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Predicting adaptively chosen observables in quantum systems

Jerry Huang, Laura Lewis, Hsin-Yuan Huang, John Preskill

Abstract

Recent advances have demonstrated that $\mathcal{O}(\log M)$ measurements suffice to predict $M$ properties of arbitrarily large quantum many-body systems. However, these remarkable findings assume that the properties to be predicted are chosen independently of the data. This assumption can be violated in practice, where scientists adaptively select properties after looking at previous predictions. This work investigates the adaptive setting for three classes of observables: local, Pauli, and bounded-Frobenius-norm observables. We prove that $Ω(\sqrt{M})$ samples of an arbitrarily large unknown quantum state are necessary to predict expectation values of $M$ adaptively chosen local and Pauli observables. We also present computationally-efficient algorithms that achieve this information-theoretic lower bound. In contrast, for bounded-Frobenius-norm observables, we devise an algorithm requiring only $\mathcal{O}(\log M)$ samples, independent of system size. Our results highlight the potential pitfalls of adaptivity in analyzing data from quantum experiments and provide new algorithmic tools to safeguard against erroneous predictions in quantum experiments.

Predicting adaptively chosen observables in quantum systems

Abstract

Recent advances have demonstrated that measurements suffice to predict properties of arbitrarily large quantum many-body systems. However, these remarkable findings assume that the properties to be predicted are chosen independently of the data. This assumption can be violated in practice, where scientists adaptively select properties after looking at previous predictions. This work investigates the adaptive setting for three classes of observables: local, Pauli, and bounded-Frobenius-norm observables. We prove that samples of an arbitrarily large unknown quantum state are necessary to predict expectation values of adaptively chosen local and Pauli observables. We also present computationally-efficient algorithms that achieve this information-theoretic lower bound. In contrast, for bounded-Frobenius-norm observables, we devise an algorithm requiring only samples, independent of system size. Our results highlight the potential pitfalls of adaptivity in analyzing data from quantum experiments and provide new algorithmic tools to safeguard against erroneous predictions in quantum experiments.

Paper Structure

This paper contains 41 sections, 34 theorems, 154 equations, 4 figures, 8 algorithms.

Table of Contents

  1. Introduction
  2. The Adaptive Model
  3. Adaptive attack on classical shadows
  4. Main Results
  5. Local observables
  6. Pauli observables
  7. Bounded-Frobenius-norm observables
  8. Proof Ideas
  9. Lower bounds for local and Pauli observables
  10. Adaptive algorithms for classical shadows
  11. Upper bound for bounded-Frobenius-norm observables
  12. Outlook
  13. Preliminaries
  14. Interaction Models
  15. Statistical Query Model
  16. ...and 26 more sections

Key Result

Theorem 1

For predicting expectation values $\trace(O_i\rho)$ of $M$ adaptively chosen local observables $O_1,\dots, O_M$, any quantum mechanism must use $N=\Omega(\min\{\sqrt{M},\log n\})$ samples. Moreover, there exists a computationally-efficient quantum mechanism using $N={\mathcal{O}}(\min\{\sqrt{M},\log

Figures (4)

  • Figure 1: Non-adaptive vs. adaptive model. The analyst queries observables $O_1,...,O_M$ to the mechanism. The mechanism is given access to samples of a quantum state $\rho$ and can conduct quantum processing and measurements in order to output predictions $\hat{o_i}$ for the expectation values $\tr(O_i\rho)$. In the adaptive model (bottom), the analyst can condition on all prior observed predictions $\hat{o}_1,...,\hat{o}_{i-1}$ in order to select the next observable $O_i$. In contrast, a non-adaptive analyst (top) cannot observe prior predictions before querying observables.
  • Figure 2: The dangers of adaptivity. Each point indicates the average error (averaged over $100$ independent runs) of the classical shadows protocol after being asked $M$ queries from the analyst for a fixed sample size of $N = 10000$. The shaded regions show the standard deviation over the independent runs. We see that after many adaptively chosen queries, the average error of classical shadows is much higher than for nonadaptive queries.
  • Figure 3: Example construction for local observables. We consider an example where the query (message) is $q^j=101$ and the randomly generated permutation is $\sigma_j = (6,7,2,0,1,5,3,4)$, where we denote each bitstring $q$ by its decimal representation. The table shows the quantum state, with each row being a computational basis state $\ket{Q}$ corresponding to a user (which is encoded in the first $\lceil \log d \rceil$ qubits of $\ket{Q}$). The permutation is encoded into the quantum state as shown in the table, e.g. the first column is $q=110$ (corresponding to $6$). The message is $q^j=101$ (corresponding to $5$) is on the $6$th entry of the permutation, so the analyst will query the single qubit observable $Z_6$.
  • Figure 4: Example construction for Pauli observables. We consider an example where the query (message) is $q^j=101$ and the randomly generated secret key is $001$. The table shows the quantum state, with each row being a computational basis state $\ket{Q}$ corresponding to a user (which is encoded in the first $\lceil \log d \rceil$ qubits of $\ket{Q}$). The secret key is encoded into the quantum state as shown in the table. We group qubits into pairs in each for clarity, which are defined by the encoding rule for the density matrix. By the encryption rule, the analyst will query the Pauli observable $Z_{b^j}$ where $b^j=011010$. One can verify that evaluating $Z_{b^j}$ on each state recovers the original message.

Theorems & Definitions (58)

  • Theorem 1: Local observables
  • Theorem 2: Pauli observables
  • Theorem 3: Bounded-Frobenius-norm observables
  • Theorem 4: Low-rank observables
  • Lemma 1: Mistake bound
  • Definition 1
  • Theorem 5: Theorem 1 in huang2020predicting
  • Theorem 6: Theorem 4 in huang2021information
  • Proposition 1
  • Theorem 7: Scalar Bernstein Inequality
  • ...and 48 more