Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates (2305.13409v4)

Abstract: We give a pair of algorithms that efficiently learn a quantum state prepared by Clifford gates and $O(\log n)$ non-Clifford gates. Specifically, for an $n$-qubit state $|\psi\rangle$ prepared with at most $t$ non-Clifford gates, our algorithms use $\mathsf{poly}(n,2^t,1/\varepsilon)$ time and copies of $|\psi\rangle$ to learn $|\psi\rangle$ to trace distance at most $\varepsilon$. The first algorithm for this task is more efficient, but requires entangled measurements across two copies of $|\psi\rangle$. The second algorithm uses only single-copy measurements at the cost of polynomial factors in runtime and sample complexity. Our algorithms more generally learn any state with sufficiently large stabilizer dimension, where a quantum state has stabilizer dimension $k$ if it is stabilized by an abelian group of $2^k$ Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest.

• The paper presents new algorithms that efficiently learn quantum states prepared with few non-Clifford gates by scaling complexity polynomially with qubits and exponentially only with the number of such gates, a significant improvement over conventional tomography.
• Two algorithms are introduced: one using entangled measurements for high efficiency requiring coherent storage, and another using single-copy measurements that is polynomially worse but more experimentally feasible.
• These efficient learning techniques have practical implications for quantum computing architectures that rely heavily on Clifford gates and for studying quantum many-body systems close to stabilizer states.

Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates

The paper "Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates" by Sabee Grewal et al. presents significant advancements in the domain of quantum state tomography, specifically targeting quantum states prepared with Clifford circuits and relatively few non-Clifford gates. The research addresses both theoretical and practical challenges in efficiently learning these states, leveraging the structured nature of Clifford gates, known for their importance in quantum error correction and fault-tolerant quantum computation, alongside a limited set of non-Clifford gates.

Overview

Quantum state tomography is foundational to verifying and utilizing quantum technologies, enabling the reconstruction of a quantum state from multiple copies of the state. The exponential growth of the computational and sample complexities with the system size in conventional quantum state tomography has been a significant barrier. The core contribution of this paper is developing algorithms whose complexities scale polynomially with the number of qubits and exponentially with the number of non-Clifford gates, thus efficiently handling states that could not be feasibly learned using traditional methods.

The algorithms presented in the paper make significant progress by reducing the dependence on physical resources to practical levels when the quantum state preparation involves a limited number of non-Clifford gates, specifically $O(\log n)$. This work is situated within the context of previous efforts that have tackled particular cases of quantum state tomography or estimation problems for specific quantum states, such as stabilizer states or matrix product states, but provides a broader applicability to cases involving arbitrary non-Clifford gates.

Main Results

The paper introduces two algorithms. The first algorithm uses entangled measurements across pairs of copies of the quantum state. It is highly efficient but requires the capability to store quantum states coherently, which can be a constraint in an online setting. The second algorithm, utilizing only single-copy measurements, scales polynomially worse with respect to runtime and sample complexity but circumvents the need for quantum state retention. Both algorithms rely fundamentally on the concept of stabilizer dimension, where the trace distance fidelity between the learned state and the true state is bounded by $O(2^t)$ when the quantum state's preparation involves up to $t$ non-Clifford gates.

Implications

The implications of these results span both theoretical and experimental domains. Theoretically, the research pushes forward our understanding of quantum state tomography's limitations and potential, especially regarding states that can be efficiently described by small stabilizer dimensions. The algorithms have potential applications in practical scenarios such as quantum computing architectures that predominantly utilize Clifford gates for their fault-tolerant properties or in quantum many-body physics where Hamiltonians can be close to stabilizer Hamiltonians perturbed by few non-Clifford elements.

Future Directions

This paper sets the stage for numerous future research avenues, from exploring more efficient methods of handling quantum state tomography across different classes of quantum states to developing even more resource-efficient learning processes that potentially relax or expand the assumptions around non-Clifford gate counts. Questionably, such research may lead to optimized protocols for near-Clifford circuits or may inspire analogous learning methods for other types of structured quantum systems. Additionally, improving upon existing pure state tomography algorithms to match or complement the presented algorithms’ capabilities could enhance overall efficiency in quantum computational tasks.

The paper, while primarily theoretical, undoubtedly enriches the toolkit available to quantum researchers and may be pivotal in the development of practical quantum devices or simulations. Its insights are crucial for both the future of quantum algorithm design, particularly in scenarios where efficiency and feasibility are bound by computation and physical constraints.