Phase transitions in parametrized quantum circuits

Abstract

Phase transitions are among the most intriguing phenomena in physical systems, yet their behavior near criticality remain challenging to study using classical algorithms. Parameterized quantum circuits (PQCs) offer a promising approach to investigating such regimes on practical quantum computers. However, in order to use it to probe critical behavior, a PQC itself should be non-trivial and exhibit a phase transition and non-analyticity -- a property that has not yet been clearly identified. In this work, we identify a mechanism for generating non-analyticities intrinsically in PQCs. As a concrete realization, we construct a class of sequential PQCs whose observable expectation value is a non-analytic function of the circuit parameter in the infinite volume limit, showing that the prepared PQC states undergo a phase transition at the non-analytic points. The entanglement and the identified order parameter have distinct behaviors in different phases, revealing a phase diagram of the PQC state. We show that classical simulation of this PQC based on tensor networks and Pauli propagation gets less efficient in the vicinity of the phase transition point, indicating a physically motivated route towards practical quantum advantage using PQCs with phase transitions.

Figures (23)

• Figure 1: (a) Illustration of a sequential parametrized quantum circuit (PQC) that exhibits a phase transition. Qubits on a 2-d lattice are initialized in a $\ket{+}$ product state. Black arrows denote $ZY$ rotations applied sequentially to the qubits. (b) Energy landscape of PQCs with (left panel) and without (right panel) phase transitions. The energy cusp singularity at $\theta=\theta_c$ signals a phase transition dividing the shaded and blank parameter regions in the left panel.
• Figure 2: Pauli propagation of the observable $Z_7Z_8$ in a PQC that consists of a $ZY$ rotation cycle $\mathcal{C}$. Each red arrow denotes a $ZY$ rotation $e^{-\mathrm{i}\theta Z_iY_j/2}$ specified in Fig. \ref{['fig:intro-figure']}(a). The vertically arranged Pauli strings together form a Pauli path contributing to the expectation $\langle Z_7Z_8\rangle_{\theta}$.
• Figure 3: Whip circuit preparing the Ising ground state. The control qubit at the center of the circle whips the remaining qubits to align with it. The circle is the Bloch sphere in the $x$-$z$ plane. The gray dumbbells indicate superposition.
• Figure 4: Non-local Pauli paths of the 2-d whip circuit. (a) Black arrows denote $ZY$ rotations that act non-trivially on the observable $Z_iZ_j$. The red-highlighted cycle corresponds to a non-local Pauli path similar to that in Fig. \ref{['fig:non_local_path']}. (b) Three non-local Pauli paths in the length-$l$ width-$w$ rectangle contributing to the expectation $\langle Z_i Z_j\rangle_{\theta}$.
• Figure 5: (a) Energy density of the Ising whip circuit and its 2nd derivative as a function of the whip angle $\theta$. Numerical curves on $L\times L$ lattices and their infinite volume extrapolated curve converge to the analytic one. $\theta_c=-\pi/4$ is the non-analytic point with the divergent derivative. (b) Entanglement entropy and entanglement spectrum of the Ising whip circuit on lattices from $2\times 2$ to $10\times 10$. They have distinguished behavior in sectors divided by the non-analytic points $\theta_c=\pm\pi/4$. (c) Boundary ($\mathcal{B}$) and lower boundary ($\mathcal{B}'$) of the 2-d lattice. The order parameter operator is the summation of Pauli-$X$ operators at the lower boundary. (d) Order parameter of the Ising whip circuit as a function of the whip angle. The transitions from zero to non-zero at $\theta_c=\pi/4+m\pi/2, m\in\mathbb{Z}$ signal spontaneous symmetry breaking.

Theorems & Definitions (4)

• Proposition 1
• proof : proof
• Proposition 2
• proof : proof