Learning Properties of Quantum States Without the I.I.D. Assumption

Abstract

We develop a framework for learning properties of quantum states beyond the assumption of independent and identically distributed (i.i.d.) input states. We prove that, given any learning problem (under reasonable assumptions), an algorithm designed for i.i.d. input states can be adapted to handle input states of any nature, albeit at the expense of a polynomial increase in training data size (aka sample complexity). Importantly, this polynomial increase in sample complexity can be substantially improved to polylogarithmic if the learning algorithm in question only requires non-adaptive, single-copy measurements. Among other applications, this allows us to generalize the classical shadow framework to the non-i.i.d. setting while only incurring a comparatively small loss in sample efficiency. We use rigorous quantum information theory to prove our main results. In particular, we leverage permutation invariance and randomized single-copy measurements to derive a new quantum de Finetti theorem that mainly addresses measurement outcome statistics and, in turn, scales much more favorably in Hilbert space dimension.

Figures (7)

• Figure 1: Illustration of a general state learning algorithm: A learning algorithm consumes $(N-1)$ copies of $\rho$ to construct a prediction $p$. Success occurs if $p$ is (approximately) compatible with the remaining post-measurement test copy $\rho_p^{A_N}$.
• Figure 2: Caricature of main results: how to lift an i.i.d. learning algorithm $\mathcal{A}$ beyond the i.i.d. setting. Left: the performance of general learning algorithms is covered by our first main result (Theorem \ref{['thm:general']}). Right: the performance of non-adaptive and incoherent learning algorithms is covered by our second main result (Theorem \ref{['thm-intro: iid-meas']}). Restricting to non-adaptive and incoherent measurement $\mathcal{M}_{\bm{r}}$ leads to much better theoretical performance guarantees. $\mathcal{M}_{\rm{dist}}$ is a measurement device with low distortion, $\bm{w}$ is calibration, $p$ is prediction, $\mathcal{A}$ is the data processing of the i.i.d. algorithm and $\mathcal{M}_{\bm{r}}^{\mathcal{A}}$ is a measurement device uniformly chosen from $\mathcal{A}$'s set of measurements. Success occurs if $p$ is (approximately) compatible with the remaining post-measurement test copies $\rho^{A_N}_{l, \bm{w},p}$ or $\rho^{A_N}_{l, \bm{r,w},p}$.
• Figure 3: A general algorithm for learning properties of quantum states in the non-i.i.d. setting. A learning algorithm $\mathcal{B}$ takes as input the $N-1$ copies of the train set and returns a prediction $p$ and a calibration $c$. Success occurs if $p$ is (approximately) compatible with the remaining post-measurement test copy $\rho^{A_N}_{c,p}$.
• Figure 4: Illustration of Algorithm \ref{['alg-non-iid']}. Algorithm \ref{['alg-non-iid']} measures a large number of the state's subsystems using $\mathcal{M}^{\mathcal{A}}_{\bm{r}}$ that represents measurement devices uniformly chosen from the i.i.d. algorithm's set of measurements (red and green parts). Then, Algorithm \ref{['alg-non-iid']} applies the data processing of Algorithm $\mathcal{A}$ to the outcomes of a part of these subsystems (green part), leading to a prediction $p$. Algorithm \ref{['alg-non-iid']} returns the remaining outcomes as calibration $\bm{w}$. Success occurs if $p$ is (approximately) compatible with the remaining post-measurement test copy $\rho_{l, \bm{r,w},p}^{A_N}$.
• Figure 5: Illustration of Algorithm \ref{['alg-non-iid-general']}. Algorithm \ref{['alg-non-iid-general']} measures a large number of the state's subsystems using the measurement device with low distortion $\mathcal{M}^{l-k}_{\mathrm{dist}}$ (red and green parts). Then, in order to predict the property, Algorithm \ref{['alg-non-iid-general']} applies the data processing of Algorithm $\mathcal{A}$ to the outcomes of a part these subsystems (green part) leading to a prediction $p$. Algorithm \ref{['alg-non-iid-general']} returns the remaining outcomes as calibration $\bm{w}$. Success occurs if $p$ is (approximately) compatible with the remaining post-measurement test copy $\rho_{l, \bm{w},p}^{A_N}$.

Theorems & Definitions (55)

• Theorem 2.1: General algorithms in the non-i.i.d. setting
• Theorem 2.2: Randomized local quantum de Finetti theorem
• Theorem 2.3: Non-adaptive algorithms in the non-i.i.d. setting
• Proposition 2.4: Classical shadows in the non i.i.d. setting
• Definition 4.1: I.i.d. states
• Definition 4.2: Permutation invariant states
• Definition 4.3: Success formulation of learning properties of quantum states
• Example 1
• Example 2
• Definition 4.4: General algorithm