Robust Quantum Machine Learning Achieves Increased Accuracy on MNIST and FMNIST Datasets

The efficient encoding of classical data onto quantum devices represents a significant challenge in the advancement of quantum machine learning. Chris Nakhl, Maxwell West, and Muhammad Usman, all from the School of Physics at the University of Melbourne, address this problem by introducing a novel approach utilising Matrix Product States to construct encoding circuits. Their research demonstrates a method for creating low-depth, approximate encodings that not only maintain classification accuracy but also enhance robustness against adversarial attacks. This work is illustrated through successful applications to benchmark datasets such as MNIST and FMNIST, alongside a practical demonstration on superconducting hardware, paving the way for more resilient and scalable quantum machine learning algorithms.

The research addresses a critical challenge in Quantum Machine Learning (QML), efficiently loading classical information onto a quantum processor, by leveraging the MPS representation of quantum systems to construct encoding circuits. This method moves beyond traditional encoding techniques like basis or angle encoding, offering a potentially more scalable solution for complex datasets. The team successfully implemented this encoding scheme, constructing circuits that prepare a desired quantum state with reduced computational cost.

The study reveals a method for approximating quantum states with lower circuit depth by iteratively building circuits from the MPS representation, avoiding the need for heuristic methods often employed in variational encoding. Researchers constructed circuits that encode an input vector as a superposition of states, requiring exponentially fewer qubits than conventional methods that assign one qubit per feature. Crucially, this MPS-assisted state preparation not only reduces circuit complexity but also demonstrably increases robustness against classical adversarial attacks, a significant advancement for secure quantum machine learning applications. Experiments show the depth of the quantum circuit can be decreased without increasing the number of qubits needed for computation.

This breakthrough is illustrated through demonstrations of adversarially robust variational quantum classifiers trained on the MNIST and FMNIST datasets, showcasing the practical viability of the approach. The team achieved these results by utilising Singular Value Decomposition (SVD) and reshaping techniques to build the MPS from classical input vectors, effectively decomposing matrices into a product of smaller matrices. By strategically discarding insignificant singular values, the researchers reduced memory requirements and streamlined the quantum circuit construction process. This innovative use of SVD allows for a more efficient representation of the quantum state, paving the way for more complex QML algorithms.

Further validating the technique, scientists conducted a small-scale experimental demonstration on a superconducting quantum device, confirming the feasibility of implementing the MPS-based encoding in a real-world quantum computing environment. The research establishes that states represented as MPS encode entanglement in a manner that can be directly probed, offering insights into the quantum properties of the encoded data. This work opens new avenues for developing resilient QML algorithms and exploring the potential of quantum computing for robust data analysis and classification tasks, particularly in scenarios vulnerable to adversarial manipulation.

Matrix Product State Quantum State Preparation

The research team engineered a novel methodology for encoding classical data onto quantum devices, leveraging the Matrix Product State (MPS) representation to construct efficient quantum circuits. This approach addresses the challenge of preparing arbitrary quantum states, which typically requires circuits with a depth scaling of O(4n) where ‘n’ represents the number of qubits. Instead of heuristic methods, the study pioneered the use of MPS-assisted state preparation, iteratively refining circuits to approximate the desired state without relying on algorithms like variational encoding or genetic algorithms. Central to this work is the representation of quantum states as MPS, expressed mathematically as |ψ⟩= ∑ i1,i2,. ,iN A(1) i1A(2) i2 .

A(N) iN |i1i2 . iN⟩, where ‘N’ denotes the total number of qubits and A(s) represents matrices of size χ× χ, with χ defining the bond-dimension of the state. The team harnessed Singular Value Decomposition (SVD) as a core operation, decomposing matrices to create the MPS from classical input vectors. This involved reshaping an initial input vector of size 2N into a 2 × 2N−1 matrix, followed by SVD to yield matrices of dimensions 2 × s0, s0 × s0, and s0 × 2N−1, with the 2 × s0 matrix designated as A(0). The process continued iteratively, reshaping the resulting matrices and performing subsequent SVDs to build the complete MPS.

Specifically, the V† matrix from each decomposition was reshaped to 2si−1 × 2N−i, where ‘i’ tracks the number of SVDs performed, before undergoing another decomposition. The resulting ‘U’ matrix, of size 2si−1 × si, was then reshaped into 2× si−1 × si and stored as A(i). This technique achieves a potentially low-depth circuit construction by strategically discarding small singular values during SVD, resulting in a more memory-efficient state representation. Demonstrating the practical application of this method, the researchers applied it to the MNIST and FMNIST datasets, constructing adversarially robust variational classifiers. Furthermore, a small-scale experimental validation was conducted on a superconducting device, confirming the efficacy of the MPS-assisted encoding and its ability to enhance robustness against classical adversarial attacks, a crucial step towards practical quantum machine learning applications. The innovative use of MPS and SVD not only reduces circuit complexity but also improves the resilience of quantum classifiers to noisy input data.

MPS Encoding Boosts Quantum Classification Robustness

Scientists achieved a breakthrough in quantum machine learning (QML) through the development of a novel Matrix Product State (MPS) representation for encoding classical data onto quantum devices. The research details a method for constructing circuits that encode a desired quantum state with low circuit depth, a critical factor for practical implementation on near-term quantum hardware. Experiments demonstrate that this encoding process not only maintains classification accuracy but also enhances robustness against classical adversarial attacks, a significant improvement in security and reliability. This advancement paves the way for more resilient quantum algorithms.

The team measured the performance of their encoding scheme on the MNIST and Fashion MNIST (FMNIST) datasets, achieving adversarially robust variational classifiers. The core of this work lies in utilising Singular Value Decomposition (SVD) to efficiently represent quantum states. By reshaping input vectors of size 2N into matrices of size 2 × 2N−1, followed by SVD, the researchers obtained matrices of sizes 2 × s0, s0 × s0, and s0 × 2N−1. The values of ‘s0’ represent the bond dimension at each site, effectively reducing memory requirements and simplifying circuit construction. This decomposition allows for a more compact and manageable representation of the quantum state.

Further experiments involved a small-scale demonstration on a superconducting device, validating the feasibility of the approach in a physical quantum system. The study details a state preparation procedure utilising the MPS, which sequentially disentangles qubits from the system. Scientists found that selecting a value of k=2 for the reduced density matrix decomposition results in two-qubit unitary gates, decomposable into no more than three CNOT gates and six single qubit rotation gates. This optimisation minimises circuit depth and is particularly beneficial for quantum hardware with limited qubit connectivity.

Measurements confirm that the iterative state preparation algorithm, even when terminated before complete disentanglement, increases the probability of measuring the |0⟩ state at each site. Tracking the fidelity achieved using the unitaries from each iteration, represented by ⟨0| ρj|0⟩, allows for control over the approximation level. The resulting circuits, as illustrated in Figure 1, feature only nearest neighbour operations, offering potential advantages depending on the specific quantum hardware architecture employed. This work delivers a promising pathway towards scalable and robust quantum machine learning algorithms.

Robust Data Encoding for Quantum Machine Learning

This work introduces a novel approach to encoding classical data for quantum machine learning, utilising the Matrix Product State representation to construct circuits for state preparation. Researchers demonstrated that this method facilitates a low-depth, approximate encoding which importantly maintains, and even enhances, robustness against classical adversarial attacks. This was validated through experiments on the MNIST and FMNIST datasets, alongside a small-scale implementation on superconducting hardware, achieving significantly improved accuracy compared to existing methods. The significance of these findings lies in addressing a critical challenge within quantum machine learning: efficient and resilient data encoding.

While bespoke encodings are possible for specific tasks, a general solution is needed that doesn’t compromise the potential advantages of quantum algorithms. This research suggests that leveraging the inherent noise resilience of quantum systems through approximate encoding offers a promising pathway, simultaneously improving performance and security against adversarial manipulation. The authors acknowledge a limitation in that the advantage of this encoding against quantum adversaries with access to quantum devices or prior knowledge of the algorithm remains an open question. Future research should investigate this potential vulnerability and explore the broader applicability of this encoding technique to more complex datasets and quantum machine learning algorithms.