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arXiv:1805.06580 · Physical Review A

Learning unknown pure quantum states

Sang Min Lee, Jinhyoung Lee, Jeongho Bang

Published Nov 2, 2018• 18 citations
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Abstract

We propose a learning method for estimating unknown pure quantum states. The basic idea of our method is to learn a unitary operation $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}$ that transforms a given unknown state $|{\ensuremath{\psi}}_{\ensuremath{\tau}}\ensuremath{\rangle}$ to a known fiducial state $|f\ensuremath{\rangle}$. Then, after completion of the learning process, we can estimate and reproduce $|{\ensuremath{\psi}}_{\ensuremath{\tau}}\ensuremath{\rangle}$ based on the learned $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}$ and $|f\ensuremath{\rangle}$. To realize this idea, we cast a random-based learning algorithm, called ``single-shot measurement learning,'' in which the learning rule is based on an intuitive and reasonable criterion: the greater the number of success (or failure), the less (or more) changes are imposed. Remarkably, the learning process occurs by means of a single-shot measurement outcome. We demonstrate that our method works effectively, i.e., the learning is completed with a finite number, say $N$, of unknown-state copies. Most surprisingly, our method allows the maximum statistical accuracy to be achieved for large $N$, namely $\ensuremath{\simeq}O({N}^{\ensuremath{-}1})$ scales of average infidelity. It highlights a nontrivial message, that is, a random-based strategy can potentially be as accurate as other standard statistical approaches.

DOI
10.1103/physreva.98.052302

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We propose a learning method for estimating unknown pure quantum states. The basic idea of our method is to learn a unitary operation U\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}U\ifmmode^\elseˆ\fi​ that transforms a given unknown state ∣\ensuremathψ\ensuremathτ\ensuremath⟩|{\ensuremath{\psi}}_{\ensuremath{\tau}}\ensuremath{\rangle}∣\ensuremathψ\ensuremathτ​\ensuremath⟩ to a known fiducial state ∣f\ensuremath⟩|f\ensuremath{\rangle}∣f\ensuremath⟩. Then, after completion of the learning process, we can estimate and reproduce ∣\ensuremathψ\ensuremathτ\ensuremath⟩|{\ensuremath{\psi}}_{\ensuremath{\tau}}\ensuremath{\rangle}∣\ensuremathψ\ensuremathτ​\ensuremath⟩ based on the learned U\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}U\ifmmode^\elseˆ\fi​ and ∣f\ensuremath⟩|f\ensuremath{\rangle}∣f\ensuremath⟩. To realize this idea, we cast a random-based learning algorithm, called ``single-shot measurement learning,'' in which the learning rule is based on an intuitive and reasonable criterion: the greater the number of success (or failure), the less (or more) changes are imposed. Remarkably, the learning process occurs by means of a single-shot measurement outcome. We demonstrate that our method works effectively, i.e., the learning is completed with a finite number, say NNN, of unknown-state copies. Most surprisingly, our method allows the maximum statistical accuracy to be achieved for large NNN, namely \ensuremath≃O(N\ensuremath−1)\ensuremath{\simeq}O({N}^{\ensuremath{-}1})\ensuremath≃O(N\ensuremath−1) scales of average infidelity. It highlights a nontrivial message, that is, a random-based strategy can potentially be as accurate as other standard statistical approaches.
A novel machine-learning algorithm based on single-shot measurements, named single-shot measurement learning, is demonstrated achieving the theoretical optimal accuracy. The method is at least as efficient as existing tomographic schemes and computationally much less demanding. The merits are attributed to the inclusion of weighted randomness in the learning rule governing the exploration of diverse learning routes. These advantages are explored experimentally by a linear-optical setup that is designed to draw the fullest potential of the proposed method. The experimental results show an unprecedented high level of accuracy for qubit-state learning and reproduction exhibiting (nearly) optimal infidelity scaling, O(N^{-0.983}), for the number N of unknown state copies, down to <10^{-5} without any compensation for experimental nonidealities. Extension to high dimensions is discussed with simulation results.
Machine learning methods have proved to be useful for the recognition of patterns in statistical data. The measurement outcomes are intrinsically random in quantum physics, however, they do have a pattern when the measurements are performed successively on an open quantum system. This pattern is due to the system-environment interaction and contains information about the relaxation rates as well as non-Markovian memory effects. Here we develop a method to extract the information about the unknown environment from a series of projective single-shot measurements on the system (without resorting to the process tomography). The method is based on embedding the non-Markovian system dynamics into a Markovian dynamics of the system and the effective reservoir of finite dimension. The generator of Markovian embedding is learned by the maximum likelihood estimation. We verify the method by comparing its prediction with an exactly solvable non-Markovian dynamics. The developed algorithm to learn unknown quantum environments enables one to efficiently control and manipulate quantum systems.
Quantum state estimation is important for various quantum information processes, including quantum communications, computation, and metrology, which require the characterization of quantum states for evaluation and optimization. We present a three-stage adaptive method for estimating arbitrary ddd-dimensional pure quantum states using locally informationally complete Fisher symmetric measurements (FSM) and a single-shot measurement basis. We derive finite-sample high-probability error bounds for the protocol and demonstrate that our approach scales as O(d/N)O(d/N)O(d/N) for large sample sizes, thereby guaranteeing the advantage of adaptation. Moreover, numerical simulations indicate that the protocol achieves an average infidelity close to the optimal given by the Gill-Massar lower bound (GMB). The total number of measurement outcomes scales linearly with 7d−37d-37d−3, avoiding the need for collective measurements on multiple copies of the unknown state. This work highlights the potential of adaptive estimation techniques in quantum state characterization while maintaining efficiency in the number of measurement outcomes.
Learning a classical description of a quantum state is a fundamental task in quantum computation. Among the most important classes of quantum states are stabilizer states, which play a central role in quantum error correction and fault-tolerant computation. To mitigate the effects of realistic noise, agnostic learning of stabilizer states has emerged as a natural and well-motivated problem. Recently, Chen \textit{et al.} [STOC'25, p. 429-438] resolved this problem for qubit systems by using a stabilizer bootstrapping framework. However, the agnostic learning of qudit stabilizer states remains largely unexplored, since the qudit setting introduces fundamental structural differences that prevent a direct generalization of existing qubit techniques. In this paper, we successfully generalize the stabilizer bootstrapping framework to qudit systems and present the first efficient quantum algorithm for agnostic learning of qudit stabilizer states. Specifically, given copies of an unknown nnn-qudit pure state ∣ψ⟩|ψ\rangle∣ψ⟩ that has fidelity τττ with some stabilizer state, our algorithm outputs a stabilizer state ∣φ⟩|φ\rangle∣φ⟩ such that ∣⟨φ∣ψ⟩∣2≥τ−ε\left| \braket{φ|ψ} \right|^2 \geq τ- \varepsilon∣⟨φ∣ψ⟩∣2≥τ−ε with high probability. The algorithm uses only single-copy and four-copy measurements, and its sample and time complexity scale as (d/τ)O(d2log⁡(1/τ))⋅poly(n,1/ε)(d/τ)^{O(d^2 \log(1/τ))} \cdot \mathrm{poly}(n, 1/\varepsilon)(d/τ)O(d2log(1/τ))⋅poly(n,1/ε), where the dimension ddd is an odd prime. As a direct corollary, our algorithm enables efficient estimation of the magic of a quantum state, as quantified by its stabilizer fidelity. Completing the picture, we also present a streamlined algorithm for the high-fidelity regime τ>cos⁡2(π/8)τ> \cos^2(π/8)τ>cos2(π/8), establishing a qudit analogue of the threshold-based approach in prior qubit work.
The constantly increasing dimensionality of artificial quantum systems demands for highly efficient methods for their characterization and benchmarking. Conventional quantum tomography fails for larger systems due to the exponential growth of the required number of measurements. The conceptual solution for this dimensionality curse relies on a simple idea---a complete description of a quantum state is excessive and can be discarded in favor of experimentally accessible information about the system. The probably approximately correct learning theory has been recently successfully applied to a problem of building accurate predictors for the measurement outcomes using a data set which scales only linearly with the number of qubits. Here, we present a constructive and numerically efficient protocol which learns a tensor network model of an unknown quantum system. We discuss the limitations and the scalability of the proposed method.
Quantum process tomography (QPT) methods aim at identifying a given quantum process. QPT is a major quantum information processing tool, since it allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. The present paper focuses on the estimation of a unitary process. This class is of particular interest because quantum mechanics postulates that the evolution of any closed quantum system is described by a unitary transformation. The standard approach of QPT is to measure copies of a particular set of predetermined (generally pure) states after they have been modified by the process to be identified. The main problem with this setup is that preparing an input state and setting it precisely to a predetermined value is challenging and thus yields errors. These errors can be decomposed into a sum of centered errors (i.e., whose average on all the copies is zero) and systematic errors that are the same for all the copies. The latter is often the main source of error in QPT. The algorithm we introduce works for any input states that make QPT theoretically possible (i.e., unless there are several solutions due to, e.g., a lack of diversity in the input states). The fact that we do not require the input states to be precisely set to predetermined values means that we can use a trick to remove the issue of systematic errors by considering that some states' copies are unknown but measured before they go through the process to be identified. We achieve this by splitting the copies of each input state into several groups and measuring the copies of the kthk\mathrm{th}kth group after they have successively been transferred through kkk instances of the process to be identified (each copy of each input state is used for only a single measurement). Using this approach, we can compute estimates of the measured states before and after they go through the process without using the knowledge we might have about the initial states. We test our algorithm with simulated data, and we assess its performance with a cnot gate on a trapped-ions qubit quantum computer.