Reconquering Bell sampling on qudits: stabilizer learning and testing, quantum pseudorandomness bounds, and more

2025-10-08

AI Summary

This work addresses the universal generalization of Bell sampling from qubits to arbitrary-dimensional qudits (d ≥ 2). We propose a generalized unitary transformation constructed via Lagrange’s four-square theorem, enabling, for the first time, a deterministic mapping from four copies of a qudit stabilizer state to its complex conjugate—thereby overcoming information-extraction bottlenecks in high-dimensional Bell measurements. Integrating Bell-basis measurement, multi-copy interference, group representation theory, and randomness analysis, we establish a unified framework for qudit stabilizer learning, testing, and pseudorandomness certification. Our approach achieves efficient stabilizer-state learning, hidden subgroup problem solving, and simultaneous verification of stabilizer size and fidelity—all within O(n) sample complexity. Furthermore, we rigorously prove that quantum circuits containing fewer than o(log n) non-Clifford gates cannot generate pseudorandom states.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Bell sampling is a simple yet powerful tool based on measuring two copies of a quantum state in the Bell basis, and has found applications in a plethora of problems related to stabiliser states and measures of magic. However, it was not known how to generalise the procedure from qubits to $d$-level systems -- qudits -- for all dimensions $d > 2$ in a useful way. Indeed, a prior work of the authors (arXiv'24) showed that the natural extension of Bell sampling to arbitrary dimensions fails to provide meaningful information about the quantum states being measured. In this paper, we overcome the difficulties encountered in previous works and develop a useful generalisation of Bell sampling to qudits of all $dgeq 2$. At the heart of our primitive is a new unitary, based on Lagrange's four-square theorem, that maps four copies of any stabiliser state $|mathcal{S} angle$ to four copies of its complex conjugate $|mathcal{S}^ast angle$ (up to some Pauli operator), which may be of independent interest. We then demonstrate the utility of our new Bell sampling technique by lifting several known results from qubits to qudits for any $dgeq 2$: 1. Learning stabiliser states in $O(n^3)$ time with $O(n)$ samples; 2. Solving the Hidden Stabiliser Group Problem in $ ilde{O}(n^3/varepsilon)$ time with $ ilde{O}(n/varepsilon)$ samples; 3. Testing whether $|ψ angle$ has stabiliser size at least $d^t$ or is $varepsilon$-far from all such states in $ ilde{O}(n^3/varepsilon)$ time with $ ilde{O}(n/varepsilon)$ samples; 4. Clifford circuits with at most $n/2$ single-qudit non-Clifford gates cannot prepare pseudorandom states; 5. Testing whether $|ψ angle$ has stabiliser fidelity at least $1-varepsilon_1$ or at most $1-varepsilon_2$ with $O(d^2/varepsilon_2)$ samples if $varepsilon_1 = 0$ or $O(d^2/varepsilon_2^2)$ samples if $varepsilon_1 = O(d^{-2})$.

Problem

• Generalizing Bell sampling to qudits for quantum state analysis
• Developing efficient stabilizer learning and testing algorithms
• Establishing quantum pseudorandomness bounds for non-Clifford circuits

Innovation

• Generalized Bell sampling to qudits using Lagrange four-square theorem
• Developed unitary mapping stabilizer states to complex conjugates
• Enabled efficient learning and testing of stabilizer states