Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2107.07378 · IEEE International Conference on Web Services (ICWS)

Best-approximation error for parametric quantum circuits

Lena Funcke, Tobias Hartung, Karl Jansen, Stefan Kühn, Manuel Schneider, Paolo Stornati

Published Sep 1, 2021• 4 citations
arXivHTMLPDF

Abstract

In Variational Quantum Simulations, the construction of a suitable parametric quantum circuit is subject to two counteracting effects. The number of parameters should be small for the device noise to be manageable, but also large enough for the circuit to be able to represent the solution. Dimensional expressivity analysis can optimize a candidate circuit considering both aspects. In this article, we will first discuss an inductive construction for such candidate circuits. Furthermore, it is sometimes necessary to choose a circuit with fewer parameters than necessary to represent all relevant states. To characterize such circuits, we estimate the best-approximation error using Voronoi diagrams. Moreover, we discuss a hybrid quantum-classical algorithm to estimate the worst-case best-approximation error, its complexity, and its scaling in state space dimensionality. This allows us to identify some obstacles for variational quantum simulations with local optimizers and underparametrized circuits, and we discuss possible remedies.

DOI
10.1109/icws53863.2021.00096

Related papers

Dimensional Expressivity Analysis, best-approximation errors, and automated design of parametric quantum circuits

Tobias Hartung, Lena Funcke, Karl Jansen +3 more

The design of parametric quantum circuits (PQCs) for efficient use in variational quantum simulations (VQS) is subject to two competing factors. On one hand, the set of states that can be generated by the PQC has to be large enough to contain the solution state. Otherwise, one may at best find the best approximation of the solution restricted to the states generated by the chosen PQC. On the other hand, the PQC should contain as few parametric quantum gates as possible to minimize noise from the quantum device. Thus, when designing a PQC one needs to ensure that there are no redundant parameters. The dimensional expressivity analysis discussed in these proceedings is a means of addressing these counteracting effects. Its main objective is to identify independent and redundant parameters in the PQC. Using this information, superfluous parameters can be removed and the dimension of the space of states that are generated by the PQC can be computed. Knowing the dimension of the physical state space then allows us to deduce whether or not the PQC can reach all physical states. Furthermore, the dimensional expressivity analysis can be implemented efficiently using a hybrid quantum-classical algorithm. This implementation has relatively small overhead costs both for the classical and quantum part of the algorithm and could therefore be used in the future for on-the-fly circuit construction. This would allow for optimized circuits to be used in every loop of a VQS rather than the same PQC for the entire VQS. These proceedings review and extend work in [1, 2].

arXiv:2111.11489•International Symposium on Lattice Field Theory (LATTICE)•May 16, 2022•0 citations

Dimensional Expressivity Analysis of Parametric Quantum Circuits

Lena Funcke, Tobias Hartung, Karl Jansen +2 more

Parametric quantum circuits play a crucial role in the performance of many variational quantum algorithms. To successfully implement such algorithms, one must design efficient quantum circuits that sufficiently approximate the solution space while maintaining a low parameter count and circuit depth. In this paper, develop a method to analyze the dimensional expressivity of parametric quantum circuits. Our technique allows for identifying superfluous parameters in the circuit layout and for obtaining a maximally expressive ansatz with a minimum number of parameters. Using a hybrid quantum-classical approach, we show how to efficiently implement the expressivity analysis using quantum hardware, and we provide a proof of principle demonstration of this procedure on IBM's quantum hardware. We also discuss the effect of symmetries and demonstrate how to incorporate or remove symmetries from the parametrized ansatz.

arXiv:2011.03532•Quantum•Mar 29, 2021•56 citations

Dimensional Expressivity Analysis of Parametric Quantum Circuits

Lena Funcke, Tobias Hartung, Karl Jansen +2 more

Parametric quantum circuits play a crucial role in the performance of many variational quantum algorithms. To successfully implement such algorithms, one must design efficient quantum circuits that sufficiently approximate the solution space while maintaining a low parameter count and circuit depth. In this paper, we develop a method to analyze the dimensional expressivity of parametric quantum circuits. Our technique allows for identifying superfluous parameters in the circuit layout and for obtaining a maximally expressive ansatz with a minimum number of parameters. Using a hybrid quantum-classical approach, we show how to efficiently implement the expressivity analysis using quantum hardware, and we provide a proof of principle demonstration of this procedure on IBM's quantum hardware. We also discuss the effect of symmetries and demonstrate how to incorporate or remove symmetries from the parametrized ansatz.

•Jan 1, 2021•53 citations

Global Optimization for Parametrized Quantum Circuits

Iosif Sakos, Antonios Varvitsiotis, Georgios Korpas +1 more

In the absence of error correction, noisy intermediate-scale quantum devices are operated by training parametrized quantum circuits (PQCs) so as to minimize a suitable loss function. Finding the optimal parameters of those circuits is a hard optimization problem, where global guarantees are known only for highly structured cases of limited practical relevance, and first-order methods can fail to find even local minima due to the presence of barren plateaus. In this work, we study the training of practical classes of PQCs, namely polynomial-depth circuits with a constant number of trainable parameters. This captures widely used PQC families, including fixed-depth QAOA, hardware-efficient ansätze, and Fixed Parameter Count QAOA. Our main technical result is a fully polynomial randomized approximation scheme (FPRAS), which, for every $ε>0$, returns an $ε$-approximate solution to the problem's global optimum with high probability, and has runtime and query complexity polynomial in $1/ε$ and the number of qubits. Unlike the standard hybrid quantum-classical training loop in variational algorithms, where the quantum device is queried repeatedly throughout the training, our approach separates the computation into two distinct stages: (1) an initial quantum data-acquisition phase, followed by (2) a classical global-optimization phase based on the trigonometric moment/sum-of-squares hierarchies. Under a standard flat-extension condition, which can be checked numerically, the method also supports the extraction of optimal circuit parameters. The existence of an FPRAS implies that the promise problem associated with the optimization of poly-depth constant-parameter PQC is in BQP. This imposes a limitation on the expressive power of the class, namely, it cannot encode combinatorial optimization problems whose objective values are separated by an inverse-polynomial gap.

quant-pharXiv:2603.21757•Mar 24, 2026•0 citations

Perspectives on Quantum Sensing and Computation for Particle Physics

Lena Funcke

<!--HTML-->Parametric quantum circuits play a crucial role in the performance of many variational quantum algorithms. To successfully implement such algorithms, one must design efficient quantum circuits that sufficiently approximate the solution space while maintaining a low parameter count and circuit depth. In this talk, we present a method to analyze the dimensional expressivity of parametric quantum circuits. This technique allows for identifying superfluous parameters in the circuit layout and for obtaining a maximally expressive ansatz with a minimum number of parameters. Using a hybrid quantum-classical approach, we show how to efficiently implement the expressivity analysis using quantum hardware, and we provide a proof of principle demonstration of this procedure on IBM's quantum hardware. We also discuss the effect of symmetries and demonstrate how to incorporate or remove symmetries from the parametrized ansatz.

•Jan 1, 2021•0 citations

Role of overparametrization in quantum approximate optimization

Daniil Rabinovich, Andrey Kardashin, Soumik Adhikary

Variational quantum algorithms have emerged as a cornerstone of contemporary quantum algorithms research. While they have demonstrated considerable promise in solving problems of practical interest, efficiently determining the minimal quantum resources necessary to obtain such a solution remains an open question. In this work, inspired by concepts from classical machine learning, we investigate the impact of overparameterization on the performance of variational algorithms. Our study focuses on the quantum approximate optimization algorithm (QAOA) -- a prominent variational quantum algorithm designed to solve combinatorial optimization problems. We investigate if circuit overparametrization is necessary and sufficient to solve such problems in QAOA, considering two representative problems -- MAX-CUT and MAX-2-SAT. For MAX-CUT we observe that overparametriation is both sufficient and (statistically) necessary for attaining exact solutions, as confirmed numerically for up to $20$ qubits. In fact, for MAX-CUT on 2-regular graphs we show the necessity to be exact, based on the analytically found optimal depth. In sharp contrast, for MAX-2-SAT, underparametrized circuits suffice to solve most instances. This result highlights the potential of QAOA in the underparametrized regime, supporting its utility for current noisy devices.

arXiv:2508.10086•Physical Review A•May 26, 2026•0 citations
In Variational Quantum Simulations, the construction of a suitable parametric quantum circuit is subject to two counteracting effects. The number of parameters should be small for the device noise to be manageable, but also large enough for the circuit to be able to represent the solution. Dimensional expressivity analysis can optimize a candidate circuit considering both aspects. In this article, we will first discuss an inductive construction for such candidate circuits. Furthermore, it is sometimes necessary to choose a circuit with fewer parameters than necessary to represent all relevant states. To characterize such circuits, we estimate the best-approximation error using Voronoi diagrams. Moreover, we discuss a hybrid quantum-classical algorithm to estimate the worst-case best-approximation error, its complexity, and its scaling in state space dimensionality. This allows us to identify some obstacles for variational quantum simulations with local optimizers and underparametrized circuits, and we discuss possible remedies.
The design of parametric quantum circuits (PQCs) for efficient use in variational quantum simulations (VQS) is subject to two competing factors. On one hand, the set of states that can be generated by the PQC has to be large enough to contain the solution state. Otherwise, one may at best find the best approximation of the solution restricted to the states generated by the chosen PQC. On the other hand, the PQC should contain as few parametric quantum gates as possible to minimize noise from the quantum device. Thus, when designing a PQC one needs to ensure that there are no redundant parameters. The dimensional expressivity analysis discussed in these proceedings is a means of addressing these counteracting effects. Its main objective is to identify independent and redundant parameters in the PQC. Using this information, superfluous parameters can be removed and the dimension of the space of states that are generated by the PQC can be computed. Knowing the dimension of the physical state space then allows us to deduce whether or not the PQC can reach all physical states. Furthermore, the dimensional expressivity analysis can be implemented efficiently using a hybrid quantum-classical algorithm. This implementation has relatively small overhead costs both for the classical and quantum part of the algorithm and could therefore be used in the future for on-the-fly circuit construction. This would allow for optimized circuits to be used in every loop of a VQS rather than the same PQC for the entire VQS. These proceedings review and extend work in [1, 2].
Parametric quantum circuits play a crucial role in the performance of many variational quantum algorithms. To successfully implement such algorithms, one must design efficient quantum circuits that sufficiently approximate the solution space while maintaining a low parameter count and circuit depth. In this paper, develop a method to analyze the dimensional expressivity of parametric quantum circuits. Our technique allows for identifying superfluous parameters in the circuit layout and for obtaining a maximally expressive ansatz with a minimum number of parameters. Using a hybrid quantum-classical approach, we show how to efficiently implement the expressivity analysis using quantum hardware, and we provide a proof of principle demonstration of this procedure on IBM's quantum hardware. We also discuss the effect of symmetries and demonstrate how to incorporate or remove symmetries from the parametrized ansatz.
Parametric quantum circuits play a crucial role in the performance of many variational quantum algorithms. To successfully implement such algorithms, one must design efficient quantum circuits that sufficiently approximate the solution space while maintaining a low parameter count and circuit depth. In this paper, we develop a method to analyze the dimensional expressivity of parametric quantum circuits. Our technique allows for identifying superfluous parameters in the circuit layout and for obtaining a maximally expressive ansatz with a minimum number of parameters. Using a hybrid quantum-classical approach, we show how to efficiently implement the expressivity analysis using quantum hardware, and we provide a proof of principle demonstration of this procedure on IBM's quantum hardware. We also discuss the effect of symmetries and demonstrate how to incorporate or remove symmetries from the parametrized ansatz.
In the absence of error correction, noisy intermediate-scale quantum devices are operated by training parametrized quantum circuits (PQCs) so as to minimize a suitable loss function. Finding the optimal parameters of those circuits is a hard optimization problem, where global guarantees are known only for highly structured cases of limited practical relevance, and first-order methods can fail to find even local minima due to the presence of barren plateaus. In this work, we study the training of practical classes of PQCs, namely polynomial-depth circuits with a constant number of trainable parameters. This captures widely used PQC families, including fixed-depth QAOA, hardware-efficient ansätze, and Fixed Parameter Count QAOA. Our main technical result is a fully polynomial randomized approximation scheme (FPRAS), which, for every ε>0ε>0ε>0, returns an εεε-approximate solution to the problem's global optimum with high probability, and has runtime and query complexity polynomial in 1/ε1/ε1/ε and the number of qubits. Unlike the standard hybrid quantum-classical training loop in variational algorithms, where the quantum device is queried repeatedly throughout the training, our approach separates the computation into two distinct stages: (1) an initial quantum data-acquisition phase, followed by (2) a classical global-optimization phase based on the trigonometric moment/sum-of-squares hierarchies. Under a standard flat-extension condition, which can be checked numerically, the method also supports the extraction of optimal circuit parameters. The existence of an FPRAS implies that the promise problem associated with the optimization of poly-depth constant-parameter PQC is in BQP. This imposes a limitation on the expressive power of the class, namely, it cannot encode combinatorial optimization problems whose objective values are separated by an inverse-polynomial gap.
<!--HTML-->Parametric quantum circuits play a crucial role in the performance of many variational quantum algorithms. To successfully implement such algorithms, one must design efficient quantum circuits that sufficiently approximate the solution space while maintaining a low parameter count and circuit depth. In this talk, we present a method to analyze the dimensional expressivity of parametric quantum circuits. This technique allows for identifying superfluous parameters in the circuit layout and for obtaining a maximally expressive ansatz with a minimum number of parameters. Using a hybrid quantum-classical approach, we show how to efficiently implement the expressivity analysis using quantum hardware, and we provide a proof of principle demonstration of this procedure on IBM's quantum hardware. We also discuss the effect of symmetries and demonstrate how to incorporate or remove symmetries from the parametrized ansatz.
Variational quantum algorithms have emerged as a cornerstone of contemporary quantum algorithms research. While they have demonstrated considerable promise in solving problems of practical interest, efficiently determining the minimal quantum resources necessary to obtain such a solution remains an open question. In this work, inspired by concepts from classical machine learning, we investigate the impact of overparameterization on the performance of variational algorithms. Our study focuses on the quantum approximate optimization algorithm (QAOA) -- a prominent variational quantum algorithm designed to solve combinatorial optimization problems. We investigate if circuit overparametrization is necessary and sufficient to solve such problems in QAOA, considering two representative problems -- MAX-CUT and MAX-2-SAT. For MAX-CUT we observe that overparametriation is both sufficient and (statistically) necessary for attaining exact solutions, as confirmed numerically for up to 202020 qubits. In fact, for MAX-CUT on 2-regular graphs we show the necessity to be exact, based on the analytically found optimal depth. In sharp contrast, for MAX-2-SAT, underparametrized circuits suffice to solve most instances. This result highlights the potential of QAOA in the underparametrized regime, supporting its utility for current noisy devices.