Dynamic parameterized quantum circuits: expressive and barren-plateau free

TL;DR

The paper proposes dynamic parameterized quantum circuits (DPQC) that incorporate intermediate measurements and feedforward to bypass barren plateaus while retaining expressive power. It develops a unifying theoretical framework via a stat-mech model to prove sufficient conditions under which DPQCs avoid BP and analyzes gradient behavior, including robustness to noise. Numerical experiments on ground- and thermal-state preparation demonstrate DPQCs can compete with state-of-the-art PQCs and produce meaningful purifications or Gibbs-like states. The authors discuss classical hardness, showing DPQCs can be worst-case hard to simulate yet average-case easy, and they highlight open questions about trainability and the occurrence of hard instances during optimization. Overall, DPQCs present a promising, flexible avenue for scalable variational quantum algorithms with potential quantum advantage, albeit with caveats and directions for future work on scaling and practical loss functions.

Abstract

Classical optimization of parameterized quantum circuits is a widely studied methodology for the preparation of complex quantum states, as well as the solution of machine learning and optimization problems. However, it is well known that many proposed parameterized quantum circuit architectures suffer from drawbacks which limit their utility, such as their classical simulability or the hardness of optimization due to a problem known as "barren plateaus". We propose and study a class of dynamic parameterized quantum circuit architectures. These are parameterized circuits containing intermediate measurements and feedforward operations. In particular, we show that these architectures: 1. Provably do not suffer from barren plateaus. 2. Are expressive enough to describe arbitrarily deep unitary quantum circuits. 3. Are competitive with state of the art methods for preparing ground states and facilitating the representation of nontrivial thermal states. These features make the proposed architectures promising candidates for a variety of applications.

Figures (13)

• Figure 1: An illustration of the dynamic parameterized quantum circuit (DPQC) architectures that we consider in this work. These circuits consist of parameterized two-qubit unitary gates $U(\bm{\theta})$, as well as parameterized nonunitary single-qubit dynamic operations, which are denoted as $\mathcal{F}(\bm{\theta})$-gates. Each such $\mathcal{F}(\bm{\theta})$ operation is a probabilistic implementation of a feedforward operation $\mathcal{F}$.
• Figure 2: An illustration of the shortest paths from a qubit measurement to a feedforward operation through the backwards light cone of the measurement. The feedforward distance of an observable, with respect to a specific DPQC architecture, is the maximum length of such paths, over all qubits on which the observable is supported. Theorem \ref{['thm:BP_DPQC_informal']} provides an upper bound on the variance of a local observable in terms of the feedforward distance.
• Figure 3: A schematic dynamic parameterized circuit on $6$ qubits with $3$ of them corresponding to ancillas.
• Figure 4: An illustration of the Hamiltonian and DPQC architecture used for ground state experiments. (a) The Hamiltonian is the perturbed toric code, for a system on a square lattice, with system qubits on the edges. (b) One layer of the ansatz consists of five sublayers. The first four are parameterized two-qubit unitary entangling gates, and the last sublayer consists of deterministic reset gates on all ancilla qubits. Gates are applied from lightest to darkest, so that all gates of the same opacity are applied in parallel. (c) Structure of the gates within each sublayer. $U_3$ denotes a generic single-qubit rotation gate with 3 Euler angles.
• Figure 5: Variational training of a DPQC architecture for a perturbed toric code Hamiltonian acting on 12 qubits. The perturbation strength is controlled by the parameter $h$. All settings were run for $100$ different seeds. The solid lines represent the average over these runs and the shaded, translucent lines illustrate the standard deviation. (a) The training dynamics of the loss function, for $h=0,0.12$ and $0.96$. The results indicate quick convergence for all trials. (b) The dynamics of the state's purity during training, depicted for the same values of $h$. Note that the purity does not explicitly enter into the loss function.

Theorems & Definitions (47)

• Theorem 1: Absence of barren plateaus in DPQCs for $k$-local Hamiltonians---informal
• Theorem 2: Noise robustness of \ref{['thm:BP_DPQC_informal']}---informal
• Definition 1: Architecture
• Definition 2: Dynamic operation
• Definition 3: Parameterized dynamic circuit
• Definition 4: Informal version of \ref{['def_ff_distance']}
• Theorem 3.A: Variance bound for $k$-local Hamiltonians---formal version of \ref{['thm:BP_DPQC_informal']}
• Theorem 3.B: Formal version of \ref{['thm:robustness-informal']}
• Lemma 4: Informal version of \ref{['lem_ham_variance']}
• Definition 5: Haar measure