Entanglement spectroscopy on a quantum computer

Abstract

We present a quantum algorithm to compute the entanglement spectrum of arbitrary quantum states. The interesting universal part of the entanglement spectrum is typically contained in the largest eigenvalues of the density matrix which can be obtained from the lower Renyi entropies through the Newton-Girard method. Obtaining the $p$ largest eigenvalues ($\lambda_1>\lambda_2\ldots>\lambda_p$) requires a parallel circuit depth of $\mathcal{O}(p(\lambda_1/\lambda_p)^p)$ and $\mathcal{O}(p\log(N))$ qubits where up to $p$ copies of the quantum state defined on a Hilbert space of size $N$ are needed as the input. We validate this procedure for the entanglement spectrum of the topologically-ordered Laughlin wave function corresponding to the quantum Hall state at filling factor $\nu=1/3$. Our scaling analysis exposes the tradeoffs between time and number of qubits for obtaining the entanglement spectrum in the thermodynamic limit using finite-size digital quantum computers. We also illustrate the utility of the second Renyi entropy in predicting a topological phase transition and in extracting the localization length in a many-body localized system

Learning the quantum algorithm for state overlap

“…Indeed, the ancillabased version of the Swap Test, shown in figure 1, continues to be the algorithm employed in the quantum computing literature (e.g. see [27,33]).…”

“…In quantum supervised learning [30,31], which subsumes quantum support vector machines [32], the Swap Test is used to assign each data vector to a cluster. The Swap Test allows one to quantify entanglement for many-body quantum states [27,33] using the Renyi order-2 entanglement, given by…”

Entanglement spectroscopy with a depth-two quantum circuit

“…As noted in Refs. [15,18] these quantities can be related to each other by the Newton-Girard Formula:…”

“…In Refs. [15,18] it was argued that an approximation to the n max largest eigenvalues of ρ can be obtained by truncating the polynomial on the right-hand-side of Eq. (11) to FIG.…”

“…This algorithm is shown in Fig. 1 [18] to compute Tr(ρ n A ) for integer n 2. Here, ρA contains k qubits and is the reduced state of |ψ (containing 2k qubits).…”