Disentangling quantum autoencoder

TL;DR

The disentangling quantum autoencoder (DQAE) is proposed to encode entangled states into single-qubit product states to provide an exponential improvement in the number of copies needed to transport entangled states across qubit-loss or leakage channels compared to unencoded states.

Abstract

Entangled quantum states are highly sensitive to noise, which makes it difficult to transfer them over noisy quantum channels or to store them in quantum memory. Here, we propose the disentangling quantum autoencoder (DQAE) to encode entangled states into single-qubit product states. The DQAE provides an exponential improvement in the number of copies needed to transport entangled states across qubit-loss or leakage channels compared to unencoded states. The DQAE can be trained in an unsupervised manner from entangled quantum data. For general states, we train via variational quantum algorithms based on gradient descent with purity-based cost functions, while stabilizer states can be trained via a Metropolis algorithm. For particular classes of states, the number of training data needed to generalize is surprisingly low: For stabilizer states, DQAE generalizes by learning from a number of training data that scales linearly with the number of qubits, while only $1$ training sample is sufficient for states evolved with the transverse-field Ising Hamiltonian. Our work provides practical applications for enhancing near-term quantum computers.

Figures (11)

• Figure 1: Disentangling quantum autoencoder (DQAE) disentangles $N$-qubit state $|\psi \rangle$ into a tensor product of $N$ single-qubit states $|\phi_j \rangle$ via unitary $U(\boldsymbol{\theta})$, and can recover the original state again via the inverse $U^\dagger(\boldsymbol{\theta})$. The DQAE can be trained in an unsupervised manner from entangled data. For transferring a quantum state via a qubit-loss channel, DQAE gives an exponential reduction in the number of copies $R$ of the state.
• Figure 2: Illustration of the product protocol: Disentangling quantum autoencoder (DQAE) to transfer quantum states across qubit loss channels with exponential advantage. Entangled input $N$-qubit state $|\psi \rangle$ is transformed by $U(\boldsymbol{\theta})$ into $\bigotimes_{j=1}^N|\phi_{j} \rangle=U|\psi \rangle$, a tensor product of $N$ single-qubit states $|\phi_{j} \rangle$. The transformed state is then sent through a qubit loss channel $\mathcal{E}$, and stored in a quantum register by the receiver. We repeat this process for $R$ copies of $|\psi \rangle$, and finally store $NR$ qubits in the quantum register. Some qubits are lost due to the channel $\mathcal{E}$ (red crosses), which we detected using the leakage detection protocol of Fig. \ref{['fig:Leakage']}. At least one of each single-qubit state survives with high probability when $R\propto\log(N)$. In this case, we collect $k=1,\dots, N$ single-qubit states $|\phi_{j} \rangle$ unaffected by loss (green circle). With entangler $U^\dagger$ we then reconstruct the original state $|\psi \rangle=U^\dagger(\boldsymbol{\theta})\bigotimes_{j=1}^N|\phi_{j} \rangle$.
• Figure 3: Leakage detection protocol preskill1998fault. If data qubit $|\psi \rangle$ is present, measurement of the ancilla yields outcome $1$ and leaves $|\psi \rangle$ unchanged. If the data qubit was lost (i.e. became an inaccessible orthogonal state $|\bot \rangle$), measurement of the ancilla yields outcome $0$.
• Figure 4: Number of copies $R$ of an $N$-qubit needed to successfully transfer at least one copy through qubit loss channel of Eq. (\ref{['eq:channel']}), where each qubit is lost with probability $q$. We show transfer with DQAE compared with unencoded transfer of the states for different qubit loss rates $q$. Dashed lines are the approximate formulas Eq. (\ref{['eq:copiesEnt']}) for unencoded and Eq. (\ref{['eq:copiesProd']}) DQAE. We demand that the protocols may only fail with a total failure probability $Q$ of at most $Q\leq 0.01$.
• Figure 5: Parameterized unitaries for disentangler unitary $U(\boldsymbol{\theta})$, which are composed of many layers of the following building blocks: a) Hardware-efficient ansatz $U_\text{HE}$ of parameterized $y$ and $z$ rotations and fixed CNOT gates arranged in nearest-neighbor fashion. b) Fermionic ansatz $U_\text{fermion}$c) Transverse-field Ising inspired ansatz $U_\text{Ising}$.