Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

How to find expressible and trainable parameterized quantum circuits?

Peter Röseler, Dennis Willsch, Kristel Michielsen

Abstract

Whether parameterized quantum circuits (PQCs) can be systematically constructed to be both trainable and expressive remains an open question. Highly expressive PQCs often exhibit barren plateaus, while several trainable alternatives admit efficient classical simulation. We address this question by deriving a finite-sample, dimension-independent concentration bound for estimating the variance of a PQC cost function, yielding explicit trainability guarantees. Across commonly used ansätze, we observe an anticorrelation between trainability and expressibility, consistent with theoretical insights. Building on this observation, we propose a property-based ansatz-search framework for identifying circuits that combine trainability and expressibility. We demonstrate its practical viability on a real quantum computer and apply it to variational quantum algorithms. We identify quantum neural network ansätze with improved effective dimension using over $6 \times$ fewer parameters, and for VQE on $\mathrm{H}_2$ we achieve UCCSD-like accuracy at substantially reduced circuit complexity.

How to find expressible and trainable parameterized quantum circuits?

Abstract

Whether parameterized quantum circuits (PQCs) can be systematically constructed to be both trainable and expressive remains an open question. Highly expressive PQCs often exhibit barren plateaus, while several trainable alternatives admit efficient classical simulation. We address this question by deriving a finite-sample, dimension-independent concentration bound for estimating the variance of a PQC cost function, yielding explicit trainability guarantees. Across commonly used ansätze, we observe an anticorrelation between trainability and expressibility, consistent with theoretical insights. Building on this observation, we propose a property-based ansatz-search framework for identifying circuits that combine trainability and expressibility. We demonstrate its practical viability on a real quantum computer and apply it to variational quantum algorithms. We identify quantum neural network ansätze with improved effective dimension using over fewer parameters, and for VQE on we achieve UCCSD-like accuracy at substantially reduced circuit complexity.
Paper Structure (25 sections, 3 theorems, 48 equations, 18 figures, 7 tables)

This paper contains 25 sections, 3 theorems, 48 equations, 18 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Metrics
  3. Expressibility
  4. Trainability
  5. Entanglement
  6. Complexity
  7. Experiments
  8. Single-objective optimization
  9. Multi-objective optimization
  10. Quantum neural networks
  11. Variational quantum eigensolver
  12. Summary and outlook
  13. Expressibility at scale
  14. Trainability error
  15. Ansatz-search setup
  16. ...and 10 more sections

Key Result

Theorem 2.1

Let $L,U\in\mathbb R$ with $L<U$ and $R:=U-L$, $\mathbf{X}=(X_1,\dots,X_m)$ be independent random gradient samples with $X_i\in[L,U]$, and $m\ge 3$. Define $Z(\mathbf{X}):=\frac{m}{R^2}s_m^2$, where the sample variance is For $k\in\{1,\dots,m\}$ and $y\in[L,U]$, let $\mathbf X_{y,k}$ be obtained from $\mathbf X$ by replacing $X_k$ with $y$, and set Then almost surely with $a=\frac{m}{m-1}$. The

Figures (18)

  • Figure 1: Single-metric ansatz search. (a) Expressibility vs number of parameters ($|\boldsymbol{\theta}|$). (b) Trainability vs number of gates ($G$). (c) Entanglement vs circuit depth ($D$). (d) Expressibility vs search space complexity. Markers denote quantum circuit ansätze (see legend), comparing the circuit found by the ansatz search ($A$) to the benchmark circuits $i^r$ from Sim2019, where $r$ is the number of repetitions. Where applicable, the horizontal band labeled Haar indicates a $\pm 3\sigma$ interval estimated from $5{,}000$ (pair) Haar-random states for the corresponding metric. Marker size encodes a proxy for circuit complexity: in (a) $\propto \sqrt{G}$, in (b) $\propto \sqrt{D}$ (depth), and in (c) $\propto \sqrt{|\boldsymbol\theta|}$. Panel (d) shows the hardware-constrained expressibility search on the 5-qubit IQM Spark device (Spark ansatz search) using its native gate set and star connectivity.
  • Figure 2: Multi-objective ansatz search. Each point corresponds to a circuit $i^r$ from the benchmark families of Sim2019 (across repetitions $r$) and the circuit discovered by our search ($A$). The horizontal band labeled Haar indicates a $\pm 3\sigma$ interval estimated from 5,000 pair Haar-random states for the expressibility metric. (a) Shows the expressibility vs. trainability and (b) expressibility vs. entanglement. For visualization purposes, we set the expressibility threshold to $0.003$, the trainability threshold to $2.5$, and the entanglement threshold to $0.85$.
  • Figure 3: QNN case study following the setup of Abbas2021PowerQNN. (a) Search outcomes in the expressibility--trainability plane for the considered search space. (b) Estimated effective dimension for the reference QNN architecture from Abbas2021PowerQNN and for the discovered PQC. (c) Training performance on the dataset of Abbas2021PowerQNN, using the same optimizer and protocol, comparing the reference architecture to our discovered PQC. For visualization purposes, we set the expressibility and trainability thresholds to $0.003$ and $2.5$, respectively.
  • Figure 4: VQE evaluation of ansätze discovered by metric-driven search. (a) Metric values of the discovered ansatz and reference circuits for $\mathrm{H}_2$ at $R=0.75\,\text{\AA}$, including expressibility, trainability, and circuit complexity. (b) VQE energy convergence for $\mathrm{H}_2$ across multiple interatomic distances using a single fixed ansatz optimized at $R=0.75\,\text{\AA}$, compared against UCCSD and hardware-efficient baselines. (c) Proof-of-concept LiH VQE at $R=1.5\,\text{\AA}$ (100 Bayesian-optimization iterations). For visualization purposes, we set the expressibility and trainability thresholds to $0.003$ and $2.5$, respectively.
  • Figure 5: (a) Bloch-sphere intuition for the single-qubit family generated by $R_z(\theta)$ under two inputs: starting from $\ket{0}$, the state remains fixed on the Bloch sphere (localized ensemble), whereas starting from $\ket{+}$ the state traces the equator as $\theta$ varies (spread ensemble). (b) Binned pairwise-fidelity histograms for the corresponding $n$-qubit product ensemble produced by applying $R_z(\theta_i)$ on each qubit $i$ starting from $\ket{+}$ (with 75 bins), compared against the discretized Haar reference as $n$ increases.
  • ...and 13 more figures

Theorems & Definitions (6)

  • Theorem 2.1
  • proof
  • Lemma 1
  • proof
  • Theorem D.1
  • proof