Vulnerability of quantum classification to adversarial perturbations

High-dimensional quantum systems are vital for quantum technologies and are essential in demonstrating practical quantum advantage in quantum computing, simulation, and sensing. Since dimensionality grows exponentially with the number of qubits, the potential power of noisy intermediate-scale quantum devices over classical resources also stems from entangled states in high dimensions. An important family of quantum protocols that can take advantage of high-dimensional Hilbert space is classification tasks. These include quantum machine learning algorithms, witnesses in quantum information processing, and certain decision problems. However, due to counterintuitive geometrical properties emergent in high dimensions, classification problems are vulnerable to adversarial attacks. We demonstrate that the amount of perturbation needed for an adversary to induce a misclassification scales inversely with dimensionality. This is shown to be a fundamental feature independent of the details of the classification protocol. Furthermore, this leads to a tradeoff between the security of the classification algorithm against adversarial attacks and quantum advantages that we expect for high-dimensional problems. In fact, protection against these adversarial attacks requires extra resources that can even scale polynomially with the Hilbert space dimension of the system, which can erase any significant quantum advantage that we might expect from a quantum protocol. This has wide-ranging implications in the use of both near-term and future quantum technologies for classification.

Figure 1

Top: In the absence of an adversarial Bob, Alice performs her classification task in three steps. First, she prepares her input quantum state $\sigma$. She then inputs $\sigma$ into her classification device, which outputs a real number $v\left(\sigma \right)$. Then, Alice computes the corresponding class label by applying some thresholding function $v\left(\sigma \right)\to h\left(\sigma \right)$ and obtains the correct class label $h_{\mathrm{correct}}$. However, Alice often wants to delegate either her state preparation of $\sigma$ or the preparation of her classification device to Bob. Even if Alice can certify the state or the device after receiving it from an adversarial Bob, she can still misclassify the input state if the dimension of $\sigma$ is high. Bottom left: Untrusted state preparation. Alice asks Bob for state $\sigma$ and Bob instead returns the state $\rho$. Alice certifies that $\rho$ is close to $\sigma$ in fidelity and gets the output $v\left(\rho \right)=v\left(\sigma \right)+\delta v\left(\sigma \right)$ from her device. Even when $\delta v$ is small, once mapped to $h$, there can be a misclassification $h_{\mathrm{incorrect}}$. Bottom right: Untrusted device preparation. Similarly, suppose Bob prepares the device implementing $\mathrm{\Lambda }$ instead of $U$ and gives it to Alice. She is able to certify the device to some precision, and obtains the output $v+\delta v$ where $\delta v$ is small. However, small $\delta v$ can still result in a misclassified label $h_{\mathrm{incorrect}}$ if the dimension of $\sigma$ is high enough.