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<p>We initiate the study of sample-optimal quantum state tomography with minimal disturbance to the samples. Can we efficiently learn a precise description of a quantum state through sequential measurements of samples while at the same time ensuring that the post-measurement states are only minimally perturbed? We quantify the accumulated cost of the protocol over <i>T</i> samples by an additive regret objective, which controls the cumulative expected post-measurement infidelity of the consumed copies. The challenge is to balance informative measurements with measurements that keep this accumulated cost small. Here, we answer this question for all pure states by exhibiting an adaptive protocol whose expected regret is <InlineEquation ID="IEq1"><EquationSource Format="TEX">\(O({d}^{3}{\log }^{2}T)\)</EquationSource><EquationSource Format="MATHML"><math><mrow><mi>O</mi><mrow><mo>(</mo><mrow><msup><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>T</mi></mrow><mo>)</mo></mrow></mrow></math></EquationSource></InlineEquation>, while the infidelity of the online estimate after <i>t</i> samples scales as <InlineEquation ID="IEq2"><EquationSource Format="TEX">\(\widetilde{O}(1/t)\)</EquationSource><EquationSource Format="MATHML"><math><mrow><mover accent="true"><mi>O</mi><mo>̃</mo></mover><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></EquationSource></InlineEquation>. We also prove a logarithmic lower bound, showing that for fixed dimension, the dependence on the number of samples is optimal up to one logarithmic factor.</p>

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Learning pure quantum states almost without regret

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月26日

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  • Josep Lumbreras,
  • Mikhail Terekhov,
  • Marco Tomamichel

摘要

We initiate the study of sample-optimal quantum state tomography with minimal disturbance to the samples. Can we efficiently learn a precise description of a quantum state through sequential measurements of samples while at the same time ensuring that the post-measurement states are only minimally perturbed? We quantify the accumulated cost of the protocol over T samples by an additive regret objective, which controls the cumulative expected post-measurement infidelity of the consumed copies. The challenge is to balance informative measurements with measurements that keep this accumulated cost small. Here, we answer this question for all pure states by exhibiting an adaptive protocol whose expected regret is \(O({d}^{3}{\log }^{2}T)\)O(d3log2T), while the infidelity of the online estimate after t samples scales as \(\widetilde{O}(1/t)\)Õ(1/t). We also prove a logarithmic lower bound, showing that for fixed dimension, the dependence on the number of samples is optimal up to one logarithmic factor.