Taming Barren Plateaus in Arbitrary Parameterized Quantum Circuits without Sacrificing Expressibility

TL;DR

This work tackles barren plateaus in parameterized quantum circuits by inserting gadget layers to form Modified Parameterized Quantum Circuits (MPQCs). The authors prove MPQCs are at least as expressive as the original circuits and, under mild locality and light-cone conditions, are provably free of barren plateaus with gradient variances bounded by a polynomial in the system size, while maintaining robustness to realistic noise. They provide a practical strategy to activate untrainable parameters and extend the approach to multi-parameter activation scenarios, enabling training of all parameters. Numerical experiments on thermal-state preparation circuits demonstrate effective elimination of barren plateaus up to 100 qubits and 2400 layers, highlighting potential for training deep PQCs on NISQ devices. The results offer a hardware-friendly path toward scalable, trainable PQCs with sustained quantum advantage potential in realistic, noisy settings.

Abstract

Quantum algorithms based on parameterized quantum circuits (PQCs) have enabled a wide range of applications on near-term quantum devices. However, existing PQC architectures face several challenges, among which the ``barren plateaus" phenomenon is particularly prominent. In such cases, the loss function concentrates exponentially with increasing system size, thereby hindering effective parameter optimization. To address this challenge, we propose a general and hardware-efficient method for eliminating barren plateaus in an arbitrary PQC. Specifically, our approach achieves this by inserting a layer of easily implementable quantum channels into the original PQC, each channel requiring only one ancilla qubit and four additional gates, yielding a modified PQC (MPQC) that is provably at least as expressive as the original PQC and, under mild assumptions, is guaranteed to be free from barren plateaus. Furthermore, by appropriately adjusting the structure of MPQCs, we rigorously prove that any parameter in the original PQC can be made trainable. Importantly, the absence of barren plateaus in MPQCs is robust against realistic noise, making our approach directly applicable to current noisy intermediate-scale quantum (NISQ) hardware. Numerically, we demonstrate the practicality of our method by modifying a commonly used PQC for thermal-state preparation. The results show that {barren plateaus are effectively eliminated} in this class of circuits with up to 100 qubits and 2400 layers, whereas the original ansatz suffers from severe gradient vanishing.

Figures (14)

• Figure 1: (a) Structure of an MPQC, where a layer of gadgets $\mathcal{G}(\bm{\theta})$ (outlined by the red dashed box) is inserted into the original PQC $\mathcal{C}(\bm{\theta})$ (indicated by the light-blue region). Each gadget $\mathcal{G}(\bm{\theta})$ (highlighted in light purple) contains an ancilla qubit initialized in $\ket{0}$, one single-qubit unitary $op$, and three two-qubit rotation gates $R_{XX}$, $R_{YY}$, and $R_{ZZ}$. The symbol $\vcenter{}$ denotes the ancilla is discarded (or measured). (b) Structure of a $T$-activating MPQC. The gates denoted by “$\cdots$” represent those in the original PQC located between $T$ and the gadget layer. In this MPQC, we specifically enlarge the gadget $\mathcal{G}\left(\bm{\theta}\right)$ acting on the same qubit as $T$, transforming it into $\mathcal{G}'_T\left(\bm{\theta}\right)$. (c) Top: Ansatz circuit for thermal state preparation, where the number of blocks equals the number of qubits $n$. Bottom: Variance comparison between PQCs and MPQCs for thermal state preparation, where the MPQCs are formed by inserting a gadget layer before the final block. Yellow and blue curves show the cost-function variances of PQCs and MPQCs, respectively, estimated via the method in Ref. shao2025diagnosing. The blue curve is omitted for $n>21$, as its values are extremely close to zero in this regime. The inset presents the gradient variance of parameters located after the gadget layer for $n=20$.
• Figure A.2: An example of an MPQC, where gadgets $\mathcal{G}\left(\bm{\theta}\right)$ drawn in blue are inserted into the original PQC. The gadget contains an ancilla qubit $\ket{0}$, one single qubit gate $op$ and three 2-qubit rotation gates $R_{XX}, R_{YY}, R_{ZZ}$.
• Figure A.3: One construction of the MPQC: all gadgets $\mathcal{G}\left(\bm{\theta}\right)$ are inserted in parallel after the $l$-th layer of the original circuit.
• Figure A.4: Trainable construction of $op$, in which we allow parameters $\theta_1$ and $\theta_2$ to be trainable.
• Figure A.5: Effect of the gadget $\mathcal{G}(\boldsymbol{\theta})$ on Pauli paths in the Heisenberg picture.

Theorems & Definitions (47)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: informal
• Definition A.1
• Theorem A.1
• proof
• Corollary A.1
• proof
• lemma A.1