Polynomial-time certification of fidelity for many-body mixed states and mixed-state universality classes

TL;DR

A polynomial-time algorithm to compute certified lower and upper bounds for the fidelity between matrix product density operators (MPDOs) is introduced, allowing for systematic improvement of the lower bounds by increasing the circuit depth and establishing a scalable framework for the verification of many-body quantum systems.

Abstract

Computation of Uhlmann fidelity between many-body mixed states generally involves full diagonalization of exponentially large matrices. In this work, we introduce a polynomial-time algorithm to compute certified lower and upper bounds for the fidelity between matrix product density operators (MPDOs). Our method maps the fidelity estimation problem to a variational optimization of sequential quantum circuits, allowing for systematic improvement of the lower bounds by increasing the circuit depth. Complementarily, we obtain certified upper bounds on fidelity by variational lower bounds on the trace distance through the same framework. We demonstrate the power of this approach with two examples: fidelity correlators in critical mixed states, and codeword distinguishability in an approximate quantum error-correcting code. Remarkably, the variational lower bound accurately track the universal scaling behavior of the fidelity with a size-consistent relative error, allowing for the extraction of previously unknown critical exponents. Our results offer an exponential improvement in precision over known moment-based bounds and establish a scalable framework for the verification of many-body quantum systems.

Figures (8)

• Figure 1: Fidelity of matrix product density operators from sequential circuit optimization. Left: $\rho = \mathrm{Tr}_{\mathcal{H}_p} |\psi_{\rho}\rangle\!\rangle\langle\!\langle \psi_{\rho}|$ is a locally purified MPDO. The upper part of the diagram represents the purification $|\psi_\rho\rangle\!\rangle$ as a two-sided MPS, where the bold vertical lines denote degrees of freedom in $\mathcal{H}_s$, and the thin vertical lines denote degrees of freedom in $\mathcal{H}_p$. Similar for $\sigma=\mathrm{Tr}_{\mathcal{H}_p} |\psi_{\sigma}\rangle\!\rangle\langle\!\langle \psi_{\sigma}|$ in the lower part. Right: Variational optimization of fidelity $F(\rho,\sigma)$ with a depth-$t$ sequential circuit with ancilla, with $U=U^{(t)}\cdots U^{(2)} U^{(1)}$.
• Figure 2: The fidelity correlator $D^{(2)}_{ij}$ of the second excited state of the critical Ising model under dephasing. We fix $|i-j|=N/2$ and vary the system size $N$. The $q=0$ solid line represents the noiseless correlator $\tilde{D}^{(2)}_{ij}$ which decays as $N^{-9/4}$. The $q=0.3$ solid line represents the exact fidelity correlator of the dephased state up to $N=14$. The dashed lines represent the lower bounds for the $q=0.3$ case obtained through different circuit structures, including sequential circuits, sequential circuits without ancilla, and FDLU circuits (with ancilla). For FDLU circuits, a depth-$1$ circuit includes one layer of even unitaries and one layer of odd unitaries. The lower bounds of $D^{(2)}$ consistently decay as $N^{-0.6}$, where the exponent is consistent with the extrapolation from the exact values with $N\leq 14$.
• Figure 3: Fidelity $F(\mathcal{N}(\rho), \mathcal{N}(\sigma))$ of the ground state and second excited state of the critical Ising model under $X$ (left) or $Z$ (right) dephasing noise with strength $q=0.3$. The lower bound on the left suggests that the noise is undecodable. The lower bound on the right shows power law $N^{-1/2}$ behavior, where the exponent agrees with the scaling analysis of the true fidelity, indicating $N$-independent relative error. We also plot the upper bound through Eq. \ref{['eq:vandegraaf']} and find that they provide stronger bounds than the superfidelity.
• Figure 4: Optimal unitary of the fidelity between an MPS and a locally purified MPDO.
• Figure 5: Sequential circuit realization of logical rotation $\bar{U} = \bar{R}_z(\alpha) \bar{R}_x(\beta)\bar{R}_z(\gamma)$ on the repetition code subspace.