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arXiv:2610.01446

Unflattening by Flattening -- How Input Distributions Shape Output Variance in Angle-Encoded Circuits

Melvin Strobl, Gabriel Mejia, Eileen Kuehn, Achim Streit

quant-phSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
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Abstract

Barren plateaus hinder training of parameterized quantum circuits by making loss gradients exponentially small in the number of qubits. For angle-encoded product states, we show how the input distribution affects output variation through algebraic input purity, i.e. the input's overlap with the circuit's dynamical Lie algebra. With the specified readouts and Haar or exact group 2-design sampling, matchgate circuits on $n$ qubits retain output variance of order $1/n$ for every pure product input. The off-diagonal family instead has zero output on computational-basis inputs at any depth and for every parameter choice. Independent uniform angles yield mean variance of the same order as matchgates. A count of diagonal Pauli strings identifies these zero-output endpoints. For any fixed dataset of nonzero inputs with at least $2n$ coordinates, we construct a classical preprocessing certificate. A shared random rotation is accepted only when the dataset's mean purity passes a computable threshold. This certifies mean output variance of order $1/n$ for the off-diagonal family before circuit execution, with a constant expected number of trials. Binary and ternary weighted encodings also recover the independent-angle mean purity from one uniform scalar input. Numerical experiments examine training behavior and extensions to larger algebras. The guarantee concerns output variance averaged over inputs, while successful learning also depends on the task and information preserved by the encoding.

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The barren plateaus phenomenon, where the gradients of parametrized quantum circuits become vanishingly small, poses a significant challenge in quantum machine learning. Foundational analyses of barren plateaus often use methods like the Weingarten formula, which, under the global Haar random assumption, effectively links the gradient variance to the Hilbert space dimension. However, many practical circuits, such as Hardware-Efficient Ans\"atze, utilize a different, non-Haar random model based on the sampling of parameters for single-qubit rotation gates. This study introduces a direct computational method for the gradient expectation and variance, specifically for parametrized quantum circuits constructed from single-qubit rotation gates. Our analysis reveals a direct, quantitative relationship between the gradient variance, circuit depth, and number of effective parameters. Numerical simulations further confirm the validity of our theoretical results. Our findings highlight the role of effective parameters as a key indicator for gradient trainability and provide a precise framework for analyzing barren plateaus in these specific non-Haar random circuits.

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Barren plateaus hinder training of parameterized quantum circuits by making loss gradients exponentially small in the number of qubits. For angle-encoded product states, we show how the input distribution affects output variation through algebraic input purity, i.e. the input's overlap with the circuit's dynamical Lie algebra. With the specified readouts and Haar or exact group 2-design sampling, matchgate circuits on nnn qubits retain output variance of order 1/n1/n1/n for every pure product input. The off-diagonal family instead has zero output on computational-basis inputs at any depth and for every parameter choice. Independent uniform angles yield mean variance of the same order as matchgates. A count of diagonal Pauli strings identifies these zero-output endpoints. For any fixed dataset of nonzero inputs with at least 2n2n2n coordinates, we construct a classical preprocessing certificate. A shared random rotation is accepted only when the dataset's mean purity passes a computable threshold. This certifies mean output variance of order 1/n1/n1/n for the off-diagonal family before circuit execution, with a constant expected number of trials. Binary and ternary weighted encodings also recover the independent-angle mean purity from one uniform scalar input. Numerical experiments examine training behavior and extensions to larger algebras. The guarantee concerns output variance averaged over inputs, while successful learning also depends on the task and information preserved by the encoding.
Deep parameterized quantum circuits may remain sensitive to a parameter change while the observables retained by a learning model barely respond. We study this separation for a fixed computational-basis measurement. For a pure-state tangent, we compare the quantum Fisher information FQF_QFQ​, the Fisher information FfullF_{\rm full}Ffull​ in the complete bitstring distribution, and the largest variance-normalized response IA\mathcal I_{\mathcal A}IA​ available to a diagonal readout space A\mathcal AA. If the joint state--tangent frame is Haar random, we prove that the two successive information fractions are independent Beta variables whose means are 1/21/21/2 and r/(2n−1)r/(2^n-1)r/(2n−1), where rrr is the centered dimension of the readout. Consequently, even the joint span of all computational-basis Pauli strings through any fixed weight kkk retain only O(nk2−n)O(n^k2^{-n})O(nk2−n) of the full-record information. Exact-statevector experiments across six circuit families show increasing finite-size agreement with this hierarchy in five nonconserving ensembles as the circuit depth grows. A number-conserving family departs strongly from the isotropic prediction even after correcting the support and readout rank, showing that rank alone is insufficient without tangent isotropy.
A coherent sum of quantum circuits introduces interference that can alter the training signal. We distinguish a single linear combination of unitaries (LCU) from a circuit that repeatedly alternates branch operations and ancilla mixing. For a single-pass LCU, independent branches with zero matrix means cannot erase the differentiated branch's gradient variance: the total variance is at least the branch variance times the fourth power of its coefficient magnitude. This bound holds at any gate position and does not require the circuit unitaries to be Haar-random. Beyond this lower bound, we derive exact loss and gradient formulas that quantify the additional contributions from interference between branches. One of these terms depends on relative coefficient phases and vanishes when the branch ensemble is invariant under multiplication by iIi \mathbb{I}iI; a four-qubit matchgate example shows that it can be nonzero. For repeated interleaving with full special-unitary ancilla control, we determine the generated Lie algebra from the associative algebra of the branches, including the available central phase directions. This control result predicts concentration only with additional assumptions on circuit sampling. Matchgate examples exhibit polynomial single-pass LCU variances and exponential concentration under joint Haar-random sampling. We also derive coefficient-selection rules and bounds for postselection, measurement error, and approximate moments.
Barren plateaus have emerged as a pivotal challenge for variational quantum computing. Our understanding of this phenomenon underwent a transformative shift with the recent introduction of a Lie algebraic theory capable of explaining most sources of barren plateaus. However, this theory requires either initial states or observables that lie in the circuit's Lie algebra. Focusing on parametrized matchgate circuits, in this work we are able to go beyond this assumption and provide an exact formula for the loss function variance that is valid for arbitrary input states and measurements. Our results reveal that new phenomena emerge when the Lie algebra constraint is relaxed. For instance, we find that the variance does not necessarily vanish inversely with the Lie algebra's dimension. Instead, this measure of expressiveness is replaced by a generalized expressiveness quantity: The dimension of the Lie group modules. By characterizing the operators in these modules as products of Majorana operators, we can introduce a precise notion of generalized globality and show that measuring generalized-global operators leads to barren plateaus. Our work also provides operational meaning to the generalized entanglement as we connect it with known fermionic entanglement measures, and show that it satisfies a monogamy relation. Finally, while parameterized matchgate circuits are not efficiently simulable in general, our results suggest that the structure allowing for trainability may also lead to classical simulability.
Variational quantum computing schemes train a loss function by sending an initial state through a parametrized quantum circuit, and measuring the expectation value of some operator. Despite their promise, the trainability of these algorithms is hindered by barren plateaus (BPs) induced by the expressiveness of the circuit, the entanglement of the input data, the locality of the observable, or the presence of noise. Up to this point, these sources of BPs have been regarded as independent. In this work, we present a general Lie algebraic theory that provides an exact expression for the variance of the loss function of sufficiently deep parametrized quantum circuits, even in the presence of certain noise models. Our results allow us to understand under one framework all aforementioned sources of BPs. This theoretical leap resolves a standing conjecture about a connection between loss concentration and the dimension of the Lie algebra of the circuit’s generators. The barren plateau problem represents one of the major bottlenecks for parametrized quantum circuits algorithms. Here, the authors study the known sources of BP using the lens of Lie algebraic theory, finding an expression of the variance of the loss function depending on the dynamical Lie algebra of the circuit.
Training the circuit of a quantum generative model is hampered by barren plateaus, where gradients vanish exponentially with the number of qubits. In the present work we propose a generative protocol based on the paradigm of quantum extreme learning machines, in which the circuit is never trained. A random latent vector is encoded into a quantum state, evolved with a fixed unitary and measured, and only a linear output layer is trained. The encoding of the latent variable determines the functions available to the model, while the unitary decides which combinations of them are seen by the trained layer. With the help of Pauli propagation we show that for a Clifford unitary each feature is a single product of sines and cosines of the encoded angles, so that a Clifford circuit, even a strongly entangling one, does not create new functions. Doping the circuit with TTT gates mixes these products and changes the nonstabilizerness of the state, but the space of functions remains the one fixed by the encoding. For brickworks of Haar-random two-qubit gates the features are no longer single products, and their variation with the latent variable decreases with the depth of the circuit. For the proposed encoding and for one- and two-dimensional distributions, generated both from expectation values and from single measurements, neither the entanglement produced by a Clifford reservoir nor the additional nonstabilizerness of Haar-random brickworks systematically improves the generator, and a reservoir with no gates at all is competitive with entangling circuits.
The barren plateaus phenomenon, where the gradients of parametrized quantum circuits become vanishingly small, poses a significant challenge in quantum machine learning. Foundational analyses of barren plateaus often use methods like the Weingarten formula, which, under the global Haar random assumption, effectively links the gradient variance to the Hilbert space dimension. However, many practical circuits, such as Hardware-Efficient Ans\"atze, utilize a different, non-Haar random model based on the sampling of parameters for single-qubit rotation gates. This study introduces a direct computational method for the gradient expectation and variance, specifically for parametrized quantum circuits constructed from single-qubit rotation gates. Our analysis reveals a direct, quantitative relationship between the gradient variance, circuit depth, and number of effective parameters. Numerical simulations further confirm the validity of our theoretical results. Our findings highlight the role of effective parameters as a key indicator for gradient trainability and provide a precise framework for analyzing barren plateaus in these specific non-Haar random circuits.