Quantum Autoencoder (QAE)

• Quantum autoencoder is a parameterized quantum protocol that compresses high-dimensional quantum states into a lower-dimensional latent space using approximate quantum adders.
• Genetic algorithm optimization is used to design explicit, minimal quantum circuits achieving high fidelities, such as 87.51% for nonorthogonal state sets.
• The protocol splits the Hilbert space into latent and trash subspaces, enabling efficient encoding, transmission, and decoding across various quantum hardware platforms.

A quantum autoencoder (QAE) is a parameterized quantum protocol or circuit designed to compress quantum information by mapping an input quantum state from a higher-dimensional Hilbert space into a lower-dimensional latent subspace, such that critical features are preserved and less relevant, redundant, or noisy degrees of freedom are discarded. The QAE generalizes the classical autoencoder paradigm to the quantum domain, with its architecture tailored to the laws of quantum mechanics. The QAE typically divides the system's Hilbert space into a latent subspace encoding the essential information and a "trash" subspace for discarded components. Decoding is realized via the inverse of the encoding operation, enabling (approximate or lossless) recovery of the original state from the latent space. Recent advances have emphasized both theoretical frameworks and practical implementations, including explicit realizations in photonic and trapped-ion systems, circuit-level optimization, denoising, and the use of genetic algorithms for circuit discovery (Lamata et al., 2017).

1. Quantum Adders as Encoders in QAEs

The core innovation in (Lamata et al., 2017) is the use of approximate quantum adders to implement compression. The target operation is a unitary $U$ that "adds" two unknown quantum states:

$$U\,|\psi_1\rangle|\psi_2\rangle|0\rangle \propto |\psi_1\rangle + |\psi_2\rangle$$

where the output Hilbert space is partitioned such that one qubit carries the compressed information while ancillary ("memory") qubits store supplementary detail. Perfect quantum adders are forbidden by quantum mechanics; instead, the protocol exploits approximate, basis-dependent adders, such as the "basis quantum adder" mapping computational basis states according to fixed rules (e.g., $U|000\rangle = |000\rangle$, $U|010\rangle = |01+\rangle$, etc.).

The performance of an approximate adder acting as QAE encoder is quantified by the fidelity:

$$F = \mathrm{Tr}(\Psi_\mathrm{id}\,\Psi_\mathrm{id}\,\rho_\mathrm{out})$$

where $$\Psi_\mathrm{id} \propto (|\psi_1\rangle + |\psi_2\rangle)$$ is the ideal normalized output and $\rho_\mathrm{out}$ is the traced (reduced) output from the actual device. Input states are often restricted, e.g. $|\psi_i\rangle = [\cos\theta_i; \sin\theta_i]$, with $\theta_i \in \{0, \pi/2\}$.

Encoding proceeds as follows:

• The adder unitary $U$ acts on the two input qubits and an ancilla.
• The "compressed" qubit can be transmitted, processed, or stored efficiently.
• Decoding is achieved by applying $U^\dagger$ with the retained memory qubits, enabling recovery of the original joint state if fidelity is high.

A concrete example tracks the expectation value $\langle \sigma_z\rangle$ of the encoded qubit against the combined input observables:

$$\langle \sigma_z \rangle_{\rho_s} = \cos^2\theta_1 \cos^2\theta_2 - \sin^2\theta_1 \sin^2\theta_2$$

2. Genetic Algorithm-Based Protocol Optimization

QAE protocols based on approximate adders require explicit gate decompositions suited to hardware platforms. (Lamata et al., 2017) employs genetic algorithms (GAs) for optimization, departing from prior gradient-descent-based approaches.

Key features of GA-based optimization:

• Circuit candidates are encoded as $g\times 3$ matrices, each row specifying a gate type, phase, and target qubit.
• The universal gate set includes $R_x(\theta)$, $R_y(\theta)$, $R_z(\theta)$, and two-qubit Mølmer-Sørensen entangling gates.
• The fitness function is based on encoder/autoencoder fidelity evaluated over a specific set of initial states.
• The configuration is "bred" via selection, recombination, and mutation to maximize fidelity.

Example performance figures include achieving 87.51% average fidelity for three nonorthogonal two-qubit states with a succinct gate sequence. The explicit gate sequence is output as a quantum circuit suitable for direct hardware implementation.

Relative to gradient-based training, GAs more efficiently traverse non-convex search spaces and are less prone to local minima. The resultant explicit gate decompositions are compatible with platforms such as trapped ions, superconducting circuits, and photonic chips.

3. Information Flow and Compression Protocol

The QAE protocol in this paradigm involves sequential steps:

1. Encoding: $U$ acts on $(|\psi_1\rangle|\psi_2\rangle|0\rangle)$, yielding an entangled state with information encoded in the outputs. Memory registers are retained.
2. Transmission or Processing: The latent qubit can undergo single-qubit operations, satisfying constraints where only such manipulations or channels are available (e.g., in quantum networking).
3. Decoding: Using stored memory qubits, the application of $U^\dagger$ reconstructs the original two-qubit state, provided fidelity losses are minimal.

This approach allows resource-efficient transmission by reducing multi-qubit quantum information to a single qubit for communication or storage, with recovery possible through decoding.

4. Comparison to Standard Quantum Autoencoder Methods

Traditional quantum autoencoders, especially those optimized by classical gradient-based routines, require iterative parameter updates post-training to maintain high performance. The adder-based QAE optimized by GA is "one-shot": the optimized circuit can be deployed directly for the intended family of states without further adjustments, provided the input distribution does not change.

Implications include:

• Resource efficiency: Compressing many-qubit quantum information into fewer active qubits.
• Platform versatility: The methods and explicit gates are natively suited to trapped-ion, superconducting, and photonic photonic processors.
• Tailored fidelity: High fidelity is maintained if the state family is well-characterized; an adder optimized for a restricted subspace (e.g., a set of $\theta$ parameters) can near-perfectly compress that subspace.
• Pragmatic suitability: Situations with only single-qubit communication channels or processing hardware can exploit such encoders for more complex quantum data.

5. Implementation Recommendations and Experimental Considerations

For practical deployment:

• Circuit depth and total number of entangling gates must be matched to hardware coherence times and connectivity constraints.
• The GA returns a minimal, explicit gate sequence, reducing the likelihood of barren plateau issues common in deep variational circuits.
• The selected gate set should be tailored to the hardware's native gate library (MS gates for ions, CNOTs for superconducting qubits, optical circuits for photonic qudits).
• Input state restrictions—compression fidelity is maximized when the likely states lie within the restricted subspace used for adder optimization.
• For experimental validation, fidelity metrics (such as state tomographic reconstruction or observable statistics) should track the performance across the span of expected inputs.

6. Theoretical and Practical Implications

The construction of QAEs via approximate adders and genetic algorithm optimization suggests a design strategy for variational quantum devices: use analytical insight to build foundational operations—approximate adders—and then employ global, non-gradient-based optimization to tailor the circuit for realistic resource constraints and specific quantum data families.

This direction promises:

• Efficient quantum memory use in communication and storage settings.
• Decomposition strategies amenable to near-term implementations.
• Cross-hardware applicability and the potential for hardware-tailored gate synthesis.
• Protocols that, once optimized for a restricted input set, offer deterministic, systematic encoding/decoding without costly retraining.

In summary, the QAE protocol combining approximate quantum adders and genetic algorithms constitutes a deterministic, architecture-agnostic, and resource-aware approach to quantum data compression, demonstrating explicit performance metrics and robust adaptability for a variety of quantum technologies (Lamata et al., 2017).