Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic estimation (2408.06967v3)

Abstract: We study the task of agnostic tomography: given copies of an unknown $n$-qubit state $\rho$ which has fidelity $\tau$ with some state in a given class $C$, find a state which has fidelity $\ge \tau - \epsilon$ with $\rho$. We give a new framework, stabilizer bootstrapping, for designing computationally efficient protocols for this task, and use this to get new agnostic tomography protocols for the following classes: Stabilizer states: We give a protocol that runs in time $\mathrm{poly}(n,1/\epsilon)\cdot (1/\tau)^{O(\log(1/\tau))}$, answering an open question posed by Grewal, Iyer, Kretschmer, Liang [43] and Anshu and Arunachalam [6]. Previous protocols ran in time $\mathrm{exp}(\Theta(n))$ or required $\tau>\cos^2(\pi/8)$. States with stabilizer dimension $n - t$: We give a protocol that runs in time $n^3\cdot(2^t/\tau)^{O(\log(1/\epsilon))}$, extending recent work on learning quantum states prepared by circuits with few non-Clifford gates, which only applied in the realizable setting where $\tau = 1$ [33, 40, 49, 66]. Discrete product states: If $C = K^{\otimes n}$ for some $\mu$-separated discrete set $K$ of single-qubit states, we give a protocol that runs in time $(n/\mu)^{O((1 + \log (1/\tau))/\mu)}/\epsilon^2$. This strictly generalizes a prior guarantee which applied to stabilizer product states [42]. For stabilizer product states, we give a further improved protocol that runs in time $(n^2/\epsilon^2)\cdot (1/\tau)^{O(\log(1/\tau))}$. As a corollary, we give the first protocol for estimating stabilizer fidelity, a standard measure of magic for quantum states, to error $\epsilon$ in $n^3 \mathrm{quasipoly}(1/\epsilon)$ time.

• The paper introduces a novel stabilizer bootstrapping framework that efficiently performs agnostic quantum state tomography with reduced sample complexity and runtime.
• The algorithms estimate a quantum state’s stabilizer fidelity and magic by leveraging iterative measurements and Bell difference sampling.
• The framework generalizes to high stabilizer dimensions and discrete product states, providing practical protocols for benchmarking quantum devices and advancing error correction.

Stabilizer Bootstrapping: A Recipe for Efficient Agnostic Tomography and Magic Estimation

The paper "Stabilizer Bootstrapping: A Recipe for Efficient Agnostic Tomography and Magic Estimation" introduces a new framework, stabilizer bootstrapping, for agnostic quantum state tomography, specifically targeting the class of stabilizer states and their generalizations. This framework provides a systematic approach for designing efficient protocols for determining the fidelity of a quantum state to a given stabilizer state, even when the target state is perturbed by noise. The primary contributions are detailed algorithms and proofs of efficiency for several classes of quantum states.

Main Contributions

1. Agnostic Tomography of Stabilizer States: The authors present an algorithm that agnostically learns stabilizer states with a sample complexity of $$O(n \log(1/\delta)) \cdot (1/\tau)^{O(\log 1/\tau)}$$ and a runtime of $$O(n^3 \log(1/\delta) \cdot (1/\tau)^{O(\log 1/\tau)})$$. This addresses an open problem about the efficient agnostic learning of stabilizer states for general fidelities, improving over previous algorithms that either ran in exponential time or required $\tau >\cos^2(\pi/8)$.
2. Estimating Stabilizer Fidelity: Leveraging the agnostic learning algorithm, the first efficient protocol for estimating the stabilizer fidelity of a quantum state is given. The sample complexity is $O(n \log(1/\delta)) \cdot (1/)^{O(\log 1/)}$ with a corresponding runtime. This result enables practical characterization of the "magic" of quantum states, crucial for benchmarking quantum devices.
3. Generalization to High Stabilizer Dimension States: For states with stabilizer dimension $n - t$, the algorithm achieves a sample complexity of $n(2^t/\tau)^{O(\log(1/))}$ and runtime $n^3(2^t/\tau)^{O(\log(1/))}$. This generalizes recent works on learning quantum circuits with few non-Clifford gates by handling agnostic settings where the fidelity constraint is relaxed.
4. Agnostic Tomography of Discrete Product States: The framework is extended to discrete product states arising from $\mu$-packing sets of single-qubit states. The proposed algorithm runs in time $(n/\mu)^{O((1 + \log (1/\tau))/\mu)}/^2$, strictly generalizing prior results that focused on stabilizer product states.
5. Improved Algorithm for Stabilizer Product States: For stabilizer product states, an optimized algorithm with a sample complexity $$\log n (1/\tau)^{O(\log 1/\tau)} + O(\log^2(1/\tau)/^2)$$ and a runtime of $n^2(1/\tau)^{O(\log 1/\tau)}/^2$ is provided. This enhances the efficiency for stabilizer product state learning compared to earlier methods.

Key Techniques and Insights

1. Stabilizer Bootstrapping: The essence of the framework lies in iteratively finding and measuring high-correlation projectors that stabilize the target state. Fidelity amplification is achieved through a recursive process involving measuring and post-selecting on stabilizer projectors.
2. Bell Difference Sampling: This sampling technique is crucial for identifying elements of the stabilizer group with high probability, underpinning the efficient implementation of the algorithms.
3. Handling High Stabilizer Dimension: For high-dimensional stabilizer states, the solution involves breaking down the problem recursively, applying stabilizer bootstrapping to progressively smaller subsystems until the stabilizer structure is revealed.
4. Quantum Information-Theoretic Guarantees: Several robust guarantees ensure that the stabilizer states and their generalizations can be learned with non-negligible probabilities, covering theoretical foundations essential for practical implementations.
5. Computational Feasibility: The algorithms are designed to achieve polynomial complexity in ideal settings while remaining efficient in broader parameter regimes that include substantial noise and lower fidelities.

Implications and Future Directions

1. Practical Benchmarking: These algorithms provide efficient tools for the practical benchmarking of near-term quantum devices, allowing detailed characterization of states that go beyond the stabilizer formalism.
2. Quantum Error Correction: Understanding and quantifying the stabilizer fidelity of quantum states is fundamental for the development of robust error correction schemes in quantum computing.
3. Quantum Resource Theories: The frameworks laid out in this paper could inform the development of new quantum resource theories, specifically around the utilization and quantification of non-stabilizer resources or "magic."
4. Cryptographic Connections: The paper hints at connections between the complexity of agnostic quantum state learning and cryptographic assumptions like Learning Parity with Noise (LPN) and Learning Subspace with Noise (LSN). Exploring these connections further could yield insights into the quantum versus classical computational dichotomy.

Conclusion

This paper makes significant strides in the field of agnostic tomography and magic estimation for quantum states. The stabilizer bootstrapping framework not only addresses open theoretical problems but also paves the way for practical algorithmic implementations. The broad applicability and efficiency improvements underscore its potential impact across quantum information science and technology. Future work will likely build on these results to explore even more complex classes of quantum states and further refine the efficiency and applicability of the stabilizer bootstrapping approach.