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Quantum autoencoders for efficient compression of quantum data
Jonathan Romero, Jonathan P. Olson, Alan Aspuru-Guzik
TL;DR
Quantum autoencoders address the need to compress quantum data under limited quantum resources. The paper trains parameterized quantum circuits with classical optimization to preserve information through a smaller latent space, demonstrating compression for Hubbard-model and molecular states while identifying entropy and channel-structure limits.
Problem
Quantum applications need tools that reduce experimental resource overhead, including methods for dimension reduction when classical compression is unavailable for quantum data.
Method
The paper trains a parameterized unitary quantum circuit by classical optimization to maximize average fidelity after encoding, discarding qubits, and decoding with reference-state ancillas.
Results
The model compresses quantum-simulation states, achieving decoded molecular wavefunction energies close to original values within chemical accuracy and compressing selected Hubbard and H4 ground states.
Takeaways & Limitations
After training, the decompression unitary can prepare states similar to the training states, and variational algorithms can use fewer active preparation parameters in the latent space.
Takeaways & Limitations
The model cannot completely capture the structure of general quantum channels, while noiseless compression is also limited by the ensemble density operator's von Neumann entropy.
Abstract
from arXiv · showhide
Classical autoencoders are neural networks that can learn efficient low dimensional representations of data in higher dimensional space. The task of an autoencoder is, given an input $x$, is to map $x$ to a lower dimensional point $y$ such that $x$ can likely be recovered from $y$. The structure of the underlying autoencoder network can be chosen to represent the data on a smaller dimension, effectively compressing the input. Inspired by this idea, we introduce the model of a quantum autoencoder to perform similar tasks on quantum data. The quantum autoencoder is trained to compress a particular dataset of quantum states, where a classical compression algorithm cannot be employed. The parameters of the quantum autoencoder are trained using classical optimization algorithms. We show an example of a simple programmable circuit that can be trained as an efficient autoencoder. We apply our model in the context of quantum simulation to compress ground states of the Hubbard model and molecular Hamiltonians.
I. INTRODUCTION
Quantum autoencoders adapt classical dimensional-reduction ideas to quantum data, where quantum-resource overhead limits applications. The paper introduces a simpler learnable circuit model and applies it to compress Hubbard-model and molecular ground states.
- Quantum applications are constrained by the amount of quantum resources realizable experimentally, making resource-reducing tools valuable.
- Classical autoencoders learn compressed latent representations by erasing selected bits while approximately reconstructing the input.
- The paper introduces quantum autoencoders for analogous machine-learning tasks on quantum systems with n+k input qubits.
- Figure 1 illustrates a 6-bit-to-3-bit latent-space autoencoder alongside a 6-3-6 quantum-autoencoder circuit.
- The model provides a simpler approach to quantum-data dimension reduction without exponentially costly classical memory.
II. QUANTUM AUTOENCODER MODEL
The quantum autoencoder uses a parameterized unitary to encode n+k-qubit input states into n qubits while discarding k qubits, then decodes them with reference-state ancillas. Training maximizes average input–output fidelity, equivalently enabling optimization through the discarded trash state under the stated condition.
- Model: The network encodes n+k input qubits, traces out k qubits, prepares reference-state ancillas, and applies a decoding evolution for comparison with the input.The discarded qubits form the trash system, while the retained qubits constitute the compressed representation.
- Model: Training seeks a parameterized unitary U_p that maximizes expected fidelity between each input state and the reconstructed output over the training ensemble.Successful autoencoding corresponds to fidelity approximately equal to 1 for all input states.
- Cost function: The circuit evaluates reconstruction by swapping the retained output with a fresh reference register and tracing over the swapped-out register.This construction compares the full input and output systems through a SWAP-based fidelity test.
- Cost function: Perfect fidelity, C1 = 1, is achievable exactly when the trash state equals the fixed reference state for every training input.Under this condition, the retained subsystem is a compressed version of the input and the complete circuit reduces to the identity map after the swap.
- Cost function: Training only on trash-state fidelity can find the ideal unitary, although the full-output and trash-state cost functions are not generally equal and satisfy C1 ≤ C2.The trash-state formulation may require fewer resources when reference-state copies are easier to prepare or input-state copies are difficult to access.
- Circuit architectures: Two programmable ansätze are considered: Circuit A uses all pairwise two-qubit gates, whereas Circuit B uses controlled single-qubit rotations plus endpoint single-qubit rotations.The four-qubit diagrams delimit each repeated unit-cell construction.
III. IMPLEMENTATION OF THE QUANTUM AUTOENCODER MODEL
The quantum autoencoder uses programmable parameterized circuits trained through a quantum–classical optimization loop. Input states are compressed, trash-state fidelity is measured against a reference, and the cost function guides parameter updates until convergence.
- Circuit design: Efficient implementation requires the unitary circuit’s gate and parameter counts to scale polynomially with the number of input qubits.A general (n+k)-qubit unitary is excluded because its parameter count scales exponentially.
- Circuit design: Programmable quantum circuits provide fixed gate networks with a polynomial number of tunable parameters that can be repeated as unit-cells.The unit-cell pattern can be repeated to increase model flexibility.
- Circuit design: Circuit A uses all pairings of general two-qubit gates and requires 15n(n − 1)/2 training parameters per unit-cell.Its two-qubit gates can be decomposed into three CNOT gates and single-qubit rotations.
- Circuit design: Circuit B combines all controlled one-qubit rotations with single-qubit rotations and requires 3n(n−1)+6n training parameters per unit-cell.The controlled rotations are organized successively by their control qubit.
- Training procedure: Training uses a quantum–classical hybrid scheme: quantum hardware prepares and measures states, while a classical optimizer updates the circuit parameters.For each training state, the input and reference are prepared, compression is applied, and fidelity is measured with a SWAP test before cost evaluation.
- Training procedure: The cost function aggregates weighted fidelities between compressed trash states and a reference state, and optimization repeats until convergence.The authors minimize log10(1 − C^2) to help prevent numerical instabilities because the cost function is upper bounded by 1.
IV. APPLICATION TO QUANTUM SIMULATION
The paper applies quantum autoencoders to compress structured ground-state wavefunctions from molecular and Hubbard Hamiltonians. Symmetries can make substantial compression possible, but achievable compression depends on the state subspace and circuit ansatz.
- Motivation: Fermionic wavefunctions occupy symmetry-defined subspaces, suggesting that quantum autoencoders can reduce the qubits needed to represent and store them.Particle-number and spin-projection constraints restrict the relevant Hilbert-space support.
- Molecular hydrogen: The hydrogen experiment trained autoencoders on six ground states and tested them on 44 states at different internuclear distances, compressing four qubits to two or one.Reference states of |0⟩⊗2 and |0⟩⊗3 were used for the two compression targets.
- Molecular hydrogen: Both circuit models achieved high fidelities for hydrogen encoding and decoded energies within chemical accuracy of 1.6 × 10−3 Hartrees.The reported chemical-accuracy threshold is also 1 kcal/mol or 43.4 meV.
- Hubbard and H4 systems: For Hubbard and H4 systems, circuit B compressed two-site Hubbard states from 4 to 2 and 1 qubits with error below 10−3, while H4 reached below 10−4 from 8 to 7 qubits.Neither circuit achieved error below 10−3 for the 4-sites Hubbard model, and circuit A did not reach that threshold for H4.
- Compression limits: The maximum lossless compression rate is fixed by the training set’s spanning subspace, so some state sets admit little or no compression.For systems with 8 fermionic modes and 4 particles, the discussed symmetry constraints imply a limit of approximately 7 qubits unless an additional symmetry is present.
V. DISCUSSION
The discussion highlights quantum autoencoders as compression and state-preparation tools while identifying entropy, channel structure, and training complexity as important boundaries.
- Quantum autoencoders can learn unitary circuits that compress quantum data, particularly for quantum simulations.The model combines a quantum circuit with classical optimization.
- A trained decompression unitary can prepare states similar to the training states from a latent-space state and reference state.This provides a state-preparation application for quantum variational algorithms.
- The specification assumes ensembles of pure input states and unitary evolution, although mixed-state inputs can be treated using purification ancillas.The purification enlarges the latent space so the ancilla qubits are not traced out.
- The defined autoencoder structure cannot completely capture general quantum channels.Specific channel instances may still support other computational tasks.
- The ensemble’s von Neumann entropy limits noiseless compression, while estimating that entropy is generally QSZK-complete.This also motivates using quantum autoencoders to estimate entropy.
- Complexity results depend on the chosen unitary family and classical optimization method, making general complexity statements unclear.
Appendix A: Molecular integrals
The appendix adopts atomic units and expresses molecular-Hamiltonian quantities through nuclear-repulsion and electronic-integral terms.
- Atomic units set the electron mass, electron charge, Bohr radius, Coulomb’s constant, and ℏ to unity.
- The molecular quantities include nuclear repulsion, represented by h_nuc.
- The electronic terms include one-electron integrals h_pq and two-electron integrals h_pqrs.