate explanation for the empirical difficulty. The results are limited to one-dimensional inputs, noise-free simulations, and specific gate parameterizations.

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